Light and Waves: A Conceptual Exploration of Physics 3031240960, 9783031240966
This book explores light and other types of waves, using this as a window into other aspects of physics. It emphasizes a
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Table of contents :
Preface
Contents
1 Theories of Light
1.1 Ancient Ideas About Light
1.1.1 Extramission Theory
1.1.2 Particle Theory
1.2 Islamic Golden Age
1.3 The Particle-Wave Debate
1.3.1 Wave Theory
1.4 Particle-Wave Duality
1.4.1 Today
1.5 Looking Ahead
1.6 Summary
1.7 Exercises
Exercises
Part IWaves
2 Properties of Waves
2.1 Introduction to Waves
2.1.1 What Is a Wave?
2.1.2 Amplitude and Wavelength
2.2 Characterizing Waves
2.2.1 Types of Waves
2.2.2 Transverse and Longitudinal Waves
2.2.3 Waves in 1, 2, and 3 Dimensions
2.3 Speed and Velocity
2.3.1 Speed and Velocity Equations
2.3.2 Speed of Light
2.3.3 Measuring the Speed of Light*
2.3.4 Speed of Light in a Medium
2.3.5 High Frequency Stock Market Trading and the Speed of Light*
2.4 Frequency and Period
2.4.1 Cars on a Road Analogy
2.4.2 Relating Velocity, Frequency, and Wavelength
2.4.3 Frequency
2.4.4 Frequency and Wavelength in a New Medium
2.5 Summary
2.6 Exercises
3 Superposition
3.1 Superposition of Waves
3.1.1 The Superposition Principle
3.1.2 Superposition with Different Frequencies
3.1.3 Constructive and Destructive Interference
3.1.4 Oscillations in Time
3.1.5 Constructive and Destructive Interference Examples*
3.1.6 Beating Patterns
3.2 Standing Waves
3.2.1 Reflection at Boundaries
3.2.2 Standing Waves from Reflected Waves and Superposition
3.2.3 Standing Waves Between Two Boundaries
3.3 Interference
3.3.1 Thin-Film Interference
3.3.2 Examples of Thin-Film Interference*
3.3.3 Interferometers and the Lack of an Aether*
3.4 Diffraction
3.4.1 Diffraction Through Holes and Around Obstacles
3.4.2 Huygens's Principle
3.5 Combining Diffraction and Interference
3.5.1 Double-Slit Experiment
3.5.2 Double-Slit Experiment Analysis*
3.5.3 Diffraction Gratings*
3.5.4 Single-Slit Experiment and Analysis*
3.5.5 The Arago-Poisson Spot*
3.5.6 Babinet's Principle
3.6 Structural Coloration*
3.7 Summary
3.8 Exercises
Exercises
4 Wave Energy
4.1 Energy and Power
4.1.1 Energy
4.1.2 Energy Units
4.1.3 Wave Energy
4.1.4 Power
4.1.5 Energy Density and Power Density
4.2 Spectra
4.2.1 Intensity Spectra
4.2.2 Continuous and Line Spectra
4.2.3 Transmission Spectra
4.2.4 Absorption Spectra*
4.3 Resonance
4.3.1 Resonance and Coupling
4.3.2 Two Resonance Examples with String Waves*
4.3.3 Resonance with Electromagnetic Waves
4.3.4 Resonance with Microwaves and Infrared Light*
4.3.5 The Tacoma Narrows and Millennium Bridges*
4.3.6 Resonance is Reversible
4.4 Non-Resonant Energy Transfer
4.4.1 Abrupt Energy Transfer to Waves
4.4.2 Energy Loss From Damping
4.5 Summary
4.6 Exercises
5 Doppler Effects, Redshifts, and Blueshifts
5.1 Doppler Effect Concepts
5.1.1 The Doppler Effect for Sound Waves
5.1.2 Cars on a Road Analogy
5.1.3 Doppler Effect Applications*
5.1.4 Red and Blue Shifts
5.1.5 Gravitational and Cosmological Redshifts*
5.2 Doppler Effect Equations
5.2.1 Moving Observer Case
5.2.2 Moving Source Equation
5.2.3 General Equation
5.2.4 Doppler Shifts for Reflections
5.2.5 Doppler Effect on Wave Power*
5.2.6 Relativistic Doppler Effect*
5.3 Supersonic Motion
5.3.1 Explanation
5.3.2 Equations for Supersonic Motion*
5.4 Summary
5.5 Exercises
6 Mechanical Waves
6.1 Pendulums
6.1.1 How Pendulums Work
6.1.2 Momentum
6.2 String Waves
6.2.1 How String Waves Work
6.2.2 Speed of String Waves
6.3 Sound Waves
6.3.1 How Sound Waves Work
6.3.2 The Speed of Sound
6.3.3 The Sound Spectrum*
6.3.4 Sonar and Medical Ultrasound*
6.4 The Physics of Music*
6.4.1 Physics Terminology for Music*
6.4.2 The Western Musical Scale*
6.4.3 Musical Intervals*
6.4.4 Major, Minor, and Non-Western Musical Scales*
6.4.5 Musical Instruments*
6.5 Water Waves
6.5.1 Capillary Waves
6.5.2 Gravity Waves
6.5.3 Phase Velocity and Group Velocity
6.5.4 Water Motion in Waves
6.5.5 Water Wave Evolution*
6.5.6 Long Wavelength Water Waves: Tsunamis, Tides, and Seiches*
6.6 Seismic Waves*
6.6.1 The Earth's Structure*
6.6.2 Earthquakes*
6.6.3 Seismic Waves*
6.7 Summary
6.8 Exercises
Part IIRays
7 Shadows and Pinhole Cameras
7.1 Shadows
7.1.1 Projection
7.1.2 Analyzing Projections*
7.1.3 Umbra and Penumbra
7.2 Eclipses
7.2.1 Solar Eclipses
7.2.2 Lunar Eclipses
7.2.3 Eclipses and Moon Phases
7.3 Light Through Small Holes
7.3.1 Pinhole Cameras and Camera Obscuras
7.3.2 Pinhole Camera Analysis
7.3.3 Multiple Pinholes*
7.4 Shadow-Based Vision*
7.5 Summary
7.6 Exercises
8 Reflection
8.1 Reflection in General
8.1.1 Why Waves Reflect
8.1.2 Requirements for Mirrors
8.2 Plane Mirrors
8.2.1 Law of Reflection
8.2.2 Corner-Cube Retroreflectors
8.2.3 Images for Plane Mirrors
8.2.4 Size of a Mirror
8.3 Concave Reflectors
8.3.1 Parabolic Reflectors
8.3.2 Concave Spherical Mirrors
8.3.3 Concave Mirror Ray Diagrams
8.3.4 Mirror Equations
8.3.5 Spherical Aberration and Coma*
8.4 Convex Spherical Mirrors
8.5 Multiple Mirrors*
8.6 Mirrors, Inversion, and Symmetry*
8.7 Fermat's Principle of Least Time*
8.8 Summary
8.9 Exercises
Exercises
9 Refraction
9.1 Refraction in General
9.1.1 Refractive Index
9.1.2 Why Waves Refract
9.1.3 Refraction From Density Gradients
9.2 Refraction at Interfaces
9.2.1 Snell's Law
9.2.2 Apparent Depth
9.2.3 Refraction and Reflection at Different Angles
9.2.4 Total Internal Reflection Examples*
9.2.5 How Much Light Gets Reflected*
9.2.6 Evanescent Waves*
9.3 Lenses
9.3.1 Types of Lenses
9.3.2 Lens Coordinates
9.3.3 Images From Lenses
9.3.4 Lens Equations
9.3.5 Cameras*
9.3.6 Vision Correction*
9.4 Multiple Lens Systems*
9.4.1 Objects and Images*
9.4.2 Microscopes*
9.4.3 Telescopes*
9.4.4 More Optical Systems*
9.4.5 Electron Microscopes*
9.5 Dispersion
9.5.1 Prisms
9.5.2 Achromatic Lenses*
9.5.3 Rainbows*
9.6 Fermat's Principle of Least Time*
9.7 Summary
9.8 Exercises
Part IIILight
10 Color
10.1 Color Vision
10.1.1 How Vision Works
10.1.2 Light and Dark Adaptation
10.1.3 Different People see Different Colors*
10.1.4 Color Vision in Animals*
10.2 Color Models
10.2.1 Color Wheel
10.2.2 Light Addition with the RGB Color Model
10.2.3 Light Subtraction with the CMYK Color Model
10.2.4 Leaf Colors in Summer and Fall*
10.2.5 HSV Color Model
10.2.6 Color Spaces*
10.3 Summary
10.4 Exercises
11 Electromagnetic Waves
11.1 Light Waves as Electric and Magnetic Fields
11.1.1 Scalars, Vectors, and Fields
11.1.2 Electric Fields
11.1.3 Magnetic Fields
11.1.4 Changing Electric and Magnetic Fields
11.1.5 Electromagnetic Waves
11.1.6 How Electromagnetic Waves Work*
11.2 The Electromagnetic Spectrum
11.3 Scattering
11.3.1 Scattering Off Large Objects
11.3.2 Scattering Off Medium Size Objects
11.3.3 Scattering Off Small Objects
11.4 Polarization
11.4.1 Electromagnetic Wave Polarization
11.4.2 Polarizers
11.4.3 Multiple Polarizers
11.4.4 Liquid Crystal Displays
11.4.5 Sources of Polarized Light
11.4.6 Polarization as Superposition*
11.4.7 Birefringence and Optical Activity*
11.5 Summary
11.6 Exercises
12 Thermal Radiation
12.1 Thermal Radiation
12.1.1 Qualitative Trends
12.1.2 Blackbodies
12.1.3 Wien's Displacement Law
12.1.4 Color Temperature
12.1.5 Stefan-Boltzmann Law
12.1.6 Remote Temperature Measurement*
12.2 Thermal Radiation Interactions
12.2.1 Radiation Coupling and Emissivity
12.2.2 Two-Way Thermal Radiation
12.3 Earth's Climate*
12.3.1 Earth's Energy Budget*
12.3.2 Greenhouse Effects on Mars and Venus*
12.3.3 Global Warming*
12.4 Summary
12.5 Exercises
Part IVModern Physics
13 Photons
13.1 The Quantum Revolution
13.1.1 Explaining Blackbody Radiation
13.1.2 Particles and Waves
13.1.3 What is a Photon?
13.2 Photon Energy
13.2.1 Planck-Einstein Relation
13.2.2 Photoelectric Effect
13.2.3 Photoelectric Effect Examples*
13.2.4 Photochemistry*
13.2.5 Compton Scattering
13.3 Photon Momentum
13.3.1 Classical Momentum
13.3.2 Photon Momentum
13.3.3 Radiometers*
13.3.4 Solar Sails*
13.3.5 Laser Tweezers*
13.3.6 Doppler Cooling*
13.4 Particle-Wave Duality
13.4.1 Quantum Interpretation of the Double-Slit Experiment
13.4.2 Photon Size*
13.4.3 Spectral Broadening of Pulses
13.4.4 Energy-Time Uncertainty
13.4.5 Particle-Wave Duality for Other Wave Types*
13.5 Summary
13.6 Exercises
14 Matter Waves
14.1 Matter Waves
14.1.1 De Broglie Relations
14.1.2 Wave Functions
14.1.3 What is Waving?
14.2 Traveling Matter Waves
14.2.1 Free Particles
14.2.2 Classical and Quantum Roller Coasters
14.2.3 Barriers and Tunneling
14.2.4 Tunneling Examples*
14.3 Diffraction and Interference
14.3.1 Electron Diffraction
14.3.2 Diffraction for Research*
14.3.3 Interference to Test Quantum Mechanics*
14.4 Standing Matter Waves
14.4.1 Electrons in a Cavity
14.4.2 Filling in Electrons
14.4.3 Molecular Vibrations
14.4.4 Structures of Atoms
14.4.5 Chemical Bonds*
14.5 Energy Level Transitions
14.5.1 Light Absorption
14.5.2 Light Emission
14.5.3 Fluorescence
14.5.4 Phosphorescence*
14.5.5 Lasers
14.6 Quantum Weirdness
14.6.1 Heisenberg Uncertainty Principle
14.6.2 Schrödinger's Cat Experiment*
14.6.3 The EPR Paradox*
14.6.4 Quantum Decoherence*
14.6.5 Macroscopic Quantum Systems*
14.7 Summary
14.8 Exercises
15 Gravitational Waves
15.1 Gravity
15.1.1 Newtonian Gravity
15.1.2 Tides
15.1.3 Gravitational Fields
15.2 Gravitational Waves
15.2.1 First Direct Detection
15.2.2 What are Gravitational Waves?
15.2.3 Frequency
15.2.4 Polarization
15.2.5 Energy
15.2.6 Momentum
15.3 Propagating Gravity
15.3.1 Speed of Gravity
15.3.2 Gravitational Near and Far Field*
15.4 Warped Space
15.4.1 The Equivalence Principle
15.4.2 Gravitational Attraction of Light
15.4.3 Curved Space
15.4.4 Ripples in Spacetime
15.5 Gravitational Wave Detection
15.5.1 Existing Observatories
15.5.2 Future Observatories
15.6 Summary
15.7 Exercises
A Numbers
A.1 Scientific Notation
A.1.1 Scientific Notation on a Calculator
A.2 More Calculator Advice
A.3 Precision
A.3.1 Determining Precision in Calculations
A.3.2 Propagating Uncertainties
A.4 Exercises
B Units
B.1 Units Are Your Friends
B.2 The Metric System
B.3 Unit Math
B.4 Unit Conversion
B.5 Exercises
C Algebra
C.1 Solving Problems
C.2 Expressions and Equations
C.2.1 Manipulating Expressions
C.2.2 Manipulating Equations
C.3 Exponents
C.4 Exercises
D Geometry
D.1 Triangles
D.1.1 Similar Triangles
D.1.2 Right Triangles and Trigonometry
D.2 Perimeters, Areas, and Volumes
D.3 Exercises
E Additional Resources
F Answers to Odd-Numbered Problems
G Figure Credits
H Useful Facts and Figures
Index
Citation preview
Steven S. Andrews
Light and Waves A Conceptual Exploration of Physics
Light and Waves
Steven S. Andrews
Light and Waves A Conceptual Exploration of Physics
123
Steven S. Andrews Department of Bioengineering University of Washington Seattle, WA, USA
ISBN 978-3-031-24096-6 ISBN 978-3-031-24097-3 https://doi.org/10.1007/978-3-031-24097-3
(eBook)
© Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Preface
Light is simultaneously familiar and mysterious. It is something that nearly every one of us sees every waking day of our lives, generally without giving it much thought. But each one of those commonplace light rays has the remarkable property of being composed of both waves and particles at the same time. Furthermore, those rays likely arose from the strange quantum mechanical transitions of individual atoms within a light bulb, or the sun, or that took place thousands of years ago in unimaginably distant stars. This combination of familiarity and mystery has led scholars to study the nature of light from antiquity to the current day, even now revealing surprising new details. This book explores all types of waves. These include light waves in particular but also string waves, sound waves, water waves, seismic waves, the bizarre matter waves of quantum mechanics, and the gravitational waves that ripple through spacetime. It also focuses on particles of matter and light, which naturally leads to explorations of velocity, forces, momentum, and other properties of physical objects. Waves carry energy, which leads to the deep concepts of energy conservation and energy transfer. Through these explorations, this book tours a substantial fraction of the field of physics. The emphasis here is on building a strong conceptual understanding of how light and waves work. Topics are illustrated with examples that are drawn from familiar experiences or that are just simply fascinating, such as the science behind bird coloration, how musical instruments work, and the causes of global warming. The language of mathematics is often clearer than the equivalent paragraphs of prose, so this book helps teach the necessary mathematical skills and then uses them to build a deeper understanding of physics. These mathematics, which never extend beyond introductory algebra, are covered in the main text and several appendices. This book was written with several audiences in mind. The first is undergraduates who are not science majors but who are required to take a science course. In my experience, these students are highly heterogeneous, with some who enjoy science but have chosen to pursue other interests, and others who prefer to avoid anything quantitative. A class on light and waves works well in this case because it is widely accessible, while retaining the interest of all students by covering topics that most have not seen before. The second audience is high school students taking a physics elective. Again, studying light and waves cuts across traditional physics
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Preface
topics, enabling the course to engage students with different backgrounds. This book is aligned with the goals of the Next Generation Science Standards, a set of high school science education standards that have been adopted by most US states, which emphasize building a conceptual understanding, presenting scientific practice frequently, and focusing on the key underlying ideas. The third audience is adults who would like to learn about light and other waves on their own, whether for personal interest or building skills. They will appreciate the fact that the text is self-contained, without requiring supplementation from lectures, homework, or online resources. Each chapter opens with a question and closes with a list of exercises. The opening questions are intended to inspire curiosity, encouraging the student to seek out answers while reading the text. The exercises at the end are divided into three categories. “Questions” test the student’s understanding of the material while also addressing common misconceptions. “Problems” generally require numerical calculation, which help develop numerical fluency, provide a sense of scale for the relevant phenomena, and build a quantitative understanding of the topic. And “Puzzles” are challenging but interesting problems that explore the topic in depth. Many of these are best approached during small-group instruction, where they serve as tasks that groups of students can discuss, puzzle over, and solve together. The back of the book includes solutions to the odd-numbered problems. The book is divided into four parts: Waves, Rays, Light, and Modern Physics. Waves introduces widely-applicable physical concepts, like resonance and superposition, which arise repeatedly throughout the rest of the book. Rays explores geometric optics, including shadows, reflection, and refraction; these topics are largely empirical but important for understanding light behaviors. Light investigates the perception and physics of light, going into topics such as color, polarization, and thermal radiation. Finally, Modern Physics applies the same physics that students have learned throughout the book to make sense of photons, basic quantum mechanics, and gravitational waves. This section highlights the many connections between quantum and classical mechanics showing, for example, how quantized matter waves are analogous to standing waves on guitar strings. There is more material here than can be reasonably covered in one quarter or even one semester. One approach is to cover all chapters, but to skip some sections within each chapter (optional sections are labeled with asterisks). Alternatively, some of the chapters could be skipped entirely. While it’s probably best to start with the Waves section, the rest of the chapters are sufficiently independent that they could be covered in pretty much any order. I thank David Boness, the physics department chair at Seattle University, who gave me an opportunity to teach a course titled “The World of Light”, from which this book evolved, and who ordered frequent encouragement. I also benefited from numerous discussions with the Seattle University physics department faculty and majors. Discussions with Bernhard Mecking and Danny Desurra were helpful as well. In addition, this book benefited substantially from feedback given to me by the students in my “World of Light” classes. Sam Harrison and others at Springer provided invaluable assistance in the final publishing stages. Although their
Preface
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contributions were less well defined, I also thank my professors and colleagues who taught me math, science, and scientific writing. These include John Finn, Jane Lipson, Ollie Zafiriou, Steve Boxer, Steve Chu, Dennis Bray, Adam Arkin, Jay Groves, Roger Brent, Herbert Sauro, and many others. Finally, my family has been a constant source of support and joy throughout this project. Seattle, USA 2023
Steven S. Andrews
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
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Properties of Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction to Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 What Is a Wave? . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Amplitude and Wavelength . . . . . . . . . . . . . . . . . . . 2.2 Characterizing Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Types of Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Transverse and Longitudinal Waves . . . . . . . . . . . . 2.2.3 Waves in 1, 2, and 3 Dimensions . . . . . . . . . . . . . . 2.3 Speed and Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Speed and Velocity Equations . . . . . . . . . . . . . . . . . 2.3.2 Speed of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Measuring the Speed of Light* . . . . . . . . . . . . . . . . 2.3.4 Speed of Light in a Medium . . . . . . . . . . . . . . . . . . 2.3.5 High Frequency Stock Market Trading and the Speed of Light* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Theories of Light . . . . . . . . . . . . . 1.1 Ancient Ideas About Light . . 1.1.1 Extramission Theory . 1.1.2 Particle Theory . . . . . 1.2 Islamic Golden Age . . . . . . . 1.3 The Particle-Wave Debate . . . 1.3.1 Wave Theory . . . . . . 1.4 Particle-Wave Duality . . . . . . 1.4.1 Today . . . . . . . . . . . 1.5 Looking Ahead . . . . . . . . . . . 1.6 Summary . . . . . . . . . . . . . . . 1.7 Exercises . . . . . . . . . . . . . . .
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Waves
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Contents
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Superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Superposition of Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 The Superposition Principle . . . . . . . . . . . . . . . . 3.1.2 Superposition with Different Frequencies . . . . . . . 3.1.3 Constructive and Destructive Interference . . . . . . 3.1.4 Oscillations in Time . . . . . . . . . . . . . . . . . . . . . . 3.1.5 Constructive and Destructive Interference Examples* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.6 Beating Patterns . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Standing Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Reflection at Boundaries . . . . . . . . . . . . . . . . . . . 3.2.2 Standing Waves from Reflected Waves and Superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Standing Waves Between Two Boundaries . . . . . 3.3 Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Thin-Film Interference . . . . . . . . . . . . . . . . . . . . 3.3.2 Examples of Thin-Film Interference* . . . . . . . . . . 3.3.3 Interferometers and the Lack of an Aether* . . . . . 3.4 Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Diffraction Through Holes and Around Obstacles . 3.4.2 Huygens’s Principle . . . . . . . . . . . . . . . . . . . . . . 3.5 Combining Diffraction and Interference . . . . . . . . . . . . . . 3.5.1 Double-Slit Experiment . . . . . . . . . . . . . . . . . . . 3.5.2 Double-Slit Experiment Analysis* . . . . . . . . . . . . 3.5.3 Diffraction Gratings* . . . . . . . . . . . . . . . . . . . . . 3.5.4 Single-Slit Experiment and Analysis* . . . . . . . . . 3.5.5 The Arago-Poisson Spot* . . . . . . . . . . . . . . . . . . 3.5.6 Babinet’s Principle . . . . . . . . . . . . . . . . . . . . . . . 3.6 Structural Coloration* . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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50 52 54 54 56 57 58 58 59 60 60 61 62 63 64 65 67 69 70
Wave Energy . . . . . . . . . . 4.1 Energy and Power . . 4.1.1 Energy . . . . . 4.1.2 Energy Units
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Frequency and Period . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Cars on a Road Analogy . . . . . . . . . . . . . . . . 2.4.2 Relating Velocity, Frequency, and Wavelength 2.4.3 Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Frequency and Wavelength in a New Medium . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.3
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4.1.3 Wave Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.5 Energy Density and Power Density . . . . . . . . . . Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Intensity Spectra . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Continuous and Line Spectra . . . . . . . . . . . . . . 4.2.3 Transmission Spectra . . . . . . . . . . . . . . . . . . . . 4.2.4 Absorption Spectra* . . . . . . . . . . . . . . . . . . . . . Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Resonance and Coupling . . . . . . . . . . . . . . . . . 4.3.2 Two Resonance Examples with String Waves* . 4.3.3 Resonance with Electromagnetic Waves . . . . . . 4.3.4 Resonance with Microwaves and Infrared Light* 4.3.5 The Tacoma Narrows and Millennium Bridges* . 4.3.6 Resonance is Reversible . . . . . . . . . . . . . . . . . . Non-Resonant Energy Transfer . . . . . . . . . . . . . . . . . . . 4.4.1 Abrupt Energy Transfer to Waves . . . . . . . . . . . 4.4.2 Energy Loss From Damping . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Doppler Effects, Redshifts, and Blueshifts . . . . . . . . . . . 5.1 Doppler Effect Concepts . . . . . . . . . . . . . . . . . . . . 5.1.1 The Doppler Effect for Sound Waves . . . . 5.1.2 Cars on a Road Analogy . . . . . . . . . . . . . 5.1.3 Doppler Effect Applications* . . . . . . . . . . 5.1.4 Red and Blue Shifts . . . . . . . . . . . . . . . . . 5.1.5 Gravitational and Cosmological Redshifts* 5.2 Doppler Effect Equations . . . . . . . . . . . . . . . . . . . . 5.2.1 Moving Observer Case . . . . . . . . . . . . . . . 5.2.2 Moving Source Equation . . . . . . . . . . . . . 5.2.3 General Equation . . . . . . . . . . . . . . . . . . . 5.2.4 Doppler Shifts for Reflections . . . . . . . . . . 5.2.5 Doppler Effect on Wave Power* . . . . . . . . 5.2.6 Relativistic Doppler Effect* . . . . . . . . . . . 5.3 Supersonic Motion . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Explanation . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Equations for Supersonic Motion* . . . . . . . 5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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78 78 79 81 81 82 84 86 87 87 88 89 90 91 93 94 94 95 96 97
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101 102 102 104 105 106 108 108 108 110 111 111 113 114 114 114 117 117 118
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Mechanical Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Pendulums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 How Pendulums Work . . . . . . . . . . . . . . . . . . . . . 6.1.2 Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 String Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 How String Waves Work . . . . . . . . . . . . . . . . . . . 6.2.2 Speed of String Waves . . . . . . . . . . . . . . . . . . . . . 6.3 Sound Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 How Sound Waves Work . . . . . . . . . . . . . . . . . . . 6.3.2 The Speed of Sound . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 The Sound Spectrum* . . . . . . . . . . . . . . . . . . . . . 6.3.4 Sonar and Medical Ultrasound* . . . . . . . . . . . . . . 6.4 The Physics of Music* . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Physics Terminology for Music* . . . . . . . . . . . . . . 6.4.2 The Western Musical Scale* . . . . . . . . . . . . . . . . . 6.4.3 Musical Intervals* . . . . . . . . . . . . . . . . . . . . . . . . 6.4.4 Major, Minor, and Non-Western Musical Scales* . . 6.4.5 Musical Instruments* . . . . . . . . . . . . . . . . . . . . . . 6.5 Water Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Capillary Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Gravity Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3 Phase Velocity and Group Velocity . . . . . . . . . . . . 6.5.4 Water Motion in Waves . . . . . . . . . . . . . . . . . . . . 6.5.5 Water Wave Evolution* . . . . . . . . . . . . . . . . . . . . 6.5.6 Long Wavelength Water Waves: Tsunamis, Tides, and Seiches* . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Seismic Waves* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 The Earth’s Structure* . . . . . . . . . . . . . . . . . . . . . 6.6.2 Earthquakes* . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.3 Seismic Waves* . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Part II 7
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123 124 124 125 126 126 127 128 128 129 131 132 133 133 133 135 136 137 140 140 141 144 146 148
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150 152 152 153 154 156 157
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165 166 166 167 168
Rays
Shadows and Pinhole Cameras . . . . . 7.1 Shadows . . . . . . . . . . . . . . . . . . 7.1.1 Projection . . . . . . . . . . 7.1.2 Analyzing Projections* . 7.1.3 Umbra and Penumbra . .
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Eclipses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Solar Eclipses . . . . . . . . . . . . . . . . . . . 7.2.2 Lunar Eclipses . . . . . . . . . . . . . . . . . . . 7.2.3 Eclipses and Moon Phases . . . . . . . . . . Light Through Small Holes . . . . . . . . . . . . . . . . 7.3.1 Pinhole Cameras and Camera Obscuras . 7.3.2 Pinhole Camera Analysis . . . . . . . . . . . 7.3.3 Multiple Pinholes* . . . . . . . . . . . . . . . . Shadow-Based Vision* . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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170 170 171 172 173 173 174 175 176 177 178
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8
Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Reflection in General . . . . . . . . . . . . . . . 8.1.1 Why Waves Reflect . . . . . . . . . . 8.1.2 Requirements for Mirrors . . . . . . 8.2 Plane Mirrors . . . . . . . . . . . . . . . . . . . . . 8.2.1 Law of Reflection . . . . . . . . . . . . 8.2.2 Corner-Cube Retroreflectors . . . . 8.2.3 Images for Plane Mirrors . . . . . . 8.2.4 Size of a Mirror . . . . . . . . . . . . . 8.3 Concave Reflectors . . . . . . . . . . . . . . . . . 8.3.1 Parabolic Reflectors . . . . . . . . . . 8.3.2 Concave Spherical Mirrors . . . . . 8.3.3 Concave Mirror Ray Diagrams . . 8.3.4 Mirror Equations . . . . . . . . . . . . 8.3.5 Spherical Aberration and Coma* . 8.4 Convex Spherical Mirrors . . . . . . . . . . . . 8.5 Multiple Mirrors* . . . . . . . . . . . . . . . . . . 8.6 Mirrors, Inversion, and Symmetry* . . . . . 8.7 Fermat’s Principle of Least Time* . . . . . . 8.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . 8.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . .
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9
Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Refraction in General . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Refractive Index . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Why Waves Refract . . . . . . . . . . . . . . . . . . . 9.1.3 Refraction From Density Gradients . . . . . . . . 9.2 Refraction at Interfaces . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Snell’s Law . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Apparent Depth . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Refraction and Reflection at Different Angles . 9.2.4 Total Internal Reflection Examples* . . . . . . .
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211 212 212 212 213 214 214 215 216 218
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219 220 221 221 222 223 225 227 230 232 232 233 234 235 235 236 236 237 238 239 241 242
10 Color . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Color Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 How Vision Works . . . . . . . . . . . . . . . . . . . . . . 10.1.2 Light and Dark Adaptation . . . . . . . . . . . . . . . . 10.1.3 Different People see Different Colors* . . . . . . . . 10.1.4 Color Vision in Animals* . . . . . . . . . . . . . . . . . 10.2 Color Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Color Wheel . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Light Addition with the RGB Color Model . . . . 10.2.3 Light Subtraction with the CMYK Color Model . 10.2.4 Leaf Colors in Summer and Fall* . . . . . . . . . . . 10.2.5 HSV Color Model . . . . . . . . . . . . . . . . . . . . . . 10.2.6 Color Spaces* . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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249 250 250 252 253 255 257 257 258 260 263 264 264 268 269
9.3
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9.6 9.7 9.8 Part III
9.2.5 How Much Light Gets Reflected* 9.2.6 Evanescent Waves* . . . . . . . . . . Lenses . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Types of Lenses . . . . . . . . . . . . . 9.3.2 Lens Coordinates . . . . . . . . . . . . 9.3.3 Images From Lenses . . . . . . . . . . 9.3.4 Lens Equations . . . . . . . . . . . . . . 9.3.5 Cameras* . . . . . . . . . . . . . . . . . . 9.3.6 Vision Correction* . . . . . . . . . . . Multiple Lens Systems* . . . . . . . . . . . . . 9.4.1 Objects and Images* . . . . . . . . . 9.4.2 Microscopes* . . . . . . . . . . . . . . . 9.4.3 Telescopes* . . . . . . . . . . . . . . . . 9.4.4 More Optical Systems* . . . . . . . . 9.4.5 Electron Microscopes* . . . . . . . . Dispersion . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Prisms . . . . . . . . . . . . . . . . . . . . 9.5.2 Achromatic Lenses* . . . . . . . . . . 9.5.3 Rainbows* . . . . . . . . . . . . . . . . . Fermat’s Principle of Least Time* . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . .
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273 274 274 275 276 277 277 279 281 286 286 288 289 291 291 291 292 294 295 297 298 300 301
12 Thermal Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Thermal Radiation . . . . . . . . . . . . . . . . . . . . . . . 12.1.1 Qualitative Trends . . . . . . . . . . . . . . . . . 12.1.2 Blackbodies . . . . . . . . . . . . . . . . . . . . . . 12.1.3 Wien’s Displacement Law . . . . . . . . . . . 12.1.4 Color Temperature . . . . . . . . . . . . . . . . . 12.1.5 Stefan-Boltzmann Law . . . . . . . . . . . . . . 12.1.6 Remote Temperature Measurement* . . . . 12.2 Thermal Radiation Interactions . . . . . . . . . . . . . . 12.2.1 Radiation Coupling and Emissivity . . . . . 12.2.2 Two-Way Thermal Radiation . . . . . . . . . 12.3 Earth’s Climate* . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Earth’s Energy Budget* . . . . . . . . . . . . . 12.3.2 Greenhouse Effects on Mars and Venus* . 12.3.3 Global Warming* . . . . . . . . . . . . . . . . . . 12.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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307 308 308 309 309 311 312 313 314 314 315 316 316 318 319 321 322
11 Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . 11.1 Light Waves as Electric and Magnetic Fields . . 11.1.1 Scalars, Vectors, and Fields . . . . . . . . 11.1.2 Electric Fields . . . . . . . . . . . . . . . . . . 11.1.3 Magnetic Fields . . . . . . . . . . . . . . . . . 11.1.4 Changing Electric and Magnetic Fields 11.1.5 Electromagnetic Waves . . . . . . . . . . . . 11.1.6 How Electromagnetic Waves Work* . . 11.2 The Electromagnetic Spectrum . . . . . . . . . . . . . 11.3 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Scattering Off Large Objects . . . . . . . . 11.3.2 Scattering Off Medium Size Objects . . 11.3.3 Scattering Off Small Objects . . . . . . . . 11.4 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 Electromagnetic Wave Polarization . . . 11.4.2 Polarizers . . . . . . . . . . . . . . . . . . . . . . 11.4.3 Multiple Polarizers . . . . . . . . . . . . . . . 11.4.4 Liquid Crystal Displays . . . . . . . . . . . 11.4.5 Sources of Polarized Light . . . . . . . . . 11.4.6 Polarization as Superposition* . . . . . . . 11.4.7 Birefringence and Optical Activity* . . . 11.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Contents
Modern Physics
13 Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 The Quantum Revolution . . . . . . . . . . . . . . . . . . . . . . 13.1.1 Explaining Blackbody Radiation . . . . . . . . . . . 13.1.2 Particles and Waves . . . . . . . . . . . . . . . . . . . . 13.1.3 What is a Photon? . . . . . . . . . . . . . . . . . . . . . 13.2 Photon Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Planck-Einstein Relation . . . . . . . . . . . . . . . . . 13.2.2 Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . 13.2.3 Photoelectric Effect Examples* . . . . . . . . . . . . 13.2.4 Photochemistry* . . . . . . . . . . . . . . . . . . . . . . . 13.2.5 Compton Scattering . . . . . . . . . . . . . . . . . . . . 13.3 Photon Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.1 Classical Momentum . . . . . . . . . . . . . . . . . . . 13.3.2 Photon Momentum . . . . . . . . . . . . . . . . . . . . . 13.3.3 Radiometers* . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.4 Solar Sails* . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.5 Laser Tweezers* . . . . . . . . . . . . . . . . . . . . . . 13.3.6 Doppler Cooling* . . . . . . . . . . . . . . . . . . . . . . 13.4 Particle-Wave Duality . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.1 Quantum Interpretation of the Double-Slit Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.2 Photon Size* . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.3 Spectral Broadening of Pulses . . . . . . . . . . . . . 13.4.4 Energy-Time Uncertainty . . . . . . . . . . . . . . . . 13.4.5 Particle-Wave Duality for Other Wave Types* . 13.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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14 Matter Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Matter Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.1 De Broglie Relations . . . . . . . . . . . . . . . 14.1.2 Wave Functions . . . . . . . . . . . . . . . . . . . 14.1.3 What is Waving? . . . . . . . . . . . . . . . . . . 14.2 Traveling Matter Waves . . . . . . . . . . . . . . . . . . . 14.2.1 Free Particles . . . . . . . . . . . . . . . . . . . . . 14.2.2 Classical and Quantum Roller Coasters . . 14.2.3 Barriers and Tunneling . . . . . . . . . . . . . . 14.2.4 Tunneling Examples* . . . . . . . . . . . . . . . 14.3 Diffraction and Interference . . . . . . . . . . . . . . . . . 14.3.1 Electron Diffraction . . . . . . . . . . . . . . . . 14.3.2 Diffraction for Research* . . . . . . . . . . . . 14.3.3 Interference to Test Quantum Mechanics*
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14.4 Standing Matter Waves . . . . . . . . . . . . . . 14.4.1 Electrons in a Cavity . . . . . . . . . 14.4.2 Filling in Electrons . . . . . . . . . . . 14.4.3 Molecular Vibrations . . . . . . . . . 14.4.4 Structures of Atoms . . . . . . . . . . 14.4.5 Chemical Bonds* . . . . . . . . . . . . 14.5 Energy Level Transitions . . . . . . . . . . . . . 14.5.1 Light Absorption . . . . . . . . . . . . 14.5.2 Light Emission . . . . . . . . . . . . . . 14.5.3 Fluorescence . . . . . . . . . . . . . . . 14.5.4 Phosphorescence* . . . . . . . . . . . . 14.5.5 Lasers . . . . . . . . . . . . . . . . . . . . 14.6 Quantum Weirdness . . . . . . . . . . . . . . . . 14.6.1 Heisenberg Uncertainty Principle . 14.6.2 Schrödinger’s Cat Experiment* . . 14.6.3 The EPR Paradox* . . . . . . . . . . . 14.6.4 Quantum Decoherence* . . . . . . . 14.6.5 Macroscopic Quantum Systems* . 14.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . 14.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . .
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370 371 373 374 377 381 384 384 385 386 388 388 389 389 391 393 394 396 397 399
15 Gravitational Waves . . . . . . . . . . . . . . . . . . . . 15.1 Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.1 Newtonian Gravity . . . . . . . . . . . 15.1.2 Tides . . . . . . . . . . . . . . . . . . . . . 15.1.3 Gravitational Fields . . . . . . . . . . 15.2 Gravitational Waves . . . . . . . . . . . . . . . . 15.2.1 First Direct Detection . . . . . . . . . 15.2.2 What are Gravitational Waves? . . 15.2.3 Frequency . . . . . . . . . . . . . . . . . 15.2.4 Polarization . . . . . . . . . . . . . . . . 15.2.5 Energy . . . . . . . . . . . . . . . . . . . . 15.2.6 Momentum . . . . . . . . . . . . . . . . 15.3 Propagating Gravity . . . . . . . . . . . . . . . . 15.3.1 Speed of Gravity . . . . . . . . . . . . 15.3.2 Gravitational Near and Far Field* 15.4 Warped Space . . . . . . . . . . . . . . . . . . . . 15.4.1 The Equivalence Principle . . . . . 15.4.2 Gravitational Attraction of Light . 15.4.3 Curved Space . . . . . . . . . . . . . . . 15.4.4 Ripples in Spacetime . . . . . . . . .
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Contents
15.5 Gravitational Wave Detection . 15.5.1 Existing Observatories 15.5.2 Future Observatories . . 15.6 Summary . . . . . . . . . . . . . . . . 15.7 Exercises . . . . . . . . . . . . . . . .
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423 423 425 426 427
A
Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Scientific Notation . . . . . . . . . . . . . . . . . . . . A.1.1 Scientific Notation on a Calculator . . A.2 More Calculator Advice . . . . . . . . . . . . . . . . A.3 Precision . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.1 Determining Precision in Calculations A.3.2 Propagating Uncertainties . . . . . . . . . A.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
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431 431 432 432 432 433 435 436
B
Units B.1 B.2 B.3 B.4 B.5
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439 439 440 441 443 444
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Algebra . . . . . . . . . . . . . . . . . . . . . . . . C.1 Solving Problems . . . . . . . . . . . . C.2 Expressions and Equations . . . . . C.2.1 Manipulating Expressions C.2.2 Manipulating Equations . C.3 Exponents . . . . . . . . . . . . . . . . . C.4 Exercises . . . . . . . . . . . . . . . . . .
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D
Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.1 Triangles . . . . . . . . . . . . . . . . . . . . . . . . . D.1.1 Similar Triangles . . . . . . . . . . . . . D.1.2 Right Triangles and Trigonometry . D.2 Perimeters, Areas, and Volumes . . . . . . . . D.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . .
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E
Additional Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
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Answers to Odd-Numbered Problems . . . . . . . . . . . . . . . . . . . . . . . . 463
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* Optional section Appendices
................... Units Are Your Friends . The Metric System . . . . Unit Math . . . . . . . . . . Unit Conversion . . . . . . Exercises . . . . . . . . . . .
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Contents
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G
Figure Credits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
H
Useful Facts and Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499
1
Theories of Light
Figure 1.1 Beams of sunlight (called crepuscular rays) over Cardigan Bay, UK.
Opening question What is a ray of light made of? (a) a stream of particles (b) a series of waves (c) energy (d) electric and magnetic fields (e) all of the above
© Springer Nature Switzerland AG 2023 S. S. Andrews, Light and Waves, https://doi.org/10.1007/978-3-031-24097-3_1
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1 Theories of Light
Light is the energy that warms you when you sit in the sun, the energy that grows the world’s plants, and the energy that powers the Earth’s wind. Light is also the stuff of rainbows, the medium by which we view the world around us, and essentially our only connection to the rest of the universe. But what actually is light? It would be nice if one could put a beam of light under a microscope and look at it, to see what it really is. However, this isn’t possible. Simply put, if light enters our eyes, then we see brightness, and if it doesn’t, then we don’t. There’s no more to see directly than that. Thus, instead of looking at light with a microscope, we have to figure out what light is by investigating its behaviors and then inferring what it must be from them. This chapter takes a historical approach in answering what light is, starting with a variety of ancient ideas and ending with the modern understanding of particle-wave duality. This history began with numerous philosophical ideas in ancient Greece, India, and China, became more advanced in medieval Egypt, and then progressed to a scientific understanding that was developed primarily in Europe and then North America during the 16th to 19th centuries. Further scientific advances are now pursued worldwide.
1.1
Ancient Ideas About Light
1.1.1
Extramission Theory
The ancient Greeks came up with perhaps the earliest theory of how light worked. They claimed that vision worked by people’s eyes emitting beams of light in what is now called the extramission theory. They believed that sight worked in somewhat the same way as the sense of touch. To determine the texture of something, for example, you reach out your hand and touch it. Your hand then feels the surface, through complicated processes, and sends signals back to your brain. Your brain interprets those signals to determine whether the surface is smooth, rough, soft, slippery, or whatever. Likewise, the ancient Greeks imagined that the eyes sent out rays that would sense the colors and shapes of objects and then report back about what they found.
Figure 1.2 (Left) Empedocles. (Right) Extramission theory, in which eye rays interact with sun rays.
1.1 Ancient Ideas About Light
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To make this more precise, Empedocles, a 5th century BCE scholar (Figure 1.2), postulated that everything was composed of fire, air, earth, and water. He claimed that Aphrodite, the Greek goddess of love, made people’s eyes out of all four elements and then lit the fire in their eyes, which shone out and constituted sight. It was soon recognized that this extramission theory would imply that people would be able to see equally well at night as during the day, which obviously is not the case. Empedocles addressed this by modifying his theory to claim that vision arose from an interaction between the rays from the eyes and rays from light sources, such as the sun. This was better, but then people realized that one can go outside and see the stars immediately without having to wait for beams of light emitted from the eyes to get there and back, showing another problem with the theory. Empedocles ignored this issue, but the subsequent Greek philosophers Plato (429–347 BCE) and his student Aristotle (384–322 BCE) addressed it by claiming that light traveled infinitely fast. This version of the extramission theory was not improved upon further. However, it was not believed universally. In fact, Aristotle himself didn’t fully believe this theory which he had helped create, but stated that “In general, it is unreasonable to suppose that seeing occurs by something issuing from the eye”1 . He thus supported the correct intromission theory, in which peoples’ eyes do not emit light, but only observe the light that shines into them. Nevertheless, Plato’s supposed authority led to the extramission theory remaining as the dominant theory of vision up until the Middle Ages. Remarkably, several important optics discoveries were made during ancient times, despite widespread belief in the extramission theory. Euclid (c. 300 BCE) described rays of light as traveling in straight lines and also wrote a treatise about vision that was largely correct in its treatment of perspective, Hero of Alexandria (c. 10–70 CE) figured out the mathematics of reflection, andPtolemy (c. 100–170 CE) quantitatively described the bending of light as it shines from air into water, called refraction. All of these philosophers explained their results within the context of the extramission theory. Meanwhile, the Chinese philosopher Mo Zi (c. 470–391 BCE), or perhaps one of his followers, made many of these same discoveries but with the correct intromission theory. Those ideas were not pursued by later Chinese philosophers and were not known in the West until much later. The extramission theory, despite being unphysical, continued to have a surprising number of adherents long after it was disproved in the Middle Ages. Leonardo da Vinci, the quintessential Renaissance man, expressed extramission views in the 1490s. Even today, children often believe in extramission until they are taught otherwise. Furthermore, recent research showed that about half of modern college students believe that eyes emit rays during vision2 . The extramission theory also survives in modern popular culture and modern fiction. For example, giving someone the “evil eye” could be interpreted as some sort
1 Lindberg, David C. (1976) Theories of Vision from Al-Kindi to Keplar, University of Chicago Press: Chicago. 2 Winer, G.A., J.E. Cottrell, V. Gregg, J.S. Fournier, and L.A. Bica (2002) American Psychologist 67:417.
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1 Theories of Light
of beam shining from the eye. Also, various superheroes and cartoon characters, from Superman to X-Men, can emit beams of light from their eyes (Figure 1.3). Finally, it should be pointed out that “ray-tracing” computer graphics, which are widely used in movie special effects, are based upon what is essentially an extramission concept: rays are drawn outward from the eye or camera lens to determine what is seen in each particular direction. Figure 1.3 Superboy using his X-ray vision to see a pie in a closed oven. From May 1950, issue #8 ©DC Comics.
1.1.2
Particle Theory
A separate particle theory of light started at about the same time as the extramission theory with a group of people called the atomists in both ancient Greece and ancient India, around the 4th or 5th centuries BCE. They claimed that everything in the universe, including light, was composed of tiny particles that were indivisible and indestructible. These particles generally differed according to shape, such as by being pointy or smooth, and could also hook onto other particles to form clusters. This theory is often traced to Democritus, a Greek philosopher born about 460 BCE, and, separately, to Kanada, an Indian philosopher who lived at roughly the some point, between the 2nd and 6th centuries BCE (Figure 1.4). Both atomistic theories had many different versions. For example, within Indian atomism, some believed that there were 4 types of atoms, others claimed 24 types, and yet others held that there were infinitely many types. Atomism supposedly explained the observation that “pure” materials, such as air, metal, and water, could be broken down, and then re-formed again. Regarding light, the atomists believed that the sun and other bright objects emitted minute atoms of light that traveled very fast, or perhaps infinitely fast, and which people
1.2 Islamic Golden Age
5
Figure 1.4 Democritus (left) and Kanada (middle), who where two early philosophers who advanced the particle view of light. (Right) Light particles.
perceived when the particles hit their eyes (this intromission explanation contradicted the extramission theory of vision that was widely believed at the time). In retrospect, the atomists’ ideas about the particle nature of light were prescient, as explained below, although this has to assigned more to chance than to scientific deduction. The particle description of light largely held through antiquity and the Middle Ages.
1.2
Islamic Golden Age
The first important discoveries about the nature of light that were based on careful analysis were made by Muslim scientists about a thousand years later. They were made during the Islamic Golden Age, which was an especially prolific period for Islamic culture, science, and mathematics and extended from about the 8th to 14th centuries. Our use of Arabic numerals and algebra date to this period. Ibn al-Haytham was an Arab scientist born around the year 965 in what is now Iraq, who later moved to Egypt (Figure 1.5). He combined the intromission theory that Aristotle introduced with Euclid’s explanation of perspective and Ptolemy’s description of refraction. He further advanced these theories, finally yielding a moderately correct explanation of how vision works. In particular, he was the first philosopher in the Western history who convincingly showed that vision occurs through the intromission method.
Figure 1.5 (Left) Ibn al-Haytham. (Right) Intromission theory, in which sunlight reflects off an object and goes into a person’s eye.
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1 Theories of Light
Al-Haytham argued that light extramission didn’t make sense because eyes that used extramission would lose an enormous amount of vision substance when looking out into space. Also, light clearly enters our eyes, by the fact that we feel pain when looking at very bright lights. He claimed, correctly, that every point on every object sends light beams out in all directions, of which a very few of them enter our eyes. His final major argument was one of simplicity: given that light is known to reflect off objects and then go to our eyes, then there’s no need for any assumption about rays being emitted from our eyes, so that assumption might as well be dropped. Thus, he concluded that the intromission theory is clearly the correct one. Al-Haytham’s work greatly influenced European science through the 17th century.
1.3
The Particle-Wave Debate
1.3.1
Wave Theory
A major flaw with the particle theory was that it did not explain refraction very well (Figure 1.6). In 1637, the French scientist René Descartes addressed this problem by hypothesizing that light was composed of waves, thus introducing the wave theory of light. It was already known that light travelled at a fast but finite speed, so he claimed that light waves bent as they moved from a faster medium to a slower medium, just as water waves and sound waves do, which were understood at the time3 . This made sense, so the wave theory started to gain acceptance. Subsequent scientists, including Christiaan Huygens and Robert Hooke, built on these ideas to formulate a mathematical theory of light waves.
Figure 1.6 (Left) Three rays of light being refracted by a plastic block. Early wave theory proponents: (Middle) René Descartes and (Right) Christiaan Huygens.
The great English physicist Isaac Newton (1643–1727), who also developed large parts of physics and much of calculus, did not agree with these new wave ideas but preferred the older particle description that had been introduced by the ancient
3 Descartes
was correct that refraction arises from different wave speeds, but incorrectly believed that light travels faster in water than in air.
1.3 The Particle-Wave Debate
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atomists (Figure 1.7). Furthermore, he conducted experiments in which he looked for the spreading of narrow light beams, with the logic that they should spread if light was made of waves. He observed no spreading, so he concluded that light was made of particles. While referring to these views in a letter to Robert Hooke, a proponent of the wave theories, he famously wrote “if I have seen further, it is by standing on the shoulders of giants.” This is typically interpreted as showing Newton’s humility, showing that even such a great scientist as himself has built upon others’ work. However, the context of this phrase suggests that it was actually a polite dismissal of the wave proponents, claiming that his particle ideas were better because they were based upon the work of more important scientists4 . Newton published his particle view of light in 1704 (not coincidentally, just after Hooke died), arguing in favor of a particle theory for light. Due to his scientific prowess across all areas of physics, this largely shifted the scientific consensus back to particles. Figure 1.7 Isaac Newton, a proponent of the particle theory of light.
Nevertheless, this did not stop the accumulation of evidence in favor of the wave theory. In 1803, Thomas Young projected light through two parallel slits and found that it bent around behind the slits and then created a pattern of light and dark stripes on a screen (Figure 1.8). This could be explained with a wave interpretation of light but not a particle interpretation. The bending of waves around the slit edges is called diffraction and the combination of the light from the two slits to create the light and dark stripes is called interference.
4 The letter reads, in part, “What Des-Cartes did was a good step. You have added much several ways, and especially in taking the colours of thin plates into philosophical consideration. If I have seen further it is by standing on the shoulders of Giants. But I make no question but you have divers very considerable experiments besides those you have published, ...” (Letter from Isaac Newton to Robert Hooke, 1675, available at https://digitallibrary.hsp.org/index.php/Detail/objects/9792)
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1 Theories of Light
Figure 1.8 (Left) Thomas Young. (Right) Pattern on light and dark lines produced by shining light through two narrow slits, arising from diffraction and interference.
While performing research on electricity and magnetism, which seemed at the time to be completely separate from research on light, the Scottish scientist James Clerk Maxwell made the remarkable discovery that electric and magnetic fields could propagate as waves (Figure 1.9). Furthermore, he calculated that these waves would propagate at same speed as that of light which suggested, correctly as it turned out, that light was an electromagnetic wave. It also led to the concept of other electromagnetic waves, including what we now call radio waves.
Figure 1.9 (Left) James Clerk Maxwell. (Right) Maxwell’s description of an electromagnetic wave.
Heinrich Hertz experimentally verified these ideas a few years later. He produced radio waves by making an electric spark with one piece of equipment, those waves propagated across his laboratory, and they then induced a spark in a completely separate piece of equipment. By the end of the 1880s, when Hertz had completed these experiments, there was no remaining doubt that light could be correctly described as being a wave. Like other types of waves, light refracted when it changed speeds, it diffracted when it went through narrow slits, and it interfered to create light and dark stripes. Also, it had been mathematically described as a wave with the theory of electricity and magnetism, and those explanations had been experimentally confirmed. At long last,
1.4 Particle-Wave Duality
9
the debate between particles and waves was over, with the wave theory being the decisive winner. Or so it seemed.
1.4
Particle-Wave Duality
In the 1890s, just a few years after Hertz had confirmed the wave nature of light, Max Planck was investigating the colors of light that are emitted by hot objects, which is called thermal radiation (Figure 1.10). After extensive effort, he came up with an equation that agreed with the experimentally observed emission essentially perfectly, where the close agreement suggested that his equation was probably correct. However, almost everyone, including Planck, thought that his equation had to be wrong because it was based on the assumption that light could only be emitted in integer amounts, which didn’t make sense.
Figure 1.10 (Left) Max Planck. (Right) Emission spectra of hot objects, such as light bulb filaments.
Albert Einstein (Figure 1.11), an unknown Swiss physicist who was unsuccessfully trying to find a faculty position at the time, didn’t worry about what made sense, but followed where the equations led. In this case, he made the bold proposal that perhaps Planck’s integer amounts of light actually represented physical particles of light. Planck disagreed, writing “The theory of light would be thrown back not by decades, but by centuries, into the age when Christiaan Huygens dared to fight against the mighty emission theory of Isaac Newton...”5 . However, Einstein wasn’t claiming that light was only particles, but instead that light was both a particle and a wave at the same time, which is now called particle-wave duality. This made even less sense, but there was strong evidence that it was correct because it immediately solved several other problems at the time. Most importantly, it solved the photoelectric effect,
5 Quoted from F. Todd Baker, “Atoms and Photons and Quanta, Oh My!: Ask the Physicist about Atomic Nuclear, and Quantum Physics”, Morgan & Claypool Publishers, 2015, p. 2-4.
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1 Theories of Light
in which ultraviolet light can knock electrons off metal but visible light cannot. Similarly, it explains why we get sunburns from ultraviolet light but not from visible light. Figure 1.11 Albert Einstein.
In another astonishing development, the French physicist Louis de Broglie argued that if light can be both a wave and a particle at the same time, then perhaps all “normal” particles are also waves. In particular, perhaps electrons, which were already known at the time to be tiny negatively charged particles, were both particles and waves. This far-fetched proposal was experimentally verified three years later when it was shown that electrons could exhibit diffraction and interference, just like other waves. The wave nature of particles was further developed over the next several decades to create the field of quantum mechanics. It enabled physicists to finally answer a long list of questions that had eluded scientists for centuries, such as why atoms emit specific wavelengths of light (e.g. why neon signs are red), how chemical bonds work, and why metals conduct electricity but non-metals don’t.
1.4.1
Today
Now, a century later, a vast amount of experimental evidence has accumulated that supports the notion of particle-wave duality for both light and matter. These results show that light and matter are non-intuitive in many ways but that it is nevertheless possible to build an understanding of how they behave. Furthermore, this understanding helps us make sense of many phenomena in the natural world, ranging from the shimmering iridescence of hummingbird feathers to the average temperature of the entire Earth. In addition, engineers have used this knowledge to build a tremendous array of modern technology, including lasers, computer chips, fiber optic communication networks, and solar panels.
1.6 Summary
1.5
11
Looking Ahead
Subsequent chapters delve further into these topics, exploring the nature of what light and waves really are. They start with an exploration of waves, including not only light but also string, sound, water, and other waves. We’ll consider waves by themselves, such as how to measure their sizes and how fast they propagate. We’ll then consider interactions between waves, including the diffraction and interference topics that were introduced above and which formed critical evidence in favor of the wave nature of light. We’ll then move on to the interactions between waves and matter, leading to the absorption and emission of waves, followed by a chapter on sound, water, and other mechanical waves. Next, the book takes a break from making sense of waves, and simply investigates what light does, by exploring shadows, reflection, and refraction. These chapters explain how mirrors and lenses work, along with rainbows and many other optical phenomena. We’ll then return to investigating waves in depth, this time focusing primarily on light. This naturally leads into quantum mechanics, a topic that can be quite advanced but is addressed on a more intuitive level here. Finally, we’ll consider the most elusive waves of all, which are waves in the gravitational field.
1.6
Summary
Light has fascinated philosophers and scientists for thousands of years (Figure 1.12). Several theories about light began in roughly the 6th century BCE. The ancient Greeks came up with the extramission theory, in which vision works through emission of light rays from the eyes. Other ancient Greeks, including Aristotle, believed instead in the correct intromission theory, in which people’s eyes only see the light that shines into them. Yet other philosophers believed in a particle theory of light, in which light (and everything else) is composed of tiny indivisible particles.
Figure 1.12 Timeline of different theories of light. Although necessarily imprecise, dark grey regions represent the dominant scientific view and light grey regions represent alternative views.
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1 Theories of Light
In the late 900s, Ibn al-Haytham provided strong evidence for the intromission theory, improved several of the optics theories that the ancient Greeks had developed, and developed the first reasonably correct explanation of human vision. In the 1600s, Descartes showed that a wave theory of light was able to explain refraction. However, Newton denounced it based on his observations that light rays travel in perfectly straight lines, which returned the general consensus back to particles. The wave theory returned to the forefront when Young and others showed that light exhibits diffraction and interference. It was then thoroughly confirmed in the late 1800s by a theoretical explanation of light waves by Maxwell and its validation by radio wave experiments. This conclusion was modified by scientific breakthroughs from the 1890s to 1920s that overturned the understandings of both light and matter. Einstein proposed that light exhibits particle and wave properties simultaneously, called particle-wave duality, which explained Planck’s thermal radiation results as well as the photoelectric effect. Also, de Broglie and others showed that normal particles, such as electrons, are also waves, meaning that they also exhibit particle-wave duality. Further development of these ideas led to quantum mechanics, which explains atomic spectra, chemical bonds, metallic behavior, and other phenomena.
1.7
Exercises
Questions 1.1. What did Ibn Al-Haytham do? (a) correctly explained how vision works (b) promoted the particle explanation of light (c) promoted the wave explanation of light (d) developed the extramission theory (e) explained particle-wave duality 1.2. What did Isaac Newton do? (a) developed key ideas in calculus (b) developed key ideas in physics (c) performed experiments on light (d) promoted the particle theory of light (e) all of the above.
1.7 Exercises
13
1.3. When do people’s eyes emit light beams? (a) when they are angry (b) when they are feeling romantic (c) whenever their eyes are open (d) at night (e) never 1.4. What is light? (a) particles, not waves (b) waves, not particles (c) either waves or particles, but never both at once (d) both waves and particles at the same time (e) still unknown 1.5. What are electrons? (a) particles, not waves (b) waves, not particles (c) either waves or particles, but never both at once (d) both waves and particles at the same time (e) still unknown 1.6. Matching. For each word on the left, give its definition from those on the right. (a) diffraction (1) waves bend when changing speed (e.g. light shining into water) (b) interference (2) waves combine, sometimes forming light and dark stripes (c) refraction (3) waves bounce off surfaces (d) reflection (4) waves bend when going past sharp edges 1.7. What were the dominant beliefs about light during the early Middle Ages (5th to 10th centuries CE)? Choose one for each part. (a) Extramission or intromission theories, (b) finite or infinite light speed, (c) particles or waves? 1.8. Would human hearing be best described as using extramission or intromission methods? 1.9. Give at least two reasons for how we know that the intromission theory is correct and the extramission theory is incorrect. 1.10. List two experimental results that support the wave explanation for light. 1.11. List two experimental results that support the modern particle explanation for light (photons).
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1 Theories of Light
Puzzles 1.12 Ostriches supposedly stick their heads in the sand because they think that they can’t be seen then. Would this work if the extramission theory were true? Discuss. (This isn’t actually true; ostriches don’t actually stick their heads in the sand.)
Part I Waves
2
Properties of Waves
Figure 2.1 Seismogram of the first 30 seconds of California’s 1989 Loma Prieta earthquake.
Opening question A lightning strike creates a flash and “clap” of thunder. Which happen? (a) Both sound and light transport energy; the sound also transports air (b) Both sound and light transport energy; no air is transported (c) The light transports energy but the sound does not (d) Neither the sound nor light transport energy, although the air moves some To better understand what light is, it helps to broaden our scope to investigate other types of waves as well. These include waves along taut strings, waves on the surface of water, and the waves of other bands of electromagnetic radiation, such as X-rays and radio waves. It also includes the wave-like properties of electrons, and the gravitational waves that ripple across the universe. All types of waves involve displacements away from a “normal” state, and all can be characterized by their wavelengths, frequencies, and speeds. However, they also vary. They have different displacement types and different displacement directions. In some cases, they have billion-fold differences between short and long waves, or slow and fast waves. Some waves can propagate through empty space, whereas © Springer Nature Switzerland AG 2023 S. S. Andrews, Light and Waves, https://doi.org/10.1007/978-3-031-24097-3_2
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Properties of Waves
others require a medium. This chapter explores these similarities and differences. In the process, it introduces the terminology and essential mathematics for waves.
2.1
Introduction to Waves
2.1.1
What Is a Wave?
Waves are disturbances that propagate. Consider, for example, shaking one end of a rope when the other end is being held fixed by a friend, a doorknob, or something else, as in Figure 2.2. As you move your hand up and down, you create a disturbance in the rope by displacing it away from the reasonably straight shape that it had initially. These disturbances then propagate away from your hand to become waves that travel along the rope. Figure 2.2 A wave on a rope.
The wave propagates because each tiny section of the rope is attached to the sections that are on each side of it. When a section gets pulled up by the section that’s behind it, it moves upward. This pulls the next section up, and that pulls the next section up, and so on. A little while later, this section gets pulled down. It responds by moving down, which pulls down on the next section, and then the next section, and so on. During this process, each section of the rope only goes up and down a small amount, but the waves propagate along the length of the rope. If you keep shaking the rope, you might notice your arm getting tired. This reflects the fact that your arm is constantly putting energy into the waves, which then carry the energy away from you and along the rope. Thus, another important property of waves is that they transport energy. Eventually, the doorknob, or whatever else the other end of the rope is tied to, might shake loose, again showing the transfer of energy from one end to the other.
2.1.2
Amplitude and Wavelength
Figure 2.3 shows a picture of a wave. This particular wave is a smooth up-and-down oscillation, called a sine wave or sinusoidal wave; we’ll consider more complicated wave shapes later on, in Chapter 3. The tops of the individual waves are called either peaks or crests and the bottoms are called troughs. The height of the peaks above the mid-level, which is the same as the depths of the troughs below the mid-level, is called the amplitude. It is typically represented
2.2 Characterizing Waves
19
Figure 2.3 Wave peaks, troughs, wavelength, and amplitude.
in math equations as A. The amplitude corresponds to the amount of energy in the wave. It is the difference between a gentle ocean swell and a storm-tossed sea; it determines the brightness of light, the loudness of sound, or the amount of shaking in an earthquake. The distance between one peak and the next peak is called the wavelength, represented with the Greek letter λ (lambda). The wavelength could also be measured from one trough to the next trough, or any other two corresponding points on the waves. It is the primary measure for the size of a wave, dictating its behavior, what it interacts with, and how we perceive it. It is the difference between long ocean swells, shorter wind-driven waves, and very short ripples; it is the pitch of a musical note, the color of light, or the type of electromagnetic radiation. The sizes of the things that produce or interact with a wave are often similar to the wave’s wavelength. For example, musical wind instruments, including everything from tin whistles to trombones, produce sound waves with wavelengths that are similar to the instrument’s length; short instruments play high notes with short wavelengths (a few centimeters), and long instruments play low notes with long wavelengths (a few meters). As we will see many times later, we will define the sizes of things, such as mirror roughness, water depth, or sizes of obstructions, based on the dimensions of the object relative to the wave’s wavelength. For example, waist-deep water is shallow when compared to 10 m waves, but deep relative to 10 cm waves.
2.2
Characterizing Waves
2.2.1
Types of Waves
There are many different types of waves. First, there is a large class of waves called mechanical waves, all of which arise from displacements within a physical object that is called the medium. String waves are a type of mechanical wave; these are waves on strings, ropes, wires, and similar media, including the rope waves described above. A plucked guitar string is another example of a string wave. These are particularly simple because they propagate in a single dimension and are easy to create and visualize. Sound waves are mechanical waves in which the air is the medium. When a stereo speaker plays music, it alternately pushes and pulls on the air to displace it away from its normal uniform pressure, leading to pressure waves that propagate away from the speaker, which are sound waves. Seismic waves, which create earthquakes, are like sound waves but have rock as their medium rather than air. They can travel through the upper levels of Earth,
20
2
Properties of Waves
where the rock is solid, or near the middle of the Earth, where the rock is molten. They can also travel across the surface of the Earth, rather like water waves. Finally, water waves are mechanical waves that have water as their medium. These are the familiar waves that we see on puddles, lakes, the ocean, and even a plain glass of water. All of these mechanical waves are transmitted through a medium, where the medium is displaced back and forth slightly as each wave passes through it, but the medium does not move as a whole. For example, sound is different from wind; sound is air moving back and forth, while wind is air moving in a constant direction. Likewise, ocean waves that crash on a beach might have been produced by a storm a thousand miles away, but the water molecules that wash against the shore were never in that storm. Instead, the storm’s energy got transmitted to those molecules through water waves. Mechanical waves clearly cannot exist outside of their media. A string wave cannot propagate past the end of the string and a seismic wave cannot propagate outside of the Earth. Explosions in deep space are always silent. Electromagnetic waves, including light, are not mechanical waves and do not require a medium. Instead, they are oscillating electric and magnetic fields, each one of which creates the other one. In their case, the “normal” state of some region of space is to have no electric or magnetic fields (or, more precisely, electric and magnetic fields that don’t change over time). Suddenly perturbing one of these fields, such as by moving a bunch of electrons from one place to another, creates a disturbance that then propagates away from that region as an electromagnetic wave. Based on analogies with mechanical waves, scientists assumed for many years that electromagnetic waves would require a medium as well, so they came up with the concept that all of space was filled with a light-transmitting medium called the luminiferous aether. Maxwell expressed this assumption by writing “The undulatory theory of light also assumes the existence of a medium”1 . However, it was troubling that this supposed aether didn’t appear to exert any friction on the planets’ orbits and then, more importantly, multiple attempts to detect the effect of the aether on the motion of light failed. Eventually, scientific consensus shifted to the conclusion that electromagnetic waves do not require a medium. We now know that outer space is truly empty, and that light waves travel through it perfectly well. Electromagnetic waves include radio waves, microwaves, infrared light, visible light, ultraviolet light, X-rays, and gamma rays, shown in Figure 2.4. For convenience, we will sometimes call all of these light, where visible light is the electromagnetic radiation that we can see and invisible light is the electromagnetic radiation that we cannot see. Matter waves are the wave natures of electrons, atoms, and other physical particles. These are not waves in matter, which are just familiar mechanical waves, but the quantum mechanical wave nature of the matter itself. In other words, an electron is both a particle and a wave at the same time, and its wave description is called a matter wave. As with light waves, matter waves are non-mechanical waves and do not require
1 James
Clerk Maxwell: “A Treatise on Electricity and Magnetism/Part IV/Chapter XX”, 1873.
2.2 Characterizing Waves
21
Figure 2.4 Electromagnetic spectrum, showing bands and wavelengths.
a medium. Matter waves are typically important for extremely small particles, like electrons, protons, and individual atoms but irrelevant for larger objects. Even a single molecule is generally large enough that its matter waves can be ignored. Although the mathematical descriptions of matter waves have been thoroughly worked out and those predictions invariably agree with experiments, the exact nature of what matter waves are still remains mysterious. Nevertheless, if we build on our statement that a wave is a disturbance that propagates, then this shows that the “normal” state for a region of space is that there is nothing there, and that the matter wave displaces this result to the possibility something might be there. We will largely postpone discussing matter waves until Chapter 14. Gravitational waves are yet another type of non-mechanical wave, and again do not require a medium2 . Gravitational waves are bizarre ripples in space itself or, more accurately, in space-time. Einstein predicted their existence in 1916, showing that they must arise whenever any object that has mass, meaning any object at all, is accelerated. This means that accelerating cars, falling boulders, and orbiting planets all produce gravitational waves. However, they are extraordinarily weak and so only become measurable when masses and accelerations are extremely large. The first experimental evidence for gravitational waves was not found until 1974, when two astronomers discovered that a pair of neutron stars that orbited each other lost energy at a rate that was consistent with it being carried away by gravitational waves. After a long search, gravitational waves were first detected directly in 2015. Again, we will largely ignore this topic for now, but will return to it in Chapter 15.
2 Note that gravitational waves should not be called “gravity waves.” This is because ordinary water waves were called “gravity waves” long before gravitational waves were conceived of, so the term “gravity waves” still refers to water waves.
22
2.2.2
2
Properties of Waves
Transverse and Longitudinal Waves
It can be helpful to categorize waves by considering the direction of the displacement relative to the direction of wave propagation. There are two options: waves are transverse if the displacement is perpendicular to the propagation direction and longitudinal if the displacement is parallel to the propagation direction (Figure 2.5). Thinking back to the rope example from before, those were transverse waves because the rope was displaced up and down while the waves travelled forward, which are perpendicular axes.
Figure 2.5 Transverse and longitudinal waves.
All transverse waves have two possiblepolarizations, describing the displacement direction. In the rope example, it was shaken up and down, so the rope was displaced vertically and those waves were vertically polarized. If it had been shaken left and right instead, then the waves would have been horizontally polarized. Figure 2.5 shows both possibilities. Light waves are another example of transverse waves, in their case because both their electric and magnetic fields, which are their displacements, are perpendicular to the direction of wave propagation (see Figure 1.9). Light waves can also be polarized either vertically or horizontally, which is defined in their case by whether the electric field goes up-and-down or left-and-right (the electric field was chosen because it’s typically more important than the magnetic field). Longitudinal waves occur when there is compression and spreading along the direction of the wave propagation. A Slinky toy that is first stretched out on a table and then given a quick push at one end forms a nice example. The quick push creates a compressed region that travels the length of the Slinky as a longitudinal wave. Longitudinal waves cannot exhibit polarization because there is only one possible axis for the displacement. Sound waves are a particularly important type of longitudinal waves. They can be created by a stereo speaker alternately pushing and pulling on air, much like a person creates Slinky waves by pushing and pulling on a Slinky. Water waves are interesting because they are both transverse and longitudinal. The transverse part is obvious; if you observe water waves, it’s clear that the displacement goes up and down while the waves move forward, so the displacement is perpendicular to the direction of propagation and they are transverse waves. Considering the longitudinal part, suppose you watch a cork or some other object that is floating on the water as waves pass by it. You would see that the cork doesn’t just go
2.2 Characterizing Waves
23
up and down but actually moves in a circle, going up, then forward, then down, and then backward, round and round. The forward and backward motions represent the longitudinal components of water waves. Seismic waves are even more complicated, exemplifying all of these wave motions. They are generated when rocks scrape past each other deep underground, at the earthquake hypocenter or earthquake focus3 . One outcome of this is that it creates sudden pressure changes in the surrounding rock, which then propagate outward through the rock as pressure waves. Except for the fact that the medium is rock rather than air, they are just like sound waves and, likewise, are longitudinal waves. These pressure waves are called P-waves, which is officially short for primary-waves because they travel faster than other seismic waves and so are usually the first to get detected; however, it can be easier to remember them by thinking of P-waves as standing for “pressure-waves.” The rock sliding motion also creates side-to-side displacements in the surrounding rock, which propagate outward as transverse waves. These are called S-waves, which officially stands for secondary-waves because they are the second wave type to get detected; again though, these can be easier to remember by thinking of the “S” as standing for “shear,” “sliding,” or “sideways.” Like all transverse waves, S-waves have two possible polarizations, vertical and horizontal. Both the P-waves and S-waves change when they get to the surface of the Earth. There, they become yet different types of waves and then travel along the Earth’s surface with both transverse and longitudinal motions, rather like water waves.
2.2.3
Waves in 1, 2, and 3 Dimensions
Transverse waves on strings, and longitudinal waves on a Slinky, areone dimensional waves, meaning that they take place along a relatively long and thin object. The fact that waves displace the string away from a straight line doesn’t affect the fact that they are still considered to be one dimensional. Water waves are two dimensional waves or surface waves because they take place at the two-dimensional surface of the water (Figure 2.6). Again, the waves displace the surface away from being totally flat, but these waves are still called two dimensional. The membrane head of a drum also supports two dimensional waves, Figure 2.6 Transverse surface wave with vertical polarization.
3 The
earthquake epicenter is the point on the Earth’s surface that is directly above the hypocenter.
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2
Properties of Waves
Table 2.1 Examples of different types of waves 1-dimensional
2-dimensional
3-dimensional
transverse
string
drum head
longitudinal transverse and longitudinal neither
Slinky
light, seismic S-wave, gravitational sound, seismic P-wave
water, seismic surface wave matter
as does the stretched rubber of an inflated balloon or essentially any other elastic surface4 . Light waves, sound waves, and seismic P- and S-waves are three dimensional waves because they travel through three-dimensional space. There are no more dimensions, so their displacements are necessarily in the same three dimensions that the waves propagate through. Table 2.1 summarizes these properties for the different types of waves.
2.3
Speed and Velocity
2.3.1
Speed and Velocity Equations
If we watch the peak of a wave as it propagates, we can see how fast it moves, which is the wave speed. Speed, or more precisely the average speed, is defined as the distance that something travels, divided by the amount of time it took, average speed =
distance . time
For example, if a car drives 80 miles in 2 hours, then its average speed is (80 miles)/(2 hours) which is 40 miles per hour. Waves are the same; if a wave peak moves 8 meters in 2 seconds, then its average speed is 4 m/s. If the object or wave moves at a constant speed during the time interval, meaning that it doesn’t slow down or speed up, then we can drop the word “average” and just call the result the “speed”. Velocity is essentially the same thing as speed, but with the addition of direction information. For example, we could say that a car has a positive velocity when driving forward and a negative velocity when driving backward. Also, if a beam of
4 Two dimensional waves can be transverse (drum head), longitudinal (no simple example), or both transverse and longitudinal (water waves). The transverse component of water waves is always polarized vertically because water flows and so cannot resist shear (sliding) displacements. However, it is possible to create a horizontally polarized transverse wave in a drum head: if you glue a handle to the middle of the drum, rotate the handle in the plane of the drum head, and then let go, the rotation motion will propagate outward as a horizontally polarized wave.
2.3 Speed and Velocity
25
light is shining straight up, then it would be correct to say that the light’s velocity is the speed of the light and it is in the direction of straight up. We will use this distinction between speed and velocity fairly casually here, but it’s worth paying attention to which way things are traveling to ensure that the numbers are used correctly. In mathematical notation, we write speed or velocity as v, the distance traveled as d, and the time duration as t. The symbol (Greek delta) shows that this is the difference of the final value minus the initial value. With these definitions, the velocity is v=
d f inal − dinitial d . = t t f inal − tinitial
(2.1)
For example, suppose a woman is watching fireworks that are 1 km away and she finds that it takes about 3 seconds between seeing the flash of an explosion and then hearing its sound. She realizes that the light travels so much faster than the sound that she can treat the light as taking no time at all. Thinking about the sound, it travels distance d = 1000 m over time t = 3 s, from which she computes that m the speed of sound is about v = 1000 3 s = 333 m/s. (A better answer would have been 300 m/s because the starting values were only known to 1 digit of precision, so the answer should also only use 1 digit of precision; this topic is explained in Appendix A.) Example. A cross-country ski racer went 15 km in 48 minutes. What was her average speed in m/s? Answer. Start by converting units to the ones you want. To do this, rewrite 15 km as 15 · 103 m, where the “kilo” prefix became 103 . Also, multiply the 48 minutes by 60 seconds/minute to get 2880 seconds. Then, use Eq. 2.1: v=
d 15 · 103 m = = 5.2 m/s. t 2880 s
This is about 12 miles/hour. The velocity equation can also be used to compute how far something travels over a given amount of time, if you know the speed. To do this, we need to solve Eq. 2.1 for d. Do so by multiplying both sides of it by t to give d = vt.
(2.2)
For example, a 5 hour flight in an airplane (t) traveling at 550 miles/hour (v) covers a distance of d = (5 hr) · (550 miles/hr) = 2750 miles.
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Properties of Waves
Yet another rearrangement allows the velocity equation to be solved for the elapsed time in terms of the velocity and distance. Do this by dividing both sides of Eq. 2.2 by v to give d . (2.3) v If an earthquake occurs 100 km away from you (d) and the seismic wave travels at 6 km/s (v), then you will feel the shaking in t = (100 km)/(6 km/s) = 17 seconds. These three forms of the velocity equation are useful because they allow any of the three values to be calculated from the other two, so you can always solve for the parameter that you don’t know. These rearrangements are simple enough that you shouldn’t need to memorize all three forms of the equation, but can just learn the first one, v = d/t, and then find the others as you need them. The fact that the three values here all have units is helpful because they can remind you about which term goes where. For example, the left side of Eq. 2.2 is d so it’s in meters, and the right side is vt so it’s in ms s, which simplifies to meters. Both sides are the same, confirming that this equation could be correct, and in fact is correct. On the other hand, if the units on the two sides of the equation were different from each other, then you would know that it was wrong. This is a good time for a brief tour of the back of this book. There are four appendices that help with math. Appendix A describes how to use scientific notation and significant figures, Appendix B covers unit mathematics and unit conversion, and Appendix C reviews basic algebra manipulations. You may want to review this material if it isn’t familiar to you. Also, Appendix D presents a few geometry results, none of which will be relevant for a while. In addition, Appendix H presents “Useful Facts and Figures”, such as the speed of light, the speed of sound, the electromagnetic spectrum, and so on, all of which will be useful when doing problems. t =
2.3.2
Speed of Light
The speed of light in vacuum, for all wavelengths and all brightnesses, is exactly 299,792,458 m/s. This is a sufficiently important value in physics that it has been assigned the letter c, c = 299, 792, 458 m/s ≈ 3.00 · 108 m/s.
(2.4)
This value is exact because the length of a meter is defined to be the distance that light travels in 1/299,792,458 seconds. More importantly, it is worth memorizing that the speed of light is very close to 3.0 · 108 m/s. In units that may be more intuitive, light travels about 186,000 miles per second, 6.7 · 108 miles per hour, or can go about 7.5 times around the Earth in one second. Using much smaller units, the speed of light is very close to 1 foot per nanosecond (10−9 seconds). Not only is the speed of light exceedingly fast, but Einstein’s theory of relativity showed that it is physically impossible for anything to travel faster than light. In fact,
2.3 Speed and Velocity
27
things that have mass, which is essentially everything except for light itself, cannot even go as fast as the speed of light. Despite being very fast, the speed of light is still slow enough to have important consequences. The fact that electrical signals cannot travel faster than light limits the speed at which computers can work. It also produces noticeable delays in telecommunication that uses satellites. In addition, light is slow enough that it takes many minutes between the time when scientists on Earth send signals to rovers on Mars and when the rovers receive those signals, and then many more minutes for the rover’s response to get back to Earth. It also means that the light that we see from stars now was emitted from them many years ago or, often, many centuries ago. In fact, the farthest galaxy that has been seen emitted its light 13.4 billion years ago, when the universe was only 400 million years old, effectively allowing us to see far back into time5 . Example. Suppose a telephone signal is transmitted using microwaves by a satellite that is in a geostationary orbit, in which the satellite is 36,000 km above the Earth’s equator. Assuming that the satellite is this distance away from both the person sending the signal and the person receiving the signal, how much delay is there between when the signal is sent and when it is received?
Answer. The signal has to travel a distance of 2 · 36, 000 km, which is 72, 000 km or 72, 000 · 103 m. Microwaves are a form of light, so they travel at the speed of light, which is 3.0 · 108 m/s. We know the distance and speed, so solve the velocity equation for time to get t = d/v (Eq. 2.3). Plugging in numbers, the time delay is t =
72000 · 103 m = 0.24 s. 3.0 · 108 m/s
This quarter second delay isn’t a lot, but it’s enough to make a conversation difficult.
5 The
galaxy, named GN-z11, was first reported in 2016. Remarkably, it is 32 billion light-years away from Earth, despite being only 13.4 billion years old and the universe being only 13.8 billion years old. The extra distance arose from the universe’s expansion. Although still unconfirmed, an object called HD1, discovered in 2022, is likely to be an even more distant galaxy, at a light travel distance of 13.5 light-years.
28
2.3.3
2
Properties of Waves
Measuring the Speed of Light*
How do we know the speed of light when it goes so much faster than essentially anything else? The answer, as is typical in science, is through a combination of good insights, clever experiments, and perseverance. Galileo, like many others, knew that one saw the flash of a distant explosion before hearing the bang, which implied that light travelled faster than sound. It was not known at the time whether light was just very fast or travelled at infinite speed. To test this, he described an experiment in 1638 in which he and an assistant took lamps to two locations that were separated by less than a mile. Galileo uncovered his lamp and started measuring time; when the assistant saw Galileo’s light, he uncovered his lamp; then, when Galileo saw his assistant’s lamp, he stopped the timer. He had hoped to measure the speed of light by combining the elapsed time with the known distance between the locations, but his result was inconclusive due to the imprecise experiment. He wrote “I have not been able to ascertain with certainty whether the appearance of the opposite light was instantaneous or not; but if not instantaneous it is extraordinarily rapid”6 . Almost 40 years later, in 1675, the Danish astronomer Ole Rømer was the first person to measure the speed of light accurately. He was attempting to improve on a different aspect of Galileo’s work, which was on the orbit of Jupiter’s moon Io. Rømer was timing the periodic emergence and disappearance of Io as it orbited around Jupiter and found that the orbit times varied slightly over the course of a year, with longer ones when Earth and Jupiter were moving apart and shorter ones when the planets were moving together. He realized that this could be explained by light traveling at a finite speed and taking longer to reach Earth when the planets were further apart. From his data, he computed that light takes 22 minutes to travel a distance equal to the diameter of the Earth’s orbit. Current data show that light actually takes about 16 minutes and 40 seconds to travel this distance, showing that Rømer was reasonably close. James Bradley, an English physicist, improved upon Rømer’s value in 1728, when he investigated a phenomenon called stellar aberration. Much as raindrops that are actually falling straight down appear to fall toward us when we are running forward, it turns out that stars that are actually overhead appear to be slightly in front of the Earth when the Earth is moving around on its orbit. By measuring this stellar aberration, which is a function of the speed of light, Bradley computed the speed of light to within 1% of the current value. Finally, more recent experiments have returned to Galileo’s more direct approach but with better equipment, generally timing light travel from one point to a distant mirror, and then back to a detector. In the first such experiment, in 1849, the French physicist Hippolyte Fizeau aimed a narrow beam of light at the teeth of a spinning gear (Figure 2.7). As the gap between two teeth swept across the light beam, it let a brief pulse of light through. This light pulse travelled 8 km across the city of Paris,
6 Galileo,
Dialogues Concerning Two New Sciences, 1638.
2.3 Speed and Velocity
29
Figure 2.7 Schematic of Fizeau’s equipment for measuring the speed of light.
was reflected by a mirror, and travelled back again. If the gear was spinning at exactly the right speed, the gap between the next pair of teeth on the gear would be at the right position just as the pulse of light returned from the mirror, so the light would get through again and would be visible. However, if the gear was spinning too slowly or too quickly, then the reflected light would hit a gear tooth and would not be visible. Fizeau was able to determine the speed of light very accurately by measuring the gear rotation rate and the distance between the mirrors. Improvements on this experiment, and other experiments, led to very precise determinations of the speed of light by the 1970s, at which point the greatest error became the definition of the meter. For this reason, the meter was redefined using the speed of light, leading to the value given in Eq. 2.4.
2.3.4
Speed of Light in a Medium
Light travels more slowly than its vacuum speed, 3.00 · 108 m/s, when it travels through a substance, such as air, water, or glass. Confusingly, this substance is called the medium that light is traveling through, using the same terminology as the medium for mechanical waves. In other words, the term “medium” has two different meanings: the one we encountered first, which was the physical object that is displaced as mechanical waves pass through it, and the new one, which is the substance that light is shining through. The medium, in this latter sense, is not required for light propagation, but slows the light down. The amount that it slows light is called the medium’s refractive index, given as n. The velocity of light in the medium, v, is v=
c . n
(2.5)
As always, c is the speed of light in vacuum, 3.0 · 108 m/s. Note that n is unitless, meaning that it doesn’t have any units at all; it’s just a number, like 5, -3, or π. In this case, n ranges from 1 to about 3 or 4.
30
2
Properties of Waves
Table 2.2 Refractive indices of several media Medium
n
vacuum air water glass sapphire diamond
1 1.0003 1.33 1.52 1.77 2.42
Table 2.2 lists the refractive index values for several media, for visible light7 : The refractive index of air is so close to 1 that we will usually ignore the difference and just treat the speed of light in air as 3.0 · 108 m/s, just like it is in vacuum. However, the other refractive index values are much larger, so do need to be considered if light is traveling in those media. Example. What is the speed of light in glass? Answer. Using the refractive index definition, v = c/n, and the refractive index value for glass of n = 1.52, the speed of light in glass is v=
3.0 · 108 m/s = 2.0 · 108 m/s. 1.52
This is 66% of the speed of light in vacuum, which is a substantial difference. The fact that light changes speed as it propagates from one medium to another can cause light rays to reflect and/or refract. We will explore reflection in Chapter 8 and refraction in Chapter 9. The refractive index concept, meaning how much waves are slowed down by a particular medium, is equally valid for all types of waves, but the “refractive index” term is typically only applied to light and other types of electromagnetic radiation.
2.3.5
High Frequency Stock Market Trading and the Speed of Light*
The book Flash Boys8 by Michael Lewis, tells an interesting story that hinges on the speed of light. It concerns several high frequency stock market trading approaches in
7 These
refractive index values are nearly independent of wavelength but not quite; the wavelength dependence is called dispersion and is discussed in Section 9.5. 8 Michael Lewis, Flash Boys, 2014, W.W. Norton and Co., London.
2.3 Speed and Velocity
31
Figure 2.8 Map of fiber optic cable from Aurora, Illinois to Carteret, New Jersey.
which traders make money off slight price differences at different stock exchanges. “Front running,” for example, is an illegal but effective way to make a lot of money. In a simple example, a trader hears of a large order to buy a stock (or other financial security) and then buys some of that stock himself before the order is actually executed. When the large order is executed a moment later, it raises the stock price. At this point, the trader sells the stock that he just bought at the higher price, making a small amount of profit in the process. With enough such trades, the trader can amass a large fortune. Front running is illegal because it defrauds legitimate investors. High frequency traders in Chicago wanted to front-run security orders that originated in nearby Aurora, Illinois (at the Chicago Mercantile Exchange) but were being executed in New Jersey (at the Nasdaq data center). These orders were being sent on commercial fiber optic cables in which the signal was carried by light traveling through glass fibers. The traders realized that the commercial fiber optic cables did not follow a straight line between the two exchanges, meaning that a new cable that more nearly followed a straight line would get a signal to New Jersey even faster because the light would have a shorter distance to travel. So, they spent about $300 million to install a private 827 mile long fiber optic cable that was as direct as possible, going under rivers and through mountains as needed (Figure 2.8). Doing this reduced the round-trip signal time from about 17 milliseconds to about 13 milliseconds. This time saving was sufficient to front-run orders, allowing the traders to make a fortune. This cable has become obsolete at this point because there are now microwave links between the two exchanges. The microwaves, which also travel at the speed of light, are even faster for two reasons. They take an even straighter route because they go through the air and never have to deviate around physical barriers. Also, and more importantly, the refractive index of air is about 1 rather than about 1.5, which means that the microwaves travel about 50% faster. These new links reduce the signal round-trip time to about 9 milliseconds. By comparison, using the speed of light and the 700.0 mile distance between the two stock exchanges, it is easy to use Eq. 2.1 to compute that the fastest possible round-trip time is 7.5 milliseconds. Similar very fast data links are still being built and upgraded for high frequency trading between all of the world’s major stock exchanges.
32
2
2.4
Frequency and Period
2.4.1
Cars on a Road Analogy
Properties of Waves
Consider the following scenario. You are standing by a road watching a steady stream of cars passing by, all going the same speed (Figure 2.9). A car passes every T seconds and the spacing between cars is λ meters. What is the cars’ speed? (Try it.)
Figure 2.9 Cars on a road, for showing relationship among wavelength, frequency, and velocity.
The solution is to think about one car. It takes T amount of time to travel the distance that is between the cars, λ. We know the distance and the time, so use the definition of velocity, v = d t , to find the velocity. The answer is v=
d λ = . t T
For example, if the spacing is 90 m and cars pass every 3 s, then the speed is 90 m 3 s = 30 m/s. The time between cars passing you, T , is called the period, where the period represents the time that each car takes to drive up to where the previous car used to be. Alternatively, the frequency is the number of cars that pass by you in some fixed amount of time, such as a second or a minute. The frequency is the reciprocal of the period, 1 . T For example, if the period is 3 seconds per car, then the frequency is 0.33 cars per second. Converting units makes this 20 cars per minute, which might be more useful. Substituting this reciprocal relationship into the previous equation leads to an equation that relates the velocity to the spacing and frequency, f =
v = λf. Using the prior numbers, we multiply the car separation, λ = 90 m, by the frequency, 0.33 cars per second, to get the speed as 90 · 0.33 = 30 m/s, exactly as before. The point of this analysis is to build a simple analogy for waves. Each car represents a wave peak and the separation between cars represents the wavelength. The variables in the above equations represent essential wave parameters. However, the fact that we derived these equations with cars shows that they are not specific to
2.4 Frequency and Period
33
waves, but instead arise simply from the definition of velocity as distance divided by time. Indeed, these equations apply to absolutely anything that progresses in a periodic fashion.
2.4.2
Relating Velocity, Frequency, and Wavelength
We’ll repeat the same calculations but with waves in mind. The wave period, T , is defined as the amount of time that it takes for one full wave to pass a fixed point or, equivalently, the time for a wave peak to travel distance λ. Alternatively, considering the example of waves on a rope, the period is how long it takes for a point on the rope to go through one full cycle, such as from up to down and then back to up. Or, for water waves, it is the time between when one water wave hits the shore and when the next wave hits the shore. In any case, combining the definition of velocity with the fact that a wave travels distance λ over time T implies that its velocity is v=
λ . T
The frequency, given by f , is the reciprocal of the period; for example, if a wave has a period of 0.5 seconds (0.5 seconds for one wave to pass), then the frequency is 2 waves passing per second. As an equation, f =
1 . T
(2.6)
Finally, combine these two equations to relate the velocity to the frequency and wavelength, yielding an equation that we’ll call the frequency-wavelength relationship9 . v = λf.
(2.7)
This equation relates the three most important properties of a wave, namely its velocity, wavelength, and frequency. This equation can also be rearranged to give the wavelength or frequency in terms of the other two parameters. For example, Figure 2.4 shows the electromagnetic spectrum with labels for both wavelength and frequency; these parameters are related through this equation using the speed of light as 3.00 · 108 m/s.
2.4.3
Frequency
Returning to the frequency, it can be listed with several different but equivalent units, all of which are somewhat counter-intuitive. The most natural ones are “waves
is no standard name for the equation v = λ f , but it is called the frequency-wavelength relationship here because this describes it accurately. 9 There
34
2
Properties of Waves
per second”, “cycles per second”, or “revolutions per minute” (rpm), in each case showing explicitly that we are counting the number of oscillations that occur every second or minute. Alternatively, it is permissible to drop the “waves,” “cycles,” or “revolutions” words, making the frequency units simply “per second” or “per minute”. These are often abbreviated as s−1 and min−1 , using the fact that a negative exponent represents division. As yet another option, there is a unit called a hertz10 abbreviated Hz, which is defined to be the same thing as the “per second” unit, 1 Hz = 1 s−1 .
(2.8)
The hertz unit is often given with a metric prefix, such as kHz for a kilohertz, which is a thousand hertz, or MHz for megahertz, which is 106 Hz. As a frequency example, the “middle C” note in the musical scale has a frequency of 262 Hz. This means that those sound waves oscillate 262 times per second. Also, a radio station that broadcasts at 99.3 MHz emits radio waves that oscillate with a frequency of 99.3 · 109 cycles per second. Much as the wavelength describes the size of a wave, the frequency describes its rate. The frequency often indicates the types of things that the wave interacts with. For example, tuning forks, used for tuning instruments and singers’ voices, are often designed to vibrate with their tines (the prongs) going together and apart at 440 times per second. This causes them to produce sound with a frequency of 440 Hz. Example. The middle C note of the musical scale has a frequency of 262 Hz. What is its wavelength? Answer. Rearrange the frequency-wavelength relation, v = λ f , to solve for frequency, giving λ = vf . This problem is about sound waves, so v = 340 m/s. Plug in f = 262 Hz to get λ=
340 m/s = 1.30 m. 262 s−1
Note how the “per second” unit in the numerator was the same as the s−1 unit in the denominator, causing both to cancel out, leaving just meters. This answer is about 4 feet long. Not coincidentally, it is about twice as long as a wind instrument that can play this note, such as a flute.
2.4.4
Frequency and Wavelength in a New Medium
Consider a beam of light that is shining from air into water. The light slows down as it enters the water due to the increased refractive index. This raises the question
10 Named
for Heinrich Hertz, who proved the existence of electromagnetic waves.
2.4 Frequency and Period
35
Figure 2.10 Cars slowing down when they drive through a bumpy region.
Figure 2.11 Wave that slows down when it goes through a slow region.
of what happens to the light wavelength and frequency. One of them must change because the frequency-wavelength relation is v = λ f and the value on the left side decreased, so one of the values on the right side must decrease as well. Which is it? To answer this question, think back to the cars on the road introduced above, but now suppose they come to a bumpy section of road where they have to go slower (Figure 2.10). The cars must enter and leave this bumpy section at the same rate. If this weren’t the case, say more cars entered than exited, then the number of cars would build up in the bumpy section. Or, vice versa, if cars left faster than they arrived, then they would have to be coming from some magical source of cars within the bumpy section. The same logic applies everywhere else along the road, whether in the bumpy section or not, showing that the car frequency must be the same at all points along the road. Thus, the frequency does not change, despite the change of velocity. Since the frequency is constant, the wavelength (i.e. the spacing between the cars) must change. Indeed, this is a common experience. When we drive into a construction area, cars slow down and become closer together; then, when we leave, cars speed up and spread out. Waves work in the same way. When light shines from air into water, the light wave frequency does not change but the wavelength becomes shorter. When the light then shines back into air, the frequency still stays the same and the wavelength increases back to where it was before (Figure 2.11). We can solve for the wavelength change using the frequency-wavelength relation. In the fast region, meaning the cars are on the fast part of the road or the light is v . Likewise, in air, the wave velocity equation gives the frequency as f f ast = λ ff ast ast in the slow region, where the cars are driving slowly or the light is going through water, f slow = λvslow . Applying the insight that f slow = f f ast allows us to set the slow two frequencies equal to each other, which leads to v f ast vslow = . λ f ast λslow
(2.9)
This can be rearranged as needed. For example, the wavelength in the slow region is λslow = λ f ast
vslow . v f ast
36
2
Properties of Waves
In words, this shows that the wavelength decreases in proportion to the ratio of the two speeds.
Example. A green laser pointer has a wavelength of 532 nm. What is its wavelength in water, where the refractive index is 1.33? Answer. The speed of light in air is v f ast = 3.0 · 108 m/s. In water, the speed is vslow = nc from Eq. 2.5, which evaluates to vslow = 2.26 · 108 m/s. Plugging these into the rearranged version of Eq. 2.9 shows that λslow = λ f ast
vslow 2.26 · 108 m/s = (532 nm) = 400 nm. v f ast 3.0 · 108 m/s
Thus, light waves are substantially shorter in water than in air. Since light wavelengths are substantially shorter in water than in air, you might be wondering how this affects their appearance. In other words, does the perceived color of light arise from its wavelength or its frequency? The direct answer to this question is to use common experience. If we put a green object underwater, it still looks green. If we then stick our heads in the water as well, it still looks green. Thus, changing the wavelength does not affect the colors we see, implying that we see color based on frequency and not on wavelength11 . From this, it would be more sensible to talk about the different colors of light based on their frequencies rather than their wavelengths, but we still typically use wavelengths for historical reasons.
2.5
Summary
Waves are disturbances that propagate. Waves carry energy but they do not transport matter. High and low points of waves are called peaks (or crests) and troughs, respectively. The heights of the peaks above the mid-level, or the depths of the troughs below the mid-level, is called the amplitude (A), which is a measure of the wave strength. The distance between two peaks or two troughs is called the wavelength (λ). The wavelength is the principal measure of the wave size and often reflects the sizes of things that the waves can interact with.
11 This
experiment isn’t actually a good test because the light always has to enter our eyes before we can detect it, so its wavelength within our eyes isn’t affected by its wavelength outside of our eyes. This also means that the entire question of whether light appears different in water or air doesn’t really make sense. Nevertheless, it’s still correct to say that the color of light arises from its frequency but for a different reason: the retinal molecules in our eyes, which detect light, respond to the light’s electric field oscillation frequency, and not to its wavelength.
2.5 Summary
37
Mechanical waves arise from disturbances in a physical medium. These include string waves, sound waves, seismic waves, and water waves. Non-mechanical waves can propagate without a medium and include light waves, matter waves, and gravitational waves. Waves can also be classified by the direction of the disturbance relative to wave propagation. Waves are transverse if the displacement is perpendicular to the direction of propagation (string, seismic S-waves, and light) or longitudinal (sound, Slinky, and seismic P-waves) if the displacement is parallel to the direction of propagation. Water waves and seismic surface waves are both transverse and longitudinal at once. Transverse waves can be polarized in either of two ways, which again describes the displacement direction, but longitudinal waves cannot be polarized. Waves can propagate in one, two, or three dimensions. Wave speed or velocity describes how fast the waves propagate, where velocity is defined as d . v= t The speed of light is very nearly c = 3.00 · 108 m/s in vacuum and the speed of sound is about 340 m/s. Electromagnetic waves travel slower in a medium (the “medium” can mean either the physical object that moves for mechanical waves, or the substance that light is shining through) than they do in vacuum. This slowing is given by the refractive index (n), using the equation v=
c . n
The time that a wave takes to undergo one full cycle is called the period (T ). The frequency ( f ) is the reciprocal of the period, f =
1 . T
The frequency can be measured in various units, of which waves per second, s−1 , and hertz (Hz) are all commonly used and all the same. The frequency is the principal measure of the rate of a wave and, as with wavelength, often indicates the things that the waves can interact with. Combining the definition of velocity with the definition of frequency gives the frequency-wavelength relationship, v = λf. This equation relates all three important wave quantities and is useful for converting between wavelength and frequency. When waves propagate from one medium to another, such as light shining from air into water, they generally change speeds. However, they do not change frequency because the waves have to enter and leave the new medium at the same rate. Combining this result with the relationship v = λ f shows that the wavelength must change in the new medium in direct proportion to the change of speed.
38
2.6
2
Properties of Waves
Exercises
Questions 2.1. If you increase the amplitude of a light wave, what changes? (a) the color (b) the brightness (c) the wavelength (d) the refractive index (e) the speed 2.2. If you increase the wavelength of a light wave, what changes? (a) the color (b) the brightness (c) the amplitude (d) the refractive index (e) the speed 2.3. Which light travels fastest? (a) laser light (b) sunlight (c) ultraviolet light (d) microwave light (e) they are all the same 2.4. Will a brighter light wave travel faster than a dimmer light wave (in vacuum)? (a) yes, if the brighter light has a shorter wavelength (b) yes, if the brighter light as a longer wavelength (c) yes, always (d) no, the dimmer light goes faster (e) no, they go the same speed 2.5. Which waves can be polarized (circle all that are appropriate): (a) light (b) sound (c) string (d) water (e) radio 2.6. Consider a green laser beam that goes from water to air. Which happens? (a) the light speed decreases (b) the frequency increases (c) the frequency decreases (d) the wavelength increases (e) the wavelength decreases
2.6 Exercises
39
2.7. Suppose you’re holding the end of a rope. What happens to the wavelength of waves on a rope if you move your hand up and down more quickly? (a) they get longer (b) they get shorter (c) no change (d) depends on the rope thickness (e) depends on the rope length 2.8. John likes to sing to his pet fish, which is in a fish tank. Sound waves travel faster in water than in air. When John sings an ‘A’ note, which has a frequency of 440 Hz, what are the waves like at the fish? (a) the frequency is 440 Hz, and the wavelength is longer (b) the frequency is 440 Hz, and the wavelength is shorter (c) the frequency is less than 440 Hz, and the wavelength is shorter (d) the frequency is greater than 440 Hz, and the wavelength is unchanged (e) the frequency is less than 440 Hz, and the wavelength is longer 2.9. Draw a picture of a wave. Label: a peak, a trough, the wavelength, the amplitude. 2.10. List the medium for each of the following types of waves, or write “no medium” if it doesn’t have one: (a) water, (b) radio, (c) sound, (d) seismic. 2.11. List 2 examples for each: (a) transverse waves, (b) longitudinal waves. 2.12. Measure your heartbeat frequency and describe how you did it. Give your answer with the correct units. Problems 2.13. This problem reviews the metric system, and helps develop a sense of scale for light waves. Use the following answers for the comparisons: “much larger” if it is more than 10x larger, “larger” if it is less than 10x larger, “similar” if the numbers are within about a factor of two, “smaller” if it is less than 10x smaller, and “much smaller” if it is more than 10x smaller. (a) What is the wavelength of green light in nm? How does it compare to the size of the following: (b) a bacterium, which is ∼2 µm long, (c) a water molecule, which is ∼0.2 nm in diameter, (d) a small protein, which is ∼2 nm in diameter, (e) a ribosome, which is a very large protein complex and ∼20 nm in diameter, (f) a cloud droplet, of which ∼20 µm is a typical size, (g) a raindrop, of which ∼2 mm is typical, (h) a soap bubble thickness, of which ∼500 nm is typical? 2.14. Lightning strikes the ground five kilometers away. (a) How long does it take the light to reach you? (b) How long does it take the sound (thunder) to reach you? (The speed of sound is about 340 m/s.) 2.15. In 1964, an earthquake near Anchorage, Alaska triggered a tsunami that traveled across the entire Pacific Ocean. It was detected in Peru, about 6500 miles away from Anchorage, 16 hours after the earthquake occurred. (a) What was
40
2
Properties of Waves
the tsunami speed in miles per hour? (b) What percent of the speed of sound is this (the speed of sound is about 760 mph)? 2.16. In 1976, the world’s fastest airplane, an SR-71 “Blackbird,” flew 3462 miles from New York to London at an average speed of 1807 miles per hour. (a) How long did it take them? (b) How many times faster than the speed of sound did the airplane fly (the speed of sound is about 670 mph at airplane cruising altitudes)? (c) Why is it relevant to compare the airplane speed with the speed of sound? 2.17. The Parker Solar Probe is an unmanned spaceship that was launched in 2018 and will fly close to the sun in order to study it. It will reach a maximum speed of 192 km/s as it nears the sun in 2024, which will be the fastest that any human-built object has moved. (a) How many times faster than the speed of sound will the spaceship fly? (b) Is it useful to compare its speed with the speed of sound; why or why not? (c) At what percent of the speed of light will the spaceship fly? 2.18. A computer central processing unit chip (CPU) runs with a clock speed of 2.7 GHz. It executes one operation in each of these clock cycles. (a) How many seconds long is one clock cycle? (b) Electrical signals travel at the speed of light. How far can an electrical signal travel in one clock cycle? (c) Wires between the CPU’s control unit and its cache memory (both on this chip), are about 2 cm long. How does this compare to how far an electrical signal can travel in one clock cycle (e.g. much shorter, shorter, similar, longer, much longer)? 2.19. It takes about 11.5 hours to fly from San Francisco to Beijing, which is about 5900 miles. (a) What is the average airplane speed in miles per hour? (b) What percent of the speed of sound is this (the speed of sound is about 670 mph at airplane cruising altitudes)? (c) What percent of the speed of light is this (the speed of light is 6.7 · 108 mph)? 2.20. (a) How far does light travel in 1 year (1 year = 3.16 · 107 s; this distance is called a light-year)? (b) The closest star to Earth is Proxima Centauri, which is 4.24 light-years away. How far is this in kilometers? (c) The fastest any spacecraft has flown while far from the sun is Voyager 1, at 17 km/s. At this speed, how many years would it take a spacecraft to get to Proxima Centauri? (d) In one or two sentences, discuss the possibility of humans traveling to other stars, based on these numbers. 2.21. (a) What is the speed of light in water, where the refractive index is 1.33? (b) What is the speed of light in glass, where the refractive index is 1.52? (c) What is the speed of light in diamond, where the refractive index is 2.42? 2.22. Infrared light travels at 7.5 · 107 m/s in Germanium (a metal-like solid that is shiny in visible light but transparent to infrared light); what is the refractive index? 2.23. Bats use echolocation to find their prey, meaning that a bat emits a click and then waits for the sound to bounce off the prey and return to the bat. By timing
2.6 Exercises
41
the returned sound, the bat can figure out how far its prey is. (a) Suppose a moth is 2 m from the bat; how much time is there between when the bat clicks and then hears the echo? (b) A typical bat sound frequency is 50 kHz. What is the wavelength of a bat click, expressed in mm? (c) How does this compare to the size of a moth’s body (e.g. much smaller, smaller, similar, larger, or much larger)? 2.24. For each of the following electromagnetic waves, list the type of radiation (e.g. visible, infrared, radio, etc.) and compute its frequency: (a) laser pointer: 635 nm, (b) dentist’s office X-ray: 0.3 nm, (c) carbon dioxide vibration: 15 µm, (d) hydrogen atom emission: 122 nm, (e) AM station: 221 m. 2.25. For the following electromagnetic waves, list the type of radiation (e.g. visible, infrared, etc.) and compute its wavelength: (a) FM station: 94.9 MHz, (b) microwave oven: 2450 MHz, (c) bluetooth: 2.45 GHz, (d) TV remote control: 3.19 · 1014 Hz. 2.26. People can hear sound waves between 20 Hz and 20 kHz. The speed of sound is 340 m/s. (a) What is the longest wavelength sound that people can hear? (b) What is the shortest wavelength sound that people can hear? 2.27. Most baleen whales (e.g. blue whales, humpback whales, and gray whales) make sounds at around 15 to 20 Hz, which can sometimes be detected across entire ocean basins. The speed of sound in seawater is about 1500 m/s. (a) What is the wavelength, in the water, for a 17 Hz whale sound? (b) How many hours would it take a whale vocalization to cross the Atlantic Ocean from the US to Europe, which is about 5500 km? 2.28. The first direct observation of gravitational waves was by the LIGO (LaserInterferometer Gravitational-Wave Observatory) collaboration in 2016. These waves started with a frequency of 35 Hz and sped up to 250 Hz. Gravitational waves propagate at the speed of light. (a) What was the initial wavelength? (b) What was the final wavelength? (c) If these were electromagnetic waves, what band of radiation would they be in? 2.29. Determine the frequency of microwave light, for which λ = 10 cm. 2.30. A surfer finds that the waves are arriving with a 9.2 second period and are moving at 5.7 m/s. What is their wavelength? 2.31. Suppose you are in a bar 2 km from where the Seattle Seahawks are playing football, and you’re watching the game on a TV. (a) The TV signal is transmitted with microwaves from the stadium, to a satellite that is 36,000 km above you, and then back down to the TV set. How long does this signal take to get to you? (b) You also hear cheering directly from the stadium when the Seahawks score a touchdown; how long does it take for the sound to reach you? 2.32. Consider a wave that has wavelength 7 mm and frequency 2 Hz. Could this be an electromagnetic wave in vacuum? Why or why not?
42
2
Properties of Waves
Puzzles 2.33. Several “musical roads” have been constructed in different countries. In each case, when a car drives over the road at a pre-specified speed, the tires vibrate on grooves that are cut into the pavement (like a rumble strip) and emit tones according to the vibration frequency. One of these, which was designed for a Honda advertisement and is in in Lancaster, California, plays part of the William Tell Overture (the Lone Ranger theme song) when driven over at 55 miles per hour. However, the engineers designed it wrong so that it is spectacularly out of tune. Given that the opening note of the song has a frequency of 349 Hz (an ‘F’ note), what pavement groove spacing should the engineers have used? 2.34. All of the planets in the solar system orbit the sun in a counter-clockwise direction, as seen when looking down at the Sun’s (and Earth’s) north pole. Most moons (including Io) also orbit counter-clockwise around their planets, and most planets (including Earth and Jupiter) rotate counter-clockwise on their axes. Would Rømer’s results have been different if (a) Io orbited clockwise around Jupiter? (b) Jupiter orbited clockwise around the Sun? (c) the Earth rotated clockwise around its axis? 2.35. The following pictures show two-dimensional waves, of which the displacement is within the plane of the surface for the first two and out of it for the last one. (a) Is the wave in picture I transverse or longitudinal, and what is its polarization? (b) Same question for picture II. (c) Same question for picture III. (d) Which of these could represent water waves, if any? Explain. (e) For each picture, what type of seismic wave is it most similar to?
(I)
(II)
(III)
3
Superposition
Figure 3.1 Ripples produced by rain falling on water.
Opening question Two stones are dropped into a pond, each producing ripples. When the sets of ripples meet, which happens: (a) The ripples reflect off each other, each going back toward where it came from. (b) The ripples add together when they overlap and then pass through each other. (c) The ripples cancel out wherever they meet, producing flatter water. (d) The ripples bend around each other, creating complex wave patterns. © Springer Nature Switzerland AG 2023 S. S. Andrews, Light and Waves, https://doi.org/10.1007/978-3-031-24097-3_3
43
44
3
Superposition
When two tennis balls collide, they bounce off each other. However, when two waves collide, they don’t. Instead, they smoothly pass through each other, temporarily combining to form new wave shapes in the process. Once they have finished passing through each other, they continue on their separate ways as if nothing had ever happened. This process is stunningly simple, arising from nothing more than the addition of the two sets of wave displacements. However, it leads to a broad array of interesting behaviors. Two sets of waves can add to create bigger waves, smaller waves, or even no waves at all. They can also add to create waves that oscillate in time but don’t move in space. The principles of wave addition explain how sound waves can curve around buildings, why oil drops on wet pavement produce colorful rings, and how holiday laser lights can create elaborate designs on walls. They even explain how light waves can add together to produce bright spots in the centers of reasonably large shadows. Perhaps the showiest examples arise in the brilliant colors of many animals, including the shimmering iridescence of hummingbirds, the vibrant blue of blue morpho butterflies, and the gaudy colors of peacock tails.
3.1
Superposition of Waves
3.1.1
The Superposition Principle
Suppose a rope is being held by one person at each end. Then, both people put pulses into the rope, where a pulse is simply a brief disturbance. The pulses propagate toward each other, as in Figure 3.2. What happens when the pulses meet?
Figure 3.2 Two people putting pulses into a rope that travel toward each other.
The answer, in a sense, is very uninteresting. The pulses don’t interact in any way at all. They don’t collide off each other, annihilate each other, break the rope, or do anything else dramatic. Instead, their displacements simply add together as the pulses pass each other, which is called the superposition principle. It is shown in Figure 3.3 for two pulses with either the same polarity or opposite polarity, where the polarity is the side of the rope that the pulse is on. Initially, the pulses are separate and moving together, then their displacements add to form either extra large or extra small displacements when the pulses are at the same place, and finally the pulses pass each other. They then continue on separately, with no remaining evidence of their prior interactions.
3.1 Superposition of Waves
45
Figure 3.3 Superposition of pulses on a rope. The two pulses have the same polarity on the left side and opposite polarity on the right side. The top row shows the pulses before they meet, the middle row shows them meeting, and the bottom row shows them after they have passed.
The superposition principle holds true regardless of pulse lengths, pulse shapes, pulse amplitudes, and pulse polarity. It also applies to all types of waves, including water waves, sound waves, and light waves. For example, Figure 3.4 shows the superposition of two sets of water waves. Here, the water displacement at any point on the surface is the sum of the displacements from each of the two sets of waves. Likewise, if two people talk at the same time, then their sound waves add together to include both sounds at once. Also, if two laser beams pass through the same point in space, then their fields add together where they cross, and the beams propagate on as if nothing had ever happened. These behaviors are sufficiently important, and different from what happens with solid objects, that the superposition principle is a defining characteristic of wave behavior1 . Figure 3.4 Superposition of water waves.
1 Despite the widespread validity of the superposition principle, which is assumed in almost all parts of this book, there are some exceptions. These arise from non-linear effects, which are simply defined as wave behaviors that violate the superposition principle. Nonlinear effects are typically negligible for low amplitude waves, but can become important when waves have large amplitudes. They are studied in the fields of nonlinear optics and nonlinear acoustics. The primary situation in which nonlinear effects are commonly observed is for water waves, where they give rise to phenomena such as wave breaking (note that some high amplitude waves are breaking in Figure 3.4) and wave development over time.
46
3.1.2
3
Superposition
Superposition with Different Frequencies
Wave superposition often gives real waves rough and complicated shapes. For example, the left side of Figure 3.5 shows multiple wave components adding together on the ocean: there are wind-driven waves, ripples from rain drops, and the boat’s wake (waves made by the boat), all added together. The displacements from these different wave components add to create the resulting water surface shape. It is somewhat possible to identify wave peaks and troughs in this photograph, but they aren’t well defined. The waves’ wavelengths and amplitudes are even less well defined. The right side of Figure 3.5 helps make sense of this situation by showing that the sum of two sine waves that have different wavelengths, each of which is a separate wave component, also creates a complicated result. However, here, it’s possible to see which features arose from which component. As in the photograph, the peaks, troughs, wavelengths, and amplitudes are well-defined for the component waves but not for the total wave.
+ =
Figure 3.5 Superposition of waves with different wavelengths. (Left) Waves from wind, rain, and boat wake all added together. (Right) Two sine waves with different wavelengths added together.
Remarkably, the summing process can also be reversed, allowing one to decompose a final wave shape into its separate components. This can be performed mathematically using a method called the Fourier transform, named for the French mathematician Jean-Baptiste Fourier, which is central to the quantitative analysis of oscillations and waves but is beyond the scope of this book. However, waves can also be decomposed experimentally in some cases. For example, white light can be separated into the different colors that compose it using a prism, producing a rainbow. Each of the colored rays leaving the prism has a single well-defined wavelength. Also, our ears decompose sound waves into their separate frequencies, detecting the high frequency components just inside the ear drum and the lower frequency components deeper into the spiral shaped tubes of the inner ear. An important point here is that all real waves, regardless of how complex their shapes might be, are nothing more than sums of simple waves, such as sine waves or smooth pulses. This means that we can focus our attention on learning how simple waves work, knowing that they can always be added back together to give complete wave shapes.
3.1 Superposition of Waves
3.1.3
47
Constructive and Destructive Interference
Consider two sets of waves that have the same wavelength. If they are in phase with each other, meaning that their peaks are in the same places and their troughs are in the same places, then they add with the peaks adding to make extra large peaks and the troughs adding to make extra large troughs. In other words, they construct a larger wave out of the two smaller waves. This is called constructive interference and is shown on the left side of Figure 3.6. Quantitatively, the new wave’s amplitude is equal to the sum of the original waves’ amplitudes. For example, if the component wave amplitudes are 1.5 cm (measured mid-level to peak top, as usual), then the new wave amplitude is 3.0 cm.
+
+
=
=
Figure 3.6 (Left) Constructive interference and (Right) destructive interference.
On the other hand, if the two component waves are out of phase, meaning that the peaks of one set of waves are at the locations of the troughs of the other set of waves, then each peak gets added to a trough and vice versa. This makes the new wave smaller than the component waves, which is called destructive interference. In the special case of equal amplitudes for the two component waves, as shown on the right side of Figure 3.6, they add to create a new “wave” that has no amplitude at all because the original peaks and troughs cancel each other out exactly.
3.1.4
Oscillations in Time
The same concepts apply to oscillations in time, such as the air pressure variations within your ear that are produced by arriving sound waves. As with spatial waves, these oscillations in time can be produced from multiple sources. Two sources that have the same frequency interfere constructively if they are in phase with each other, and destructively if they are out of phase. As another way of seeing this, the x-axes of the graphs in Figures 3.5 and 3.6 were neither shown nor labeled. They could represent the position, for example showing how addition of separate sets of waves on a string forms the total string shape. They could also represent the time, for example showing how different sound sources add together to create the total sound at your ear.
48
3.1.5
3
Superposition
Constructive and Destructive Interference Examples*
An example of destructive interference arises in ship design. Much of the energy required to power a ship gets carried away by the wave energy of the ship’s wake, so ships that make smaller wakes are more efficient. However, by their nature, ships have to move large volumes through the water, which unavoidably creates large wakes. Several naval architects came up with a clever solution to this problem in the early 1900s, adding a “bulbous bow” to the front of a ship, a large bump in the hull that’s just below the waterline; Figure 3.7 shows an example. The bulbous bow creates a second wake that is out of phase with the main ship’s wake. Intuitively, it might seem that creating a second wake would require even more power, but it actually requires less total power because the two wakes interfere destructively to create a smaller total wake. Modern bulbous bows decrease ship fuel use by about 12 to 15%. Figure 3.7 Bulbous bow on a ship.
Noise-canceling headphones are another example of destructive interference. In their cases, the headphones sense the sound outside of each ear pad with a microphone and then play the same sound on a speaker on the inside of the ear pad, but with reversed wave polarity. These new waves interfere destructively with the original waves to create smaller total waves. This produces a tiny quiet zone around each ear. Rogue waves form yet another example of wave superposition. These are individual water wave peaks that are more than twice as tall as the surrounding peaks, such as in the classic woodblock print shown in Figure 3.8. Mariners had reported isolated giant waves for many years, but scientists were skeptical of their existence due to a lack of definitive proof. The lack of evidence undoubtedly arose in part because many of the ships that encountered rogue waves didn’t return to tell about them. However, this changed in 1995 when the so-called Draupner wave was detected at an oil platform in the middle of the North Sea and was thoroughly recorded. It was found to be 85 feet tall from trough to peak, while all of the surrounding waves were less than half as tall. There is still no scientific consensus about what causes rogue
3.1 Superposition of Waves
49
waves, but wave superposition clearly plays a major role2 . The idea is that many different waves move past each other in the ocean all the time and they add through superposition, as always. In the rare situation that all of these waves happen to have a peak at the same location and at the same moment, meaning that they interfere constructively, they add to produce an exceptionally tall wave. This rogue wave then dissipates as quickly as it began, as the different component waves continue on in their separate directions. Figure 3.8 “The Great Wave off Kanagawa,” a woodblock print by Katsushika Hokusai, which shows a very large wave that would presumably be classified as a rogue wave.
3.1.6
Beating Patterns
Let’s consider one more wave addition problem, this time with component waves that have slightly different wavelengths, or slightly different frequencies, shown in Figure 3.9. Now, the waves are in phase for a while, and then out of phase, and then back in phase, and so on. This causes them to alternate between interfering constructively and destructively to produce what’s called a beating pattern. The total wave has a reasonably well-defined wavelength that is similar to the original ones, but its amplitude gradually rises and falls.
+ = Figure 3.9 Waves with nearly the same wavelengths add to produce a beating pattern.
2 Rogue
waves also arise partly from non-linear effects.
50
3
Superposition
Beating patterns arise frequently when tuning musical instruments. If the sound waves from two instruments have slightly different frequencies, then they alternate between constructive and destructive interference to produce a noticeable pulsing sound. A musician will typically adjust one of the instruments to make the beating frequency slower and slower until it’s finally completely gone. At this point, the instruments produce the same frequency and are said to be in tune. As another example, surfers speak of ocean waves arriving at the shore in “sets” of large waves, separated by “lulls” of small waves. Again, this is a beating pattern, caused by the addition of waves with slightly different wavelengths. It is popularly rumored that the seventh wave is always the largest. If so, then this implies that the beating period is about 14 waves long, so there are about seven waves from a lull up to the biggest wave of the set, and then seven smaller waves down to the next lull.
3.2
Standing Waves
3.2.1
Reflection at Boundaries
When waves reach the end of whatever they are traveling through, they encounter a boundary. This could be waves on a rope reaching a doorknob, sound waves crossing the room and hitting a wall, water waves crashing into rocks on the shore, or light waves hitting a mirror. At this boundary, there are a few possibilities. First, the energy that was carried in the wave could be transferred to the boundary itself, which is absorption. Alternatively, some or all of the energy could bounce back off the boundary to create a new wave that is now propagating in the opposite direction, which is reflection. In this latter case, there are again two options. The new wave could reflect back with the same polarity as the the original wave, meaning that its displacement is on the same side as before, or it could have the opposite polarity, meaning that it is flipped over. These two possibilities are shown in Figure 3.10. The particular boundary conditions that dictate whether the same or opposite polarity will be produced are not particularly important, but it is worth realizing that both polarities are possible3 .
3.2.2
Standing Waves from Reflected Waves and Superposition
Consider the case in which there is a steady procession of incident waves traveling in one direction, say toward the right, they reflect off a boundary, and then the
3 For
a rope, “hard” boundaries, such as a rigid doorknob, produce the opposite polarity; on the other hand, “soft” boundaries, such as a ring looped around a pole, produce the same polarity. These are shown in Figure 3.10. Boundaries also occur where the wave speed changes, such as where a heavy rope (slow) is tied to a light rope (fast), or where light goes from air (fast) into water (slow). In these cases, the reflected wave has the same polarity if the new medium has a faster wave speed and the opposite polarity if the new medium has a slower wave speed.
3.2 Standing Waves
51
Figure 3.10 Reflection of pulses off boundaries that have two different boundary conditions. The one on the left produces an opposite polarity while the one on the right produces the same polarity.
reflected waves return going toward the left. The two sets of waves add through the superposition principle, as always. The incident and reflected waves have the same wavelength, so they undergo constructive or destructive interference. More precisely, they alternate between interfering constructively when the incident and reflected waves are in phase with each other, and then interfering destructively when the incident and reflected waves are out of phase. Figure 3.11 shows the resulting wave pattern, called standing waves, which look like waves that simply rise and fall over time but without propagating4 . Standing waves are large during constructive interference moments and small during destructive interference moments. Figure 3.11 Standing waves in a rope. The solid line shows the position of the rope at a particular moment and the other lines show rope positions at other times.
Considering standing waves on a rope, some parts of the rope don’t go up and down but are always stationary, which are the nodes of the standing waves. In contrast, the parts of the rope that go up and down the most are the antinodes. If we look at the rope at a single moment, it will have a peak at one antinode, a trough at the next antinode, and then a peak at the antinode beyond that. The wavelength is always measured from peak to peak, or from trough to trough, so this means that one wavelength spans two antinodes, or two nodes. Equivalently, the distance between adjacent nodes is half of a wavelength. There is almost always either a node or an antinode at the boundary where the wave reflects. For example, there is a node at the boundary on the right side of Figure 3.11 and an antinode at the left side of Figure 3.12. Whether there is a node
4 The term “standing waves” can also refer to waves that propagate at the same speed as a moving medium but in the opposite direction, causing the waves to appear stationary. Waves in river rapids are common examples. They are quite different from the standing waves that this section focuses on.
52
3
Superposition
or an antinode depends on the boundary conditions and, again, is not particularly important5 . Figure 3.12 Water waves hitting a concrete seawall and reflecting off it, with an antinode at the seawall. This forms exceptionally rough water called clapotis. Note the smoother water in the background, past where the seawall ends.
3.2.3
Standing Waves Between Two Boundaries
Standing waves are particularly interesting when they have boundaries at both ends. Examples include standing waves on a jump rope, on a guitar string, and within a flute. There are also standing waves in microwave ovens; in this case, the pattern of nodes and antinodes causes food to heat unevenly. Other examples include standing electron waves in radio antennas, standing light waves in lasers, and even standing water waves that produce the oceans’ tides. The region between the two boundaries, where the standing waves are, is called the cavity. Figure 3.13 shows several standing waves in cavities. This figure uses boundaries that produce nodes, which has the important consequence that there must be a node at both ends of each standing wave, as shown. As a result, only standing waves with specific wavelengths can fit in the cavity. Standing waves with other wavelengths are impossible because they wouldn’t have a node at each end. One possible standing wave, shown at the bottom of the figure, has half of its wavelength equal to the cavity length. As an equation, this is λ2 = L, where λ is the wavelength and L is the cavity length. The middle of the figure shows another possibility, now with a full wavelength equal to the cavity length; here, λ = L. The top of the figure shows that three half-waves could equal the cavity length, and this pattern obviously continues. Generalizing these results, the wavelengths that fit in the cavity are those that are are those with n λ2 = L, where n, called the mode number, is any positive integer (e.g. 1, 2, 3, etc.). Rearranging this equation gives the result that the waves that can fit into a cavity of length L must have wavelength
5 Boundaries
antinodes.
that reverse wave polarity have nodes and those that preserve wave polarity have
3.2 Standing Waves
53
Figure 3.13 The first three harmonics for waves in a cavity.
2L . (3.1) n These standing waves are the harmonics or normal modes of the system. The n = 1 normal mode is also called the fundamental mode or first harmonic and is often the most important mode. The n = 2 normal mode is the second harmonic or first overtone, the n = 3 normal mode is the third harmonic or second overtone, and so on. These terms are listed on the left side of Figure 3.13. Each normal mode has a different frequency, which leads to a list of frequencies that are the natural frequencies of the system. As usual, we can convert between wavelength and frequency using the frequency-wavelength relationship, v = λ f (Eq. 2.7). Rearranging this to f = λv and substituting in the standing wave wavelengths from above shows that the natural frequencies are λ=
f =
nv . 2L
(3.2)
This shows that the n = 1 mode, meaning the fundamental mode, has the lowest v , which is called, not surprisingly, the fundamental possible frequency. It is f = 2L frequency. There is no possible way to have a standing wave in this cavity with a lower frequency. However, specific higher frequencies are possible by changing to higher harmonics. From Eq. 3.2, the second harmonic has twice the frequency of the fundamental, the third has three times the frequency, the fourth has four times the frequency, etc.6 Thus, there are two essential results for standing waves in cavities. First, standing waves in a cavity can only have specific wavelengths and specific frequencies. Waves with other wavelengths, and thus other frequencies, don’t fit in the cavity and so cannot exist as standing waves. Second, there is a minimum frequency for standing waves in a cavity, which is the fundamental frequency. One can get higher frequencies by choosing higher harmonics, but it is impossible to get a lower frequency.
6 Both this frequency result and Eq. 3.2 assume that the wave velocity is independent of wavelength, which is called nondispersive. This is true for string waves, sound waves, light waves in vacuum, and many other types of waves, but is not always true. Water waves are a notable exception.
54
3
Superposition
Example. Compute the fundamental and higher harmonic frequencies for standing waves in a flute, which is a sound wave cavity with a length of 0.650 m. Graph the frequency as a function of mode number. ●
1500 frequency, f
Answer. Use Eq. 3.2 with v equal to the speed of sound, which is 340 m/s, and L = 0.650 m. Using n = 1 gives the fundamental frequency as 262 Hz (the middle C note, as we’ll see in Chapter 6). Using n = 2 gives the second harmonic as 523 Hz, n = 3 gives the third harmonic as 785 Hz, n = 4 gives the fourth harmonic as 1046 Hz, etc. Note that the frequencies increase linearly with the mode number.
● ●
1000 ● ●
500 ● 0 0
1
2 3 4 5 mode number, n
6
Example. Violins have 4 strings, each with a vibrating length of 32.8 cm. One of them, (the D string) has a fundamental frequency of 293.7 Hz. What is the wave velocity on this string?
Answer. Rearrange Eq. 3.2 to solve it for the velocity, giving v=
2L f . n
The question asks about the fundamental frequency, so n = 1, and the length is L = 32.8 cm = 0.328 m, from which we compute that v = 193 m/s. This is about 430 miles/hour! (Note that we are not computing the velocity of the sound wave from the string, which is 340 m/s as always, but the velocity for wave propagation along the string.)
3.3
Interference
3.3.1
Thin-Film Interference
We are all familiar with colorful oil sheens on wet roads, shown on the left side of Figure 3.14. Often arising from only a single drop of oil, they have colorful stripes that circle the drop’s initial location. Their colors do not arise from the oil itself, which is almost perfectly clear, but from constructive and destructive interference in a phenomenon called thin-film interference. The right side of Figure 3.14 shows how it works using light with a single wavelength, in this case red light. Some of the incident light reflects off the top of the oil, some continues into the oil and reflects off the interface between the oil and water,
3.3 Interference Figure 3.14 Thin-film interference from a thin layer of oil on water. (Left) A single oil drop on a wet road. (Right) Light rays, with waves showing constructive interference.
55
air oil water
and the rest continues on into the water. The two rays of reflected light, from the top and bottom of the oil layer, have the same wavelengths and overlap each other, but may have a phase shift, meaning that their peaks and troughs are offset from each other. The phase shift arises from the fact that the ray that reflects off the top of the oil has a polarity reversal and, more importantly, the second reflected ray traveled a slightly longer distance. The phase shift produces either constructive or destructive interference. If the oil thickness happens to be just right so that the wave peaks of the two reflected rays line up with each other, then they interfere constructively and produce a bright reflection. In a slightly thicker part of the oil, the second reflected ray has to travel slightly farther, so the the waves are out of phase, producing destructive interference and no reflected light. With yet thicker oil, the second ray travels another half wavelength farther, putting the reflected waves back in phase again. The end result is a series of bright and dark rings, with a new bright ring every time the oil is enough thicker that the second reflected ray travels one more wavelength through the oil. White light is a mixture of all colors, each with different wavelengths, so their rings of constructive and destructive interference occur at different locations. For example, if blue light interferes constructively with some oil thickness, then green light is going to require slightly thicker oil to interfere constructively due to its longer wavelength. These different locations for constructive and destructive interference produce the colored rings. In a sense, the rings act like a very precise contour map for the thickness of the oil. A few details are interesting to consider. First, where does the light go if there is destructive interference? The answer is that it’s not reflected, so all of it must get transmitted through the oil and into the water. This is remarkable when you think about it because it says that the reflected light waves cancel each other out and, somehow, put their energy back into the transmitted beam. Second, why do the colored rings include silver, brown, and magenta rather than the bright colors of rainbows? The answer is that several different colors can interfere constructively with the same oil thickness, which then combine to create these mixed colors. Finally, why did we ignore additional light reflections within the oil layer? The answer is that they are very weak and so can be largely ignored, but would need to be considered in a more careful treatment.
56
3.3.2
3
Superposition
Examples of Thin-Film Interference*
Thin-film interference arises for all transparent films that are up to a few light wavelengths thick. For example, thin-film interference produces colors in soap bubbles and the wings of wasps and flies, shown in the left and middle of Figure 3.15. It is also used in antireflective coatings to reduce reflections on eyeglasses, camera lenses, binoculars, and other optics, shown on the right side of the figure. Antireflective coatings typically use a thin layer of magnesium fluoride, which is reasonably durable and effective at reducing reflections over most of the visible range. However, it is less effective at the blue end of the visible spectrum, leading to the characteristic purple color of high quality optics.
Figure 3.15 Thin-film interference for a soap bubble, wasp wings seen on a white and black background, and forming purple reflections on a camera lens (the orange dot in the middle is light coming through the lens).
Stacking multiple thin layers on top of each other enables more choices about which reflected waves should interfere constructively and which destructively. This approach is used for optical interference filters, for which it is often possible to choose exactly which wavelengths are transmitted and which are absorbed. Although not typically called thin-film interference, the same phenomenon happens when a thin layer of air is sandwiched between two pieces of glass. Again, the light reflects off two interfaces and then interferes either constructively or destructively. This produces a series of bright and dark rings called Newton’s rings (Figure 3.16). Or, with white light, it produces a series of colored rings, much like one sees with a thin film of oil. Figure 3.16 A diagram of Newton’s rings, and the rings observed with yellow light.
3.3 Interference
57
It’s appropriate that Newton’s rings are named for Isaac Newton because he studied them in detail, correctly using the curvature of a piece of glass to find that the thickness of air at the first dark ring was “the 1/88739 part of an inch”7 , which is about 286 nm. However, it’s also ironic because Newton was a strong proponent of the particle theory of light, arguing firmly against the very wave explanations that explain the rings. Rather than using his results to be the first person to measure the wavelengths of visible light, he was so convinced of his particle explanation that he interpreted his experimental results as the light particles having alternate “fits of easy reflexion” and “fits of easy transmission”8 . It wasn’t until Young’s work nearly 100 years later, and Young’s re-analysis of Newton’s own data, that the wave nature of light was shown and the wavelengths accurately determined.
3.3.3
Interferometers and the Lack of an Aether*
The essential aspect of thin-film interference is that a light beam is split into two rays, they travel separately, and then those rays are recombined. Upon recombination, the rays interfere constructively if they are in phase with each other, or destructively if they are out of phase. An interferometer does the same thing but uses much longer pathlengths. In the Michelson interferometer shown in Figure 3.17, light from a source enters from the left, is split into two rays by a beam splitter (often just a partially silvered piece of glass), and those two rays each go to a mirror and back. The two rays of the beam then recombine back at the beam splitter, where part of the light goes back toward the source and part goes toward a screen. If the two “arms” of the interferometer are exactly the same length, to within less than a wavelength of light, then the two rays are in phase when they recombine, so they interfere constructively and produce a bright spot at the screen. However, extension of one arm by just a quarter wavelength of light changes the result to destructive interference. Another quarter wavelength extension produces constructive interference, then a little more produces destructive Figure 3.17 Michelson Interferometer. Light enters from the left, is split at the beam splitter, goes to the two mirrors, recombines at the beam splitter, and then goes left toward the source and forwards to the screen.
7 Newton, 8 Newton,
Opticks, p. 202. Opticks, p. 281.
58
3
Superposition
interference again, and so on. The result is that this is an extremely sensitive way to measure tiny changes in distances. Albert Michelson and Edward Morely, two American physicists, developed this type of interferometer in 1887 and then used it to discover that light propagates without a medium. In accordance with the standard assumption at the time, they expected that light did require a medium, called the aether, when they designed the experiment. They reasoned that the speed of the Earth through the aether, as the Earth both rotates and orbits the sun, would create an aether “wind.” This would cause light to move slower when it propagates in the same direction as the Earth’s motion, due to the headwind, than when going perpendicular to the Earth’s motion, where there’s only a side-wind. This effect would be measurable using an interferometer because slight delays of one of the two light beams would affect the constructive or destructive interference. To their surprise, they found that changing the instrument orientation with respect to this hypothetical aether wind made absolutely no difference. From this, they were forced to conclude, correctly, that there is no aether wind and, by extension, that light does not require a medium to propagate. Their result that the speed of light does not depend on the observer’s motion is troubling when one thinks about it critically. For example, suppose two different observers are moving at two different speeds and they measure the speed of the same light beam relative to themselves. Simple logic says that their results should differ by the difference in their own speeds, but Michelson and Morley’s experiment showed that their results must be exactly the same. This puzzle could only be solved by redefining the very concepts of space and time, which then led to Albert Einstein’s development of special relativity 13 years later.
3.4
Diffraction
3.4.1
Diffraction Through Holes and Around Obstacles
All waves that propagate in two or three dimensions, including water, sound, seismic, and light waves, have the property that their wavefronts bend around obstacles. This is called diffraction. Figure 3.18 shows water waves diffracting through a gap between two peninsulas, forming nearly semicircular wave fronts on the far side. The same thing happens for other types of waves. Sound waves diffract around the edges of buildings, enabling us to hear people even if we cannot see them. Radio waves diffract around hills, and light waves diffract through pin holes. There are two main diffraction trends for waves that go through a hole, both of which are shown in Figure 3.19. First, smaller holes cause the waves to spread out more. Waves barely spread out when the hole is much larger than the wave’s wavelength, they spread more when the hole size is similar to the wavelength, and they spread as much as possible, into complete semicircles or hemispheres, when the hole is smaller than the wavelength. Second, larger holes transmit more wave energy. Holes that are much larger than the wavelength transmit almost all of the wave energy, holes that have similar sizes to the wavelength transmit less energy,
3.4 Diffraction
59
Figure 3.18 Diffraction of water waves through a narrow gap (São Martinho do Porto, Portugal).
and holes that are much smaller than the wavelength transmit essentially no energy at all. The low transmission with very small holes isn’t just because of their small areas, but because waves barely interact with holes or objects that are much smaller than their wavelengths. Figure 3.19 Diffraction effects of large and small holes.
Microwave ovens demonstrate these trends nicely. They have a sheet of perforated metal in the door which has holes that are about 3 mm in diameter. These holes are very large when compared to visible light waves (∼500 nm), so visible light waves go through these holes without energy loss and without the rays being bent, allowing you to see your food. However, the holes are very small when compared to microwave wavelengths (∼12 cm) so essentially none of that energy can escape through the holes, thus protecting you. Any microwaves that did make it through the holes would spread out fully due to diffraction, but that’s unimportant because so little energy actually gets out.
3.4.2
Huygens’s Principle
Christiaan Huygens showed that diffraction can be explained as a result of wave superposition. In what’s now called Huygens’s principle, he showed that wave propagation can be understood by assuming that each point on a wavefront of the actual waves, called the “primary waves,” behaves as a source that radiates “secondary waves” in all directions9 , as shown on the left side of Figure 3.20. The secondary waves then add together through superposition to create new primary waves; the
9 Actually,
they only radiate in all “forward” directions, which is not explained well by the theory
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locations where they interfere constructively form the trough and peak of the next primary wave, and the locations where they interfere destructively are the in-between points, with small total displacements. Figure 3.20 Huygens’s principle applied to plane waves, diffraction through a small hole, and diffraction through a large hole.
Huygens’s principle agrees with common sense in that it focuses on local interactions. Instead of thinking of waves as disturbances that propagate for long distances, it shows that each point in space, such as a point on the water’s surface, only responds to the secondary waves that come into that point from nearby. That point then sends out new secondary waves, which again act at other nearby locations. To apply this to diffraction, consider water waves diffracting through a narrow gap, shown in the middle of Figure 3.20. The water in the middle of the gap rises and falls in response to the incoming waves. According to Huygens’s principle, it then acts as a secondary wave source, emitting circular secondary waves into the water on the far side of the gap. They don’t get added to any other secondary waves in this case, because those got blocked by the barrier, so the result is just a single set of circular wavefronts. Alternatively, suppose the waves came to a wide gap, shown on the right side of the figure. Now, Huygens’s principle shows that many secondary waves add together, thus increasing the transmitted energy. However, the secondary waves get cut off at the ends of the gap, causing the primary waves to bend around. All of these results agree with observations.
3.5
Combining Diffraction and Interference
Yet more interesting behaviors arise when light diffracts to create multiple sets of spreading waves, and then those diffracted waves interfere with each other. These behaviors are described for light waves here but apply to all types of waves because, again, they are just a consequence of the superposition principle.
3.5.1
Double-Slit Experiment
In Thomas Young’s double-slit experiment, illustrated in Figure 3.21, light waves go through two narrow openings, spread out behind each one due to diffraction, and then interfere with each other. If these interfering light waves then shine on some sort of a screen, they produce bright spots or Fringes where the waves interfere constructively
3.5 Combining Diffraction and Interference
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and dark spots where they interfere destructively. This experiment, first performed in 1801, provided strong evidence in favor of the wave theory of light. Figure 3.21 Diagram of double slit experiment.
In slightly more detail, there is always a bright spot on the screen directly behind the center of the two slits, called the “central maximum”. This makes sense because both light paths have exactly the same distance to this point, so there are exactly the same number of waves in the two paths, so the waves are in phase. This produces constructive interference and thus a bright spot. Dark spots on either side of the central maximum arise because the two light pathlengths differ by a half wavelength there, so the waves are out of phase and interfere destructively. Additional fringes represent larger pathlength differences, with each bright spot at a location where the pathlength difference is an integer number of wavelengths.
3.5.2
Double-Slit Experiment Analysis*
A quantitative explanation for this experiment is not as complicated as it might seem. Figure 3.22 shows a diagram of the experiment, focusing on the position of one of the bright spots. It is called the m’th maximum, where m is some integer; in the diagram, m = 2. At this m’th maximum, the fact that there is constructive interference implies that the lower light path is exactly mλ longer than the upper light path. Placing this extra distance at the left end of the lower path produces the small right triangle that is shown at the left side of the figure in purple. This triangle has a side length of mλ and hypotenuse d, where d is the separation between the slits. Using the definition opposite , of the sine function as hypotenuse sin θ =
mλ . d
The figure also shows a large right triangle with black lines. It has a base of L, which is the distance between the slits and the screen, and height D, which is the distance between the central maximum and the m’th maximum. A little geometry analysis shows that the angle at the left end of this triangle is essentially the same as the one we labeled θ, so this is θ as well (one way to see this is to imagine making D smaller
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Figure 3.22 Diagram of double slit experiment. Light paths are shown in blue.
and smaller and seeing that both θ values will shrink at the same rate; or, make D bigger and bigger and seeing that both θ values will increase at the same rate). The opposite tangent function is ad jacent , so D . L For small angles, sin θ ≈ tan θ, enabling these equations to be combined to give tan θ =
mλ D = . d L
(3.3)
This equation describes where all of the bright spots appear and how far apart they are. For example, it shows that D is proportional to λ, so changing to a longer wavelength leads to more widely separated dots. Also, a little rearrangement shows that D is also proportional to L, so moving the screen farther away also leads to more widely separated dots. However, perhaps counter-intuitively, it shows that D is inversely proportional to the slit separation, d. This means that moving the slits farther apart causes the dots to come closer together. In other words, if you want widely separated dots, then you have to make the slits very close together.
3.5.3
Diffraction Gratings*
An obvious extension of the two-slit experiment is to consider what happens if there are more slits, such as 3, 4, or even hundreds of equally spaced slits. It turns out that the result is essentially the same as the double-slit experiment but with smaller and brighter spots (left side of Figure 3.23). This is the basis of optical components called diffraction gratings, which are widely used to separate white light into its component colors, using the fact that different wavelengths have their bright spots at different locations (right side of Figure 3.23). Quantitatively, the same equation that we derived for the double-slit experiment, Eq. 3.3, applies to diffraction gratings, too, where d is now the separation between slits. To see this, if the pathlength difference to a bright spot from two adjacent slits is exactly one wavelength, then the pathlength difference from the next slit is one more wavelength, and the slit beyond that is one more wavelength, and so on. Thus, all of these waves interfere constructively at the same locations as for the double-silt experiment.
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Figure 3.23 (Left) Diagram of a diffraction grating. (Right) A beam of white light shining into a diffraction grating from the left and being redirected into multiple beams.
More generally, essentially any regular array of objects can serve as a diffraction grating for waves with wavelengths that are similar to the array spacing. For example, compact discs (CDs), used for storing music and data, have regular arrays of “pits”; these rows of pits are 1600 nm apart, which is close enough to the wavelength of visible light for CDs to produce colorful rainbows through visible light diffraction and interference. At a much smaller size scale, the atoms in crystals are regularly spaced with about 0.5 nm separations, which makes them good diffraction gratings for Xrays. Using them in this way is called X-ray diffraction, which is a tremendously useful technique for determining the precise structures of crystals, whether of table salt or large proteins. Christmas laser lights, which might shine spots all over someone’s house, or Santa Claus pictures on a wall, are also based on diffraction gratings. They produce more complex pictures using more complex grating patterns, but the central principles of diffraction and interference are the same.
3.5.4
Single-Slit Experiment and Analysis*
Light shining through a single slit also produces a series of bright spots because the waves that go through different parts of the slit interfere with each other. The central maximum is extra bright and wide in this case, and is again flanked by a row of small bright spots on either side. This pattern can be analyzed with essentially the same diagram as before, now in Figure 3.24. The key insight for this analysis is that each minimum in the interference pattern (the dark regions) will occur where there is constructive interference between the rays traveling at the two extreme edges of the slit. To see this, suppose the top and bottom rays differ by one wavelength, so they interfere constructively. In this case, the ray that goes through the center of the slit is off by half of a wavelength, so it interferes destructively with the ray at the top edge. Next, a ray that is a little below the center will interfere destructively with one that is a little below the top edge. More generally, every ray that goes through the top half of the slit will interfere destructively with the ray that is exactly half a slit width below it. This means that
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Figure 3.24 Diagram of single slit experiment.
there is destructive interference for all rays, which makes this an intensity minimum. Other intensity minima are similar, leading that the conclusion that constructive interference for the extreme rays produces intensity minima on the screen. We already did the mathematics for constructive interference of the extreme rays in the doubleslit experiment, so we just copy over that equation to give the result that the single-slit intensity minima are at mλ D = . a L
(3.4)
The only replacement is that a is the slit width. Again, note that a narrower slit produces more widely spaced spots and vice versa.
3.5.5
The Arago-Poisson Spot*
Despite the interference fringes that Thomas Young’s double slit experiment revealed in 1801, many scientists were still not convinced that light was a wave. To help address the issue, the French Academy of Sciences held a competition to explain the properties of light in 1818. One of the entrants was Augustin-Jean Fresnel, who submitted an explanation of diffraction based on Huygens’s principle and Young’s double slit experiment. Siméon Poisson, an eminent French mathematician, was one of the judges and a firm believer in the particle theory of light. To try to prove that Fresnel’s submission must be incorrect, he argued that the wave theory of light would predict that the shadow of a perfectly round object would have a bright spot in the exact center of the shadow. This is because the light waves would diffract around the object and interfere on the far side. In the exact center of the shadow, all of the diffracted waves would be in phase with each other, so they should interfere constructively and thus produce a bright spot. Poisson claimed that this was an absurd result, showing that light could not be a wave and thus Fresnel’s submission must be incorrect.
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Dominique-François-Jean Arago10 , who was the head of the competition committee, performed Poisson’s experiment. To the astonishment of the scientific community, Arago observed a bright spot at the center of the shadow exactly as Poisson had predicted (Figure 3.25). This finally convinced most scientists at the time that light was in fact a wave and led to Fresnel winning the competition. However, it didn’t sway Poisson, who still believed in the particle theory. Figure 3.25 Shadow of a small solid round ball, a few millimeters in diameter, hanging from a magnetized needle. It shows diffraction and interference effects, including Arago’s spot at the center of the shadow.
3.5.6
Babinet’s Principle
Jacques Babinet, a 19th century French physicist and science popularizer, discovered another intriguing result for diffraction, now called Babinet’s principle. It is that the diffraction pattern of an opaque object is identical to that of a hole that has the same shape. For example, the diffraction pattern that arises from a single narrow slit is the same as that for an equally narrow wire; both have a bright central spot and multiple bright fringes on either side. Also, the diffraction pattern for a fine wire mesh, such as the bug netting in many tents, is the same as that for a series of narrow slits, such as a diffraction grating. A simple laboratory exercise, which only requires a laser pointer, uses Babinet’s principle to measure the size of a human hair. Here, laser light is shone at a hair, and the fringe spacing in the resulting diffraction pattern is entered into Eq. 3.4 to give the hair diameter. You may have noticed that photographs of stars that were taken with large telescopes usually have bright points coming out of the brighter stars, which are called diffraction spikes. These spikes are explained by Babinet’s principle. They are caused by struts in the telescope that hold the so-called secondary mirror in place in front of the telescope’s primary mirror. Starlight that passes close to the struts diffracts around their edges and creates bright streaks in the image, which are the diffrac-
10 Arago
had a remarkably adventurous early career in science, including being detained by Spain as a suspected French spy while he was actually conducting surveying work. Later on, he became the Prime Minister of France. See http://lousodrome.net/blog/light/2015/07/07/fresnel-and-thepoisson-spot/.
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tion spikes. From Babinet’s principle, the diffraction spike patterns are the same as those that would arise from a matching arrangement of narrow slits. For example, Figure 3.26 shows that the Hubble Space Telescope has struts in a “+” shape and that these create matching “+” shaped diffraction spikes11 . Furthermore, looking along each row of spikes shows that it exhibits a single-slit interference pattern, with a bright central region and then a series of bright spots (they blur together some due to the mixture of wavelengths in white starlight), again arising from diffraction and Babinet’s principle.
Figure 3.26 (Left) A view into a model of the Hubble telescope showing the primary mirror at the back, the back side of the secondary mirror as a black circle, and two of the struts that hold the secondary mirror in place. (Right) Image of the star Proxima centauri taken by the Hubble Space Telescope. Note the diffraction spikes.
As yet another example of Babinet’s principle, the diffraction pattern for a small hole is a circular disk that is surrounded by concentric rings, known as the Airy disk12 , shown on the left side of Figure 3.27. The same Airy disk pattern appears as colored rings around the moon if there are thin clouds at nighttime, shown in the right panel of the figure. This corona arises from diffraction of moonlight around the water droplets or ice crystals that make up the cloud. The moonlight diffracts around these droplets or crystals, causing light to come to our eyes from slightly different angles, creating the rings.
11 Each strut creates a line that is perpendicular to itself in the image, on both sides of the star. Thus,
a vertical strut creates a horizontal line and vice-versa. Telescopes that have “Y”-shaped struts, such as the James Webb Space Telescope, also have a full line from each strut, this time creating 6-pointed diffraction spikes. 12 Named for the English astronomer Sir George Airy, who first described it mathematically.
3.6 Structural Coloration*
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Figure 3.27 Left: An Airy disk formed by laser light shining through a small hole. Right: corona around the moon. Both are caused by diffraction.
3.6
Structural Coloration*
Look around yourself and observe the colors. Most of them, presumably, arise from pigments, which are chemicals that absorb some wavelengths of light while either transmitting or reflecting the others. For example, clothes are colored by pigmented dyes, walls are painted with pigmented paint, printed paper is colored with ink pigments, and our own skin colors arise from natural skin pigments. However, some members of the natural world, including both animals and plants, do not achieve their colors from pigments but from interference effects. This can produce spectacular results, such as the vibrant colors of parrot feathers, the shimmering iridescent patches on hummingbirds, and the metallic gold colors of some beetle wing covers. Structural coloration is the general term for biological colors produced from interference effects rather than pigments. It generally arises from one or more of the mechanisms that we introduced above, or a combination of them13 . The first mechanism for structural coloration is thin-film interference, whether with one or multiple film layers; Figure 3.28 shows several examples. In addition to the colors in wasp and fly wings that we saw before, it also gives buttercup flower
Figure 3.28 Structural coloration in a buttercup, hummingbird, and shell, all arising from thin-film interference.
13 A
good reference is: Sun, Jiyu, Bharat Bhushan, and Jin Tong (2013) “Structural coloration in nature” Royal Society of Chemistry Advances 3:14862-14889.
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petals a brilliant yellow gloss, in their case arising from a layer of cells that combine thin-film interference with yellow pigment. It creates iridescent feathers in pigeons, grackles, and hummingbirds, all of which alternate layers of proteins with pigments. It also produces the silver color of sardines and a prominent silver stripe on neon tetras, a popular aquarium fish. These fish alternate layers of cytoplasm, which is essentially just the fluid within cells, and guanine crystals, which have a high index of refraction and hence produce high light reflection at each interface. Many types of beetles, including the aptly-named jewel beetles and Japanese beetles, a common invasive species in eastern North America, have metallic gold, green, or other colored wing-covers. Again, these arise from multiple thin layers. A final example is the iridescent mother-of-pearl that one finds on the insides of oyster and other shells, yet again arising from multiple thin layers. The second mechanism is diffraction gratings, shown in Figure 3.29. These are less common but help produce the colors of some beetles, and wasps. They also create the colors of some butterflies, including the brilliant blue color of the blue morpho butterfly.
Figure 3.29 Structural coloration in a blue morpho butterfly, with a detail of a scale on the right, arising from diffraction grating effects.
Thirdly, it is possible to combine multilayer thin-film interference with diffraction gratings, producing a regular pattern of both layers and stripes at the same time, for which Figure 3.30 shows several examples. This is called a photonic crystal. Many birds have evolved to produce photonic crystals in their feathers using regular patterns of air-filled pockets within protein structures. They are responsible for most blue colors in birds, including on blue jays, Steller’s jays, eastern bluebirds, and blue parrots. They also provide the colors in peacock tails. How can one tell if color arises from pigments or structure? The answer is to run various tests on a colored sample, such as a butterfly scale, to see if the changes are more consistent with pigments or structural coloration. For example, if the color remains after grinding the scale into a fine powder, then it’s probably a pigment. Or, if the color remains after the scale is bleached, which degrades most chemicals but often leaves structures intact, then it’s probably structural. One can also test for various interference and diffraction effects, such as by seeing whether the color changes with viewing angle or after the scale changes its dimensions slightly after being dried out or swelled up with water. For those with the necessary equipment, it’s also possible
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Figure 3.30 Structural coloration in (L) an eastern bluebird, (M) a macaw parrot, and (R) peacock feathers, arising from photonic crystals.
to observe the detailed structure of the scale with an electron microscope to see if it has a regular series of layers or grooves that might indicate structural colors. Remarkably, this approach can even work on prehistoric bird feather fossils14 , where it shows how they may have looked.
3.7
Summary
When multiple waves overlap each other, the total displacement is the sum of the displacements of the individual waves, which is called the superposition principle and is a defining property of waves. Adding waves that are in phase with each other leads to constructive interference and produces a higher amplitude wave, while adding out of phase waves leads to destructive interference and produces a lower amplitude result. Component waves with slightly different wavelengths, or frequencies, alternate between constructive and destructive interference to produce a beating pattern. Waves that reflect at boundaries either maintain or reverse their polarity, depending on the boundary conditions. Either way, the incident and reflected waves alternate between interfering constructively and destructively, creating standing waves. These waves don’t move in space but oscillate in time, with nodes where wave displacements are always zero and antinodes where displacements are maximum. One wavelength is the distance between two nodes. Standing waves between two boundaries can only have specific wavelengths because other wavelengths cannot fit in the cavity while obeying the boundary conditions; these conditions typically require either a node or an antinode. A consequence is that standing waves in cavities can only have specific frequencies. The lowest frequency is called the fundamental mode or first harmonic, while ones with higher frequencies are called higher harmonics or overtones. The possible standing wave wavelengths and frequencies in a cavity are
14 Vinther, Jakob, et al. “Structural coloration in a fossil feather” Biology Letters 6:128-131 (2010).
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2L nv and f = . n 2L Thin-film interference, such as seen in oil sheens, arises when an incident light ray reflects at multiple sequential boundaries and then the different reflected waves interfere with each other. Similarly, interferometers split and then recombine incident light rays, again with either constructive or destructive interference. Diffraction is the bending of waves as they go through holes and past obstacles. When waves diffract through holes, smaller holes result in greater wave spreading and less wave energy transmitted. Essentially no energy gets through holes that are much smaller than a wavelength. Diffraction can be explained using Huygens’s principle, which states that every point on a wave acts as a source of secondary waves and those secondary waves interfere to produce the next primary waves. Light that shines through two nearby slits, in the double-slit experiment, produces a row of bright spots, or fringes, which arise from alternating regions of constructive and destructive interference. The bright spots are at locations λ=
mλ D = . L d Diffraction gratings are conceptually similar but have many uniformly spaced slits, so they transmit more light and produce smaller bright spots. Diffraction and interference through a single slit produces a slightly different spot pattern. Diffraction around a round obstacle leads to a tiny point of light in the center of the obstacle’s shadow, called the Arago-Poisson spot. Babinet’s principle states that diffraction patterns produced by obstacles are the same as those produced by openings that have the same size and shape. While most animals and plants are colored with pigments, some exhibit structural coloration. In most cases, their colors arise from thin-film interference, diffraction gratings, or a combination of these, called photonic crystals. Almost all natural iridescence arises from structural coloration, as do the colors in most blue bird feathers.
3.8
Exercises
Questions 3.1. What is the cause of Arago’s spot, the bright light in the center of a shadow of a circular object? (a) Interference around the object, and then diffraction at the center (b) Light diffraction around the object, followed by destructive interference at the center (c) Light diffraction around the object, followed by constructive interference at the center
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(d) Destructive interference of different light wavelengths (e) Light particles that collide off each other, with most collisions at the center 3.2. Which of the following is an example of Babinet’s principle? (a) Only standing waves of specific frequencies can exist in a string of fixed length (b) The double slit experiment always produces a bright spot at the center (c) Every point on a wave can be thought of as being a source that creates secondary waves (d) The diffraction pattern for an object is identical to that for the same shape hole (e) Interference between waves reflecting off two sides of a thin film produces bright colors 3.3. You’re attending a party with loud music and are standing reasonably close to one of the two speakers. You find that the music is noticeably quieter if you move by just a few feet in some direction. What might make it quieter? (Choose all that are appropriate.) (a) Constructive interference of sound from the two speakers (b) Destructive interference of sound from the two speakers (c) Constructive interference of sound from the speaker and its reflections off the walls (d) Destructive interference of sound from the speaker and its reflections off the walls (e) None of these; interference doesn’t affect sound volume 3.4. What is the primary reason why microwave ovens have a turntable for spinning food? (a) To let you see all sides of your food (b) It stirs the food as it warms up (c) Microwave ovens have hot and cold spots; the hot spots are at standing wave nodes (d) Microwave ovens have hot and cold spots; the hot spots are at standing wave antinodes (e) It keeps microwaves from leaking out of the oven 3.5. Which types of waves can form standing waves? (a) only string waves (b) string and light waves (c) only mechanical waves (d) only electromagnetic waves (e) all types of waves
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3.6. Many freeways have concrete walls next to them to reduce noise in adjacent communities. However, it is still possible to hear traffic noise when standing behind a wall. Why is that? (a) sound waves diffract around the top of each wall (b) sound waves go through cracks in the walls (c) constructive interference of sound waves (d) sound waves reflect off the walls (e) sound waves travel faster in concrete than in air 3.7. A friend is looking at the interference fringes in the double-slit experiment that are created by a laser. She wants to change the experiment to yield a wider spacing between the fringes. Which of the following will help her? (Choose all that are appropriate.) (a) make each slit wider (b) make the slits farther apart (c) make the slits closer together (d) use a laser with a shorter wavelength (e) place the viewing screen farther from the slits 3.8. You are trying to determine if a butterfly scale is colored due to structural coloration or colored pigments. Which tests would support structural coloration? (Choose all that are appropriate.) (a) A microscope image shows a regular array of fibers (b) The color goes away when bleach is added (c) Placing the scale in water, where it swells, shifts the color to longer wavelengths (d) Crushing the scale into a fine powder has no effect on the color (e) The scale has different colors on different parts of it 3.9. Draw diagrams of waves that show (a) superposition, (b) constructive interference, and (c) destructive interference. 3.10. Noise-canceling headphones are headphones with a microphone on the outside of each earpiece that senses the incoming sound waves. Electronics in the headphones then emit the same sound through a speaker on the inside of the earpiece but with the polarity reversed. Explain how this reduces the amount of noise that you hear. 3.11. A thin film of oil on water strongly reflects blue light. (a) Draw a diagram that shows the cause of this. (b) Should the oil be made thicker or thinner to reflect green light? 3.12. If you watch the colors in a soap bubble that’s in the sun, you will notice that it’s colorful for a while, then it turns totally transparent, and then it pops. Explain why it turns transparent just before popping.
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3.13. Suppose you’re the only person on a small suspension bridge, and you want to jump up and down to create the first overtone standing wave in the bridge. (a) Where, along the bridge, would jumping be totally ineffective? (b) Where would jumping be most effective? 3.14. (a) Sketch a diagram showing the interference pattern that is observed for the double-slit experiment. (b) Sketch a similar diagram, but showing the result that one would expect to see if light were only a particle and not a wave, so the superposition principle did not apply. Problems 3.15. Diffraction. (a) Draw a diagram that shows diffraction through a small hole. Now suppose you’re in a room which has a 1 meter wide door, which is open, and music is playing outside. (b) What is the behavior of very low notes with ∼10 m wavelength (not go in the room, go in the room and travel in a straight line across it, or go in the room and fill the room)? (c) What is the behavior of medium notes with ∼1 m wavelength? (d) What is the behavior of very high notes (∼10 cm wavelength)? 3.16. A violin E string is 32.8 cm long and has a fundamental frequency of 659 Hz. (a) What is the wave velocity on this string? (b) How many times faster than the speed of sound is this? 3.17. Consider an acoustic guitar with 65 cm long strings. The A string is tuned to produce a note with a fundamental frequency of 110 Hz. (a) What is the wavelength of the fundamental mode? (b) What is the wave velocity on the string? (c) What is the frequency of the first overtone? (d) The musician now shortens the vibrating part of the string by pressing it against a guitar fret (without changing the wave velocity) to make its fundamental frequency 123 Hz (a ‘B’ note); how long is the vibrating part of the string now? 3.18. Ahmed wants to measure the thickness of one of his hairs using interference. He shines a laser with a 650 nm wavelength onto the hair and observes fringes on the wall, 3 meters away. He determines that the distance between the two brightness minima that are on either side of the central bright spot is 57 mm. How thick is the hair? 3.19. Consider a bathroom shower that is 1.5 m long and 1.5 m wide, and suppose you’re singing in this shower. (a) What sound wavelength has the lowest resonant frequency, meaning that it’s the fundamental frequency? (b) Using the fact that the speed of sound in air is 340 m/s, what is the frequency of this sound wave? 3.20. The waves in the following diagram are neither in phase nor out of phase, but are in between with what’s called a phase shift. (a) Add them using the superposition principle. (b) Is the wavelength the same or different from the original waves? (c) Is the final amplitude the same as for constructive interference, or destructive interference, or in between?
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+ =
3.21. The waves in the following diagram have exactly a factor of two difference in wavelengths. (a) Add them using the superposition principle. (b) Is the repeat interval in the final wave, meaning the distance where it exactly repeats itself, the same as that of the longer or shorter wavelength component wave? (c) Is the wave shape the same as that of the longer wavelength or shorter wavelength component wave, or neither?
+ =
Puzzles 3.22. Consider two radio antennas that are separated by exactly one wavelength. They broadcast waves with the same frequency and are in phase with each other. (a) Make a diagram (a top view, in which each antenna is a dot) that shows where the broadcasted radio waves interfere constructively and where they interfere destructively. Hint: use a drawing compass. (b) Now assume the towers are separated by two wavelengths and make a new diagram of where they interfere constructively and destructively. 3.23. A friend has a clever idea of building a flashdark, which is just like a flashlight except it shines darkness instead of light. His plan is that it will emit light that that is out of phase with the existing light, so the light waves will cancel out and produce darkness. Give some reasons for why this wouldn’t actually work. 3.24. Suppose one guitar string oscillates at 196 Hz and another at 197 Hz. (a) If they are in phase at time 0, then how long, in seconds, will it take be for them to be in phase again? (b) What is the beating frequency of these strings? (c) What would the beating frequency be if the second string were changed to 198 Hz? (d) What if it were changed to 196.5 Hz? (e) Based on these results, what is the beating frequency when the frequency difference is f ?
4
Wave Energy
Figure 4.1 Wave hitting a breakwater in Crete, Greece, showing its immense amount of energy.
Opening question Plants use chlorophyll molecules for photosynthesis, which are green. Which is not true? (a) chlorophyll primarily absorbs red and blue light (b) chlorophyll transmits green light (c) the electrons in chlorophyll can oscillate with the frequency of blue light waves (d) chlorophyll primarily absorbs green light (e) red and blue light waves transfer energy to chlorophyll.
© Springer Nature Switzerland AG 2023 S. S. Andrews, Light and Waves, https://doi.org/10.1007/978-3-031-24097-3_4
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4 Wave Energy
Waves transport energy, and exchange energy with the things around themselves. For example, light bulbs convert electrical energy to electromagnetic wave energy, those waves propagate across the room, and they deliver their energy into the colored surfaces that absorb them. Likewise, water waves are produced by wind over the ocean surface, they transport this energy across the ocean, and they deliver their energy into beaches on the far shore. As yet another example, plucking a guitar string produces string waves. They vibrate until they are absorbed by the body of the guitar, where their energy gets converted to sound waves. After propagating through the air, they are absorbed by soft objects such as curtains and rugs. All of these interactions involve energy transfer into waves, energy transport as waves, and then energy transfer out of waves. Energy exchange into and out of waves occurs in a few main ways, including either abruptly or gradually, and either with or without matching frequencies. Gradual energy transfer with matching frequencies, called resonance, is particularly interesting. Among other things, it is responsible for most of the sounds that we hear and the colors that we see.
4.1
Energy and Power
4.1.1
Energy
Energy is surprisingly difficult to define because it is not something that one can see, feel, or measure directly. Instead, it is best described as a fundamental physical quantity that is always conserved. That is, energy can be transferred between things, or it can be converted between different forms, but the total amount of energy never changes. This “law of conservation of energy” is one of the bedrock principles of physics1 . Figure 4.2 shows the energy conversions that occur when a book is dropped on the floor. Figure 4.2 A book’s energy being converted between different forms, but always conserved.
1905, Einstein showed that mass and energy are equivalent, related by the equation E = mc2 , so the law of conservation of energy is now more correctly called the law of conservation of massenergy. Energy, or mass-energy, is conserved because of deep connections between conserved quantities and symmetries of nature that were discovered by Emmy Noether in 1915. The relevant symmetry of nature in this case is the observation that the laws of physics do not change over time, which is called time-translation symmetry.
1 In
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The book starts with gravitational potential energy. Potential energy is often described as stored energy. More precisely, it is energy that is held in an object’s position relative to other objects or in an object’s internal structure. The potential energy in this case is gravitational, meaning that it’s the energy of the book’s height above the floor. Other forms of potential energy include elastic potential energy, such as the energy of a stretched rubber band, chemical potential energy, such as the energy in food, gasoline, or dynamite, and electric potential energy, such as the energy released by a spark or lightning. When the woman lets go of the book, its potential energy gets converted to kinetic energy. Kinetic energy is the energy of moving objects, whether of books, electrons, or planets. Kinetic energy is larger for objects that are moving faster and/or that are heavier. As the book lands on the floor, its kinetic energy gets converted to vibrations in both the floor and the book, some of which produce sound waves that then get absorbed by the walls of the room. In the process, the energy dissipates to billions of molecules that all move in uncorrelated ways. This is called thermal energy, meaning the energy of heat. It would be difficult to measure, but the book, floor, and surrounding room would warm up by an amount that’s equivalent to the book’s prior kinetic energy. Energy conversion into thermal energy is special because it is irreversible, meaning that it only proceeds in one direction. A slightly warmed up book won’t convert its thermal energy into kinetic energy, causing it to suddenly leap up into the air. This irreversibility of energy transfer leads to the identification of thermal energy as a “degraded” form of energy that is less useful than other forms. However, it’s not totally useless, as seen by the fact that steam engines are able to convert some of their thermal energy into kinetic energy.
4.1.2
Energy Units
Energy can be measured with many different units. The metric unit, which is typically the best unit for calculations, is the joule, abbreviated as J, where 1 J is about how much potential energy an apple has when it rolls off a table, and then how much kinetic energy it has just before it hits the floor2 . Or, a joule is roughly how much energy you exert when you lift a textbook up by 6 inches. A joule can also be expressed in metric base units of kilograms, meters, and seconds, with the conversion that 1 J = 1 kg m2 s−2 . Another common energy unit is the calorie, where 1 calorie equals 4.184 J and is the energy required to heat 1 milliliter of water by 1◦ C. Food calories, which are the energy units used on food packaging, are different yet. A food calorie, or kilocalorie, is equal to 1000 calories, making 1 food calorie equal to 4184 J.
2 The joule unit is named forJames Prescott Joule (1818–1889), an English physicist, mathematician, and brewer, who discovered the relationship between mechanical work and heat, which led to the law of conservation of energy.
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Wave Energy
Some gyms offer “battle ropes” for building fitness because people exert a lot of energy when making waves in ropes (Figure 4.3). It’s typically sufficient to say that the person’s energy simply becomes wave energy. However, one can also be more specific about the partitioning of the wave energy between potential energy, meaning the rope’s height above the floor, and kinetic energy, meaning the rope’s vertical speed. Clearly, this partitioning varies at different points along the rope. Figure 4.3 A woman exercising by transferring her energy into waves in “battle ropes”.
The same considerations also apply to other types of waves. For example, water waves involve both the potential energy of having some water displaced upward and the kinetic energy of water molecules moving. Similarly, sound waves involve the potential energy of compressed air and the kinetic energy of moving air molecules. In contrast, light waves are purely electric and magnetic potential energies. Regardless of how the energy is partitioned, the energy in a wave is always proportional to the square of the wave amplitude, E ∝ A2 .
(4.1)
This agrees with our prior description of the amplitude as representing the energy in a wave (Section 2.1.2). As we said before, amplitude determines the energy of an ocean wave, the brightness of a light, or the loudness of a sound. Quantitatively, Eq. 4.1 shows that the woman exercising with battle ropes in Figure 4.3 would get four times as much exercise if she doubled the amplitude of her waves.
4.1.4
Power
Waves often arrive in a continuous sequence, making it useful to consider the rate of energy flow. This rate is called power, quantified as the amount of energy converted
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in some amount of time, E . (4.2) t Here, E is the amount of energy converted and t is the amount of time being considered. Power is measured in units of watts, abbreviated W, which is the number of joules of energy in each second3 . As equations, 1 W = 1 J/s = 1 kg m2 s−3 . As an example, a 25 W light bulb converts 25 joules of electrical energy (E) into light during every second (t). Applying these results to waves, wave power is how fast waves transmit energy. Waves that have higher amplitudes contain more energy, from Eq. 4.1, so they also transmit more power. Also, waves that propagate faster are able to move energy faster. Combining these two effects shows that wave power increases with greater wave amplitude and velocity as P=
P ∝ A2 v.
(4.3)
Again, there are more detailed equations for specific wave types, but this result is sufficient for our interests. Example. Sven consumes 2500 food calories of energy per day. What is his average power input? Answer. Convert units from food calories per day to watts,
?W=
food cal. hr. min. 4184 J 1 day 1 1 2500 · · · · = 121 J/s. cal. 24 1 day 1 food hr. 60 min. 60 s
Sven’s power input is 121 J/s, which is 121 W. Total energy is always conserved, so Sven’s average power output is also about 121 W. This accounts for his physical work, such as hiking uphill, and also how much his body heat warms his surroundings.
4.1.5
Energy Density and Power Density
Often, we only detect waves over a small area, so it makes sense to consider their energy or power only within that area. We do so with the energy density or power density, which represent the energy or power within a fixed area. For example, suppose someone taps the middle of a drum head with a drumstick, transforming 2 J of kinetic energy to wave energy. As the wavefront spreads out along
3 Watt
units are named for James Watt (1736–1819), a Scottish engineer and chemist who made substantial improvements to the design of steam engines.
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the taut membrane, shown in Figure 4.4, that 2 J of energy is conserved, but it spreads out into a ring that has a larger and larger circumference. This implies that there must be less and less energy within any one centimeter of the wavefront. To make this quantitative, the total amount of energy, E, spreads out over the circumference of a circle, which has length 2πr , where r is the radius. The energy density, ρ2D (Greek letter rho), is the ratio of the energy and the length that it is spread over, making it ρ2D =
E . 2πr
(4.4)
Plugging in numbers, the same 2 J of drumstick energy has a density of 3.2 J/m when the wave radius is 0.1 m, and a density of 1.6 J/m when the wave radius is 0.2 m. In the latter case, 1.6 J/m represents 1.6 joules of energy for each meter of wavefront length. Figure 4.4 A pulse created at the center of a taut membrane at t = 0 and expanding outward, decreasing in amplitude as it expands while conserving total energy.
Reversing the mathematics gives the energy in some region from the energy density. As an equation, one multiplies the energy density by the size of the region, E = ρ2D L,
(4.5)
where L is the length of the region. Continuing with the prior example, if the wavefront on the drum has an energy density of 1.6 J/m, and 1 cm of the wavefront gets absorbed by some object, then that object absorbs (1.6 J/m) · (0.01 m) = 0.016 J of energy. The same concept of energy spreading applies in three dimensions, but with the replacement of the circumference of a circle with the surface area of a sphere. The surface area of a sphere is 4πr 2 so the energy density over some amount of area on the surface is E . (4.6) ρ3D = 4πr 2 For example, suppose a camera flash emits 76 J of light energy that spreads out equally in all directions. If there’s a wall that’s 3 meters away from the flash (and facing it), then the light energy density that hits this wall would be 76 J over 4π(3 m)2 = 0.67 m2 . Dividing gives the result that about 0.67 J of light energy would hit each square meter of the wall. As another example, this time using power rather than energy, suppose a 25 W light bulb emits light in 3 dimensions. The power density that it transmits to some surface that is r distance away from it would be 25 W divided by the area 4πr 2 .
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Example. A loudspeaker uses 170 W to broadcast sound in all directions. What is the sound power density at a person who is 50 m away? How much sound power hits her eardrum, which has a surface area of 65 mm2 ? Answer. Compute the power density with Eq. 4.6, but with power instead of energy. ρ3D =
P 170 W = = 0.0054 W/m2 . 2 4πr 4π(50 m)2
The sound power that hits the eardrum is this power density value times the eardrum area: P = (0.0054 W/m2 ) · (65 · 10−6 m2 ) = 3.5 · 10−7 W.
4.2
Spectra
4.2.1
Intensity Spectra
In 1666, Isaac Newton famously showed that white sunlight could be split into the colors of the rainbow using a prism. Further, he also showed that these colors could be recombined with another prism to form white light again. These results showed for the first time that white light is not especially pure, as was previously believed, but a mixture of many different colors. Other types of waves can have energy at multiple wavelengths simultaneously as well, as we saw in Chapter 3 when exploring wave superposition. During a stormy day on the ocean, for example, one might observe long wavelength ocean swells, medium wavelength wind waves, and short wavelength raindrop ripples all at once. Likewise, a marching band produces long sound waves from the bass drums, medium sound waves from the clarinets, and short sound waves from the flutes. It can helpful to show the amount of wave power in the different wavelengths with an intensity spectrum, where the different intensities are displayed as a graph. For example, Figure 4.5 shows the intensity spectrum for a fluorescent light bulb. In this spectrum, the x-axis shows the light wavelength, which is also depicted using colors, while the y-axis represents the light’s brightness. This spectrum shows that fluorescent light bulbs emit some light at all colors, but have particularly strong emission at particular wavelengths in the red, green, and violet ranges. The green emission, peaking at 547 nm, has the highest peak, implying that the light bulb emits more light here than at any other wavelength. The red peak is nearly as high, but is also a lot narrower than the green peak, which gives it a smaller peak area. This area difference implies that this light bulb emits much more total green light than total red light. Figure 4.6 shows another intensity spectrum, now for sound waves. This time, the x-axis is in frequency units, ranging from about 30 Hz to 16 kHz. The y-axis
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Figure 4.5 Spectrum of a fluorescent light bulb.
Figure 4.6 An audio equalizer showing a sound spectrum.
represents sound intensity, here showing that the sound was most intense at 314 Hz, but was also substantial at other frequencies. This y-axis uses units of decibels (dB), which is a non-linear scale that’s often used for sound. Here, an increase of 3 dB represents twice as much wave power and an increase of 10 dB represents 10 times as much wave power. However, there is no consensus on what 0 dB represents. It often represents the quietest sound that a human can hear, but it also sometimes represents the loudest sound that some particular piece of audio equipment can work with, or something else. For this graph, 0 dB was defined as the loudest sound that the sensor4 could work with, so all of the measurements have negative decibel values.
4.2.2
Continuous and Line Spectra
Intensity spectra can often be classified into line spectra and continuous spectra, depending on whether they are spiky or smooth. Line spectra, which are spiky, arise when most of the power is confined to a few wavelengths or, equivalently, a few frequencies. These frequencies almost always represent the natural frequencies of the system that emits the waves (Section 3.2.3).
4 The
image is from an iPhone app called Spectrum.
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As an example, Figure 4.7 shows a spectrum for the sound emitted from plucking a guitar string, showing that most of the sound energy is confined to a series of specific frequencies. The string that was plucked in this case has a fundamental frequency of 82.4 Hz, observed in the first peak, and has overtones at 165 Hz, 247 Hz, 330 Hz, and so on, which are the positions of the subsequent peaks. As an interesting curiosity, the third harmonic emits more wave power than the fundamental, but a person listening to the combined tone would still perceive its pitch as that of the fundamental frequency5 . Figure 4.7 Spectrum from a guitar playing the E2 note.
Figure 4.8 Emission spectrum from hydrogen atoms.
intensity
Figure 4.8 shows another line spectrum, in this case representing the light emitted when an electric current runs through hydrogen gas (like a neon sign, but with hydrogen instead of neon). Again, nearly all of the energy is emitted at discrete wavelengths and, again, these wavelengths correspond to the natural frequencies of the system. Here, they are the natural frequencies of electron matter waves in hydrogen atoms. Spectral data on atoms and molecules, investigated in the field of spectroscopy, are extremely helpful for investigating the physics of these systems because they show the frequencies of the atoms’ and molecules’ normal modes.
0 400
500 600 700 wavelength (nm)
Continuous spectra arise when wave power is spread over a wide range of wavelengths. This typically occurs when a system either doesn’t have well defined natural
5 Our ears hear all of the frequencies that are emitted, including the harmonics. Our auditory systems then combine this information to figure out what the fundamental frequency must have been, even if there wasn’t much power in the fundamental frequency itself. The inferred fundamental frequency is then what we perceive as the pitch of the sound.
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frequencies, or has so many different natural frequencies that they all blend together. For example, the left side of Figure 4.9 shows the seismic wave intensity from an earthquake. The underground rocks don’t have well defined natural frequencies, so the spectrum is continuous. The right side of the figure shows the visible portion of the spectrum for an incandescent light bulb (traditional light bulbs with a filament), which is another continuous spectrum. It shows that incandescent light bulbs emit energy throughout the visible region, but with more red and yellow light than green or blue light. Although not shown here, the emission continues on into the infrared, where most of the power is actually emitted. In this case, the light bulb filament has atoms with many natural frequencies, which all blend together to create this continuous spectrum.
Figure 4.9 (Left) Power spectrum of an earthquake. (Right) Light intensity spectrum of a “soft white” incandescent light bulb.
4.2.3
Transmission Spectra
Transmission spectra show the fraction of light that is transmitted through some sample. Transmission is typically quantified as the percent of the light’s power that makes it through the sample, as opposed to being absorbed. As an equation, it is %T =
Ptransmitted · 100%, Pincident
(4.7)
where Pincident and Ptransmitted are the powers of the incident and transmitted light. Figure 4.10 shows transmission spectra for glass from clear, green, and amber beer bottles. They show, much as one would expect, that clear bottles transmit most visible light, green bottles transmit violet and green light, and amber bottles transmit only small amounts of yellow and red light. All glass samples block light below 300 nm because glass itself is opaque in the UV range. Brewers care about these spectra because ultraviolet and short-wavelength visible light causes beer to decompose and gain unpleasant flavors, whereas longer wavelength light does not, so they prefer bottles that block the problematic light.
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Figure 4.10 Transmission spectrum of 2.8 mm thick beer bottle glass.
The left panel of Figure 4.11 shows infrared transmission spectra for several gases that are in the Earth’s atmosphere. The x-axis represents the light wavelength, as usual, but measured in micrometers (often called microns); the far left end of the scale is 1 µm, which is 1000 nm and is close to the visible region. The right panel shows the transmission spectrum of the complete atmosphere, this time with the x-axis starting at 0 µm. It shows that the atmosphere is opaque in the UV, reasonably transparent in the visible range (about 0.4 to 0.7 µm), and then opaque again in many spectral bands throughout the IR region. We can determine which molecules create these spectral bands by comparing their positions with the individual gas transmission
Figure 4.11 Transmission spectrum of (left) gases in the Earth’s atmosphere and (right) the complete Earth’s atmosphere.
spectra in the left panel. The results (including from spectra that aren’t shown here) are listed at the bottom of the right panel. For example, the atmosphere is opaque in much of the ultraviolet region due to oxygen and ozone molecules (O2 and O3 , respectively), is opaque between about 5.5 and 7.5 µm due to water vapor (H2 O), and is opaque at around 2.8 µm, 4.3 µm, and beyond about 14 µm due to carbon dioxide (CO2 ). These latter regions are growing slightly over time due to increased carbon dioxide in the atmosphere, which is the main cause of global warming (we’ll revisit this topic in Section 12.1).
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4.2.4
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Absorption Spectra*
Transmission spectra can be hard to use for quantitative work because: (1) they are difficult to adjust to account for thicker or thinner samples, such as if the beer bottle glass discussed above were made thicker or thinner, (2) are difficult to combine, such as finding the light transmission through both the green and amber glass together, and (3) are hard to read when transmission values are very small; for example, 0.1% transmission and 1% transmission are different by a factor of 10 but look about the same on a transmission spectrum. The solution is to reverse the y-axis of the transmission spectrum, so it shows absorption rather than transmission, and to then rescale it using the logarithm. The result is the absorption coefficient, A, defined as A = − log
%T . 100%
(4.8)
The absorption coefficient is a unitless value in which absorption of 0 means that no light is absorbed, 1 means that 90% of the light is absorbed, 2 means that 99% of light is absorbed, 3 means that 99.9% of light is absorbed, and so on6 . Figure 4.12 shows the absorption spectrum for β-carotene, which is the orange pigment in carrots and pumpkins, where the left y-axis is marked in absorption units. It shows that β-carotene molecules absorb green, blue, and violet light, while transmitting red, orange, and yellow light. We see the colors that are not absorbed, so this spectrum shows that β-carotene should appear orange, as in fact it does. Figure 4.12 Absorption spectra of β-carotene. The left axis shows absorption for a concentration of 10 µmol/L and a thickness of 1 cm. The right axis shows the extinction coefficient in units of M−1 cm−1 .
Because of the mathematical rescaling, absorption values increase linearly with the sample thickness and the concentration of molecules in the sample. This means that doubling the sample thickness also doubles the absorption coefficient. Similarly,
6 Despite being unitless, absorption values are sometimes listed with “optical density units” or ODs.
However, this does not change either the numerical values or their interpretation.
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doubling the sample concentration also doubles the absorption coefficient. This linear relationship is given by the equation A = cl. (4.9) where (Greek letter epsilon) is called the molar extinction coefficient for the sample, l is the sample pathlength, and c is the sample concentration7 . The extinction coefficient gives an absolute measure of how colored the molecule is, independent of the concentration or pathlength. The right axis of Figure 4.12 is marked in extinction coefficient units. In addition to being linear with sample thickness and pathlength, the absorption coefficient for a mixture of samples is simply the sum of the absorption coefficients for the individual samples.
4.3
Resonance
4.3.1
Resonance and Coupling
Energy is often transferred between oscillating systems through the process of resonance. More precisely, resonance is the transfer of energy from one oscillating system, called the driver, to one of the natural frequencies of another oscillating system, called the driven system. Two oscillating things that interact with each other are in resonance, or they resonate, when they have the same frequencies and they are out of resonance when they have different frequencies. For example, consider a father (the driver) who is pushing his daughter on a playground swing (the driven system), shown in Figure 4.13. The girl and her father
Figure 4.13 A father pushing his daughter on a swing. She gains energy through resonance.
are in resonance when the father pushes at intervals that match the swing’s natural frequency, where this is simply the swing’s oscillation rate when it isn’t being pushed.
7 The sample concentration is measured in moles of molecules per liter of water (or other solvent), where 1 mole/liter is called 1 molar and abbreviated 1 M. A mole is about 6.022 · 1023 molecules. More practically, if you calculate the molecular weight of the molecules, then that many grams of the sample is equal to 1 mole. See any introductory chemistry book for details.
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Admittedly, it is hard to push a swing at the incorrect interval, but one can imagine it. Doing so clearly would not be effective for achieving large swinging arcs, or for making the child happy. In this example, the father uses resonance to transfer energy from his oscillatory pushing to the girl’s oscillatory swinging. As a wave example of resonance, consider walking along while carrying a cup of coffee. If your footsteps have the same frequency as the natural frequency of the coffee sloshing back and forth in the cup, then they are in resonance. In this case, energy gets transferred from your oscillatory body motion to standing waves in the cup, creating larger amplitude waves and then spilled coffee. A solution is to walk either slower or faster, so that your footsteps are out of resonance with the standing waves. The strength of the interaction between the two oscillating systems is called coupling. In the swing example, coupling is how hard the father pushes on the swing. If the father doesn’t push very hard, meaning that the coupling is weak, then the swing isn’t going to go very high. At the opposite extreme, if the father pushes very hard, meaning that the coupling is strong, then he will transfer a great deal of energy to the swing and it will go much higher. In the coffee example, the coupling is given by the firmness of your footsteps and the rigidity of your arm. This means that other ways to prevent spilling coffee would be to walk more gently or to hold your arm less rigidly, both of which decrease coupling between the cup and your footsteps. Thus, to transfer energy through the process of resonance, two conditions need to be met: an oscillating driver needs to resonate with a natural frequency of a driven system and there needs to be sufficient coupling.
4.3.2
Two Resonance Examples with String Waves*
A simple example of resonance arises in a laboratory experiment that explores string standing waves, shown in Figure 4.14. A string extends from a post at one end to a pulley at the other end, and then down to a weight which holds the string at a constant tension. The string is also clamped onto a peg that oscillates up and down, called the “driver”. Experimentally, it’s seen that standing waves appear in the string if the driver oscillates at particular frequencies, and not for other frequencies.
Figure 4.14 Resonance for standing waves on a string.
To make sense of this result, the string between the driver and pulley constitutes a cavity, so only specific standing waves can fit in this space (see Section 3.2.3). If
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the driver resonates with one of those standing wave’s natural frequencies, then it transfers energy to the string through resonance. If not, then it doesn’t, and the string just shakes a little without exhibiting any significant standing waves. When it does resonate, the amplitude of the string wave corresponds to the degree of coupling between the driver and the string, which is determined by how far the driver peg oscillates up and down. With a little skill, one can tune the driver to produce the fundamental mode or any of about 10 of its overtones. Standing waves on a violin string are subtly different. The string and cavity are essentially the same, but the driver is a bow that is drawn over the string. This driver does not have a well-defined frequency but instead sticks and slips at irregular intervals, causing it to drive many different frequencies at once. Of these frequencies, the ones that match the string’s natural frequencies resonate with the string and transfer energy to it while the others have very little effect.
4.3.3
Resonance with Electromagnetic Waves
Electromagnetic waves often transfer their energy to atoms, molecules, or other physical objects through resonance as well. In almost all cases, the waves’ oscillating electric field alternately pushes and pulls on electrically charged particles, which then leads to energy transfer if the light frequency is in resonance with the object’s natural frequency. The light’s magnetic field exerts oscillating forces too, but they are much weaker and so can be ignored in most cases8 . The amount of energy transfer (or the probability of energy transfer for quantum mechanical systems, discussed in Chapter 13) is determined by the coupling between the light and the object, where this coupling depends on the amount of electric charge and how far the electric charges are able to move. As an example, a radio antenna is essentially just a wire that conducts electricity. If one were to move some of the electrons in this wire from one end to the other and let them go, they would oscillate back and forth several times before coming to a rest, much like the child on a swing. This oscillation can also be driven by radio waves that have the same frequency, meaning that the waves are in resonance with the electrons going back and forth in the antenna. If this antenna is attached to a radio, then the radio amplifies the electron motions, does some more signal processing, and converts the result to sound waves for people to hear. Electrons within molecules are similar, but travel much shorter distances than the ones in antennas so they have much faster natural frequencies, typically at the frequencies of visible or ultraviolet light waves (∼ 1014 −1016 Hz). β-carotene molecules form a nice example. Figure 4.15 shows that they have a long and straight structure, and a main chain that alternates between single and double bonds, which
8 One of the few cases where the magnetic field matters is in a phenomenon called circular dichroism, where a sample absorbs different amounts of the two polarizations of circularly polarized light. See Andrews and Tretton “Physical principles of circular dichroism” Journal of Chemical Education 97: 4370–4376 (2020).
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together allow electrons to move back and forth relatively unimpeded much as they do in radio antennas. As we saw in Figure 4.12, β-carotene molecules absorb blue and violet light, with an absorption peak at around 450 nm. This corresponds to a frequency of around 6.7 · 1014 Hz, implying that light waves with this frequency resonate with electron motions in β-carotene molecules. Also, this is the natural electron oscillation frequency in these molecules9 . Figure 4.15 Chemical structure of β-carotene, one of several similar carotene molecules.
4.3.4
Resonance with Microwaves and Infrared Light*
Other bands of the electromagnetic spectrum behave similarly, but resonate with other molecule normal modes. Microwaves resonate with molecular rotations, typically in the GHz frequency range (1012 s−1 ). Water molecules and parts of sugar molecules tend to couple particularly strongly to microwaves because they have electrically charged atoms that electric fields can push or pull on (Figure 4.16). Microwave ovens use this resonance to heat food; the ovens emit microwaves, the microwaves drive rotations of water and sugar molecules in the food through resonance, and then these spinning molecules bump into other molecules to transfer energy elsewhere. In contrast, molecules that cannot rotate, such as frozen water molecules in chunks of ice, or the crystals in a porcelain plate, don’t resonate with microwaves and so are not heated10 . Also, fat molecules, such as those in vegetable oil, don’t get heated either, but in their case it is because they have minimal electric charges on their atoms so they don’t couple well with electromagnetic waves.
Figure 4.16 Rotation of a water molecule, showing the three rotational axes. These rotational frequencies are different, but all in the GHz range. The oxygen atom (red) carries a slight negative charge while the hydrogen atoms (gray) carry slight positive charges. The oxygen atom only moves in two of the modes, so those two couple to the microwaves while the third does not.
9 This explanation is qualitatively correct in many ways, and lends useful insight, but it also ignores some essential details that arise from quantum mechanics, including that different electrons do not move independently. See Section 14.4.1 for the quantum mechanics solution to this problem. 10 Ice melts in a microwave oven primarily due to microwave absorption by liquid water on the surface of the ice, which then gets warm and conducts heat into the ice.
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At higher frequencies, infrared light resonates with molecule vibrations, in which the atoms within a molecule oscillate back and forth much like balls on springs (Figure 4.17).
Figure 4.17 The three vibrational modes of a water molecule (shown with exaggerated displacements). Each has a distinct frequency, but all are in the infrared region. The oxygen atom (red) carries a negative charge while the hydrogen atoms (gray) carry a positive charge.
Again, the light’s electric field pushes and pulls on charged atoms, now driving molecule vibrations. This energy transfer can be felt with a heat lamp, which warms your body by emitting infrared light which then resonates with vibrational modes of molecules within your skin. As before, molecules only couple with infrared light waves if their atoms have electric charges. Water molecules have charged atoms so they absorb infrared light strongly, whereas the nitrogen and oxygen molecules in the air do not have charged atoms so they don’t absorb infrared light significantly. It is worth noting that electromagnetic waves are absorbed in all of these examples by the normal modes of an atom, molecule, or antenna, and not necessarily by any single electron. Correspondingly, the absorbed energy gets put in the motions of these normal modes, much like energy can be put in the vibrations of a guitar string.
4.3.5
The Tacoma Narrows and Millennium Bridges*
Resonance is a major concern in engineering and architecture because resonance with wind, people, or earthquakes can create dangerous oscillations in structures. For this reason, all modern bridges and tall buildings are designed to not have natural frequencies that correspond with frequencies that they are likely to encounter, and to have sufficient damping (see below) to reduce any oscillations that do arise. Nevertheless, mistakes inevitably occur. The original Tacoma Narrows Bridge, shown in Figure 4.18, was completed in 1940. It was a suspension bridge that spanned the Tacoma Narrows, in Washington State. It was one of the longest suspension bridges in the world when it was completed, but nevertheless most of the technology that it used had been proven in other long suspension bridges, including San Francisco’s Golden Gate Bridge and New York City’s George Washington Bridge. The Tacoma Narrows Bridge was distinctive for its exceptionally narrow and thin bridge deck, which were seen as benefits at the time for giving the bridge a graceful look, but proved to be its undoing. Once the bridge deck was completed, even before the rest of the bridge was finished, the deck was found to move up and down whenever the wind blew, leading the construction workers to nickname the bridge “Galloping Gertie.” The vertical motion arose from resonance between oscillations from the wind and the bridge’s natural oscillation frequency. Attempts to control this motion largely failed.
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Figure 4.18 The first Tacoma Narrows Bridge, exhibiting “fluttering” oscillations.
On November 7, 1940, just four months after the bridge had opened, a storm brought 40 mile per hour winds to the Tacoma Narrows. This is a strong wind, but would not be remarkable normally. As before, resonance between wind effects and the bridge’s natural frequencies led the bridge to oscillate vertically, but with much greater amplitude this time. After a while, the bridge suddenly changed from this vertical motion into a twisting “fluttering” motion. This was again powered by the wind, but subsequent analyses showed that it was not technically resonance because it did not arise from a correspondence between wind and bridge frequencies. The bridge only survived this fluttering for about 45 minutes before breaking apart and collapsing into the Tacoma Narrows, where it still lies underwater. This failure had an enormous impact in the physics and engineering communities, in part because a local camera shop owner, Barney Elliott, filmed the oscillations and collapse (Figure 4.18). His film has served as a cautionary tale and teaching tool for generations of students. The lessons learned from this bridge collapse, such as the finding that thin decks tend to resonate in high wind, were applied immediately to other bridges and are still important to bridge design. However, bridges still exhibit unexpected resonance on occasion. The Millennium Bridge, shown in Figure 4.19, is a footbridge that spans the River Thames in central London. Figure 4.19 The Millennium Bridge.
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Despite efforts to avoid resonance problems during its design, the bridge swayed back and forth dramatically on its opening day, in June 2000, when there were up to 2000 people on the bridge at once. It turned out that this oscillation arose from resonance between the natural swaying frequency of the bridge and the pedestrians pushing left and right with their footsteps. This is surprising because one wouldn’t normally expect 2000 people who were walking normally to synchronize their footsteps. They did in this case though because slight bridge swaying caused people to step sideways a little to brace themselves; these steps pushed the bridge in the opposite direction, so people braced themselves again, and so on. This led to positive feedback, where a little swaying led to factors that amplified the swaying. The bridge was closed again, retrofitted with dampers to reduce the oscillations, and then reopened two years later. The bridge has not exhibited substantial resonance since then.
4.3.6
Resonance is Reversible
The physics of resonance is reversible, meaning that if two systems are in resonance, then the energy can be transferred in either direction. Also, the amount of coupling is the same in both directions. However, the terminology is not always reversible. This is because the resonance process is defined as the transfer of energy from one oscillating system to a natural frequency of another oscillating system and some oscillating systems don’t have natural frequencies. This reversibility is nicely demonstrated in yet another classic example of resonance, which is that it’s possible to shatter a wine glass using sustained sound with the correct frequency. Here, the sound waves are the driver and the wine glass is the driven system, where the sound waves are driving oscillations of the wine glass rim bending in and out. At resonance, there can be sufficient energy transfer from sound waves to the wine glass that the oscillations can become big enough to break the glass. In the reverse process, a simple tap on the wine glass creates the same rim oscillations. Those oscillations transfer their energy to sound waves, forming the characteristic ring sound of the wine glass. This energy transfer from wine glass to sound is identical to the previous one from sound to wine glass. However, this latter energy transfer is not technically resonance because there is no natural frequency for sound waves in the air. Our other examples of resonance can also be run in reverse. The girl on the swing can be slowed down through resonance, a radio antenna can broadcast radio waves as well as receive them, and atoms and molecules can emit light in addition to absorbing it. The similarity of the atomic absorption and emission has enabled researchers to figure out the sun’s composition. In 1814, the German physicist Joseph von Fraunhofer observed that the sun’s spectrum has many dark lines in it, shown in the upper panel of Figure 4.20, including two particularly prominent dark lines in the yellow region of the spectrum at around 589 nm. The explanation, figured out 40 years later by Gustav Kirchoff and Robert Bunson, is that deeper layers of the sun emit light at
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Figure 4.20 (Top) The sun’s spectrum. Black lines arise from absorption of some light frequencies through resonance with specific atom or molecule normal modes. (Bottom) The emission spectrum of sodium atoms. The agreement between the sodium emission wavelength and some black lines in the solar spectrum shows that those black lines arise from absorption by sodium atoms in the sun’s atmosphere.
all wavelengths, but some of that light gets absorbed by atoms in more outer layers of the sun, causing the dark bands in the spectrum. Furthermore, they knew that hot sodium atoms emit yellow light11 at 589 nm, at exactly the same wavelengths as the dark bands in the sun’s spectrum. They realized that atoms absorb and emit light at the same wavelengths, which we now know is because resonance is reversible, from which they inferred the presence of sodium in the sun’s atmosphere.
4.4
Non-Resonant Energy Transfer
4.4.1
Abrupt Energy Transfer to Waves
Waves can also be created by sudden events. For example, plucking a guitar string, throwing a rock into a still pond, and clapping one’s hands are sudden events that create large but spatially localized displacements. These displacements propagate outward as waves. These abrupt energy transfers do not produce a well-defined wavelength or frequency because the initial displacement is only a localized pulse, rather than regularly repeating waves. It can be visualized by considering the shape of the water surface the moment after a falling stone puts a hole in it; this hole has a diameter, but nothing that resembles a wavelength. Similarly, abrupt sound events, such as hand clapping, also don’t have well-defined frequencies so one hears a bang, click, or snap, rather than a clear tone. An equivalent way of saying that the wavelength is not well-defined is to say that there are many different wavelengths at once that are all added together. This follows from the discussion of the superposition of waves with different wavelengths (Section 3.1.2), where we found that sine waves with different wavelengths can add to
11 This yellow emission is easily observed by dropping grains of table salt, which is sodium chloride,
into the flame from a stove or lighter.
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create complex waveforms. Or, vice versa, complex waveforms can be decomposed into multiple sine waves with different wavelengths. In this case, an abrupt pulse can be decomposed into many wavelengths and, correspondingly, many frequencies. Using the terminology introduced above, abrupt pulses create waves with continuous spectra. The wide range of wavelengths can be seen directly with the water waves that result from a stone falling in a pond. The stone puts a hole in the water, meaning it creates many different wavelength waves at once. In water, it turns out that waves with different wavelengths propagate at different speeds (called dispersion, which we will discuss in Section 6.5), so the different wavelengths separate into different rings as they propagate outward, with long waves on the outside and short waves on the inside. This wave pattern visually shows the many different wavelengths. Electromagnetic waves can be created abruptly as well. An example is an electric spark, which creates a sudden displacement in the local electric field. As with the prior examples, sparks produce a wide range of different wavelengths at once, in their case covering most of the radio frequencies. These radio waves are apparent if you listen to AM radio, tuned to any frequency you want, during a thunderstorm; each lightning stroke produces a crackle on the radio12 .
4.4.2
Energy Loss From Damping
Oscillations, including waves, typically lose energy over time due to friction, which is called damping. Damping is a non-resonant type of attenuation, using the definition that attenuation is simply the loss of energy over time. Returning to the playground swing example, if the father stops pushing his daughter on the swing, she will continue swinging for a while but will gradually slow down due to the damping that is caused by friction in the swing-set. Her swinging energy is not lost of course, because energy is conserved, but transformed into heat. Damping usually works for a broad range of frequencies. For example, water waves that wash up on a gently sloped sand beach lose their energy to damping, regardless of the wave frequency. It is also always irreversible, meaning that energy can be transformed from oscillations to heat but never from heat to oscillations. Damping is desirable in some situations. For example, vehicles need to have springs in their suspensions so that the wheels can roll over bumps while keeping the passengers relatively still, but these same springs also need to be damped to prevent the people from bouncing up and down excessively. Figure 4.21 shows a diagram of a motorcycle suspension including both a spring and damper. Also, engineers added dampers to reduce oscillations in the original Tacoma Narrows suspension bridge, which proved to be inadequate, and in the Millennium Bridge, where they were
12 The
visible light from a spark has a different source, arising from the moving electrons exciting air molecules, which then emit visible light when they relax.
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Figure 4.21 Shock absorber for a motorcycle, showing the spring and damper (the piston that is inside the spring).
successful. Tall building typically include dampers to prevent excessive swaying in the wind. All mechanical waves lose energy through damping, whether through air resistance, friction within the medium, or turbulent flow in water waves. In each of these cases, the wave energy gets converted into the random molecule motions that constitute thermal energy. Individual molecules can lose energy through damping as well. In a microwave oven, for example, water molecules are resonantly excited using microwaves, but then they transfer their energy to other molecules through collisions, which is damping.
4.5
Summary
Energy is a conserved quantity, meaning that the total amount doesn’t change, but it can change forms or be exchanged between objects. Energy forms include kinetic, potential, and thermal energy, of which conversion into thermal energy is typically irreversible. Energy units include joules, calories and food calories. When energy is transferred continuously, such as from a hot light bulb filament into visible light, the energy transfer rate is called power and is measured in watts. Wave energy is proportional to square of the wave amplitude. Waves that propagate in 2 and 3 dimensions, such as water and light waves, generally spread out from their sources, so their energy density decreases as their wavefronts expand. Intensity spectra depict the amount of energy at each wavelength or frequency. They can often be classified as line spectra, where the energy is localized to a few distinct wavelengths, or continuous spectra, where the energy is distributed over a range of wavelengths. Transmission spectra show how much light is transmitted through some sample, such as a piece of colored glass. Absorption spectra are conceptually similar, showing how much light is absorbed by some sample, but use different axis scaling. Sharp peaks in all types of spectra typically represent the normal modes of the system that is being investigated.
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Energy transfer between oscillating systems often occurs through resonance, in which an oscillating driving system adds energy to a normal mode of a driven system that has the same frequency. The coupling quantifies the strength of the interaction between the systems. In the example of a father pushing his daughter on a swing, the father’s pushing and the daughter’s swinging are in resonance and the coupling is how hard the father pushes. Light absorption often occurs through resonance, with the oscillating electric field of the light driving oscillations of electrons or other electrically charged objects. This largely explains energy transfer from radio waves to radio antennas, microwaves to molecule rotation, infrared light to molecule vibration, and visible and ultraviolet light to electron excitation. In each case, resonance occurs when the light and object have the same frequency and the coupling is determined by the electrical charge quantity and how far those charges move. Energy transfer between systems that have the same frequency is reversible, but is only called resonance if the driven system has a natural frequency. Energy can also be transferred to an oscillating system through an abrupt event, such as plucking a guitar string, which typically adds energy to a wide range of frequencies at once. Vice versa, energy can be transferred out of an oscillating system and into heat through damping, which again generally applies to a wide range of frequencies. Damping is irreversible.
4.6
Exercises
Questions 4.1. Which types of waves transmit energy? (a) waves don’t transmit energy (b) only electromagnetic waves (c) only electromagnetic and water waves (d) everything except sound waves (e) all waves transmit energy 4.2. Chlorophyll is a molecule found in plants that is essential for their photosynthesis. It is a pigment that creates the green color of plants. Which of the following are true? List as many as are appropriate. (a) chlorophyll primarily absorbs green light (b) chlorophyll absorbs red and blue light, and transmits green light (c) chlorophyll has no intrinsic color but is an example of structural coloration (d) the electrons in chlorophyll resonate with green light waves (e) chlorophyll absorbs energy from red and blue light
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4.3. It is possible for a sustained high musical note to break a wine glass through resonance. (a) What two periodic processes are in resonance with each other? (b) Is this favored by strong or weak coupling? (c) Is this favored by strong or weak damping? 4.4. After a rock is dropped in a calm pool, the resulting water waves spread out in circles and get smaller. List two reasons why they get smaller (both were topics of this chapter). 4.5. We saw that the energy density of water waves decreases as the waves spread out in proportion to r −1 , where r is the distance travelled, and the energy density of light waves decreases in proportion to r −2 . How does the energy density of string waves depend on the distance that they travelled, given again as a power of r ? 4.6. Lycopene is a plant pigment that is similar to β-carotene, except that the molecule is slightly longer. (a) Would you expect it to absorb higher frequency or lower frequency light? (b) What color would lycopene appear to be? (c) Lycopene is the major color pigment of either tomatoes or bananas; which one is it, based on your color prediction? 4.7. Anechoic chambers are rooms in which the walls are covered with surfaces that absorb all sound waves. Would you expect this absorption to work through resonance or damping? 4.8. Suppose an engineer designed a perfect light damper, which efficiently damped all light waves that hit it. What color would it appear to be? 4.9. Sketch the intensity spectrum for a green laser pointer. 4.10. In some Western movies, a person puts his ear on a train track to listen for an approaching train, because it’s easier to hear a train through the rails than through the air. Why might this be? Problems 4.11. On average, each person in the United States uses about 12,000 kWh (kilowatthours) of electricity per year. (a) What type of unit is kWh (e.g. energy, power, etc.)? (b) Convert this energy use rate to joules per year. (c) Convert this value to watts. 4.12. Ocean wave energy has long been recognized as an abundant source of renewable energy, so some devices have been developed for generating electrical power from wave power. One such device (an RM3 floating-point absorber
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wave-energy converter13 ) produced an average of 350 kW of power when the wave height, measured trough to peak, was 4.25 m and the wave speed was 16.7 m/s (water wave speeds depend on the wavelength, discussed in Section 6.5). How many times more power would this device have produced if (a) the waves had been only 1 meter high? (b) the waves were 6 m high? (c) the waves were moving at 20 m/s? 4.13. Saturn is about 10 times farther from the sun than the Earth is. How many times weaker is the sunlight on Saturn? (Measured, for example, in watts of power per square meter at the planet’s surface.) 4.14. The Shanghai Superintense Ultrafast Laser Facility (SULF) built a laser that produced light pulses with 10.3 petawatts of power (peta = 1015 ). Each pulse lasted for about 23 fs (femto = 10−15 ). (a) Worldwide electricity consumption is about 2.4 TW on average (tera = 1012 ). How many times more power was in the laser pulse? (b) How much energy was in the laser pulse? (c) How many times more energy is in a single packet of sugar (11 food calories)? 4.15. A 15 W LED light bulb is 2 m from a wall. (a) What is the power density of the light on the closest part of the wall? (b) A 2 mW red laser pointer is also 2 m from that wall, producing a red dot on the wall that has a radius of 2 mm. What is its power density? (c) Which is brighter at that spot, the light bulb or the laser pointer? 4.16. Shown below is a water wave spectrum, measured by an oceanographic buoy near Oahu, Hawaii. It shows the energy at different wave frequencies. (a) What are the peak frequencies of the two swells? (b) What are these two wave periods? (c) Which swell has a higher peak energy? (d) Which swell has a higher total energy?
13 Muljadi, Eduard and Yi-Hsiang Yu (2015) “Review of marine hydrokinetic power generation and
power plant” Electric Power Components and Systems 43:1422.
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4.17. Shown below is the solar radiation spectrum. The yellow curve is the sunlight measured in space, above the atmosphere, and the red curve is the sunlight measured at sea level. Ignore the black curve. (a) At what color is the sun brightest both above the atmosphere and at sea level? (b) Sunlight is usually described as white; can you explain why? (c) Estimate the atmospheric transmission coefficient (in percent) at 300 nm, 500 nm, 1400 nm, and 1600 nm.
Puzzles 4.18. Linus plays a note on his piano, holding the piano key down so the note plays for a long time. (a) Would the sound exhibit a line or continuous spectrum? (b) What are some causes of attenuation? (c) If his piano were underwater instead of being in air, would the attenuation be stronger or weaker? (d) Would the widths of the peaks in the sound spectrum be wider or narrower when the piano is underwater? (Hints: consider how damping affects spectra, and whether more or fewer frequencies need to be added to produce the new waveform.) 4.19. Shown below is the Uranus reflectance spectrum (similar to transmission, but for reflected light instead). (a) Using this spectrum, what color would Uranus appear to be? (b) What are the dominant colors of the light that the atmosphere is absorbing? (c) Estimate the visible albedo of Uranus, meaning fraction of visible light energy (380 to 740 nm) that is reflected.
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Figure 5.1 This photograph, called The Hubble eXtreme Deep Field, shows about 5,500 galaxies, many of which are billions of light-years away. It was collected from about 23 days of exposure time by the Hubble Space Telescope. The colors arise, in part, from relative motions between those galaxies and our solar system.
Opening question A train blows its whistle as it speeds past you. Which one is true? (a) you hear a higher whistle pitch than if the train were stopped (b) you hear a lower whistle pitch than if the train were stopped (c) you hear a high whistle pitch as the train approaches and a low pitch as it recedes (d) you hear a high whistle pitch when the train is far and low when it is near (e) the whistle pitch is unaffected by the train’s motion.
© Springer Nature Switzerland AG 2023 S. S. Andrews, Light and Waves, https://doi.org/10.1007/978-3-031-24097-3_5
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A common experience is to stand next to a road, and hear that a fast-moving car makes a high pitched sound as it approaches and then a low pitched sound as it recedes. Similar effects arise with emergency vehicle sirens, train whistles, passing airplanes, and other moving sources of sound. Also, if you are riding on a moving train that is approaching a crossing, you hear that the warning bells have a high pitch as you approach them, and then a low pitch after you pass. These effects, which apply to all types of waves, occur because waves pass by an observer at a different frequency if the wave source or observer is moving. They are widely used to measure how fast things are moving, whether fluid in blood vessels, cars on roads, or stars in galaxies. In some cases, the source or observer can be moving faster than the wave speed. This leads to a range of different behaviors, such as motorboats that skim over the water surface rather than pushing through the water, airplanes that create sonic booms, and nuclear reactors that emit an eerie blue glow.
5.1
Doppler Effect Concepts
5.1.1
The Doppler Effect for Sound Waves
In 1842, the Austrian physicist Christian Doppler (Figure 5.2) proposed that the observed colors of binary stars, in which two stars orbit each other, would be affected by the stars’ motions. He argued that the stars would emit the same color light the entire time, but that the motion of the stars would affect the light wavelengths that we observe. While he was correct, as explained below, it turns out that this effect is much too small to observe with the human eye. However, his description of how observed wave frequencies are affected by motion, now named after him as the Doppler effect, is an important result with many practical applications. Figure 5.2 Christian Doppler, Austrian physicist (1803–1853).
The current definition of the Doppler effect is that it is the change of frequency of a wave due to relative motion between the source and the observer. It applies to all types of waves, including sound waves, water waves, light waves, and others.
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The Doppler effect is particularly noticeable with sound waves. For example, if an emergency vehicle that is using its siren drives past, the sound always has a higher than normal pitch as the vehicle approaches, transitions to a lower pitch as it passes by, and then maintains the lower pitch as it recedes. Or, vice versa, if you are in a car and driving past a stationary horn, the sound again has a higher than normal pitch as you approach, transitions to a lower pitch as you drive by, and then maintains its lower pitch as you drive away. In both cases, the observed sound frequency depends only on the relative velocity between the source and observer — the pitch is higher when the source and observer are approaching, and lower when they are receding. The pitch does not depend on how far the source and observer are from each other. To understand how the Doppler effect works, consider the moving observer case first, in which the sound source is stationary and the observer moves either toward or away from it, shown in Figure 5.3. As is normal, the waves expand outward in space away from the source with a uniform separation between the wave peaks in all directions. The person in black is standing still, so he observes the waves at the frequency they were emitted with, regardless of where he stands. However, the person in blue is running toward the wave source. He is running into the wave peaks at the same time as they are moving past him, so he encounters the wave peaks more rapidly than normal. As a result, he encounters waves at a higher frequency, so he hears a higher pitch. Vice versa, the person in red is running away from the source, going in the same direction as the waves, so wave peaks pass her more slowly than normal. She observes this by hearing a lower pitch. moving observer hears low frequency
moving observer hears high frequency stationary observer hears source frequency
Figure 5.3 Doppler effect from moving observers. The source is stationary, at the center of the circles, and three people observe the sound.
The moving source case is similar, but here the sound source is moving and the observer is stationary, shown in Figure 5.4. Focusing on the blue person first, whom the vehicle is approaching, the source emits each wave peak a little closer to him than it did for the previous wave due to the vehicle’s motion. This means that each wave peak takes less time to reach him than the previous one did. This decreases the time between wave peaks, so he receives an increased wave frequency and hears the sound with a higher pitch. For the red person, the vehicle emits each wave a little farther from her than it did for the previous wave, so each wave takes longer to get to her than the previous one did. This increases the time between successive wave peaks and decreases the wave frequency, which she observes by hearing a lower pitch.
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Figure 5.4 Doppler effect from a moving source. The source is moving to the right and sending out sound waves, shown with black circles, with a constant frequency.
Thus, if an observer and source are approaching each other, regardless of which one is doing the moving, then the observer hears the sound with a higher pitch than it was emitted with. Vice versa, if an observer and source are moving away from each other, then the observer hears the sound with a lower pitch than it was emitted with.
5.1.2
Cars on a Road Analogy
The same results can also be seen in the traffic analogy that was introduced in Section 2.4.1, in which a steady procession of cars are driving along a road and a person standing next to the road counts the number of cars passing by over some time period to find the car frequency. Each car represents a wave peak. Now, consider bicyclists on the same road, who move more slowly than the cars. If a cyclist rides in the same direction as the cars, shown with the red person in Figure 5.5, the cars pass her less frequently than they would if she were stationary. This is a Doppler effect, where the “source” is some parking lot that the cars are driving out of, the cars represent the waves, and the cyclist is a moving observer. Here, the observer is moving away from the source, so the observed frequency is lower than the frequency that a stationary person would observe. Vice versa, a cyclist who rides against the traffic, shown with the blue person in the figure, passes cars more frequently than if he were stationary. In this case, the observer, meaning the cyclist, is moving toward the source, so the observed frequency is increased.
Figure 5.5 Doppler shift analogy with cars on a road. The red cyclist is riding in the same direction as the cars, so cars pass her at a slower rate than they would if she were stationary. The blue cyclist is riding against the cars, so cars pass him at a faster rate.
As before, the observed frequency increases if the observer moves toward the source, and decreases if moving away.
5.1 Doppler Effect Concepts
5.1.3
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Doppler Effect Applications*
Doppler effects are often simple ways to measure the speeds of things remotely, meaning that one can be relatively far from the object that’s being investigated. One example is medical Doppler ultrasound, in which a probe emits sound waves that reflect off various structures in a person’s body and are then detected back at the probe. By testing for frequency shifts between the emitted and detected sounds, the probe can measure motion within the body, such as blood flow within a person’s kidney (Figure 5.6), fluid flow in a person’s brain, the motion of muscles, or even fluid flow within a fetal heart. The fact that these are non-invasive measurements, with nothing actually inserted into the body, makes them safe, simple, and relatively inexpensive. Figure 5.6 Doppler ultrasound measurement of blood flow in an adult kidney, showing flow toward the viewer in red, and away in blue.
The same principles apply in environmental fluid flow applications as well, leading to the development of “acoustic Doppler velocimetry,” which is essentially just another name for Doppler ultrasound. This approach is used to map river flow speeds within river channels and to investigate water flow over and around coral reefs1 . Doppler radar is conceptually similar, but makes use of the Doppler effect with microwaves instead. Unfortunately, the term “Doppler radar” has become synonymous with weather radar, in which radio waves are bounced off falling raindrops to map out where storms are; however, this is largely a misnomer because weather radar systems rely primarily on the strength and directions of the microwave echos and
1 See, for example, Lane, et al. “Three dimensional measurement of river channel flow processes using acoustic Doppler velocimetry” Earth Surface Processes and Landforms: The Journal of the British Geomorphological Group 23:1247 (1998); Reidenbach et al. “Boundary layer turbulence and flow structure over a fringing coral reef” Limnology and Oceanography 51:1956 (2006).
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not on the Doppler effect2 . On the other hand, actual Doppler radar is widely used for measuring the speeds of things that are outdoors. For example, police use radar guns to measure car speeds. In this case, the radar gun emits a microwave signal at some particular frequency (typically about 10.5 GHz), the microwaves reflect off a moving car and change frequency in the process due to the Doppler effect, and then the reflected waves get received by the radar gun. The gun electronics compute the frequency shift and, from that, the car’s speed. Similar Doppler radar devices measure airplane motion, the speeds of balls in baseball games, and even the speeds of some of the asteroids and comets that share the solar system with us.
5.1.4
Red and Blue Shifts
The colors of stars arise primarily from their surface temperature, with hotter stars being more blue and cooler ones being more red. Slight spectral shifts also arise from the stars’ motion toward or away from the Earth due to the Doppler effect, where increased frequencies are called blueshifts and decreased frequencies are called redshifts. Figure 5.7 shows the redshift for a sun-like star that is moving away from the Earth at 2000 km/s, which is much faster than most actual stars3 . Even with this very fast speed, the redshift is too small to affect the overall star color perceptibly, but is seen instead as shifts in absorption and emission lines. Stars are made of the same hydrogen, helium, and other atoms that we have here on Earth, so we can determine the light frequencies that the stars absorb or emit very accurately. Differences between these and the observed values represent Doppler shifts (and other effects that are typically much less important, described below). By measuring redshifts of distant galaxies, Edwin Hubble, for whom the Hubble Space Telescope is named, made an important astronomical discovery in 1929. He found that most galaxies have redshifts, and that these redshifts are larger for more distant galaxies. This implies that those galaxies are moving away from us, which means that the universe is expanding. More recent work, discussed below, has confirmed and extended his results. Other work on red and blue shifts has shown that some stars have shifts that oscillate, implying that the stars are alternately approaching and receding. This result often indicates that a planet orbits the star, called an exoplanet because it’s outside our solar system. The planet itself is typically too small to see, but its gravitational
2 Weather radar systems can also use the Doppler effect to measure raindrop speeds to determine distant wind speeds and drop sizes, but these are not the results that are typically reported. 3 The fastest known star, as of 2019, was S5-HSV1, which is about 29,000 light-years from Earth and moving at 1755 km/s; it appears to have been flung out of the Milky Way galaxy due to an interaction with the black hole at the center of the galaxy. Since then, several much faster stars were discovered orbiting close to the black hole at the center of the galaxy, moving at up to 24,000 km/s. A galaxy called GN-z11 is much faster yet, moving away from us at over 98% the speed of light, giving it so much redshift that its emitted visible light appears in the mid-infrared.
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Figure 5.7 Simulated redshift. The sun’s spectrum, including the Fraunhofer absorption lines (Section 4.3.6), is shown on top. The bottom shows the spectrum that one would see if the sun, or a similar star, were moving away from us at 2000 km/s, showing redshifted absorption lines.
force on the star causes the star to wobble around in a circle, as it orbits the center of mass between itself and the planet. The period of the wobble indicates the planet orbital period (from which its distance to the star can be calculated) and the amount of wobble indicates the planet’s mass. Alternatively, asChristian Doppler first proposed, oscillating red and blue shifts may represent a binary star system, in which two stars orbit each other. In this latter case, it’s sometimes possible to see the effects for the two stars separately. Yet another application for red and blue shifts is for determining rotation speeds of the sun, asteroids, other stars, or galaxies. Figure 5.8 illustrates this effect, showing a non-rotating star on the left, and a star with a vertical rotational axis on the right. For the rotating star, we see one half of it moving toward the Earth, producing blueshifted light, while the other half is moving away, producing redshifted light. These red and blue shifts can still be detected even if we can’t see the different parts of the star separately, because the detected light adds together to cause the observed emission lines to be broader than normal. The rotation rate can then be computed from the line width. Figure 5.8 Redshifts and blueshifts for a non-rotating star on the left and a rotating star on the right (the axis is vertical, with the left side approaching and the right side receding). Spectra show how emission lines spread out, when averaged over the entire star. These effects are grossly exaggerated here since real shifts are typically much less than 1 nm.
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Gravitational and Cosmological Redshifts*
Electromagnetic and gravitational waves have the additional curiosity that there are three distinct mechanisms for creating redshifts and blueshifts. First, there are the Doppler shifts described above. Second, gravitational fields, such as the strong fields near white dwarf stars and even the weak gravitational field of the Earth exert a force on light which creates gravitational redshifts and gravitational blueshifts. The gravitational force doesn’t change the speed of light but does affect its frequency, shifting this light to lower frequencies when it leaves a massive object and to higher frequencies when it approaches a massive object. These red and blue shifts are explained for light waves by the rescaling of time and space that arises in the theory of general relativity. Alternatively, they make a little more sense when thinking of light as photons (Chapter 13), where photons gain energy as they fall toward a massive object and lose energy when they travel away from a massive object. Third, redshifts arise from the expansion of the entire universe, leading to the cosmological redshift. Imagine a very distant star that emits a yellow light wave. As the wave propagates through space, the space itself expands, rather like a rubber sheet that is being stretched, which stretches the wave out slightly. After being stretched very gradually for enough millions of years, the yellow waves get stretched into orange light waves, thus exhibiting the redshift. This redshift is generally not significant for stars within the Milky Way galaxy but is significant for light that reaches us from other galaxies and is the primary indication we have that the universe is expanding. The discovery that the universe is expanding also implies that it must have expanded from something that was much smaller than it is now. Extrapolating the current expansion rate backward in time shows that the universe must have begun from a single point approximately 13.8 billion years ago, during an event now called the Big Bang. Although the cosmological redshift and Doppler redshift are distinct phenomena, they are effectively the same thing if the universe is expanding at a constant rate. In other words, distant galaxies exhibit only a single amount of redshift, and this redshift can be interpreted as arising from either stretching light waves or a velocity difference between source and observer. As a result, the Doppler shift equations given below apply reasonably well to the cosmological redshift for the entire observable universe. However, this only applies if the universe is expanding at a constant rate, which appears to have been reasonably true for the last several billion years, but is not a perfect approximation, and was wildly untrue during the universe’s “inflation” phase, which occurred less than a second after the Big Bang.
5.2
Doppler Effect Equations
5.2.1
Moving Observer Case
There are several different equations for computing Doppler shifts, depending on the situation. The moving observer case applies, obviously, when the wave source is
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stationary and the observer is moving. It can also be sufficient whenever the source and observer’s speeds are much less than the wave speed4 ; in particular, it is almost always adequate for electromagnetic waves because they propagate so much faster than essentially anything else. Consider sound waves with the moving observer situation, as in Figure 5.3. The observed frequency, f o , is the sum of the frequency with which the waves were emitted at the wave source, f s , and the extra frequency that arises from the observer’s motion, f , so f o = f s + f . The extra frequency, f , is the Doppler shift and is the number of waves that the observer passes through each second that arises from his or her own speed. To find this number, define the observer’s speed as vo , in meters per second. Convert this to the number of waves passed through per second by realizing that λ gives the number of meters per wave, so dividing the speed by λ gives the observer’s speed as vλo waves per second. Thus, the observed frequency is fo = fs ±
vo . λ
For the “±” symbol, use the upper symbol if the observer is moving toward the source and the lower symbol if the observer is moving away. This answer is complete, but some rearrangements make it more useful. Multiplying the top and bottom of the second term by f s and then simplifying using the frequency-wavelength relation, c = λ f s , where c is the wave speed (regardless of the wave type), gives the observed frequency as fo = fs ±
f s vo . c
Factoring out the source frequency gives the moving observer equation, vo . fo = fs 1 ± c
(5.1)
To check that the signs are correct, suppose the observer is moving toward the source, meaning that we use the upper symbol. In this case, 1 + vco is greater than 1, so the observed frequency is larger than the emitted frequency. This means that an approaching observer hears a higher pitch, which agrees with our prior discussion. Vice versa, if the observer is moving away from the source we use the lower symbol, making 1 − vco less than 1, so the observed frequency is below the emitted frequency. This also agrees with our prior results, further supporting this equation. The Doppler shift term can be written by itself as f = ± fs
4 As
v , c
(5.2)
a practical matter, “much less” means a factor of 10 difference for quick calculations and a factor of 100 if better accuracy is desired.
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where v is the difference between the observer and source velocities, v = vo −vs . Again, use the upper symbol if the observer and source are approaching each other and the lower symbol if they are moving apart. This equation is completely correct in the moving observer case, but can also be used even the source is moving, so long as it is much slower than the wave speed. Example. A steady procession of cars is traveling along a road at 20 m/s with cars passing a stationary object every 4 s. If a bicyclist travels in the same direction as the traffic at 10 m/s, what is the frequency of cars passing the cyclist? If this cyclist turns around and bikes against traffic, what is the frequency of cars passing? Answer. The Doppler effect here arises solely from the moving observer, so we use the moving observer equation, Eq. 5.1. The car period is T = 4 s, so their frequency is f s = T1 = 0.25 Hz. The “wave speed” is the car speed, so c = 20 m/s. The observer velocity is 10 m/s, either with or against traffic. Plugging these numbers into the equation gives fo = fs
10 m/s vo = (0.25 Hz) 1 ± . 1± c 20 m/s
This evaluates to 0.38 Hz with the upper sign, representing the case where the cyclist is riding against traffic, and 0.13 Hz with the lower sign, representing the case where the cyclist is riding with traffic. These results show that one encounters cars more frequently when traveling against traffic than with it, in agreement with common experience.
5.2.2
Moving Source Equation
Consider sound waves with the moving source situation, as in Figure 5.4. Again, we want to know what the observed frequency is in terms of the source frequency and, now, the source velocity. Skipping the derivation this time5 , the moving source equation is c . (5.3) fo = fs c ∓ vs As before, use the upper symbol if the source is moving toward the observer and the lower symbol if it’s moving away. For example, if the source is moving toward
= 0 and the next at t = Ts . They are detected at d distance ahead of the source, which is at times dc and Ts + d−vcs Ts , so the observed wave period is To = Ts (1 − vcs ). The reciprocal gives the frequency.
5 To derive this, suppose the source emits one wave at t
5.2 Doppler Effect Equations
111
c the observer, using the upper symbol gives the latter factor as c−v , which is greater s than 1, so the frequency increases. This agrees with the discussion above. Vice versa, c , which is less than 1, meaning that the a receding source leads to a factor of c+v s frequency decreases; again, this agrees with expectations.
5.2.3
General Equation
The moving observer and moving source equations, Eqs. 5.1 and 5.3, can be combined to give a general equation that applies if either the source or observer are moving, or even if both are moving at once, c ± vo . (5.4) fo = fs c ∓ vs The rules with the symbols are essentially the same as before: use the top symbol in the numerator if the observer is moving toward the source and vice versa, and use the top symbol in the denominator if the source is moving toward the observer and vice versa. If the medium is moving, such as sound propagating through air on a windy day, then the source and observer speeds need to be measured relative to the medium. Example. On a calm day, you are driving on a road at 30 m/s and a truck is coming toward you at 40 m/s. It’s honking its horn, which has a frequency of 160 Hz. What frequency do you hear as you approach the truck, exactly as it passes, and as it recedes? Answer. Both source and observer are moving, so use Eq. 5.4. Here, vo is your speed, vs is the truck’s speed, and f s is the horn frequency, giving fo = fs
c ± vo c ∓ vs
= (160 Hz)
340 m/s ± 30 m/s . (340 m/s) ∓ (40 m/s)
Use the upper signs for the approaching truck, which evaluates to an observed frequency of 197 Hz, and the lower signs for the receding truck, which evaluates to 131 Hz. The truck is neither approaching nor receding exactly as it passes, so there’s no Doppler effect at that moment; this means that the observed frequency is 160 Hz as it passes.
5.2.4
Doppler Shifts for Reflections
Many Doppler shift situations involve reflections, or echos, such as ultrasonic Doppler imaging and police radar guns. The procedure for handling them mathematically is simple: treat the surface where the reflection is taking place first as
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an observer that receives a signal, and then as a source that re-emits the signal at the same frequency, computing Doppler shifts for the two segments separately. In practice, these two shifts are always very nearly the same, and in the same direction, with the result that reflections create a doubled Doppler shift. For example, suppose a stationary police radar gun emits microwaves at 10.5 GHz, they reflect off a car that’s approaching at 60 m/s, and they reflect back to the radar gun. We want to find the total Doppler shift. Microwaves are electromagnetic radiation, and 60 m/s is much less than 3 · 108 m/s, so we use Eq. 5.2. The velocity difference is 60 m/s and the car is approaching, so the car “observes” the microwaves with the frequency shifted upward by 2100 Hz. The car then reflects the waves at this higher frequency. However, it’s now acting as a moving source, so the waves get Doppler shifted up by another 2100 Hz, leading to a total Doppler shift of 4200 Hz by the time the reflected waves are received back at the radar gun6 . The same approach works if the source is moving and the reflective surface is stationary. For example, suppose a train is driving toward a cliff where it will enter a tunnel, and blows its whistle. The passengers on the train hear the whistle directly, and then hear the echo of the whistle off the cliff. In this case, the sound is emitted at the original frequency from the moving source and is Doppler shifted to a higher frequency as it hits the cliff, with the cliff acting as a stationary observer. The cliff then reflects the sound, emitting it at this higher frequency. Those reflected waves are then heard by passengers on the train, who are moving into them and acting as moving observers, causing a second Doppler shift to higher frequency. To solve this quantitatively, suppose the train’s speed is vtrain . This is likely greater than 1% of the speed of sound, 340 m/s, so we use Eq. 5.4. The total shift is c c + vtrain fo = fs c − vtrain c c + vtrain , = fs c − vtrain
where c is the speed of sound, f s is the train whistle frequency, and f o is the frequency that the passengers hear from the echo. In the first equality, the first factor in parentheses represents the moving source for the emitted sound, and the second factor represents the moving observers. These factors get combined in the second equality.
6 Radar guns are able to measure Doppler shifts precisely, despite their being a tiny fraction of the 10.5 GHz radar frequency, by adding the reflected waves to the original waves to create a beating pattern (Section 3.1.6). They then use the beat frequency to find the Doppler shift.
5.2 Doppler Effect Equations
5.2.5
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Doppler Effect on Wave Power*
Although usually overlooked, the Doppler effect also changes the observed wave power. However, it’s rarely noticeable because the changing distance between the source and observer typically has a much larger effect. In the moving observer case, the observer’s motion clearly has no effect on the waves themselves. They have the same amplitude and travel at the same speed as always. However, an approaching observer (blue person in Figure 5.3) encounters waves faster than if he were stationary, so he encounters more wave energy in every second, meaning that he observes a greater wave power. This greater wave power arises solely from the increased wave frequency, and not from getting closer to the source. Likewise, a receding observer (red person in Figure 5.3) encounters waves more slowly, so she observes fewer waves per second and hence less wave power. These effects are proportional to the observed frequency, implying that the observed power is a factor of ffos greater than it would be for a stationary observer at the same distance from the source. In addition, we found before (Section 4.1.5) that the power 1 density at distance r from a 3D wave source is ρ3D = Ps 4πr 2 (ρ3D represented energy density before and power density now, but the ideas are the same). Combining these results shows that a moving observer measures power density ρ3D = Ps
fo . 4πr 2 f s
Substituting in the moving observer equation from Eq. 5.1 gives ρ3D = Ps
vo 1 . 1 ± 4πr 2 c
(5.5)
This shows that a moving observer hears a louder sound when approaching a noise source than when receding, when measured at the same distance away. The moving source situation (Figure 5.4) is essentially the same with regard to the frequency effect, so the observed power density is again proportional to ffos . Conceptually, the power density also decreases with distance away from the source 1 as Ps 4πr 2 . However, we don’t care about the actual distance to the source in this case, but the distance to where the source was when it emitted the sound that we’re observing. If it’s approaching, then that distance is farther than the current distance to the source, making the sound quieter than one would expect. Skipping the derivation, the result is fs . ρ3D = Ps 4πr 2 f o Substituting in the moving source equation from Eq. 5.3 gives ρ3D = Ps
1 4πr 2
c c ∓ vs
.
(5.6)
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This contrasts the moving observer case, showing that a stationary observer hears a quieter sound when a moving source is approaching than when receding, when measured at the same distance away.
5.2.6
Relativistic Doppler Effect*
Electromagnetic waves have the remarkable property of traveling at the same speed relative to any observer, regardless of the observer’s speed. This shows that electromagnetic waves don’t travel through a medium and, further, requires that the very notions of space and time get redefined; this was the focus of Einstein’s work on relativity. However, things stay reasonably simple in the vast majority of cases where light sources and observers move much slower than the speed of light. In those cases, Doppler shifts occur as described above and can be computed with one of the equations given above. In the rare event that the relative velocity between a source and an observer is a significant fraction of the speed of light, as occurs in high energy physics experiments and the speeds of the most distant galaxies, Doppler shifts need to be computed from the relativistic Doppler shift equation. It is 1−β , (5.7) fo = fs 1+β 7 where β = ∓ v c and c is the speed of light . This Doppler shift has the same trends as normal, where approaching objects are blueshifted and receding ones are redshifted, but the effects are exaggerated. At normal speeds, which aren’t close to the speed of light, this equation approaches the non-relativistic case that was given in Eq. 5.2.
5.3
Supersonic Motion
5.3.1
Explanation
In some cases, it’s possible for an object that emits waves to travel faster than the waves themselves. This is called supersonic motion when considering sound waves. The left side of Figure 5.9 shows a diagram of a military airplane flying faster than sound. The plane emits sound in all directions as it flies but continually gets ahead of the sound that it emitted previously. This produces a cone of expanding waves that
7 Eq. 5.7 is for radial motion, meaning motion of the source and observer toward or away from each
other. Because spatial concepts need to be redefined when considering relativity, there is a separate relativistic Doppler effect equation for tangential motion, in which the source and observer pass by each other.
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115
Figure 5.9 (Left) Sound waves from a supersonic source. (Right) Photograph of the shadow of a supersonic bullet, where diagonal lines are shock waves.
all add up to produce a very high amplitude displacement at its front edge, which is called a shock wave. Note that the shock wave does not occur at the moment that a plane becomes supersonic, but is produced continuously as it flies along. The right panel of Figure 5.9 shows multiple shock waves created by a bullet that is flying at nearly twice the speed of sound. The shock waves have a cone shape called the Mach cone, named for the Austrian physicist Ernst Mach, who first investigated them. Suppose a woman is standing on the ground as a supersonic plane flies overhead. She does not hear the plane approach at all, because the plane is faster than its sound. She still does not hear the plane when it is overhead, because the sound waves emitted at that moment have not propagated away from the airplane yet. Finally, the shock wave hits her when the sound has had time to reach the ground, at which point she hears all of the sound in a single moment, called a sonic boom. Afterward, the woman hears the plane receding as a low rumbling sound, where the emitted sound has been strongly Doppler shifted to lower frequency due to the plane’s speed8 . As another example of supersonic motion, most guns fire bullets at supersonic speeds because faster bullets carry more kinetic energy. The shock waves off the bullets create the distinctive “crack” sound of a gunshot. Also, a skilled performer can make the tip of a long whip move supersonically, again creating a crack sound. Less dramatically, many boats often travel faster than water waves, and don’t even have to go very fast to do so. As a typical example, when a standard recreational motorboat is driven at slow speeds (such as when driving in a “no wake” zone), it slowly pushes its way through the water and produces a fairly small wake (left side of Figure 5.10). When it then speeds up to go faster than water wave speed, the boat rises up noticeably to ride more over the water and less through it, which is called planing. A strong wake expands outward from the boat at this point, which is analogous to the shock waves of supersonic motion9 (right side of Figure 5.10). Surfboards, kite boards, and water skis plane in the same manner.
8A
person in the plane has a very different experience. According to military pilot reports, there is no distinctive change when an airplane transitions from subsonic to supersonic, except that the jet engines burn fuel a lot faster. The noise from these engines is transmitted through the airplane’s frame and then through the air within the plane, just like normal, so things don’t sound any different. 9 This analogy is valid but the situations are not identical because water wave speeds depend on their wavelengths, whereas sound waves do not.
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Figure 5.10 (Left) A motorboat going slower than its planing speed, which is analogous to subsonic motion. (Right) A motorboat planing, which is analogous to supersonic motion.
The fact that nothing can travel faster than the speed of light would seemingly imply that there is no analogy for supersonic motion for light waves. However, this is not quite true. What Einstein actually showed was that nothing can travel faster than the speed of light in vacuum, but he didn’t say anything about traveling faster than light that has been slowed down by traveling in a medium, such as glass or water. In particular, nuclear reactions often emit electrons into water that move faster than the speed of light in water. These fast electrons disrupt the electrical charges in the water slightly, which then create an electromagnetic shock wave in the water that is very much like the supersonic shock wave. The electromagnetic wave creates light called Cherenkov radiation, after the Soviet scientist Pavel Cherenkov who discovered it. This radiation creates the characteristic blue glow of underwater nuclear reactors (Figure 5.11). Figure 5.11 Cherenkov radiation.
5.4 Summary
5.3.2
117
Equations for Supersonic Motion*
For quantitative work with supersonic motion, most of the equations given above still apply, but with some exceptions that can generally be figured out by referring to Figures 5.3 and 5.9. In the moving observer case, now with a supersonic observer, Eq. 5.1 is still valid for an approaching observer but gives a negative result for a receding observer. This negative result shows that the observer is going through the waves in the reverse order from normal. The order of the waves doesn’t normally matter, so the observed frequency would be the absolute value of what the equation returns. In the moving source case with a supersonic source, Eq. 5.3 is still valid for a receding source. However, it returns a negative result for an approaching source. This time, the answer is that there are no waves at all ahead of a supersonic source, so there is no observed frequency.
5.4
Summary
The Doppler effect is a shift of a wave’s observed frequency relative to the frequency it was emitted with, arising from the motions of the wave source and/or wave observer. Doppler shifts increase the observed frequency when the source and observer are moving toward each other, such as the high-pitched sounds of approaching cars. Vice versa, they lower the observed frequency when the source and observer are moving away from each other, such as the the low-pitched sounds of receding cars. These effects depend only on the velocities of the wave source and observer. Doppler shifts are widely used for remote sensing, including in medical imaging, environmental fluid mechanics, and with radar to measure car speeds. Electromagnetic Doppler shifts, called redshifts for decreasing frequencies and blueshifts for increasing frequencies, are used in astronomy to quantify the velocities and rotation rates of stars and galaxies. Comparing observed atomic absorption and emission lines against their known frequencies yields the Doppler shift, then giving the star or galaxy velocity. Redshifts also arise from gravitational effects and cosmological expansion. Doppler effects can be computed from several equations. For a moving observer (Eq. 5.1), vo . fo = fs 1 ± c The Doppler shift is (Eq. 5.2), f = ± fs
v . c
For a moving source (Eq. 5.3), fo = fs
c . c ∓ vs
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These combine to give the general equation (Eq. 5.4), fo = fs
c ± vo . c ∓ vs
This applies if the source and/or observer and/or medium is moving. Echos are treated by computing Doppler effects for the two segments separately, typically yielding a shift that is about double its value without the echo. Doppler effects also affect wave power. The relativistic Doppler effect applies if the source or observer is moving a significant fraction of the speed of light. Supersonic wave sources travel faster than the waves, creating a cone-shaped shock wave that propagates along behind the moving object. Observers hear it as a sonic boom when it passes by them. Particles can move faster than light waves if the light is moving through a medium such as water or glass, creating Cherenkov radiation.
5.5
Exercises
Questions 5.1. A woman paddles her kayak on a lake on a windy day. When do the waves hit the boat with the highest frequency? (a) (b) (c) (d) (e)
When she’s in the middle of the lake. When she’s near the shore. When she kayaks downwind, going in the same direction as the waves. When she kayaks upwind, going into the waves. They always hit at the same frequency.
5.2. Many stars appear to have different colors. For example, Betelgeuse is red and Sirius is blue (both are in or near the Orion constellation). What is the primary cause of this color difference? (a) They emit different color light, due to different surface temperatures (b) The Doppler effect (c) Red and blue shifts from star motion (d) Gravitational redshift (e) Cosmological redshift 5.3. Marvin the Martian has a green helmet. If the helmet appears violet, what is Marvin doing? (a) Traveling very fast away from Earth (b) Traveling very fast toward Earth (c) Traveling very fast past the Earth (d) Trying to blow up the Earth (e) Not moving, but he’s a long way from the Earth
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119
5.4. In 2004, the Cassini spacecraft was orbiting Saturn when it released a lander called Huygens, which flew down to Saturn’s moon Titan. The mission almost failed because the engineers didn’t design the equipment to account for the Doppler shift between the fast-moving Cassini and the stationary Huygens. They fixed this by making sure that Cassini’s velocity was perpendicular to the direction from Cassini to Huygens. Why did this work? Problems 5.5. A stationary bat emits an ultrasonic sound with a frequency of 40.0 kHz. This “click” reflects off a moth that is flying toward the bat at 6.0 m/s. (a) What sound frequency does the moth hear? (b) What sound frequency does the bat hear in the echo off the moth? 5.6. You are on a train, traveling at 15 m/s, that is about to drive into a tunnel that is at the base of a large cliff. The train blows its whistle, which has a frequency of 600 Hz. (a) What frequency do you hear from the whistle directly? (b) What frequency does a stationary person who is ahead of the train hear? (c) What frequency does a stationary person who is behind the train hear? (d) What frequency do you hear in the echo off the cliff? 5.7. A galaxy called NGC 1357 has been observed to have a redshift: light that it emits at 656.2 nm is observed at 660.8 nm. In this problem, compute numbers to 4 significant figures. (a) What color is this light? (b) What are the two frequencies? (c) How fast is the galaxy moving away from us? 5.8. Low pressure sodium lamps are a common type of street light; these are inexpensive, yellow, and sometimes make a buzzing sound. The light comes from excited sodium atoms that emit light primarily at 589.0 nm. They are very hot, with a temperature of about 4000 Kelvin (3700◦ C or 6700◦ F). Because they are so hot, the atoms move very fast, with typical speeds of about 2000 m/s. In this problem, use 6 significant figures. Compute: (a) the wavelength of light that you detect from an atom moving toward you at 2000 m/s, (b) the wavelength of light that you detect from an atom moving away from you at 2000 m/s, (c) the difference between these two wavelengths, which is called the Doppler broadening of the spectral line width. 5.9. Tzipor is attending an air show. An airplane, which is emitting noise at about 120 Hz, flies past her at 415 m/s. (a) What frequency does she hear, if any, as the airplane approaches? (b) What frequency does she hear, if any, as the airplane departs? 5.10. The Helios-2 spacecraft was launched by the United States in 1976 to study the sun. In flying reasonably close to the sun (inside Mercury’s orbit), it set a speed record of about 253,000 km/hr. Its communications radio broadcast at 2295.37 MHz. Assuming it was flying directly away from the Earth at the time of its speed record and that the Earth was effectively stationary (both of which are reasonably accurate) what was its radio frequency received at Earth?
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5.11. A gravitational wave was detected in 2015 that had a frequency that sped up over time, ending at about 250 Hz. It arose from black holes that were about 1.3 billion light-years away and that were moving away from the Earth at about 28,000 km/s due to the expansion of the universe. What was the ending frequency that the black holes actually emitted (the gravitational redshift is small enough that it can be ignored)? 5.12. US-708 is one of the fastest known stars, traveling at about 1200 km/s relative to the Earth. Part of this speed is across the sky, which does not produce a Doppler effect, and part is away from the Earth, which does produce a Doppler effect. The latter “radial” speed is 708 km/s away from the Earth. What is the observed wavelength of 550 nm light that is emitted from the star? 5.13. When a car drives past Sergeant Murphy, he hears the pitch shift from 520 Hz to 345 Hz. (a) What pitch did the car emit? (b) How fast was the car going? Puzzles 5.14. Bats use echolocation to find their prey, which are primarily moths. (a) How might Doppler effects interfere with their echolocation abilities? (b) How might bats use Doppler effects to their advantage? (c) How might moths use Doppler effects to evade bats?10 5.15. Alex and Sal are standing still on top of Mt. Washington, NH, where the wind is blowing at 231 miles per hour (103 m/s). Alex is directly upwind of Sal. (a) When Alex sings a note at 220 Hz, what frequency does Sal hear? (b) When Sal sings at 220 Hz, what frequency does Alex hear? (c) If Sal were to measure the wavelength of Alex’s singing, what would he measure? (d) If Alex were to measure the wavelength from Sal, what would he measure? (e) What would be different if the wind speed were 350 m/s? 5.16. This is a continuation of the same problem, but with the additional information that Alex and Sal are standing 2 meters apart. (a) If Alex speaks with 1 W of sound power, what power density does Sal receive, in W/m2 ? (b) If Sal speaks with 1 W of sound power, what power density does Alex receive? 5.17. Angular Doppler effect. Consider a jumprope that is tied at one end to a tree branch and held by a person at the other end, who is swinging it around in a circle in the normal sort of way. In more technical language, this is a circularly polarized string wave. To find the rotational frequency, you mount a camera beyond the tree branch and take a video of the rope as it goes around; later, you analyze the video to determine that the rope’s period is 0.8 s. (a) What is the rope’s rotational frequency? (b) Suppose the camera mount spins the
10 Here are some references for those who want to learn more about this topic: Wikipedia “Doppler
shift compensation”; Jones, “Echolocation” Current Biology 15:R484 (2005); Yin and Müller, “Fast-moving bat ears create informative Doppler shifts” Proc. Natl. Acad. Sci. USA 116:12270 (2019).
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camera in the same direction as the string (i.e. around the same axis as the string rotation) at 1 Hz. Now what rope frequency would you measure from the video? (c) What would the measured frequency be if the camera mount spun the camera in the opposite direction at 1 Hz? (d) More generally, if the string frequency is f s and the camera rotates at f c in the string’s direction, what is the observed frequency f o ? (e) What does a negative observed frequency mean?
6
Mechanical Waves
Figure 6.1 Several types of mechanical waves.
Opening question A hurricane is far out to sea. Of the waves that it makes, which reach the shore first? (a) The long wavelength waves (b) The short wavelength waves (c) The tall waves (d) The low amplitude waves (e) They all arrive at the same time © Springer Nature Switzerland AG 2023 S. S. Andrews, Light and Waves, https://doi.org/10.1007/978-3-031-24097-3_6
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Mechanical Waves
Imagine relaxing on a beach, watching waves lap against the shore, while listening to a friend playing his guitar. This scene includes the mechanical waves of everyday life — water waves, a vibrating guitar string, and sound waves. Mechanical waves are relatively easy to study because they involve displacements in physical media and travel at comprehensible speeds, unlike the extraordinarily fast speed of light waves. These properties make them simple in some respects, but also give them a wide range of rich and interesting behaviors. Mechanical waves include string, sound, water, and seismic waves. String waves are particularly simple, making them ideal for building an understanding how all types of waves work. Sound waves enable us to have a sense of hearing, and are the essential component of music. Water waves are often the most familiar type of wave, but are also notable for their high level of complexity, such as the fact that water waves with different wavelengths propagate with different speeds. Seismic waves have interesting behaviors too, with some seismic waves behaving like sound waves, others like string waves, and yet others like surface waves. And yet, all of these waves arise from similar forces and momentum influences.
6.1
Pendulums
6.1.1
How Pendulums Work
Pendulums are not waves, but are a nice starting point for building an understanding of how waves work. Consider a simple weight that swings back and forth at the end of a string, as in Figure 6.2. Why does the weight swing back and forth? To answer this question, consider the moment when the weight is at the extreme left end of its swing, when it is fully stopped and is about to start moving toward the right (left end of the figure, t = 0). At this moment, gravity is pulling the weight down and the string is pulling the weight
Figure 6.2 A pendulum at four different time points, where T is the pendulum’s period. Blue arrows show the forces acting on the weight, red arrows show the net forces, and green arrows show the weight’s momentum.
6.1 Pendulums
125
up at an angle, along the direction of the string1 . The up and down components of these two forces largely cancel out, leaving an overall force that is mostly toward the right. So, the weight swings to the right. This net force to the right continues as the weight swings right, causing it to move faster and faster. When the weight reaches the bottom of its arc (second panel, t = T /4), called the equilibrium position, the string pulls straight up and gravity pulls straight down, so it’s no longer being pulled forward. However, the weight doesn’t stop moving, but coasts on toward the right due to its inertia, which is its tendency to continue moving in a constant direction. Then, as the weight swings upward on the right side, the force from the angled string starts to pull the weight back toward the left. This slows the weight down, eventually stopping it at the end of its arc (third panel, t = T /2). The net force continues to pull the weight toward the left, so it starts to swing back toward the middle, now moving left. Again, the weight’s inertia carries it past the equilibrium point (right end of the figure, t = 3T /4) and on up the left side of the arc, where it stops (back to left end of figure, t = T ). This process repeats over and over again. There are two influences here: a restoring force pulls the weight toward the equilibrium position, and the weight’s inertia causes it to keep moving past the equilibrium position and on up the far side. These influences are out of phase with each other, with the restoring force strongest at the ends of the arc and the inertial influence strongest in the middle of the arc. Nearly all oscillations arise from similar dynamics, in which a restoring force pulls the system back toward equilibrium and an inertial influence causes the system to coast past the equilibrium state and on to the other side. This applies to pendulums, oscillating springs, and all types of mechanical waves.
6.1.2
Momentum
Inertia, the tendency of things in motion to keep on moving in the same direction, is an important concept but doesn’t have a mathematical definition. This role is filled by momentum, which is essentially the same thing as inertia, but is quantitative. The momentum of a moving object is p = mv,
(6.1)
where m is the object’s mass, and v is its velocity (note that momentum uses a lower case p, whereas upper case P implies power). This equation shows that momentum is greater for both heavier and faster objects. For example, a high speed train has a great deal of momentum because it is both heavy and fast, whereas a falling feather has very little momentum because it is light and slow.
1 The
only things strings can do is pull, and that pull is necessarily in the direction of the string.
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Mechanical Waves
Example. Which has more momentum: a 20 g rifle bullet moving at 750 m/s or a 5.0 kg bowling ball moving at 3.0 m/s? Answer. Use Eq. 6.1 for each: bullet:
p = mv = (0.020 kg) · (750 m/s) = 15.0 kg m/s
bowling ball: p = mv = (5.0 kg) · (3.0 m/s) = 15.0 kg m/s They have exactly the same momentum because the bullet is fast but light, while the bowling ball is heavy but slow. They would deliver the same impacts to a target, if these impacts could be spread over the same areas.
6.2
String Waves
6.2.1
How String Waves Work
The same principles apply to string waves. For simplicity, consider a lightweight string that has beads evenly spaced along its length, as in Figure 6.3. Suppose that waves are propagating from left to right on this string. We’ll focus on the position of the bead that is shown in red, as it rises and falls in response to a passing wave. At the starting time (t = 0) the red bead is at a wave trough. It just went down to this point and has come to a stop before turning around and going back up. The strings on the two sides of this bead are both pulling the bead up; they are also pulling outward, but the outward forces cancel each other out, leaving just a net upward force. This accelerates the bead up, making it go up faster and faster. Once the bead reaches the middle height, called the equilibrium position, the strings on its sides pull in opposite directions to produce no net force (t = T /4). However, the bead continues up due to its inertia. As it continues coasting up, the strings on its sides change their angles to start pulling it downward, eventually stopping the Figure 6.3 Waves on a thin string that has equally spaced beads along it. The string only moves up and down slightly, as waves propagate from left to right. One bead is colored red.
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bead at the moment when it becomes the next wave peak (t = T /2). This downward force continues, accelerating the bead back down. It then coasts past the equilibrium position (t = 3T /4) and continues on to become a wave trough (t = T ), where the cycle repeats. These dynamics are identical to those of the pendulum. A restoring force always pulls the bead back toward its equilibrium position, pulling it up when it’s below the equilibrium position and down when it’s above the equilibrium position, and the bead’s inertia causes it to coast beyond the equilibrium position and on to the far side. Each bead on the string is moving up and down with these same dynamics, each responding only to the forces exerted on itself. The only thing that’s new here is that the restoring force is not created by some external influence, but by the bead’s neighbors. As a result, the motions of the different beads on the string are inextricably linked together, which leads to propagating waves.
6.2.2
Speed of String Waves
The speed of the string waves depends on the strengths of the restoring and inertial influences. The restoring force arises from the tension in the string, meaning how tightly the ends of the string are pulled on. This tension tries to restore the string to a straight line shape, with greater tension providing a greater restoring force. This greater restoring force increases the bead acceleration, which then speeds up the bead oscillations, causing the waves to move faster. Tension, given here with Ftension , is a force, which is measured in pounds in the English system and newtons (N) in the metric system (in base units, 1 N = 1 kg m s−2 ; also, 1 pound is about 4.45 newtons). The inertial influence arises from the bead masses and is quantified with the momentum equation, Eq. 6.1. Increasing the bead masses increases the inertia, making the beads harder to speed up and also harder to slow down. Thus, heavier beads cause the waves to propagate more slowly. The bead masses can be quantified individually, but it makes more sense at this point to smear the beads out along the string, to create a continuous string that has its own mass. After doing so, the inertial parameter becomes the mass per unit length along the string, given by μ (Greek letter mu). A mathematical treatment of these restoring forces and inertial influences shows that the speed of waves on a string is vstring =
Ftension . μ
(6.2)
This equation shows that the wave speed increases with higher tension and decreases with higher string mass per unit length, both as we predicted. This equation also shows that the speed of string waves does not depend on the wave’s frequency or wavelength. This important property is described by saying that string waves are nondispersive.
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A more general way of expressing the speed of waves on a string is with the equation restoring property v= , (6.3) inertial property which turns out to be remarkably general, applying to all types of mechanical waves. Example. In Section 3.2.3, we found that the wave speed on a violin “D” string is 193 m/s. If its mass per unit length is 1.3 g/m, what is its tension? Answer. Solve Eq. 6.2 for the tension force and then plug in numbers, 2 = (0.0013 kg/m) · (193 m/s)2 = 48.4 N. Ftension = μvstring
This is a force of 10.9 pounds, meaning that the same tension (and same musical note) could be achieved by holding the string vertically and hanging a 10.9 pound weight from it.
6.3
Sound Waves
6.3.1
How Sound Waves Work
One of the simplest ways that sound waves can be created is with a speaker, which alternately pushes and pulls on the air in front of it. Each time the speaker pushes, it pushes air molecules closer together in the region right in front of itself, creating slightly higher air pressure. This layer of higher pressure air molecules then pushes on the next layer of air, and that pushes on the layer of air beyond that, and so on. Vice versa, when the speaker pulls back, it decreases the air pressure in front of itself, causing molecules beyond that region to move back into this low pressure zone, and the same for subsequent air layers. In this process, the speaker causes the air molecules to oscillate back and forth in sound waves that propagate away from the speaker and across the room. Figure 6.4 shows two illustrations of a sound wave. The top panel shows air molecules that are closer together or farther apart (this is greatly exaggerated from reality). If this panel could show the individual air molecules moving, they would oscillate back and forth, going left and right. The bottom panel shows a graph of the air pressure over several waves, where the wave peaks are regions with higher pressure and the troughs are regions with lower pressure. Focusing on a single air molecule, it behaves very much like the red bead on a string in Figure 6.3. When the air pressure is higher on the left side of the molecule than on the right, it gets pushed to the right. Its inertia then carries it on in the same direction, even when the pressure difference goes away. A little later, the pressure becomes higher on the molecule’s right side, which then brings it to a stop and pushes
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Figure 6.4 Diagram of a sound wave, showing air density differences, which correspond to air pressure differences.
it back toward the left. Again, its inertia carries it on in the same direction, until it is brought to a stop by higher pressure on the left side. Then, the cycle repeats again. As before, these dynamics arise from restoring forces and inertial influences, where the restoring forces arise from air pressure always pushing molecules in the direction that equalizes air pressure, and the inertial influences arise from the momenta of moving air molecules.
6.3.2
The Speed of Sound
The restoring and inertial influences combine for sound waves in essentially the same way as they did for string waves, giving the speed of sound as γh.c. Pair vsound = . (6.4) ρair The Pair factor is the air pressure, which supplies the restoring force (again, don’t confuse the different “P”s; Pair is air pressure, not power or momentum). Air pressure can be measured with many possible units including pounds per square inch (psi) or atmospheres (atm), but the preferred metric unit is pascals (Pa); in base units, 1 Pa = 1 kg m−1 s−2 . Also, 1 atm = 101,325 Pa, which is typical atmospheric pressure at sea level, and 1 psi ≈ 6895 Pa. Next, ρair is the air density, which supplies the inertial influence. Air density is measured here in kg/m3 , where typical air density at sea level is about 1.23 kg/m3 . The final term in Eq. 6.4 is γh.c. (Greek letter gamma), which is simply a constant that is equal to 1.40 for Earth’s air2 . Note that both the Pair and ρair terms represent the average air conditions, while ignoring the slight variations caused by the waves. As with the string wave speed, Eq. 6.4 shows that the speed of sound increases with greater restoring forces and decreases with greater inertial influence.
2 The γ
h.c. parameter is called the heat capacity ratio, which is the ratio of the heat capacity measured at constant pressure to the heat capacity measured at constant volume (the “h.c.” subscript is not standard, but is used here to reduce confusion with other γ parameters). Its value depends on the shapes of the gas molecules. At typical temperatures and pressures, γh.c. is 5/3 for monatomic gases such as helium and argon, 7/5 for gases that are composed of linear molecules, such as nitrogen, oxygen, and carbon dioxide, and 8/6 for gases that are composed of nonlinear molecules such as water and methane.
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A problem with Eq. 6.4 is that changes that affect the air density also tend to affect the air pressure at the same time, so these parameters aren’t independent of each other. The solution is combine equation 6.4 with an equation called the ideal gas law, which relates the air pressure and temperature3 . Doing so gives the speed of sound with the more useful equation, vsound =
γh.c. k B T . m air
(6.5)
Here, k B is Boltzmann’s constant, which has the value 1.38 · 10−23 m2 kg s−2 K−1 , and T is the absolute temperature, which is the temperature measured in kelvin (K) units. The temperature in kelvin is the temperature in Celsius plus 273.15 (convert from Fahrenheit to Celsius with: ◦ C = 59 (◦ F − 32◦ )). Thus, 20◦ C is the same as 293.15 K. The important thing about the absolute temperature is that 0 K is the coldest that anything can possibly be because, at this temperature, all atoms and molecules come as close as physically possible to stopping their motions. The final parameter in the equation, m air , is the mass of one air molecule. Taking the average of the nitrogen and oxygen masses that are in the air on Earth, m air is about 4.8 · 10−26 kg. Example. Compute the speed of sound at sea level, where Pair = 1 atm, ρair = 1.23 kg/m3 , and T = 15◦ C, using both Eqs. 6.4 and 6.5. Answer. Pair 1.40·(101325 Pa) Eq. 6.4: vsound = γh.c. = = 340 m/s ρair 1.23 kg/m3 −23 m2 kg s−2 K−1 )·(288.15 K) kB T = 1.40·(1.38·10 4.8·10 = 341 m/s Eq. 6.5: vsound = γh.c. −26 kg m air Both equations give essentially the same answers, and ones that agree with the speed of sound that we’ve been using elsewhere. It is helpful to look at Eq. 6.5 to see what the speed of sound does and does not depend on. First of all, the speed of sound depends on the air temperature, going faster when the air is hotter and slower when the air is colder. For example, a typical home freezer is about −20◦ C, which is 253 K. Plugging numbers into Eq. 6.5 shows that the speed of sound in a freezer is about 320 m/s, which is substantially slower than the speed of sound in the room around it, due to the cold temperature. The speed of sound also depends on the mass of the air molecules, going faster when the molecules are lighter (because they have less inertia). Based on this, sound must propagate much faster through helium party balloons than in the surrounding air.
Pair V = N k B T , where Pair is the pressure, V is the volume, N is the number of molecules, k B is Boltzmann’s constant, and T is the absolute temperature.
3 The ideal gas law is
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131
In contrast, the speed of sound does not usually depend on the air pressure, despite the impression given by Eq. 6.4 (the air density depends on pressure, too, so the effects cancel out). Also, similar to string waves, the wave speed does not depend on the wavelength or frequency. This means that sound waves are nondispersive.
6.3.3
The Sound Spectrum*
Humans can hear sound waves with frequencies between about 20 Hz and 20 kHz. The 20 Hz waves are extremely low tones, which are often more felt than heard, while the 20 kHz tones are high piercing sounds. Frequencies below 20 Hz are called infrasound and frequencies above 20 kHz are called ultrasound. Other animals can perceive other sound ranges, shown in Figure 6.5. For example4 , elephants can hear sounds down to about 16 Hz, while some of the larger whales can presumably hear even lower frequencies, based on the finding that they sometimes vocalize down to about 10 Hz. Low frequency waves are attenuated less than higher frequency waves, so using low frequencies enables these animals to communicate over very long distances; elephants can communicate over more than 2 miles and whales over more than 100 miles. Birds cannot produce infrasound, but some can hear infrasound. In particular, pigeons can hear down to about 2 Hz, which may help them navigate by making them able to hear distant ocean waves breaking or air turbulence over distant mountains. At the other end of the spectrum5 , bats can hear up to 115 kHz and porpoises can hear up to 150 kHz. Both of these species perform echolocation, in which Figure 6.5 Hearing ranges of some animals. The human range is shaded.
4 Infrasound references. Elephants: Heffner, R. and H. Heffner “Hearing in the elephant (Elephas maximus)” Science 208:518, 1980. Whales: Richardson, Greene, Malme, Thomson, Marine Mammals and Noise, Academic Press, 1995. Pigeons: Heffner, H.E., et al. Behavior research methods 45:383, 2013. 5 Ultrasound references: Bats and porpoises: Wikipedia “Hearing range” and sources therein; bat hearing range varies by species, sometimes getting up to 200 kHz, but their most sensitive range is about 15 to 90 kHz. For porpoises, also see Mourlam, M.J. and M.J. Orliac, “Infrasonic and ultrasonic hearing evolved after the emergence of modern whales” Current Biology 27:1776, 2017. Moths: Moir, H.M., et al. “Extremely high frequency sensitivity in a ‘simple’ ear” Biology letters 9:20, 2013. Mice: Wikipedia “Ultrasonic vocalization” and references therein.
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they emit short clicks or chirps and then listen for echos to determine where things are around themselves. Using high frequencies for echolocation means that they use short wavelengths, and these short wavelengths enable them to detect small things. Some insects can hear even farther into the ultrasound. In particular, some moths that are hunted by bats can hear up to about 300 kHz, enabling them to hear the bats’ ultrasound chirps and then fly away. Interestingly, mice also have very high hearing ranges, which again helps them avoid predators. In their case, their high hearing ranges enable them to communicate with each other using so-called ultrasonic vocalizations, which are above the hearing ranges of cats and other mouse predators.
6.3.4
Sonar and Medical Ultrasound*
Much as bats and some marine mammals use sound for echolocation, people also use sound to create images, of which sonar and medical ultrasound are examples. Sonar is used to image underwater objects, such as the seafloor and enemy submarines. Its development began in 1913 and saw major progress during and after the World Wars. In passive sonar, one simply listens for sounds emitted by the object of interest, such as motors or propellers, and then tries to guess what the source must be. An advantage of passive sonar, at least in military applications, is that no noise is emitted by the sonar device, allowing stealthy operation. In contrast, in active sonar, a transmitter emits a “ping” and then listens for echos. Use of low frequencies, typically in the 1 to 5 kHz range, enables imaging of objects that are several miles away due to low sound attenuation in the water. However, these low frequencies have long wavelengths which provide poor spatial resolution. On the other hand, high frequencies, sometimes up to the MHz range, offer good spatial resolution but relatively poor range. In addition to military applications, sonar is being used increasingly for finding schools of fish that can be caught. This is remarkably effective and, unfortunately, is contributing to the overfishing of most fish species. Medical ultrasound is conceptually similar to active sonar, again having a transducer that emits sound and a receiver that detects echos. In this case, it is used to image internal body structures such as a person’s tendons, muscles, internal organs, or blood vessels. Also, obstetric ultrasound is widely used for checking on fetuses (Figure 6.6). Medical ultrasound uses much higher frequencies than sonar, typically from 1 to 18 MHz, in order to provide greater spatial resolution. Again, the lower frequencies are better for longer ranges, while the higher frequencies give better spatial resolution. Ultrasound imaging can also be combined with Doppler effect measurements (section 5.1.3) to determine blood flow rates within people’s hearts or blood vessels. Ultrasound imaging has the advantages of being noninvasive, relatively cheap, offering real-time images, and having no adverse health effects.
6.4 The Physics of Music*
133
Figure 6.6 Ultrasound image of a fetus in the womb at 12 weeks of pregnancy.
6.4
The Physics of Music*
6.4.1
Physics Terminology for Music*
Scientists describe musical notes as having three qualities: pitch, loudness, and timbre. Pitch describes the sound wave frequency, with low pitches for low frequencies, and high pitches for high frequencies. Loudness is how loud the note is, so it is the amplitude of the sound wave. Timbre describes the quality or character of the note, describing whether it sounds like a violin, flute, trumpet, human voice, or something else. The timbre reflects the shape of the sound waves, shown in the middle column of Figure 6.7. Every wave shape corresponds with a specific spectrum (Section 3.1.2), and vice versa, so timbre also reflects the sound spectrum produced by a particular instrument, shown in the right column of Figure 6.7. Additionally, timbre describes how the sound changes over time, such as the difference between a continuous note and a tap on a drum.
6.4.2
The Western Musical Scale*
Legend has it that the ancient Greek philosopher Pythagoras walked past a blacksmith shop at one point, and noticed that some pairs of hammers made tones that sounded good together whereas others made tones that sounded bad together. He also realized that there was a mathematical relationship between the hammer sizes and the tones they created. Those observations led him to develop a musical scale that was based on mathematics and that was developed around the tones that sound good together. Whether the details of this story are true or not, it is known that the Western musical scale, meaning the notes that one finds on a piano and all other
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Figure 6.7 Waveforms of some musical instruments, showing different timbre.
Figure 6.8 Scientific pitch notation, showing the names of notes on a keyboard and in written music.
Western instruments, was originally developed in ancient Greece and was based on mathematical principles. The modern Western musical scale, which is quite similar to what Pythagoras purportedly developed, is divided into octaves, each of which represents a factor of 2 in frequency. An octave is not divided into 8 notes, as one might reasonably assume from its name, but into 7 natural notes, named A, B, C, D, E, F, and G. These are the white keys on a piano, shown in Figure 6.8. An octave can also be divided more finely into 12 semitones, which is called the chromatic scale. This adds five new notes, called A, C, D, F, and G, where the symbol is called a sharp and indicates that the note is a half-step above the corresponding natural note6 . These added notes are the black keys on a piano.
than expressing these notes as a half-step above the given natural note with a symbol, they can also be expressed as a half-step below the next natural note with a symbol, which is called a flat. For example, A is the same note as B. 6 Rather
6.4 The Physics of Music*
135
Figure 6.9 Frequencies of a wide range of musical notes.
Scientific pitch notation extends these note names by giving each octave a number, with octave 0 at the low frequency end of what people can hear and octave 8 at the high frequency end. Each octave starts with C and goes up to the next B, before starting over again at the next C. For example, there is a note near the middle of the piano keyboard called middle C, which is usually the first note that is taught to beginning piano players; it is labeled C4 in scientific pitch notation. It starts the 4th octave, so the note below it is B3 and the note above is D4 ; see Figure 6.8. Orchestra musicians traditionally tune their instruments, and singers tune their voices, to the A4 note, which is 5 whole notes above middle C. This note has been standardized to have a frequency of 440 Hz, thus giving it the alternate name of A440. Defining this as a standard pitch then sets the frequencies for all the rest of the notes, of which a wide range are listed in Figure 6.9. Note in this table that A4 indeed has a frequency of 440 Hz. Also observe that an octave represents a factor of 2 in frequency so, for example, A3 is 220 Hz and A5 is 880 Hz. These same factors of 2 can be seen with other octaves as well, such as going from C1 at 32.7 Hz to C2 at 65.4 Hz.
6.4.3
Musical Intervals*
As Pythagoras supposedly observed, notes tend to sound better together, which is described as consonant, if their frequency ratios involve small integers. For example, the frequency ratio for two notes that are an octave apart is 2:1 (or, equivalently, 1:2). An example arises in the first two notes of “Somewhere Over the Rainbow.” Intervals of multiple octaves also sound nice, where the ratios are 3:1, 4:1, etc. A perfect fifth, in which the frequency ratio is 3:2, is also consonant. The first few notes of “Twinkle, Twinkle, Little Star” are an example. A fifth is, confusingly, an interval of 4 notes (e.g. C4 and G4 ) and is called perfect if the frequency ratio is very nearly 3:2. Other consonant intervals are the perfect fourth, with a 4:3 ratio, the major third with a 5:4 ratio, and the minor third with a 6:5 ratio. The Western musical scale was developed to give many consonant intervals. Consonance is generally good when considering all 12 semitones, and improves when notes are selectively removed. For example, essentially all of the 7 natural notes are consonant with each other (e.g. “Happy Birthday to You”). Dropping two more
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notes produces a pentatonic scale, which can be even more selective about choosing consonant intervals (e.g. the Beatles’ “Ob-La-Di, Ob-La-Da”). The opposite of consonance is dissonance, in which two notes do not sound good together. Dissonant notes typically have frequency ratios with larger integers. For example, two notes that are separated by one semitone have reasonably large frequency ratios and so are somewhat dissonant. For example, the E4 and F4 notes have a frequency ratio of 1.059, which is not close to any fraction with small integers. Extreme cases of dissonance are rarely used in anything other than avant-garde music, but mild dissonance is often used to create tension. Examples include the opening notes of the Beatles’ “A Hard Day’s Night” and many of the chords of heavy metal music. The so-called Wolf interval was a particularly dissonant interval in the musical scale during the 17th and 18th centuries, but has been avoided more recently by slightly adjusting the frequencies that are used in the Western musical scale.
6.4.4
Major, Minor, and Non-Western Musical Scales*
The vast majority of Western music is organized around a “home note”, which is called the tonic. This is often the note that the music starts or ends on. Most music also selects most of its notes from one of two heptatonic scales, meaning scales that have seven notes per octave. In the major scale, the seven notes start with the tonic note and are then spaced in the pattern W-W-H-W-W-W-H, where W represents a whole step, meaning two semitones, and H represents a half step, meaning one semitone. For example, music is said to be in the key of C-major if the tonic note is C, and the other notes are D, E, F, G, A, and B. If you look at the positions of the black and white keys on the piano keyboard in Figure 6.8, you can see that these notes, starting with C, are separated in the W-W-H-W-W-W-H pattern. One can also start up one note, making D the tonic note. Using the same pattern of note spacings, looking at the piano keyboard shows that the other notes are E, F, G, A, B, and C. Any of the other 10 starting points are valid as well. Regardless of which note is chosen as the tonic, every note of the major scale has strong consonance with the tonic note. The minor scale is essentially the same, but has the spacing W-H-W-W-H-W-W. For example, the key of A-minor has the notes A, B, C, D, E, F, and G. These are the same notes as the key of C-major, but the music has a different sound, typically giving it more pensive mood, because A is the tonic in this case. The difference comes from the notes having less consonance with A than they do with C. Many non-Western musical systems have histories that are fully as rich as that of Western music but based on very different sounds. Nevertheless, the physics of consonance and dissonance are the same, often leading to similar concepts of octaves and intervals. In some cases, their musical scales can be mapped to the same 12 semitones, but using different groups of notes. For example, the Algerian scale, which is used in traditional northern African music, has the note spacing W-H-WHH-H-WH-H, where WH represents 1.5 steps. Also, the Persian scale, used in Iran
6.4 The Physics of Music*
137
and elsewhere in the Middle East, has note spacing H-WH-H-H-W-WH-H. Both of these can be played on a standard piano keyboard. Yet other musical traditions use notes that aren’t part of the Western scale at all. For example, the Japanese ritsu and ryo scales, which are used in Buddhist chants, are pentatonic scales that use slightly different frequencies than the notes in the Western scale. Doing so gives them a greater level of consonance. Traditional Indian music uses 7 notes per octave, as is typical with Western music, but also tuned to different pitches. These notes are also sometimes further divided to give about 22 notes per octave7 . Other cultures have yet more musical scales.
6.4.5
Musical Instruments*
Essentially all musical instruments produce tones through the excitation of standing waves in the instrument. Those standing waves then couple to the outside air to create pressure waves, which we hear as sound. However, instruments differ in what they use as the standing wave medium, whether strings, air, a membrane, or the instrument itself. Chordophones are stringed instruments, creating sound through the vibration of their strings. Examples include guitars, banjos, violins, pianos, and harps. The name arises from the fact that all of these instruments can play multiple notes at once, called a chord. However, chordophones also include stringed instruments that only play a single note, such as the Vietnamese dan bau, which has only one string. The normal modes of a chordophone are the standing waves on strings. Their frequencies were shown before (Eq. 3.2) to be f =
nv , 2L
(6.6)
where n is the mode number (1, 2, 3, ...), v is the string wave speed, and L is the string length. When a string is plucked, the pitch that one hears corresponds to the string’s fundamental mode (n = 1), but higher harmonics are present as well and contribute to the timbre (see Figure 6.7). These instruments’ vibrating strings are typically supported by a “bridge” that sits on top of the front face of the instrument’s body. In acoustic instruments, meaning not electric ones, this front face serves as a “sound-board”, which moves in and out with each string vibration. This pushes air in and out of holes in the body, which creates sound waves. Electric guitars are different. Their string motions are detected with a set of pickups that are located near where the strings are strummed, each of which detects the string motion only at its particular position along the string length and then sends the signal to an amplifier. The nodes and antinodes for the different string harmonics have well-defined locations along the string, so the location of a pickup determines which
7 Traditional
sical music”.
Indian music is highly developed and quite complicated. See Wikipedia “Indian clas-
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Figure 6.10 The first three normal modes of open and closed pipes. Red lines show standing pressure waves. Frequencies were computed by combining the standing wave wavelength (e.g. 2L for the open pipe fundamental) with the frequency-wavelength relation, v = λ f .
harmonics it detects. Pickups that are closer to the center of the string detect more of the fundamental frequency, which is generally described as giving a “warmer” tone, whereas those that are closer to the end detect more overtones, providing a “brighter” tone. Aerophones are wind instruments, creating sound through air vibrations that are nearly always within a tube. Examples include flutes, clarinets, trumpets, harmonicas, organs, and didgeridoos. Turbulent air typically provides the initial vibrations and these vibrations resonate with the normal modes of air oscillating back and forth in the instrument’s tube. As the air oscillates in the tube, it pushes and pulls on air at the tube’s open end, creating sound waves that propagate outward. In simple cases, the tube ends can be classified as open or closed, which then determines the boundary conditions for the standing waves. For example, a flute has two open ends, one of which the musician blows across. The air pressure cannot change substantially at the tube ends, because they are open to the outside air, so these locations have to be pressure wave nodes, shown in the left column of Figure 6.10. As with standing waves on strings, the fundamental frequency arises when the wavelength is twice the flute length (there is a node at either end of the flute and an antinode in the middle, so there is half of a wave within the flute’s length). Overtones arise for higher harmonics of this fundamental mode. In contrast, a clarinet has one open end and one closed end, this time with the musician blowing into the closed end. There is actually some air flow through the “closed” end, but it’s called closed because it is not open to the outside air pressure. The pressure wave within the clarinet has a node at the open end again, but now has an antinode at the closed end, as shown in the right column of Figure 6.10. These boundary conditions allow a standing wave that is only one quarter of the full wavelength, making a clarinet’s musical range about an octave below a flute’s, despite the fact that the instruments are about the same length. The closed end also means that clarinets only play the odd harmonics of the fundamental, meaning that their frequencies are 3 times higher, 5 times higher, and so on; again, see the right column of Figure 6.10.
6.4 The Physics of Music*
139
Figure 6.11 Normal modes of membranes. The fundamental is mode (0,1), at the lower left.
These concepts of open and closed end tubes don’t always apply well in practice because most instrument shapes are not straight pipes. In a saxophone, for example, the tube width is not uniform but expands along the length of the instrument. Also, the bell shapes at the ends of many aerophones, and brass instruments in particular, raise the frequencies of the low resonances. As part of this, brass instruments sometimes play a pedal tone, meaning that all of the overtones are present for a note but not the fundamental mode itself. In this case, our ears seem to fill in the gap for us, so we hear a pedal tone as though it has the pitch of the fundamental frequency, despite the fact that there aren’t actually any sound waves at that frequency. Membranophones produce sound with a vibrating membrane. Drums are the principal example, although kazoos also qualify. As with other instruments, drum heads also have normal vibrational modes, shown in Figure 6.11. The fundamental mode, which is the lowest frequency normal mode, involves the entire drum head flexing in and out repeatedly. Higher frequency normal modes have some parts of the drum head going up while others go down. These higher frequency modes always have nodes on the drumhead, much as there are nodes in string wave overtones. However, drum normal modes differ from string ones in that their frequencies are not integer multiples of the fundamental. In other words, drum overtones are not harmonics. As a result, people often don’t hear distinct pitches from drums. Nevertheless, a few drums do give specific pitches. For example, the djembe is a West African drum that is played with the hands. When the player strikes the drum, the membrane vibrations excite air vibration modes within the drum shell, which is a shaped cavity under the drum head. The normal modes of these air vibrations are essentially the same those of a tube with one open end and one closed end. Separately, timpani are orchestral drums with specific pitches. In their case, the player strikes near the edge of the drum so as to put most of the energy into specific vibrational modes that happen to harmonize well, often giving musical fifths and octaves. Finally, idiophones produce sound through vibrations of the instrument itself, without the use of strings or membranes. Examples include xylophones, marimbas, bells, cymbals, and mbiras. Yet again, these instruments produce sounds that arise from normal modes, this time of the instrument. The Caribbean steel pan, which is made from the end of an oil drum, is a particularly intriguing idiophone. The drum is hammered into a bowl shape, then with specific notes hammered into different regions of the pan. Each region vibrates
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reasonably independently of the rest of the instrument, allowing one to hear distinct notes. However, these regions do couple with the other regions to some extent, creating an intricate set of harmonics that together give the instrument its distinctive quality.
6.5
Water Waves
Water waves arise from the same general principles as string waves and sound waves. The equilibrium state for water is a flat surface, and restoring forces act to pull the water back to this state. Meanwhile, inertial influences cause any water that is moving to keep on going. Thus, displacements don’t go away, but oscillate back and forth, creating propagating waves.
6.5.1
Capillary Waves
A complication is that there are two types of restoring forces, each with different behaviors. The first restoring force is produced by the water’s surface tension, which causes the water surface to act like an elastic membrane, like the head of a drum. Surface tension arises from the fact that the molecules at the water’s surface are strongly attracted to the water molecules next to and under them, but not to the air above them. This pulls these surface molecules downward against the rest of the water, creating a skin on the water’s surface that is denser than the rest of the water. It also exerts sideways pulls on these molecules, creating a tension force in the surface that pulls the water surface taut. This latter tension always acts to flatten the water surface, which then provides a restoring force for waves. The resulting waves are called capillary waves or, informally, ripples, and are shown in Figure 6.12. The restoring force is strongest where the water surface is curved tightly, so these effects are only important when wavelengths are shorter than about a centimeter. Figure 6.12 Capillary waves expanding from a point where a drop fell in the water.
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Capillary waves propagate with speed
vcapillar y =
2πγs.t. ≈ ρwater λ
4.57 · 10−4 m3 /s2 . λ
(6.7)
Here, γs.t. is the surface tension, which has a value of about 0.072 N/m for water8 , ρwater is the density of water, which is about 1000 kg/m3 , and λ is the wavelength. The wavelength dependence is important. It arises from the restoring force being greater when the water surface bends more sharply, which happens when the wavelengths are shorter. The result is is that capillary waves are dispersive, meaning that waves with different lengths travel at different speeds. This dispersion can be seen in Figure 6.12 by the fact that the short waves are on the outside of the circle of ripples, due to their sharp bends and fast speeds, while the long waves are on the inside, due to their gentle bends and slow speeds.
6.5.2
Gravity Waves
The other restoring force for water waves is the Earth’s gravity, which acts by pulling water downward. Gravity also produces a flat surface for the equilibrium state, and again pulls water that is away from this state back toward being flat. The waves that result, which are the water waves that we typically think about, are called gravity waves9 . Figure 6.13 shows gravity waves that were created by a bird that just swam up to the water surface. Gravity has a much stronger influence than surface tension on waves that are over about a centimeter long, allowing these longer waves to be considered as simply gravity waves. Gravity waves act differently if they are in deep or shallow water, where the difference doesn’t depend on the absolute water depth but on the depth relative to the wave’s wavelength. More precisely, the water depth is compared to λ/2. For example, if the water depth is a wavelength or more, then it is substantially more than λ/2 and so can be considered deep. On the other hand, if the depth is only a quarter of the wavelength or less, then it is substantially less than λ/2 and so is shallow. In between these values, the waves combine deep and shallow water dynamics. The wave speed for deep water gravity waves is gλ (6.8) ≈ (1.56 m/s2 )λ. vdeep = 2π
confuse this γs.t. value with the heat capacity ratio, γh.c. , that was used in Eqs. 6.4 and 6.5; the “s.t.” subscript is not conventional, but used here to reduce confusion. Water has an unusually large surface tension. This affects how it pours, allows small insects and paper clips to float on the surface, and helps it make droplets. 9 Don’t confuse gravity waves with gravitational waves, which are esoteric waves of gravity itself, emitted by things like black holes orbiting each other. They’re the subject of Chapter 15. 8 Don’t
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Figure 6.13 Waves produced by a bird (a coot) that just surfaced.
Here, g is the acceleration from Earth’s gravity, which is 9.8 m/s2 , and λ is the wavelength. Both the restoring force and inertial influence are proportional to the water density in this case, so the density terms canceled out and none remain in this final result. Whereas capillary wave speeds are faster for shorter wavelengths, gravity wave speeds have the opposite dependence, going faster for longer wavelengths. This dispersion occurs because the restoring force is proportional to λ, due to longer waves displacing more water. Figure 6.13 shows this dispersion by the fact that the long waves are on the outside of the circle, because they traveled faster, while the short waves are on the inside, because they traveled slower. Figure 6.14 shows a graph of wave speeds for both capillary and deep water waves. It shows that the speed decreases with increasing wavelength for short waves, which are capillary waves, reaches a minimum of about about 0.23 m/s for waves that are 1.7 cm long, and then increases with wavelength for longer wavelengths, which are now gravity waves10 . These speeds increase without limit, provided the water stays deeper than λ/2. As always, wave velocity, frequency, and wavelength relate through the frequency-wavelength relationship, v = λ f , enabling one to compute the water wave frequency from the wavelength and velocity. The top axis of Figure 6.14 shows these frequency values. Shallow water gravity waves propagate more slowly than deep water waves because of constraints introduced by the bottom. Their speed is vshallow =
10 The
gd,
(6.9)
line in the figure combines the influences of the surface tension and gravity restoring forces
for waves of all wavelengths by showing the equation v =
2πγs.t. ρwater λ
+
gλ 2π .
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Figure 6.14 Speed of water waves in deep water as a function of wavelength. The top axis shows the wave frequency.
where g is the acceleration due to gravity and d is the water depth. The wavelength does not appear in the equation, showing that shallow water waves are nondispersive11 . Because they are nondispersive, one would expect that an initial disturbance would propagate outward without separating into long and short waves, as is observed for a pulse on a string or a brief burst of sound12 . However, what typically happens instead is that the height of the wave itself affects the water depth, which then affects the wave speed. For example, when an ocean swell approaches the seashore and “feels” the bottom, the wave peaks start to travel faster than the wave troughs. This causes the waves to tilt forward and then break, creating surf (Figure 6.15). Example. Waves on a deep lake have a wavelength of 2.0 m. (a) What is their speed away from shore, where the water is 10 m deep? (b) What is their speed when they approach the shore, where the water is 0.2 m deep? (c) What is their wavelength near the shore? Answer. The wavelength is more than a few cm, so these are gravity waves. In part (a), the water depth is much larger than λ/2, so these are deep water waves. Using Eq. 6.8, vdeep ≈
(1.56 m/s2 )λ =
(1.56 m/s2 ) · (2.0 m) = 1.77 m/s.
For part (b), we don’t know the precise wavelength, but it doesn’t matter, and it’s undoubtedly long enough that λ/2 is much greater than the 0.2 m depth.
11 If
the water depth is in between the deep water and shallow water extremes, the wave speed is gλ tanh 2πd v = 2π λ . 12 Nondispersive water waves pulses are called solitons. However, they aren’t simple shallow water waves, but instead arise from a cancellation of dispersive effects and nonlinear effects, which then remove all dispersion.
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Thus, use Eq. 6.9, vshallow =
gd = (9.8 m/s2 ) · (0.2 m) = 1.40 m/s.
For part (c), the deep water frequency comes from the frequency-wavelength relation, vdeep 1.77 m/s = f = = 0.89 Hz. λdeep 2.0 m The frequency doesn’t change when the waves enter shallow water, so use the frequency-wavelength relation again to find the wavelength, λshallow =
vshallow 1.40 m/s = = 1.57 m. f 0.89 Hz
The wave peaks get closer together as the waves enter shallow water and slow down.
Figure 6.15 Waves entering shallow water and then breaking.
6.5.3
Phase Velocity and Group Velocity
An interesting result happens when waves spread out around an initial disturbance, such as from a rock that’s thrown in the water. This disturbance creates waves of many wavelengths, each of which propagates at its own speed. As the waves move outward, they also interfere with each other. It turns out that these waves interfere constructively at one radius away from the circle’s center, have partial constructive
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interference at nearby radii, and interfere destructively everywhere else. This produces a “group” of waves near the radius where constructive interference occurs, and reasonably flat water elsewhere (see the discussion about beating patterns in Section 3.1.6 for a similar result, but with only two different wavelengths). Remarkably, the wave group expands outward from the center at the different speed than the individual waves that make it up. In technical terminology, the wave group expands with the group velocity and the individual waves expand at their phase velocities, where these latter speeds are the ones that were presented in all prior equations. For deep water waves, and if there is a fairly narrow range of wavelengths involved, it turns out that the group velocity is exactly half of the phase velocity, vdeep,gr oup
vdeep 1 = = 2 2
gλ . 2π
(6.10)
The difference between group and phase velocities can be seen by carefully watching waves spread out from a rock thrown in the water. Most of the waves appear to be concentrated in a ring around the initial disturbance, which is the wave group. As this ring of wave moves outward, individual wave peaks rise up at the inside of this group, move through it, and then disappear as they get beyond it. Group and phase velocities can also be observed in boat wakes. If you paddle a canoe on a busy lake, for example, the canoe is typically fast enough to overtake a motorboat wake that’s moving in the same direction, meaning that the canoe is faster than the waves’ group velocity. However, the canoe is slower than the phase velocity of the component waves, so it gets passed by individual wave peaks as it crosses the wake. Each of these wave peaks appears to form at the back of the wake, grows as it moves through the wake, surfing the canoe forward a little in the process, and then dissipates as it leaves the front of the wake. The classic ’V’ shape of boat and duck wakes, shown in Figure 6.16 and called the Kelvin wake pattern, also arises from constructive and destructive interference. Again, the wake moves at the group velocity, while the waves within it move at the faster phase velocity. Capillary waves exhibit the same phenomenon of having different phase and group velocities but with the opposite dependence. In their case, the wave group moves faster than the individual waves, vcapillar y,gr oup
3vcapillar y 3 = = 2 2
2πγs.t. . ρλ
(6.11)
If you throw a pebble into a pond and watch carefully, you will see that the circular ripple pattern moves faster than the waves themselves. Here, wave peaks appear to rise up at the front of the ripple, move back through it, and then dissipate at the back of the ripple.
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Figure 6.16 Wake behind a duck (a hooded merganser).
Figure 6.17 Motion of a few individual water molecules over a few wave periods. The left panel shows deep water and the right panel shows shallow water. Note the combination of circular motion and net transport.
6.5.4
Water Motion in Waves
The left panel of Figure 6.17 shows trajectories for a few individual water molecules in deep water waves. It shows that the molecules don’t simply move up and down, but go approximately around in circles, moving forward at each wave peak and backward at each wave trough. For this reason, water waves are not transverse waves, despite their appearance, but a combination of transverse and longitudinal waves. The radii of the molecules’ circular trajectories are equal to the wave amplitude at the surface and are smaller lower down, decreasing nearly to zero at the wave base, which is about half the wavelength, or λ/2, below the surface. About 96% of the wave energy is contained in the water above the wave base. One effect of this circular motion is that water waves are not sine waves. Instead, they have the shape of a trochoid curve, which has long troughs and sharp peaks and is produced by rolling motion. For example, Spirograph drawing toys produce
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Figure 6.18 A trochoid curve being created with a Spirograph drawing set.
Figure 6.19 The top panel shows the forces on molecules with red arrows. The bottom panel shows the same forces with red arrows, and the molecule velocities in black arrows.
trochoid curves, as shown in Figure 6.18. Likewise, a reflector on a bicycle wheel also maps out a trochoid curve as the bicycle rolls along. In shallow water, the molecules’ circular motion gets flattened out by the water bottom, leading to water molecule motions that are more elliptical or, in extreme cases, simply back-and-forth. At the very bottom, the water always goes back and forth with each wave, stirring up sand or other material on the bottom as it does so. The circular dynamics of deep water waves arise from the combination of restoring and inertial influences. The upper panel of Figure 6.19 shows the restoring force directions for gravity waves using red arrows, which don’t depend on the direction of wave motion. To see that these forces make sense, suppose the waves are in a highly viscous liquid such as molasses. Here, inertia does not matter and the fluid simply flows in the direction of the local forces. For this wave shape made out of molasses, the molasses would slowly flatten itself out, moving along the directions shown with the red arrows. Returning to a wave made of water, suppose the wave is traveling to the right and consider a single molecule that starts in a wave trough. Initially, the force on this molecule would point up, then right as the next wave caught up to it, then down when it’s at the wave peak, and finally left as the wave is leaving, before going back to up again. In other words, the force on this molecule would rotate over time. Similar rotating forces arises in many physical situations. For example, a ball on a string that is being swung around in a circle experiences an inward force along the direction of the string. However, the ball does not move toward the center but goes around
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the circle instead, perpendicular to the force, due to its inertia. Another example is the moon orbiting the Earth, where gravity always pulls the moon toward the Earth but the moon’s inertia carries it around in a circular orbit, again perpendicular to the direction of the force. The same thing happens in the water wave. The rotating force always pushes a water molecule toward the center of a circle, but its inertia carries it around the perimeter of the circle. The lower panel of Figure 6.19 illustrates the directions that specific water molecules are moving with black arrows. In addition to this circular motion, Figure 6.17 also shows some net motion of water molecules in the direction of the waves. This mass flow of water is called Stokes drift, due to its explanation in 1847 byGeorge Stokes, an Irish fluid dynamicist. Stokes drift causes material floating at the ocean surface, such as logs or toy boats, to be pushed along by the waves. In deep water, there is a gentle return flow of water beneath the wave base that counteracts the mass flow within the waves. In shallow water, as the waves approach the shore, this return flow is impeded by the bottom, but it still has to be present somehow. In some cases, the return flow is an offshoredirected water current that flows right along the bottom, under the incoming waves, called an undertow. Alternatively, the water can move sideways to join rip currents, which are narrow currents of water that flow away from the shore, cutting through the incoming waves in the process13 .
6.5.5
Water Wave Evolution*
Most water waves are generated by wind blowing over the water’s surface. Not surprisingly, there is some friction between the moving air and the water, which pushes on the water. But, how does this create the waves that we see, whether the little ripples that are produced by a gust, or the choppy waves on a lake, or the big ocean swells that roll in from distant storms? Scientists have investigated these questions for centuries, leading to a reasonably good understanding by now, although one that is still far from complete14 . Consider a lake that is completely still, with a glassy smooth surface. This system has what’s called translational symmetry because every part of the water’s surface is just like every other part. Then, a gust of wind blows over it. The gust is turbulent, meaning that there are lots of random eddies in the wind that have different speeds and that swirl in different directions. The layer of air that’s just above the water is particularly turbulent. As shown on the left side of Figure 6.20, this turbulence pushes down on the water by different amounts in different places, which creates a slight unevenness in the water’s surface, breaking the translational symmetry.
13 Despite
common misconceptions, undertows don’t pull swimmers underwater and are generally quite safe. However, rip currents can be dangerous if swimmers don’t realize that they can escape them by simply swimming parallel to the beach, to get back to regions with an onshore Stokes drift. 14 See Pizzo, Deike, and Ayet, “How does the wind generate waves?” Physics Today 74: 38-43 (2021); Huang, Long, and Shen, “The mechanism for frequency downshift in nonlinear wave evolution” Advances in applied mechanics 32: 59-117C (1996).
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Figure 6.20 The left panel shows wind blowing over water, where air turbulence near the surface creates an uneven water surface. The right panel shows wave growth by wind.
Next, the wind converts the slightly uneven water surface into little waves through positive feedback, in which the bigger bumps on the water surface grow bigger, which then causes them grow bigger yet. The right side of Figure 6.20 illustrates this wave growth. It shows that the wind pushes on the upwind side of each wave, which pushes some water up the wave. This transported water stay near the wave top because the airflow then separates from the water at the wave crest, producing just some weak eddies on the downwind side of each wave that interact minimally with the water surface. Meanwhile, the airflow expands back downward, where it encounters the next wave and pushes water up toward its crest. These effects are stronger with higher amplitude waves, with the net effect that the initially slightly uneven water surface builds up into a series of wind-driven waves. Continued and/or faster wind causes these waves to continue to grow larger, in terms of both height and wavelength. Several other types of waves build through similar processes. For example, stream beds often have current ripples in the sand that are typically around a centimeter tall. They build up because the moving stream water pushes sand up the upstream side of each ripple, and then the water flow separates from the surface at the top of the ripple, so the suspended sand settles back down near the top of the ripple. Blowing snow creates a regular pattern of snow drifts in very much the same way. Yet another example is the the ski moguls that are found at alpine ski areas. In this case, skiers consistently ski between the moguls, always pushing snow onto the mogul that’s downhill of themselves. This positive feedback process causes the moguls to grow over time. A curious aspect of water waves occurs after the wind stops and the waves continue propagating. It turns out that the waves continue evolving, with their energy continually being transferred into lower frequency waves with longer wavelengths. For example, a storm in the South Pacific Ocean might create large waves with a 10 second period, but then those same waves are found to have a 20 second period when they reach the Oregon coast several days later. This frequency downshift arises from complicated non-linear effects that are poorly understood. In particular, wave breaking, in which the wave crest curls over into white foam on the wave top, clearly plays an important role in frequency downshift but is also very hard to study due to the complicated interactions that are involved.
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6.5.6
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Long Wavelength Water Waves: Tsunamis, Tides, and Seiches*
Several phenomena can produce very long wavelength water waves, sometimes on the order of hundreds of kilometers. Recalling that water is considered shallow if its depth is less than about λ/2, this means that all water is shallow when compared to these waves. Even the ocean, which is 10 km deep at its deepest point, is shallow when compared to a 100 km long wave. Tsunami are created by a sudden displacement of a large amount of water, typically created by undersea earthquakes, landslides, or eruptions. For example, the 2011 Japan tsunami was created when an earthquake lifted part of the seafloor by several meters15 . That raised the water above it too, causing essentially a step in the ocean surface, which then propagated as a long wavelength wave. The wave amplitude was only few meters initially, and then decreased as the wave spread out from its source, so ships on the ocean didn’t notice it. However, the tsunami was extremely destructive when it reached land. The wave base for shallow water waves is the bottom of the water, so the entire depth of the water moves back and forth as the wave passes, all the way down to the ocean bottom, which represents a great deal of energy. This energy gets compressed into much less water depth as a tsunami approaches the shore, which can raise the wave amplitude up to create towering breakers. However, more often, the long wavelength of the tsunami causes it to rise and fall slowly at the shore, flowing far onto land when each peak arrives and pulling far out to sea when each trough arrives. The 2011 Japan tsunami exhibited both behaviors, depending on the the part of the shore where it hit, creating waves in some places that reached up to 40 m high and others that travelled 10 km inland. Tsunamis travel very quickly. From √ the shallow water wave speed equation (Eq. 6.9), a tsunami’s speed is vshallow = gd; taking the depth as 3.5 km, which is about the average ocean depth, its speed evaluates to 185 m/s, which is about 410 miles per hour, or the speed of a commercial airplane. This means that tsunamis cross entire oceans in less than a day. Because of their long wavelengths, tsunami periods are quite long, typically on the order of 10 minutes. In some cases, the first thing that people observe is that the ocean water pulls far back out to sea, which is a tsunami trough. If they’re smart, they run to high ground at this point, because the tsunami peak will arrive several minutes later and flow far onto land. This tends to repeat multiple times as successive waves hit the shore. Tides are another type of long wavelength gravity waves, observed as the ocean surface rising and falling periodically, often with two high tides and two low tides each day. They are standing waves that are driven by the gravitational influences of the sun and moon on the oceans, of which the moon has more effect because it is
15 The 2011 T¯ ohoku earthquake and tsunami occurred on March 11, 2011, about 72 km offshore of
northern Japan. It was an exceptionally large event, with a magnitude 9.0 earthquake and a very large tsunami that washed away many villages, killed about 20,000 people, and caused meltdowns at the Fukushima Daiichi nuclear power plant. See Wikipedia “2011 T¯ohoku earthquake and tsunami”.
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Figure 6.21 Tide ranges in different parts of the world.
much closer (see Section 15.1.1). These driving forces are not powerful enough to create large tides on their own but, instead, the tides we observe arise from resonance between these driving forces and the natural frequencies of the seawater sloshing back and forth within an ocean basin16 . Figure 6.21 shows the tide ranges in different parts of the world, where red regions have large tides and blue regions have small tides. These variations occur because ocean basins with different shapes have different normal modes, causing them to resonate differently with the sun’s and moon’s driving forces. An interesting aspect of the figure is that the tides have nodal points (the white dots at the centers of the white lines). As with nodes in string standing waves, there are no tidal displacements at these nodes. Instead, the water sloshes in circles around them, rising up against the shore on one side of the circle and pulling away on the other side of the circle. The circular motion arises from a combination of the ocean basin shapes and the fact that the moon orbits the Earth at an angle to the equator, causing its driving force to push the water around in giant circles. Seiches are also long standing waves, where the water sloshes back and forth across the whole body of water. They can occur in small bodies of water such as dishpans and swimming pools, but are normally only called seiches when they occur in large bodies of water, such as lakes and enclosed seas. As with tsunamis, something has to displace the water first, but then seiches differ from tsunamis in that they
16 A
fascinating book about the science of tides and the surrounding culture is Tides: The Science and Spirit of the Ocean by Jonathan White, 2017, Trinity University Press, San Antonio, TX.
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Figure 6.22 Water level over time at the the two ends of Lake Erie, during April, 2020. The gray line is for the west end of the lake (the Fermi power plant, near Detroit, Michigan) and the green line is for the east end (Buffalo, New York).
involve multiple cycles of standing waves rather than a single event that spreads and dissipates. Figure 6.22 shows data from a typical seiche event, which in this case is water sloshing back and forth across the length of Lake Erie. In this event, a strong west wind blew over the lake on April 13, pushing the lake water toward the east end of the lake. This produced the large spikes in the green and gray curves, of which the green line shows the water level at the east end, in Buffalo, NY, and the gray line shows the water level at the west end, near Detroit. After the wind stopped, the lake sloshed back and forth for about three more days, with about a 14 hour standing wave period.
6.6
Seismic Waves*
6.6.1
The Earth’s Structure*
The Earth is composed of four major layers, shown in Figure 6.23. Its center is a solid core that is made of mostly iron, called the inner core. Nuclear reactions from radioactive material within the Earth, plus some residual heat from the Earth’s formation, heat the inner core to about 5700 K, which is similar to the temperature of the surface of the sun. It is surrounded by the liquid outer core, which is also made of mostly iron. This layer is heated from below and cooled on its top, rather like a pot of water on a stove, causing the liquid iron to slowly flow in giant swirls, which is the main cause of the Earth’s magnetic field. Outside of that is the mantle, a thick layer of rock that is basically solid, but softened sufficiently by the high temperature and pressure that it is able to flow over geologic timescales. Finally, the crust, which is only tens of kilometers thick, is a rocky layer that floats on top of the mantle; in
6.6 Seismic Waves*
153
Figure 6.23 The Earth’s inner structure, with color showing approximate temperature. Seismic wave paths are shown for P-waves in blue, for S-waves in pink (adjacent to some blue arrows), and for surface waves in yellow. Seismograms are shown for selected locations.
places, oceans float on top of the crust. The temperature and pressure are low enough in the crust that the rock is brittle. Essentially all life on Earth, including ourselves, lives within a few meters of the top of the crust or the top of the oceans. Earth’s crust, and the upper portion of the mantle, are divided into seven large tectonic plates, along with many small plates. These plates behave somewhat like rafts of bubbles on a cup of hot cocoa, constantly being pushed around by the underlying mantle flow, creating mountains where they collide, and creating valleys where they get pulled apart, in a process called continental drift. At some points in the Earth’s history, the continents have drifted together to form large supercontinents, and other times, like now, they are separated and spread out around the Earth’s surface. Because the rock in the crust is brittle, the plates don’t always slide against each other smoothly as they drift, but they sometimes stick for a while before breaking loose, at which point they produce seismic waves that are felt as earthquakes.
6.6.2
Earthquakes*
The specific location where the rocks stick and then break loose is the earthquake’s hypocenter or focus. It is typically within about 70 km of the surface, where the rocks are cool enough to be brittle and able to stick and slip, in contrast to down deeper where the rock is softened by temperature and pressure. Nevertheless, earthquake
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foci do occur down lower on occasion, sometimes as deep as 700 km below the surface. The location on the Earth’s surface that is directly above the earthquake’s focus is called the epicenter, which literally means “upon the center.” Earthquake strengths were quantified on the Richter magnitude scale from its development in 1935 up to the 1970s. After this, the scale was replaced by the more precise “moment magnitude scale”, but the values were intentionally kept fairly similar, so informal earthquake reports often still talk about “Richter magnitudes” even though this isn’t technically correct. Regardless of what it’s called, the scale is a measure of total earthquake energy that uses a logarithmic scale17 , meaning that every time the value increases by 1, the energy increases by a factor of about 31. For example, a magnitude 8 earthquake is 31 times more powerful than a magnitude 7 earthquake. Small earthquakes are fairly common, with thousands per year worldwide in the magnitude 5 range, but large ones are rare, leading to only occasional magnitude 8 or 9 earthquakes.
6.6.3
Seismic Waves*
As we saw in Section 2.2.2, earthquakes produce two types of waves that propagate outward from the focus, P-waves and S-waves. P-waves, shown in Figure 6.23 with blue arrows, are named primary waves because they are always the first to arrive at any specific location, due to their being the fastest waves. They are longitudinal waves in rock pressure, just like sound waves are longitudinal waves in air pressure. As longitudinal waves, P-waves do not have a polarization and can propagate through liquids, such as the Earth’s outer core, exactly as sound can be transmitted through air or water. P-waves typically create little damage, making them useful for early warning systems. S-waves, shown in Figure 6.23 with pink arrows (next to some blue arrows), are named secondary waves because they arrive after P-waves, due to their slower speed. They are shear waves, meaning that they have a sideways displacement, and are transverse waves. As such, they have two polarizations, like string waves. They cannot travel through liquids, such as the outer core, because liquids flow rather than rebounding in response to shear forces. Once P-waves or S-waves reach the ground surface, they can change into any of several types of surface waves, shown in Figure 6.23 with yellow arrows, all of which propagate along the surface somewhat like water waves. However, unlike water waves, the restoring force for all seismic waves is the elasticity of the rock rather than gravity. Also, seismic surface waves can include side-to-side (shear) displacements. There are two major types of surface waves, which are Love waves, in which the ground surface shifts horizontally in transverse waves but not vertically, and
17 The
scaling √ is a little unusual here, where a magnitude increase of 1 corresponds to an energy increase of 1000. In more detail, the moment magnitude is Mw = 23 log10 (M0 ) − 18.1 3 , where M0 is the energy released in Joules.
6.6 Seismic Waves*
155
Rayleigh waves, or “ground roll”, in which the ground surface moves both vertically and horizontally in waves that combine longitudinal and transverse motions. Both types of surface waves, which were named for their discoverers18 , travel slower than the P- and S-waves, but are responsible for most earthquake damage. As with other mechanical waves, seismic waves arise from restoring forces and inertial influences. Rock is extremely incompressible, making the restoring forces extremely large, but rock is also dense, which leads to large inertial influences. The high incompressibility turns out to have a larger effect, so seismic waves propagate very quickly, with P-waves going around 8000 m/s through the crust. They go even faster deeper in the Earth due to the higher pressure, propagating at up to about 14,000 m/s at the mantle’s inner edge (on the other hand, they are slower in the inner and outer cores due to the change of chemical composition from rock to iron). These speed changes can be seen as refraction effects in Figure 6.23, where the paths always bend toward the direction where the waves travel slower. S-waves travel about 60% as fast as P-waves, and surface waves travel somewhat slower yet, in both cases due to lower restoring forces. Seismic wave data, measured on a seismograph instrument and producing a graph called a seismogram (see the insets in Figure 6.23), can be very informative. First of all, the distance between a measurement point and an earthquake’s focus can be determined from a seismogram, by finding the time difference between the P-wave and S-wave arrival times and then computing distance from the known difference in travel speeds. With seismograms from different locations, it’s possible to combine the distances to precisely locate the earthquake’s epicenter and depth. The amount of shaking recorded in the seismogram can then be used to find the earthquake’s magnitude. Also, seismograms provide information on the Earth’s structure because seismic waves reflect off boundaries between the layers, so their arrival times help show both the presence and locations of the boundaries. In addition, S-waves are never observed on the opposite side of the Earth from an earthquake, in the “S-wave shadow zone”, which provided the key evidence that the outer core is liquid. Example. Elsa feels a gentle earthquake and then much stronger one 3 seconds later. How far is she from the earthquake focus? Answer. She correctly guesses that the first earthquake was the P-wave and the second was the S-wave. She’s also probably fairly close to the epicenter, so she uses the P-wave speed in the crust of v P = 8000 m/s; the S-wave speed is 60% of this, which is v S = 4800 m/s. Defining the distance to the focus as x, the arrival times were tP =
18
x vP
and
tS =
x . vS
Augustus Edward Hough Love (1863–1940) was an English mathematician who focused on elasticity and waves. John William Strutt, 3rd Baron Rayleigh (1842–1919), also known as Lord Rayleigh, was also an English mathematician, in his case focusing on fluid dynamics.
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The difference is
tS − t P = x
1 1 − vS vP
Mechanical Waves
.
Solving for the distance gives −1 1 1 1 −1 1 − x = (t S − t P ) − = (3 s) = 36000 m. vS vP 4800 m/s 8000 m/s
Thus, the earthquake focus was about 36 km away from her.
6.7
Summary
Mechanical waves include string, sound, water, and seismic waves, which have many things in common. Waves, and oscillations more generally, work through a combination of restoring and inertial influences. The restoring influence always acts to reduce the displacement, while the inertial influence causes changes in the displacement to keep on changing. Mechanical wave speeds relate to these influences as v=
restoring influence . inertial influence
Restoring forces depend on the wave type, being tension for string waves, air pressure for sound waves, surface tension for water capillary waves, gravity for water gravity waves, and rock elasticity for seismic waves. Inertial influences, which are quantified as momentum, arise from the mass of the medium, whether string, air, water, or rock. Waves are nondispersive if the wave speed is independent of the wavelength, as it is for string, sound, and most seismic waves, and dispersive if the wave speed depends on the wavelength, as it does for most water waves. Sound waves propagate faster in warm air than cold air because the air molecules move faster. This speed is computed from the absolute temperature, measured in kelvin (K), which is a temperature scale in which 0 K is the coldest temperature possible. Humans can hear sound waves between about 20 Hz and 20 kHz, while some animals can hear higher or lower frequencies. Low frequency sound waves generally travel farther due to less damping, while high frequency sound waves can be preferable for echolocation because their shorter wavelengths offer better spatial resolution. Musical notes are described by pitch, loudness, and timbre. Scientific pitch notation names musical notes with an octave value, then a note pitch within the octave that is given with A through G, and then sometimes a sharp or flat symbol. The scale is standardized so the A4 note, also called A440, has a frequency of 440 Hz. All musical instruments, whether chordophones (strings), aerophones (winds), membranophones (drums), and idiophone (xylophones, etc.), produce notes that result from the instrument’s normal modes, which then couple to sound waves in the air. Notes
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generally sound better together, called consonant, if the ratio of their frequencies involves small integers, and are otherwise dissonant. Capillary water waves are dispersive with faster propagation for short waves, whereas deep-water gravity waves are dispersive with faster propagation for long waves. This dispersion is visible in the ripples that spread from a sudden disturbance in the water. It also causes groups of waves, such as those in a boat’s wake, to propagate faster or slower than the individual waves within the group. The individual molecules in a gravity wave mostly go around in circles, but also move forward slightly at each wave. Shallow-water gravity waves, meaning that the water depth is significantly less than half of the wavelength, are nondispersive. Tsunamis, tides, and seiches have very long wavelengths, so they are shallow water waves and often travel very fast. The Earth is composed of an inner core, outer core, mantle, and crust. The tectonic plates that make up the crust and upper mantle move around through continental drift and sometimes stick and break loose, which creates earthquakes. Seismic waves include P-waves, which are fast longitudinal waves, S-waves, which are slower transverse waves, and Rayleigh and Love surface waves, which are even slower and cause most earthquake damage. S-waves cannot propagate through liquids, from which we can infer that the outer core is liquid and the other Earth layers are solid.
6.8
Exercises
Questions 6.1. Which of these changes would lead to the fastest waves on a string? (a) (b) (c) (d) (e)
less tension, lighter string less tension, heavier string more tension, lighter string more tension, heavier string just more tension, mass doesn’t matter
6.2. What is the speed of sound on the moon? (a) The same as on Earth, 340 m/s (b) Faster than 340 m/s (c) Slower than 340 m/s (d) It depends on the temperature (e) There is no sound on the moon 6.3. What factors are essential for mechanical waves to propagate? (Choose all that apply.) (a) Things in motion keep on moving, even without external forces. (b) Things in motion always come to a stop. (c) There need to be forces that promote displacements in the medium. (d) There need to be forces that reduce displacements in the medium.
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6.4. Swiftlets are tropical birds that live in caves and navigate with echolocation using clicks. The caves are sufficiently crowded that hundreds of birds fly in the same space, all echolocating at once. Which properties of a bird’s click could it vary, in order to distinguish its own echo from all of the other noises? (Choose all the apply.) (a) Pitch (b) Dispersion (c) Timbre (d) Inertia (e) Loudness 6.5. Which of the following waves are dispersive? (Choose all that apply.) (a) String (b) Sound (c) Capillary water waves (d) Gravity water waves in deep water (e) Gravity water waves in shallow water (f) Seismic P-waves 6.6. Suppose that sound waves were dispersive, with long waves traveling faster than short waves. (a) Describe how this would affect conversations and music. (b) What would a plucked guitar string sound like? 6.7. For standing waves on strings, it was shown that the second harmonic is one octave above the fundamental frequency. (a) Are all string wave harmonics an integer number of octaves above the fundamental frequency? (b) Which harmonic is two octaves above the fundamental frequency? 6.8. The speed of waves on a membrane, such as the head of a drum, was not given here. From your knowledge of other wave speed equations, make some predictions about wave speeds on membranes: (a) does the wave speed increase or decrease with greater membrane tension? (b) does the wave speed increase or decrease with a heavier membrane? 6.9. A sand grain falls in a puddle and makes ripples that spread outward. (a) Are the long wavelength ripples on the inside or outside of the expanding circle? (b) A boulder falls into a lake and makes waves that spread outward. Are the long wavelength waves on the inside or outside of the expanding circle? 6.10. A tennis ball is floating in a lake and is just beyond reach. Can you retrieve it by throwing rocks in the water just beyond it, and letting the waves push it toward you? Explain.
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Problems 6.11. Suppose a friend builds her own electric guitar and she decides to put the pickups exactly in the middle of the string. (a) Which harmonics will the pickups detect and amplify? (b) What are the frequency ratios of these harmonics relative to the fundamental frequency (e.g. 2:1). (c) Would these harmonics be consonant or dissonant? 6.12. A person sings the A440 note. Then, he breathes in helium from a party balloon tries to sing the same note, but it comes out one octave higher. (a) Treating his vocal system as an open ended pipe, how many times faster was the speed of sound after he breathed in helium? (b) Using the speed of sound equation, how many times faster is the speed of sound in helium than in air? Use: γh.c. is 1.4 for air and 1.67 for helium, and m air is 4.8 · 10−26 kg for air and 6.6 · 10−27 kg for helium. (c) Why are these values different? 6.13. The “hull speed” of a boat is the speed of a water wave that has the same wavelength as the boat length. Boats generally have a hard time traveling faster than their hull speed. (a) What is the hull speed for a 5 meter canoe in deep water? (b) What about for a 100 meter long ship in deep water? (c) What about for a 10 cm long duck in deep water? (d) What about for a 5 meter canoe in 10 cm deep water? 6.14. Many American √ boat designers know the equation that the boat’s hull speed is v = 1.34 L W L where v is the speed in knots and L W L is the boat length in feet, measured at the waterline. Is this the same equation as the deep water wave speed for wavelength λ = L W L ? Use the conversions 1 knot = 0.514 m/s and 1 foot = 0.3048 meters. 6.15. The 2011 Japan tsunami was triggered by an earthquake 70 km offshore. (a) How long after the earthquake happened was the earthquake’s P-wave felt in Japan, which traveled at 6 km/s? (b) The frequency of this sound wave is about 1 Hz, what was the wavelength? (c) Suppose the tsunami wavelength was 100 km, and the ocean is about 100 m deep in that region. What was the tsunami wave speed? (d) How long did it take the tsunami to reach Japan? 6.16. Two people are talking on a string telephone that is 14 m long. A string telephone has a cup at each end and a string that connects them, which carries the sound as string waves, shown below. The people pull on the string with about 2 pounds of force (9 N), so the string tension is 9 N. The string has a mass density of 1.2 g/m. How long does it take sound to go from one person to the other through the telephone?
6.17. A pipe organ is an aerophone with open ended tubes, with essentially the same physics as the flute. (a) For normal room temperature, where the speed of
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sound is about 340 m/s, how long is the pipe for the A440 note? (b) Suppose someone wanted to install an outdoor pipe organ at the South Pole that would be in tune when the temperature is −80◦ C (193 K). How long would the pipe be for the A440 note for that organ? 6.18. A whirly tube is a musical toy. It is a corrugated plastic tube that is open at both ends, 0.74 m long, and makes noise when swung in a circle. Faster swinging excites higher overtones and thus produces higher pitches. (a) What is the whirly tube’s fundamental frequency? (b) What is the frequency of the second harmonic? (c) What is the frequency of the third harmonic? (d) What note is fundamental frequency, using scientific pitch notation? (e) What note is the second harmonic? 6.19. At the surface of the sun, the temperature is 5778 K, the average atomic mass is about 1.90 · 10−27 kg (mostly atomic hydrogen and some helium), and the heat capacity ratio is γh.c. = 53 . The pressure varies from about 80 Pa to about 12 kPa. (a) What is the speed of sound? (b) How many times faster is this than the speed of sound on Earth? 6.20. This problem investigates seiche periods. (a) What is the fundamental frequency of a standing wave in a lake, for wave speed v and lake length L? (b) Assuming that a seiche is a shallow water wave, compute the wave period for lake length L, lake depth d, and gravitational acceleration g. This result is known as Merian’s formula, originally derived by J.R. Merian in 1828. (c) Use this period to compute the seiche period for Lake Erie, which is about 288 km long and 19 m deep. (d) Compare your result to the actual value of 14.1 hours (is your prediction reasonably close? what are the major approximations that led to differences?). 6.21. People determine the directions where sounds come from with two methods. We hear an intensity difference between our ears for short waves and a phase difference between our ears for long waves. (a) Why is the intensity difference more useful for short waves? (b) Supposing an adult’s ears are 20 cm apart and the sound comes directly from one side, how many wavelengths is this separation for a 100 Hz sound wave? (c) How many wavelengths is this for an 5000 Hz sound wave? (d) Why can we localize low notes but not high notes using a phase difference? 6.22. A whitewater river flows between two boulders, creating a series of stationary waves. These waves don’t move relative to the land, but propagate upriver at the same speed as the water flows downriver (they are often called “standing waves”, but are completely different from the standing waves that occur in a cavity, discussed in this book). If the wavelength is 3 m and the water is 4 m deep, what is the river flow speed?
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Puzzles 6.23. The world’s largest earthquake occurred on May 22, 1960 in Chile, with a magnitude of 9.5. How many times more energy was released here than Los Angeles’s 1994 Northridge earthquake, which had a magnitude of 6.7? 6.24. Imagine you are holding a rope up by one end. At any point in the rope, the tension at that point arises from the weight of the rope that is below that point. This implies that the tension is high at the top and decreases to zero at the bottom. Suppose you put a pulse in the rope at the top, which then propagates downward. Would the pulse take infinite time to reach the bottom? Discuss. 6.25. A pipe of length L is closed at both ends. What are the natural frequencies for sound waves in this pipe as functions of L and the speed of sound? 6.26. The chromatic musical scale has 12 semitones in each octave. Assuming they are equally spaced (called equal temperament tuning), meaning that the frequency ratio between one note and the next one is a constant, what is the value of this constant?
Part II Rays
7
Shadows and Pinhole Cameras
Figure 7.1 Shadow of vines on snow. The shadows present an interesting perspective on the vines and also reveal the shape of the snow surface.
Opening question Suppose you are standing on a sidewalk on a sunny day and looking down at your shadow. Which of the following are always true, which are never true, and which are sometimes true? (a) The area of the shadow is larger than you are (b) The shadow of your head is sharper than the shadow of your ankles (c) The middle of the shadow is darker than the edge of the shadow (d) The shape of the shadow arises, in part, from the shape of the Sun © Springer Nature Switzerland AG 2023 S. S. Andrews, Light and Waves, https://doi.org/10.1007/978-3-031-24097-3_7
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Shadows are simple but intriguing. The mere reduction of light in some places, which is all that shadows are, is often able to reveal shapes and textures of objects that would otherwise go unnoticed. If you look around yourself, wherever you are, you will see that shadows create many of the cues that help your brain figure out where objects are relative to you, and relative to each other. If the lighting were truly even so that there were no light or dark regions, then everything would look very flat, without a distinct three-dimensional structure. This chapter starts our unit on light rays, in which light is assumed to travel in perfectly straight lines, or occasionally in smooth curves as it refracts. These assumptions are not strictly true, as we saw in our investigations on diffraction and interference, but are generally adequate to describe the shapes of shadows, the reflection of light off mirrors, and the refraction of light as it transitions between air and glass.
7.1
Shadows
7.1.1
Projection
We start by investigating shadow sizes and shapes, seeing how they depend on the object, the light source, and the surface that the shadow appears on. This will help explain many things that you’ve likely observed, even if you weren’t consciously aware of them. It will also introduce ideas that extend far beyond the simple study of shadows. Consider the shape of your shadow on the ground when you’re relaxing outside on a sunny day, such as the shadow in Figure 7.2. Your shadow isn’t the same shape every time you look at it, of course, but depends on your position, the angle of the ground, and the sun’s angle above the horizon. If you’re standing on level ground, your shadow is small when the sun is nearly overhead, a reasonable likeness of yourself when the sun is lower in the sky, and very elongated when the sun is close to the horizon. These are all effects of projection where, in this case, your threedimensional shape is being projected onto the ground’s two-dimensional surface. Figure 7.2 Shadow of a person bicycling, showing projection onto flat ground.
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Figure 7.3 Projections in a shadow.
The projection of an object’s shadow can be divided into two stages. First, the object blocks the sunlight to create a three-dimensional shadow that extends throughout the space behind the object. Then, some surface cuts through this shadow to create the two-dimensional projection, which is the shadow that you see. Figure 7.3 shows two types of projections. In anorthographic projection, a screen that is perpendicular to the sun’s rays is placed behind the object. The other type is an oblique projection, in which the surface is not perpendicular to the light rays. Here, the shadow is elongated and enlarged. The type of projection can affect the size and shape of the shadow. For example, a round ball casts a perfectly circular shadow in an orthographic projection but a larger and elliptically-shaped shadow in an oblique projection. Also, orthographic projections always create the smallest possible shadow for a given sun angle. To see this, imagine tilting the screen in Figure 7.3 up or down, and observe that the shadow fits on the screen as drawn but would extend off the edge of the screen with either tilt direction.
7.1.2
Analyzing Projections*
Shadows can provide an easy way to measure the heights of tall objects, such as flag poles, trees, or fences. For example, the left side of Figure 7.4 shows a diagram of a post and its shadow, where the shadow is an oblique projection. The figure is a ray diagram because it also shows the light ray that just skims the top of the post and ends at the tip of the shadow. Figure 7.4 (Left) Ray diagram for a shadow on flat ground. (Right) Ray diagram for an orthographic projection.
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The post and shadow create a right triangle, where the light ray that is shown forms the triangle’s hypotenuse. From triangle trigonometry (Appendix D), the post height and shadow length relate to the sun’s angle above the horizon with tan θ =
h soblique
,
(7.1)
where h is the post’s height, soblique is the shadow length, and θ is the sun’s angle. This equation can be rearranged to yield any of the three variables in terms of the other two. Although typically less useful, orthographic projections can be investigated in a similar manner, as shown on the right side of the figure. This time, the shadow is perpendicular to the light ray rather than to the post. The triangle shown with a solid line is the important one. Here, the triangle base is the shadow, which has length sor tho , while the triangle hypotenuse is the post, which has height h. We know that the angle that’s shown at the bottom of this triangle is equal to θ because: the sun’s angle above the horizon is θ from the right corner of the dashed triangle, which implies that the left corner of the dashed triangle has angle 90◦ − θ, which then implies that the bottom corner of the solid triangle has angle θ. Also, this makes sense, because both θ values would shrink if the sun were lower in the sky, and vice versa. From trigonometry, these parameters relate to each other as cos θ =
sor tho . h
(7.2)
Again, this can be solved for any one value if the other two are known. Example. A post is 5 m tall and the sun is 32◦ above the horizon. How long are the (a) oblique and (b) orthographic shadows? Answer. m (a) Rearrange Eq. 7.1 and plug in numbers: soblique = tanh θ = tan5 32 ◦ = 8.0 m. (b) Rearrange Eq. 7.2 and plug in numbers: sor tho = h cos θ = (5 m) cos 32◦ = 4.2 m. As always, the orthographic projection is shorter than the oblique projection. It’s also interesting to see that the orthographic projection is shorter than the post, while the oblique projection is longer than the post.
7.1.3
Umbra and Penumbra
Returning to your own shadow, notice that its edge is not completely sharp, but slightly fuzzy. Furthermore, the shadow of your head is generally fuzzier than the
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shadows of your ankles (in Figure 7.2, the shadows of the wheel spokes are sharper than the shadow of the rider’s back). Figure 7.5 helps explain this by showing the shadow of a ball on a wall using an orthographic projection. The center of the shadow, meaning the entire shadow inside the fuzzy edge, is called the umbra, which is Latin for shadow1 . All of the light is blocked within the umbra, meaning that an ant within the umbra would not be able see any part of the light source. The surrounding region of the shadow, in which only some of the light is blocked, is called the penumbra, which is Latin for “almost shadow.” The penumbra is the fuzzy edge of the shadow, transitioning from the darkness of the umbra at its inside edge to the full brightness of the light at its outside edge. An ant within the penumbra would be able to see part of the light source but not all of it. There isn’t a name for the region beyond the penumbra, but all of the light reaches the wall there. Figure 7.5 Shadow of a ball on a wall, showing the umbra and penumbra.
The umbra has a larger diameter than the ball in this figure. This is because the ball is larger than the light source, so the rays of light that define the edges of the umbra diverge away from each other. Vice versa, if the ball happened to be smaller than the light source, then the rays that define the edges of the umbra would converge inward and the umbra would be smaller than the ball. In fact, there would be no umbra on the screen at all if the ball were small enough. There would still be an umbra in the space behind the ball, but it would end before it got to the screen. In contrast, the penumbra is always larger than the object, as can be seen from the rays in Figure 7.5. Back to your own shadow, its fuzzy edge is the penumbra, which is created by the width of the sun in the sky. Also, the shadow of your head has a larger penumbra than that of your ankles because your head is farther from the ground.
1 The
word umbrella is Italian for “little shadow,” from a climate in which umbrellas are used to create shade rather than to stay dry in the rain.
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Shadows and Pinhole Cameras
Eclipses
Eclipses are shadows that are cast by planets or moons. In particular, for our own planet and moon, eclipses occur whenever the Moon casts a shadow on the Earth, or the Earth casts a shadow on the Moon.
7.2.1
Solar Eclipses
A solar eclipse, shown in Figure 7.6, occurs when we can’t see the Sun, meaning that the Moon has crossed between the Sun and Earth and we’re in its shadow.
Figure 7.6 (Left) Diagram of a solar eclipse, not to scale. (Right) Photograph of the Moon’s shadow on the Earth during the August 2017 solar eclipse from the International Space Station.
The Moon’s shadow, like all shadows, has an umbra and penumbra. The Moon is smaller than the Sun, so the diameter of its umbra is smaller than that of the Moon. By coincidence, the distance between the Earth and Moon is almost exactly equal to the length of the Moon’s umbra (correctly shown in the diagram), which is equivalent to saying that the Sun and Moon appear to have the same size in the sky when observed from Earth2 . The Moon’s umbra reaches the Earth during some solar eclipses but doesn’t quite for others, where the variation arises from the fact that the Moon’s orbit is not perfectly circular, so the distance between the Earth and Moon varies slightly over time. The left panel of Figure 7.7 shows a total solar eclipse, in which the Moon’s umbra reaches the Earth and people within it observe that the Moon blocks all of the light from the bright portion of the Sun. This allows them to see the Sun’s dim glowing outer atmosphere, which is much larger than the bright part of the Sun but is too dim to see normally. The middle panel shows a partial eclipse, observed by a person in the penumbra. Here, the Moon doesn’t quite line up with the Sun and so only blocks part of the Sun’s light. Finally, the right panel shows an annular eclipse,
diameters of the Sun and Moon are each about 0.5◦ of arc in the sky. You can observe this by comparing their apparent sizes to the width of your thumb, held at arm’s length, which is about 2◦ (obviously, be careful when looking at the Sun). Both the Sun and Moon diameters are about a quarter of a thumb width.
2 The
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Figure 7.7 A total, partial, and annular solar eclipse.
informally called a “ring-of-fire” eclipse, in which the Moon’s umbra does not quite reach the Earth. Here, people in the center of the eclipse see a ring of sunlight around the Moon. People in this portion of the penumbra, in which the light source is seen on all sides of the opaque object, are sometimes said to be in the antumbra. Solar eclipses are exciting to observe but fairly rare. There are two reasons for this. First, the Moon’s orbit is tilted about 5◦ from Earth’s orbit, with the result that the Moon doesn’t go directly between the Sun and Earth each time it passes by. Instead, the Moon’s shadow usually passes above or below the Earth, falling on the Earth only about once or twice per year. Secondly, even when eclipses do happen, the Moon’s umbra is so small that it only passes over a tiny fraction of the Earth’s surface. As a result, one generally has to wait for a long time to see a solar eclipse that is reasonably close to home. For example, the United States (which has an area of 2% of the Earth’s surface) had a total solar eclipse in 2017, will have an annular eclipse in 2023, a total eclipse in 2024, and another total eclipse in 2033.
7.2.2
Lunar Eclipses
In a lunar eclipse, shown in Figure 7.8, the Moon seems to disappear briefly because it passes through the Earth’s shadow3 . They are classified as a penumbral lunar eclipse when the Moon only passes through the Earth’s penumbra, a partial lunar eclipse when part of the Moon passes through the Earth’s umbra, and a total lunar eclipse when the entire Moon passes through the Earth’s umbra. Figure 7.9 shows that the Moon becomes a dark red color as it enters or leaves the Earth’s umbra, which is sometimes given the dramatic term of a “blood moon”. It is not as sinister as it sounds, but arises from the fact that the few rays of light that do get to the Moon have to pass quite close to the Earth, typically going through the Earth’s atmosphere. As we will see in Section 11.3, our atmosphere preferentially scatters blue light, which creates both blue skies and red sunsets. Some of the unscattered red light rays also keep on going out of the Earth’s atmosphere and on into space;
3 The solar and lunar eclipse names would be more logical if they were called an Earth eclipse and a Moon eclipse (or lunar eclipse), in both cases describing what the shadow lands on. However, that’s not the convention. Instead, the terms reflect our Earth-centric view of the solar system, in which we describe what appears to go dark to us.
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Figure 7.8 Diagram of a lunar eclipse, not to scale.
if the Moon is behind the Earth, they then illuminate the Moon with red light. In more colorful terms, the Moon is lit up during lunar eclipses by the sunsets on Earth. Lunar eclipses, like solar eclipses, occur once or twice per year. However, each one can be observed by anyone on the night side of the Earth, which makes them much more common. Figure 7.9 Three pictures of the Moon during a lunar eclipse showing its passage through the Earth’s umbra.
7.2.3
Eclipses and Moon Phases
It’s interesting to think about how eclipses relate to the phases of the Moon. A solar eclipse only happens when the Moon is between the Earth and Sun, making it an extreme version of a new moon. Vice versa, a lunar eclipse happens when the Moon is on the far side of the Earth as the Sun, making it an extreme version of a full moon. Alternatively, from the point of view of a hypothetical Moonling, the Earth has phases. This being would observe that the Earth is full during what we call a solar eclipse, and the Earth is new during what we call a lunar eclipse.
7.3 Light Through Small Holes
7.3
Light Through Small Holes
7.3.1
Pinhole Cameras and Camera Obscuras
173
A pinhole camera (Figure 7.10) is the simplest camera that one can build. It is a fully darkened box, typically the size of a shoebox or smaller, that lets light in through a tiny hole on one side of the box and has film on the other side.
Figure 7.10 Pinhole camera.
It can be understood with the same types of ray drawing methods that we used above to investigate the umbras and penumbras of shadows. In this case, the light rays travel in straight lines and all have to go through the same pinhole, which means that they create an image of the external scene on the film. This image is inverted from the original scene, with both up and down reversed, and left and right reversed. If the image is observed directly rather than being recorded on film, the same thing is called a camera obscura4 (Figure 7.11). These were particularly popular in the 17th and 18th centuries and were often made large enough for a person to stand inside. Figure 7.11 Camera obscura.
4 Some
camera obscuras replace the pinhole with a lens for greater light-gathering capability, although this does not affect the function substantially.
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Although simple to understand and fun to build, pinhole cameras have the drawback that the pinhole transmits very little light. This requires very long exposure times, often ranging from minutes to hours. Using a larger pinhole lets more light in, which helps the dimness problem, but causes fuzzy images. This is because the light rays from any single point in the scene can go through different parts of the hole and then on to different parts of the film. This is sort of like a penumbra, in which light rays from a large light source arise from different locations and then create a fuzzy penumbra at the edge of a shadow. Vice versa, smaller pinholes decrease fuzziness but at the cost of letting less light in. Pinholes that are too small also suffer from another problem, which is that light is not just rays but also waves, and these waves diffract around the sides of the pinhole and then spread out inside the camera to create fuzzy images again. This problem becomes significant when the pinhole diameter is similar to the light wavelength (Chapter 3). It turns out that pinhole cameras produce the sharpest images when the hole diameter is about √ (7.3) d = 2 λL, where λ is the light wavelength and L is the distance between the pinhole and film. For example, a 10 cm long pinhole camera should have a hole that is about 0.5 mm across for the sharpest images.
7.3.2
Pinhole Camera Analysis
As we did for shadows, we can draw ray diagrams of pinhole cameras and camera obscuras that help us find the sizes of distant objects. Figure 7.12 shows an example. Here, the object that is being observed, technically called the object, gets projected onto the back side of the pinhole camera to create an image. Figure 7.12 Ray diagram of a pinhole camera.
The light rays shown in the diagram create two triangles. These are similar triangles, meaning that they have the exact same shape, despite having different sizes. Their shapes are the same because they have the same interior angles; for example, the two interior angles that are next to the pinhole, which are marked with angle markings in the diagram, are equal to each other because the rays are straight lines. This is useful because it means that the heights and widths of the triangles are
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directly proportional to each other. As an equation, ho hi = , do di
(7.4)
where do and h o are the object’s distance and height, and di and h i are the image’s distance and height, with both distances being measured from the pinhole5 . This equation enables us to find any one value if we know the other three. Example. In Figure 7.12, the candle is 20 cm high and 60 cm from the pinhole, and the camera is 30 cm long. How tall is the candle’s image? Answer. We rearrange Eq. 7.4 to solve for h i , and plug in numbers, hi =
h o di (20 cm)(30 cm) = = 10 cm. do 60 cm
In words, the image is half as far from the pinhole as the candle, so it’s also half as high.
7.3.3
Multiple Pinholes*
If a pinhole camera has multiple pinholes, then each pinhole creates its own image. Figure 7.13 shows an interesting version of this. This picture was taken during a partial solar eclipse, when the Sun was partially covered by the Moon. The gaps between the leaves of a tree acted as many small pinholes, each of which created a separate image of the Sun on the ground.
Figure 7.13 Multiple images of the Sun during a partial solar eclipse, created by gaps between leaves on a tree acting as many small pinholes.
5 The image distance, d , is called the pinhole camera’s focal length. However, this term is confusing i
because the light does not come to a focus at the image. In contrast, the same “focal length” term is also used for curved mirrors and lenses, where the light does come to a focus at the image.
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Figure 7.14 Examples of shadow-based vision. From left to right: (1) A euglena cell, showing the eyespot toward the top of the figure, (2) A planarian worm, with cup eyes, (3) A chambered nautilus, with pinhole eyes, and (4) A zebra ark clam, which has primitive compound eyes (not visible in photograph). The bottom row shows diagrams of these eyes, with photoreceptors in orange, nerve fibers in blue, and shadows in gray.
7.4
Shadow-Based Vision*
While we tend to think of vision as a sense that is unique to the more advanced animals, the truth is that almost all organisms have some way of seeing. Plants, fungi, and even bacteria are able to sense light with photoreceptor proteins and then respond to those signals. Furthermore, many of these organisms can also determine where the light is coming from, often using visual systems that are based on shadows. Figure 7.14 shows several examples. Starting at the left end of the figure, euglena are widespread single-celled organisms that both swim and photosynthesize, thus exhibiting both animal and plant-like behaviors. They have a few photoreceptors near the front end of their cells that are located slightly behind a small patch of red pigment; together, these are called the eye spot. The pigment is important because it casts a shadow on the photoreceptors, which the cells can then use to orient themselves toward where the light is coming from. This enables them to swim toward the light, called phototaxis, which gives them more light for photosynthesis. Next, planaria are widely-distributed flatworms that are often studied in biology laboratories, and are slightly more sophisticated. Their photoreceptors line the inside of cup-shaped depressions. Light that comes from one side of the worms illuminates the photoreceptors that are on the far side of the depressions, while the photoreceptors on the close side remain in the cup’s shadow. This improves the worm’s sensitivity to the light source direction, enabling them to escape bright places and go to relative safety of dark places (called negative phototaxis). The chambered nautilus, which is a carnivorous marine snail that lives on coral reefs in the Indian and Pacific Oceans, takes this cup idea a little farther by nearly closing off the opening to yield eyes that work like pinhole cameras. As with humanmade pinhole cameras, nautilus eyes see images of their surroundings, but with
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relatively poor resolution and with low light collection ability. Nevertheless, their vision is evidently good enough for them to find their prey successfully. Finally, some clams, such as the tropical Arca zebra clam, take a different approach to shadow-based vision. They have tubes that are pointed in different directions and photoreceptors at the bottom of each one, each of which reports on whether there is light in its particular direction. This vision helps them determine whether a potential predator is nearby, indicating that it’s time to shut their shell. It is widely believed that the advanced camera-type eyes that humans have evolved from pinhole eyes like those of the nautilus6 . This evolution would have required a few relatively simple steps, each of which would have offered a significant vision improvement. The interior of the eye would need to be filled with a transparent substance that would improve the focus of the light onto the photoreceptors (it would also, likely, reduce the risk of infection). Next, the transparent substance at the eye opening would need a higher refractive index so that it could act as a lens. And, finally, muscles would be needed to adjust the focus. Indeed, sophisticated animal eyes may have evolved independently as many as 40 different times, and have been widespread ever since life diversified during the Cambrian explosion, 540 million years ago. Insect eyes are quite different from ours, having many small tubes, each with a lens at the front and photoreceptors at the back. They likely evolved from something that was more similar to the clam eyes shown here.
7.5
Summary
Shadows and pinhole cameras can be understood with ray optics, meaning that light is assumed to travel in straight lines (or refract in smooth curves), while the wave and particle explanations of light are ignored. Shadows represent projections of an object onto a surface, where the projection is orthographic if the surface is perpendicular to the light rays and oblique otherwise. Shadow sizes relate to the object’s height and light source angle with tan θ =
h soblique
and
cos θ =
sor tho . h
Shadows consist of an umbra, where all light is blocked, and a penumbra, where only some light is blocked. The umbra is larger than the object if the object is larger than the light source, and vice versa, while the penumbra diameter is always larger than the object.
6 See Schwab, I. R. “The evolution of eyes: Major steps. the Keeler lecture 2017: Centenary of Keeler Ltd.” Eye 32:302–313, 2018; Lamb, Trevor D., Shaun P. Collin, and Edward N. Pugh, “Evolution of the vertebrate eye: opsins, photoreceptors, retina and eye cup” Nature Reviews Neuroscience 8: 960–976, 2007.
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Solar eclipses occur when the Moon casts a shadow on the Earth. The eclipse is total in the Moon’s umbra, partial in the Moon’s penumbra, and annular in the center of the shadow when the Moon’s umbra does not reach the Earth. Lunar eclipses occur when the Earth casts a shadow on the Moon. They can be total, partial, or penumbral, depending on the Moon’s location in the Earth’s shadow. The Moon appears red during lunar eclipses due to the scattering of blue light by the Earth’s atmosphere and the remaining red light illuminating the Moon. Pinhole cameras and camera obscuras are darkened boxes with small holes that let light in. The light that goes through the hole creates an image of the external scene on the far side of the box. The image is inverted and can be either magnified or reduced, depending on the relative distances of the object and image. The object and image heights and distances are proportional to each other, with hi ho = . do di The image is always slightly fuzzy due to light entering the hole from different angles and, if the pinhole is on the order of a light wavelength or less, to diffraction effects. Many simple organisms have shadow-based vision systems which determine where the light is coming from by creating shadows over some of their photoreceptors. These simple vision systems likely illustrate steps along the evolutionary pathways that led to human and insect eyes.
7.6
Exercises
Questions 7.1. Which shapes are possible for the orthographic shadow of a cube on a flat surface? (Choose all that apply.) (a) Triangle (b) Square (c) Rectangle (d) Pentagon (e) Hexagon 7.2. Lunar eclipses following the 1982 eruption of the Mexican volcano El Chichón were notably darker than normal. What could have caused this? 7.3. Suppose you observe a lunar eclipse that happens to occur exactly at sunset, so you can see the Sun in one direction and the Moon in the opposite direction. If you wave, and watch with a very good telescope, will you be able to see your shadow on the Moon? Why or why not?
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7.4. The Moon is not completely black during a total solar eclipse, but is dimly illuminated (likewise, it’s often possible to see the dark side of a new moon). Where is this light coming from? 7.5. Three celestial objects, such the Sun, Moon, and Earth, are said to be in syzygy when they are lined up in a perfect line. Are the Sun, Moon, and Earth always, sometimes, or never in syzygy during eclipses? 7.6. When Venus crosses between the Earth and Sun, called a Venus transit, it appears as a small black dot on the face of the Sun. During this time, is the Earth in Venus’s umbra or penumbra? 7.7. Suppose an astronaut is on the Moon during a total lunar eclipse. What would this astronaut observe when facing the Earth? 7.8. Do the planets (e.g. Venus, Mars, and Jupiter) have phases, like the Moon? Explain.
Problems 7.9. It’s a sunny day, you have a meter stick, and you want to measure the height of a flagpole that is casting a shadow on level ground. (a) When the meter stick is vertical, its shadow is 85 cm long. What is the sun’s angle? (b) The flagpole’s shadow is 7.4 meters long. How tall is the flagpole? 7.10. Suppose the sun is 45◦ above the horizon and is casting a shadow of a 50 foot tall flagpole. (a) How long is the orthographic projection of this shadow? (b) How long is the oblique projection on level ground? 7.11. Suppose the sun is shining onto a spherical ball, which is casting a shadow. Treat the sun as a small light source. (a) What is the shape of the orthographic projection? (b) What is the shape of the oblique projection? (c) Which has larger area? 7.12. Consider a light that is casting a shadow of a ball on a wall. (a) Under what conditions, if any, would the shadow have no umbra? (b) Under what conditions, if any, would it have no penumbra? 7.13. You are casting an orthographic shadow of a ball onto a wall. (a) How can you make the umbra larger than the ball (or is it impossible)? (b) How can you make the umbra smaller than the ball (or is it impossible)? (c) How can you make the radius of the penumbra larger than the ball’s radius (or is it
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impossible)? (d) How can you make the radius of the penumbra smaller than the ball radius (or is it impossible)? 7.14. Suppose you have a pinhole camera which is 5 cm long between the hole and the film. You photograph a 1 cm long bug that is 10 cm in front of the pinhole. (a) Draw a diagram of this. b) What is the length of the bug’s image on the film? (c) Is the image rightside up or upside down? 7.15. Miss Wimpole is sitting in her camera obscura and admiring the image of her parent’s folly (a fake medieval ruin). While wondering how tall it is, she notices that a sheep is grazing at its base. She estimates that the sheep is three feet tall and she measures its image as 0.25 inches. Her camera obscura is 5 feet across, from hole to screen. (a) How far away is the folly? (b) The folly’s image is 4 inches tall. How tall is the folly? 7.16. Suppose you take a picture of a candle with a pinhole camera that has 0.2 mm diameter hole. The candle is 1 meter from the camera and emits about 18 mW of visible light (12 lumens of light with about 683 lm/W). (a) What is the light power per square meter, at 1 meter away from the candle? (b) How much light power goes through the pinhole? (c) You take a 1 second exposure. How much energy goes through the pinhole?
Puzzles 7.17. A friend makes a pinhole camera with a 0.2 mm pinhole, but he doesn’t make the pinhole perfectly round. Instead, it’s closer to a square shape. Will this affect the image and if so, how? 7.18. If you look down on the solar system, looking at the Earth’s north pole, you will see that the Earth’s orbit around the sun, its rotation on its axis, and the Moon’s orbit around the Earth are all counter-clockwise. (a) If the Moon is waxing, meaning that a larger fraction appears to be lit by the sun each night, then does the left or right appear to be bright for an observer in the Northern Hemisphere? (b) What about for an observer in the Southern Hemisphere? (c) Is a waxing moon high in the sky in the morning or evening for a Northern Hemisphere observer? (d) How about for a Southern Hemisphere observer?
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Reflection
Figure 8.1 A mirror sculpture titled C-curve by Anish Kapoor, exhibited in the South Downs Chalk Hills, UK.
Opening question Which of the following mirrors would make your reflection look smaller than your actual size? (a) A flat mirror (b) A convex mirror (c) A concave mirror (d) A very small flat mirror (e) A very large flat mirror © Springer Nature Switzerland AG 2023 S. S. Andrews, Light and Waves, https://doi.org/10.1007/978-3-031-24097-3_8
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Mirrors have fascinated people for thousands of years, from the ancient Greeks who developed myths about the god Narcissus falling in love with his own reflection, to King Louis XIV of France who built his fabled Hall of Mirrors at Versailles, and on to many modern artists who incorporate mirrors in their artwork. Mirrors present images of the world around us that are familiar for the most part, yet also changed in intriguing ways. Light reflection off mirrors, or other suitably large and flat surfaces, follows the law of reflection, which states that rays reflect off surfaces at the same angle that they arrived with. This rule is simple and even intuitive, but can lead to surprisingly complicated behaviors, especially when mirrors are curved. Reflection is often considered in the context of visible light, as it is throughout most of this chapter, but also occurs for other types of waves. Electromagnetic waves across the spectrum, ranging from radio waves to X-rays, are reflected by mirrors in modern telescopes. Sound and water waves are reflected by concrete walls, and seismic waves are reflected by transitions in the Earth’s rock. All of these reflections follow the same principles.
8.1
Reflection in General
8.1.1
Why Waves Reflect
Waves reflect when they reach boundaries in their media. Consider a pulse that propagates along a rope and reaches a doorknob, which the rope is tied to, and which is the boundary. The pulse in the rope pulls the doorknob up or down but the doorknob doesn’t move. This causes the rope to exert a large force on the doorknob and, correspondingly, causes the doorknob to exert an equally large force back on the rope. This second “reaction” force starts a new pulse that propagates back along the rope, which is the reflection. A sound wave reflecting off a brick wall is similar. In that case, air molecules exert a force on the wall when they collide with it, so the wall exerts an equal force back on the air molecules. This reflects both the individual air molecules and any sound waves that are propagating through the air. Light wave reflection off a shiny metal surface is similar as well. There, the electric field of the light wave pushes and pulls on the electrons in the metal which causes those electrons to move. The moving electrons then emit a reflected light wave. As another view of these same behaviors, the initial incident wave, whether a rope pulse, sound wave, or light wave, cannot transfer its energy either into or past the boundary. The only direction the energy can go is back toward where it came from, which is the reflected wave. Waves also reflect at boundaries within a medium at locations where the wave speed changes. Returning to the rope example, suppose a pulse that is sent down a thin rope suddenly arrives at a thick rope (Figure 8.2). The thick rope is heavier, so the wave speed is slower there (see Eq. 6.2), and the wave reflects at the boundary. This is only partial reflection, with part of the pulse getting reflected back along the thin rope and the rest continuing on along the thick rope.
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Figure 8.2 A incident pulse on a thin rope being partially reflected from the point where the thin rope meets a thick rope. The knot has essentially no effect.
The same thing happens for light at a glass or water surface. Again, there is an abrupt change in the wave speed due to the change in refractive index and, again, part of the energy is transmitted and part is reflected1 .
8.1.2
Requirements for Mirrors
Two conditions have to be met for a surface to act like a mirror. First, the surface needs to be big enough, in this case meaning that it has to be larger than the wave’s wavelength. For example, ocean waves reflect well off large concrete walls, but not off posts in the water that have much smaller diameters than the wave’s wavelength (Figure 8.3).
Figure 8.3 Left: Waves reflecting off a concrete seawall. Right: Waves not reflecting off a sign post, because its diameter is smaller than the wavelength.
This condition is also illustrated by the interactions of electromagnetic waves with raindrops. Raindrops are larger than visible light waves, so we can easily see both individual drops and approaching rainstorms. However, they are somewhat smaller than the microwaves that are used by weather radar, which have wavelengths of a few centimeters, so those waves interact weakly with raindrops. Their interaction is strong enough that the radar can still “see” rain, but weak enough that it can also penetrate through rainstorms and detect what’s on the far side of them. Finally,
1 We saw partial reflection at interfaces before in the context of thin film interference (Section 3.3.1)
and will revisit it again in Sections 9.2.3 and 11.4.5.
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raindrops are much smaller than radio or television waves, making it possible for us to listen to the radio or watch television even if it’s raining outside. Recall from the discussion on diffraction and interference (Section 3.4) that very little wave energy propagates through holes in a barrier that are much smaller than a wavelength. The requirement of a sufficiently large surface for reflection is essentially the same thing, but turned around. Now, the observation is that very little wave energy is reflected off objects that are much smaller than a wavelength. The second requirement for a surface to act like a mirror is that the surface needs to reflect adjacent rays of light toward the same direction. This means that the surface needs to be flat, implying that any bumps and grooves need to be much smaller than a wavelength to prevent light from being scattered to many different directions. Surfaces that are flat enough, such as bathroom mirrors and glass windows, are shiny and produce what’s called a specular reflection. In contrast, rough surfaces and other surfaces that scatter light, such as white walls, white paper, ground glass, and brushed metal surfaces, do not act like mirrors but instead produce what is called a diffuse reflection. Aluminum foil provides a nice example of this effect. Both sides of the foil are just plain aluminum, but the shiny side is shiny because its bumps are smaller than light wavelengths, whereas the matte side is dull become its bumps are larger than light wavelengths. Some surfaces exhibit both diffuse and specular reflections. A clean china dinner plate, for example, has a smooth surface that produces specular reflection but also has an underlying porcelain structure that scatters light and and produces a diffuse reflection.
8.2
Plane Mirrors
8.2.1
Law of Reflection
Consider a standard mirror, which we’ll assume is large enough to reflect light waves and smooth enough for specular reflection. The law of reflection states that any ray of light that hits a mirror always reflects with the same angle that it arrived with, as shown in Figure 8.4. It helps to introduce some terms. The surface normal is the direction that is perpendicular to the mirror surface at the location where the ray hit it. The mirror is horizontal in the figure so its surface normal points straight up. The incident ray is the incoming ray of light and the reflected ray is the ray of light after reflection. The incident angle, θi , and reflected angle, θr , are the angles between these rays and the surface normal. These angles are small if the light rays are nearly perpendicular to the surface and can increase up to nearly 90◦ as the rays are made more parallel topg
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Figure 8.4 A ray being reflected from a surface.
the surface, approaching what is called a grazing reflection. Using this terminology, the law of reflection is θr = θi .
(8.1)
This relationship was first stated formally about 2300 years ago by Euclid2 (despite his incorrect assumption that light emanated from people’s eyes) but was undoubtedly known long before that. It is the basis of all of the reflection behaviors that are observed from flat mirrors, multiple mirrors, curved mirrors, and other types of mirrors. They are the focus of the rest of this chapter.
8.2.2
Corner-Cube Retroreflectors
To demonstrate how one uses the law of reflection, suppose we add a second mirror that is perpendicular to the first one, as in Figure 8.5, and a light ray enters with incident angle of 60◦ to the first mirror. Where does it go? Figure 8.5 Ray diagram for a corner cube retroreflector. 60° 60°
30° 30°
Using the law of reflection, its reflected angle must be 60◦ . Then, the fact that the two mirrors are perpendicular to each other implies that the reflected ray must hit the second mirror with an incident angle of 30◦ . Using the law of reflection again gives the next angle of reflection as 30◦ . Finally, adding up all of these direction changes
2 See
R.A. Houstoun “The law of refraction” Science Progress in the Twentieth Century (19191933), 16(63):397-407 (1922).
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shows that the light ray gets turned around exactly 180◦ between when it enters and when it leaves the two mirrors, meaning that the answer to our question is that it gets redirected right back toward where it came from. Remarkably, generalizing the incident angle from 60◦ to arbitrary angle θ shows that the light ray always returns toward where it came, regardless of the initial angle. Perpendicular mirrors like this are called a retroreflector. In practice, they typically use three mutually perpendicular mirrors rather than two, in order to redirect light on all three spatial dimensions, and are then called corner-cube retroreflectors. Corner-cube retroreflectors are used widely (Figure 8.6). They are placed on ocean buoys and sailboats to increase their visibility on large ships’ radar systems, where they work by reflecting radar waves, in the microwave frequency range, back toward the large ships. They are used in bicycle reflectors to reflect car headlight beams back toward the car driver. Surveyors measure distances and directions by reflecting laser beams from retroreflectors. Finally, American Apollo and Soviet Lunokhod astronauts placed several corner cube retroreflectors on the Moon so that researchers can measure the distance to the Moon by timing how long laser beams take to get there and back3 . In all of these cases, the retroreflectors are used to redirect incoming rays back toward where they came from.
Figure 8.6 Corner-cube reflector examples. (A) The top of the red portion of the buoy is a corner cube radar reflector, (B) Close-up of a bicycle reflector, (C) A surveyor with a retroreflector, (D) Corner cube retroreflector left on the Moon by the Apollo 14 mission.
8.2.3
Images for Plane Mirrors
Returning to a single mirror, Figure 8.7 shows a candle in front of a standard flat mirror, more technically called a plane mirror. This candle is called the object. When looking into the mirror, it appears that there is a second candle located behind the mirror but, of course, it’s actually just a reflection. It is called the image. Note
3 The results show that the Moon is receding from the Earth by about 3.8 cm per year. It is receding because the Moon creates tides on the Earth that pull the Earth backward, thus slowing the Earth’s rotation. Correspondingly, these tides pull the Moon forward, which speeds it up. As the Moon goes faster, it orbits slightly farther from the Earth.
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187
that the image appears to be as far behind the mirror as the object is in front of the mirror. From common experience, we know that this image location doesn’t depend on the observer’s position, but is always as far behind the mirror as the object is in front of it. Figure 8.7 Reflection of a candle in a plane mirror.
To better understand plane mirrors, it helps to ask some questions about the image. (1) Where is the image relative to the mirror? (2) How large is the image compared to the object? And (3) is the light actually coming from the image or not? These questions are fairly straightforward with plane mirrors but will become more complicated when we get to curved mirrors. There are three ways to answer them. One is to just know what the answers are, the second is by drawing a ray diagram, and the third is with math equations. Different methods are easier in different situations, so it helps to have some familiarity with each. For the candle in front of the plane mirror, we already know the answers to the questions from common experience. First, we know that the image location is as far behind the mirror as the object is in front of the mirror. Next, we know that the image candle is the same size as the actual candle, meaning that the image is neither magnified nor reduced. That is, the image is farther away from us, but it appears to be a second candle that’s the same size as the original one. Finally, we know that the light that we see from the image doesn’t actually come from the image’s location because that location is behind the mirror. The light only appears to come from the image but doesn’t really, so it is called a virtual image. Now that we have the answers, let’s find them again using a ray diagram, shown on the left side of Figure 8.8. As a little advice, ray diagrams may look simple, and are simple, but are surprisingly hard to draw without practice. Focusing on just the candle flame, it emits rays of light in all directions, each of which bounces off the mirror according to the law of reflection. If you place your eye in one of those reflected rays, say the red ray in the figure, you see that the flame appears to be back along the direction of where that ray came from; you don’t see the mirror, because it’s shiny, and you don’t see that the light ray bends, but instead you see that the candle flame appears to be somewhere behind the mirror, shown with the dashed line. To determine how far behind the mirror, you need to either open your second eye or move the first one (blue ray in the figure). Seeing the same image with a different
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line of sight makes it appear at a different angle, which is called parallax and lets you determine the image’s distance. By looking backward along both rays, you see that they appear to come from a single point that is directly behind the mirror, and is as far behind the mirror as the candle is in front of the mirror. All other rays from the flame that reflect off the mirror appear to come from this same point too, which is the location of the flame’s image.
Figure 8.8 Ray diagrams for reflection from a plane mirror.
To make this ray diagram, draw any two rays outward from the candle flame, each of which reflects off the mirror according to the law of reflection. The rays diverge after reflecting, so they need to be extended back behind the mirror to find the point where they cross. This point is where they appear to come from and is the location of the image. The rays emitted from the candle didn’t cross here but had to be extended backward instead, so it is a virtual image. Repeating this process for the base of the candle (right side of Figure 8.8) shows that the image of the base is also directly behind the mirror and is as far behind the mirror as the base is in front. Filling in the rest of the candle’s image then shows that the candle’s image is upright, meaning that it’s not upside-down, and is the same size as the object. These last two results can be combined by saying that the image’s magnification is 1, where the fact that this number is positive implies that the image is upright and the magnitude of the number shows the scaling of the image relative to the original object. By drawing yet more rays, this time from the front or back of the candle, you can confirm that the spiral stripes on the candle are also drawn correctly. Finally, the third method for locating the image is with equations. The distance from the mirror to the object is called the object distance and labeled do , and the distance from the mirror to the image is called the image distance and labeled di . Each distance is defined as positive if it is on the shiny side of the mirror and negative if it is on the back of the mirror, here meaning that do is positive and di is negative. With this, the image location is di = −do .
(8.2)
The image magnification, M, is the relative height of the image when compared to the object. Denoting these heights as h i for the image and h o for the object, the magnification is
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M=
hi . ho
(8.3)
The image and object are the same size in this case, so M = 1.
8.2.4
(8.4)
Size of a Mirror
Ray diagrams can also be used to determine how much of the mirror is necessary to see an image. Figure 8.9 shows a portion of the prior ray diagram. This time, it shows that the entire image can be seen from a particular eye location using just those rays of light that are between the red and blue arrows. Figure 8.9 Another ray diagram for reflection from a flat mirror, now for one eye. The orange part of the mirror shows the part of the mirror where the image appears.
These light rays bounce off the region of the mirror that is marked in orange; all other portions of the mirror could be covered in masking tape without blocking the view of the image. In a sense, it is as though the eye were looking at an actual candle through a window, where the marked region of the mirror is the portion of the window that they eye is looking through. The region of the mirror, or window, is clearly smaller than the size of the image. By imagining the eye moving around, the size and placement of this region can be seen to depend on the position of the eye.
8.3
Concave Reflectors
8.3.1
Parabolic Reflectors
Suppose you have a beam of light with parallel rays, such as sunlight, and you want to reflect the entire beam to a single point. As an initial attempt, it would make sense to use several plane mirrors and to orient each one, using the law of reflection, to reflect the light toward the desired point, as shown in the left side of Figure 8.10. This would be pretty good, but the light at the desired point, which is called the focus, would be blurry. You could then improve on the result by using smaller and
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smaller mirrors. Eventually, it would turn out that the mirror surface would approach a smooth parabolic curve (i.e. the graph of the equation y = ax 2 ), as shown in the right side of Figure 8.104 . Figure 8.10 Parallel rays entering a parabolic mirror and converging at the focus. Left: flat mirrors approximating a parabola. Right: a parabolic mirror.
focus
focus
optical axis
Parabolic mirrors are special because they convert light between a beam of parallel rays and a point. If rays enter the mirror both parallel to each other and parallel to the mirror’s optical axis, which is the line of symmetry down the middle of the parabola, then they get reflected toward the mirror’s focus. Vice versa, parabolic mirrors can redirect light from the focus to a beam of parallel rays. Parabolic mirrors that focus from beams to points include satellite dishes, which focus signals from satellites into a microwave receiver, parabolic microphones, which focus wildlife or sporting sounds toward a microphone, and solar cookers, which focus sunlight toward a pot to be heated (left and middle images in Figure 8.11). A flashlight works in the other direction; it has a bright light bulb at the focus and uses a parabolic mirror to redirect the light into a beam of parallel rays (right image in Figure 8.11).
Figure 8.11 Some parabolic reflectors. (1) a satellite dish, operating at microwave frequencies, (2) a parabolic microphone, which reflects sound waves, (3) a solar cooker, reflecting mostly visible and infrared sunlight, and (4) a flashlight for creating a beam of visible light.
4 See
problem 8.17 for the proof that a parabola is the correct shape for this behavior.
8.3 Concave Reflectors
8.3.2
191
Concave Spherical Mirrors
Concave spherical mirrors are similar to parabolic mirrors, but are shaped like portions of spheres rather than portions of parabolas. They are also cheaper to make and easier to analyze. As with parabolic mirrors, beams of parallel rays that shine into the mirror along the optical axis get reflected toward a point called the mirror’s focus, which is typically labeled “F” in diagrams (see Figure 8.12). The light doesn’t focus perfectly, as it did for a parabolic mirror, but still focuses reasonably well, provided that the mirror is only a small portion of a sphere. Figure 8.12 Parallel rays reflect at a spherical mirror and converge at the focus, which is halfway between the mirror surface and mirror center.
f
F
C
optical axis
R
A spherical mirror’s focus turns out to be exactly halfway between the mirror surface and the center of the mirror’s curvature, which is often labeled C. This means that the mirror’s focal length, which is the distance between the mirror surface and the focus, is exactly half of the mirror’s radius of curvature (the radius of the sphere). As an equation, R , (8.5) 2 where f is the focal length and R is the radius of curvature. This relationship implies that tightly curved mirrors have short focal lengths and flatter mirrors have longer focal lengths. Considering an object in front of a concave spherical mirror, we can repeat the questions that we asked for plane mirrors. (1) Where is the image relative to the mirror? (2) How large is the image compared to the object? And (3) is the light actually coming from the image or not? As before, three ways to answer these questions are to just know the answers, to determine the answers from ray diagrams, and to compute them with equations. Unfortunately, everyday experience is less helpful this time. Starting with the method of just knowing the answers, they are listed in Table 8.1. f =
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Table 8.1 Relationship between object and image for concave mirrors Object position
Image
Image position
Orientation and size
Configuration name
from 0 to F at F from F to C at C from C to ∞ at ∞
virtual none real real real real
from 0 to −∞ parallel beam from ∞ to C at C from C to F at F
upright, magnified
magnifying mirror point-to-beam
8.3.3
inverted, magnified inverted, same size inverted, reduced inverted, zero
inverted image beam-to-point
Concave Mirror Ray Diagrams
Next, let’s make sense of them using ray diagrams. The left panel of Figure 8.13 shows the reflection of a candle in a concave mirror, where the candle is outside of the mirror’s center of curvature. Its image is clearly inverted. It’s hard to tell with only the single viewpoint of the photograph, but the image is actually floating in space in front of the mirror. The middle panel of the figure illustrates this, showing that rays from the candle flame reflect off the mirror while obeying the law of reflection, as always, and then converge in front of the mirror at a point that is between the focus and the mirror’s center. If you were to put a piece of paper here, all of the converging rays would make a bright spot on the paper. Because the light is really here, as opposed to just appearing to come from this location, it is called a real image. A similar ray diagram for the base of the candle shows that its location is also in front of the mirror; it’s at the same distance from the mirror as the flame’s image, but on the optical axis. Filling in the rest of the candle’s image shows that the complete image is both upside-down and smaller than the original object, so it is described as being inverted and reduced.
F
C
optical axis
Figure 8.13 The image of a candle with a concave mirror shown with many rays (left) and just two principal rays (right).
Without a piece of paper at the image location, as is the case in the photograph, all of the rays still go to the image location but then continue on through it and diverge again. If you look back along these rays, the fact that they all come from the same point makes it appear that the image is an actual candle that is floating in space at the image’s location.
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The ray diagram in the middle panel is difficult to draw because each reflection angle has to be measured carefully. The solution is to use only two rays, which is sufficient for locating the image, and to pick rays that are especially easy to draw, as shown in the right side of the figure. There are four easy-to-draw rays for curved mirrors, called principal rays (Figure 8.14). Before drawing them, first draw the mirror and the optical axis, making sure that the mirror center is actually at the center of the mirror circle and the focus is halfway between the mirror and the center. Don’t worry if the ray goes beyond the edge of the mirror as it is drawn; these rays are only used to locate the image, so just extend the mirror as needed and then reflect the ray. Figure 8.14 shows these rays in order from easiest to hardest to draw.
Figure 8.14 The four principal rays for a concave spherical mirror.
In the same order, the rays are: (Red) The center of the mirror surface is vertical, so a ray that bounces off this point reflects with the same angle as its incident angle. This is just the law of reflection, but is easier with a vertical surface. (Green) A ray that goes through the center of the sphere, regardless of its direction, hits the mirror perpendicular to the mirror surface. Thus, it is reflected back through the center. (Blue) An incident ray that is parallel to the optical axis is reflected through the focus. This arises from the definition of the focus. (Purple) A ray that goes through the mirror’s focus will be reflected parallel to the mirror’s axis. Again, this arises from the definition of the focus. Figure 8.15 uses these principal rays in ray diagrams for objects at different distances from a concave mirror. It starts with an object that is closer to the mirror than the focus, which I call the “magnifying mirror” configuration; it may be familiar as a shaving or makeup mirror, both of which provide a magnified reflection of one’s face. In this configuration, the rays from the object spread out after they hit the mirror.
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Looking back along the rays shows that they all appear to come from a virtual image that is behind the mirror, upright, and enlarged. This configuration is similar to a plane mirror in that it has an upright virtual image behind the mirror, but the image is now enlarged.
Figure 8.15 Concave mirror ray diagrams.
Next, if the object is at the focus, then all of the emitted rays end up being parallel to each other. We this saw before when considering parabolic mirrors. In this “pointto-beam configuration”, the rays never cross, on either side of the mirror, so there’s no image but a parallel beam instead. Moving the object outside of the focus returns to the “inverted image” configuration that we considered first, which has a real inverted image. The image is magnified if the object is inside of the center, the same size as the object if the object is at the center, and reduced if the object is outside of the center. Note the symmetry between the object and the image: if you put the object at the image location, then the image goes to the place where the object was. Finally, not shown in Figure 8.15, if the object is moved off toward infinite distance away, then the rays all converge at the mirror’s focus. Again, this agrees with what we saw for parabolic mirrors.
8.3.4
Mirror Equations
The third method for answering questions about images is with equations. Themirror equation relates the mirror focal length, f , the object distance, do , and the image distance, di with 1 1 1 + . = f do di
(8.6)
As for the plane mirror (Eq. 8.2), the do and di distances are positive for locations that are on the shiny side of the mirror and negative for locations that are behind the mirror. For the most part, the object distance, do , is always positive because it doesn’t make sense to consider reflections of physical objects that are behind a mirror (however, object locations can be behind mirrors if one is considering reflections from multiple mirrors, which is discussed later). The image distance, di , is negative for the “magnifying mirror configuration”, in which the image is behind the mirror, and
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positive for the “inverted image” configuration, where the image is in front of the mirror. Finally, the mirror focal length also follows the rule that it’s measured from the shiny side of the mirror, meaning that f is always positive for concave mirrors. The image magnification is M=
di hi =− . ho do
(8.7)
As before (Eq. 8.4), M is the magnification, h i is the height of the image, and h o is the height of the object. Example. A concave mirror has a 30.0 cm focal length and the object is 70.0 cm in front of it. Where is the image and what is its magnification? Answer. Solve the mirror equation, Eq. 8.6, for
1 di
:
1 1 1 = − . di f do The mirror is concave, so f is positive, and the object is in front of the mirror, so do is positive. With these, plug in numbers to get d1i = 30.01 cm − 70.01 cm = 0.019 cm−1 . Then, take the reciprocal to get di =
1 = 52.5 cm. 0.019 cm−1
The magnification equation, Eq. 8.7, gives M =−
di 52.5 cm = = −0.75. do 70.0 cm
Thus, the image is 52.5 cm in front of the mirror and its magnification is -0.75, meaning that it’s reduced in size and inverted. These results agree with the fifth row of Table 8.1 and with the ray diagram for this situation, which is shown in Figure 8.13.
8.3.5
Spherical Aberration and Coma*
All modern large telescopes focus light using concave mirrors. Mirrors are better than lenses because they are easier to support, have one surface that needs polishing rather than two, and have the same focal length for all colors of light.
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Inexpensive telescopes that use mirrors typically use spherical mirrors because they are cheap to produce. However, as mentioned above, spherical mirrors don’t quite redirect all light from a beam of parallel rays to the same point, with the result that their images are slightly fuzzy, an artifact called spherical aberration. Better telescopes address this by using parabolic mirrors. Their images don’t exhibit spherical aberration but are perfectly focused, provided that the object is exactly in line with the optical axis. Objects that aren’t exactly along the optical axis again have fuzzy images, now arising from the rotational asymmetry of the parabolic shape, which is called coma. No single mirror can solve both problems at once. Spherical mirrors are symmetric so they don’t have coma but they do have spherical aberration, whereas parabolic mirrors don’t have spherical aberration but do have coma, and any other shape necessarily has some combination of the two aberrations. The American astronomer George Ritchey and the French astronomer Henri Chrétien came up with a clever to solution to this problem in the early 1910s by using not one but two curved mirrors, thus giving them more parameters that they could adjust. Their design, now called the Ritchey-Chrétien design, reflects the light off a large primary mirror and then a small secondary mirror, both of which have hyperbolic shapes (Figure 8.16). The camera film or digital sensor is located at focus of the second mirror, where the light is almost completely free of both spherical aberration and coma. Most modern large telescopes use this design, including the Hubble Space Telescope, the Spitzer Space Telescope, and Hawaii’s Keck Observatory.
Figure 8.16 Diagram of a Ritchey-Chrétien telescope.
The James Webb Space Telescope is even more advanced. It takes this concept of two curved mirrors a step farther by having a three-mirror anastigmat, which has three curved mirrors. This enables a wide field of view with even less aberration.
8.4 Convex Spherical Mirrors
8.4
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Convex Spherical Mirrors
The final mirror shape that we will consider is convex spherical mirrors, used primarily for providing a wide-angle view. For example, convex mirrors are used for car rear view mirrors, at blind corners in hallways, at blind curves on streets, and in stores to deter shoplifting (left side of Figure 8.17). Convex mirrors always produce a virtual image that is located behind the mirror, and that is upright and reduced. This is illustrated in the ray diagram shown in the right side of Figure 8.17. Here, the object is now to the left of the mirror, but still on the shiny side, and the image is behind the mirror on the right.
F
C
Figure 8.17 Convex mirrors. (Left) A convex mirror on a road. (Right) Ray diagram of a convex mirror.
Remarkably, the same principal rays that we used for concave mirrors also apply to convex mirrors. They are: (Red) reflection off the mirror at the optical axis, (Green) reflection of a line that goes through the center of the sphere, and straight back out again along the same path, (Blue) a ray that starts parallel to the optical axis and then reflects with the reflected ray coming from the focus, and (Purple) a ray that goes toward the focus and then reflects to become parallel to the optical axis (not shown). These rays spread out after reflecting off the mirror, so they don’t form a real image. Instead, they need to be extended backward to find the location of the virtual image behind the mirror. This location always turns out to be between the back side of the mirror surface and the focus, and is always closer to the mirror than the object is. Interestingly, the symmetry that we saw before between object and image apply here as well; if the object and image are reversed, then this becomes the “magnifying mirror” configuration of a concave mirror. The mirror and magnification equations given before, Eqs. 8.6 and 8.7, apply here as well, with the sole change that the focal length, f , is now negative. This is because the the focus is now on the back side of the mirror instead of the front side.
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Example. A convex mirror has a 20.0 cm radius of curvature and the object is 5.0 cm in front of it. Where is the image and what is its magnification? Answer. The focal length is half the radius of curvature, so it is 10.0 cm; however, the mirror is convex, so f = −10.0 cm. Solve the mirror equation, Eq. 8.6, for d1i : 1 1 1 = − . di f do The object is toward the shiny side of the mirror so do is positive. Plugging in numbers gives d1i = 10.01 cm − 5.01cm = −0.30 cm−1 . Then, take the reciprocal to get 1 = −3.3 cm. −0.30 cm−1 The magnification equation, Eq. 8.7, gives di =
M =−
di −3.3 cm =− = 0.67. do 5.0 cm
Thus, the image is 3.3 cm behind the mirror and its magnification is 0.67, meaning that it’s upright but reduced in size. These results agree with the ray diagram for this situation, in Figure 8.17.
8.5
Multiple Mirrors*
You might reasonably expect that multiple mirrors would be substantially more complicated than single mirrors. However, it turns out that they can be analyzed relatively easy. In short, the rule is that the image from one mirror is the object for the next mirror. Also, the total magnification is the product of the magnifications from the individual mirrors. As an example, Figure 8.18 shows a sailor in a submarine who is looking at a seagull through a periscope, which is composed of two plane mirrors. If the periscope is 2 m long and the gull is 1 m in front of the upper mirror, where does the sailor see the gull? Starting with the upper mirror, the gull is the object, and it’s a plane mirror, so the image is located equally far behind the mirror as the object is in front. This produces the image shown above the periscope. It’s a little confusing here because the mirror is at 45◦ rather than horizontal or vertical, but you can extend the drawing of the mirror as far as needed to see that the image is in fact directly behind the line of the mirror. More precisely, Eq. 8.2 states that di = −do for a plane mirror, so the image is 1 m above the upper periscope mirror. This image is now the object for the second mirror, which is the lower mirror on the periscope. It is another plane mirror, so the new image is as far behind that mirror as the object is in front. Quantitatively, the object is 3 m above the mirror (2 m of periscope plus 1 m distance above the top),
8.6 Mirrors, Inversion, and Symmetry*
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so the image is 3 m behind the mirror. The magnification is 1 for both mirrors, and 1 × 1 = 1, so the final magnification is also 1. Thus, the sailor sees the gull straight ahead, at a position that is 3 m behind the lower mirror, and the image is the same size as the actual gull. Figure 8.18 A sailor in a submarine looking through a periscope at a seagull.
first image
first second reflection reflection object
second image
The approach of taking the image from one mirror as the object for the next mirror is exactly the same for curved mirrors. Note that it is worth being careful to measure the object and image distances from the mirror that is currently being considered.
8.6
Mirrors, Inversion, and Symmetry*
Returning to the simple case of a single plane mirror, you might wonder whether the image in a mirror is the same as the object, or is somehow reversed. And if it is reversed, then how is it reversed? From common experience, it seems that mirrors reverse left and right. That is, if you look at yourself in a mirror, you see that the image of your head is still on top, and your feet are on the bottom, implying that it didn’t reverse up and down. However, the image of your left hand appears to be the image person’s right hand and vice versa, suggesting that the mirror reversed left and right. Similarly, if you hold a book up to the mirror, the text is backward from left to right, but not flipped vertically. This doesn’t make sense though, because a mirror is just some shiny metal on a piece of glass, so there is no way that it could possibly know which axis it is supposed to reverse. Also, if you tip the mirror onto its side, that doesn’t change the image. The only reasonable conclusion, which is correct, is that mirrors don’t actually reverse left and right. Instead, they preserve left and right, just like they preserve top and bottom. The mirror appears to reverse left and right because when you turn around to face someone who is behind you, you actually reverse your left and right sides. If you were to stand in the place of your image person, you would walk around behind the mirror, and then rotate around your vertical axis to face the back of the mirror; this rotation creates the reversal of left and right.
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To show this point but with a different example, consider a plastic foosball player, as shown in Figure 8.19. When turning around, foosball players don’t rotate about a vertical axis like we do, but instead they rotate about a horizontal axis that goes along their arms. As a result, when a foosball player wants to talk face-to-face with a teammate, the teammate would need to flip over. Now, if this foosball player were Figure 8.19 Foosball players.
to look in the mirror, and was able to think, he would think that the image he was seeing would appear just like he was looking at a teammate, except that the person in the image would be the other way up. Thus, the foosball player would say that mirrors preserve left and right, but reverse up and down. As before though, the truth is that mirrors don’t reverse up and down either; instead, foosball players reverse up and down when they turn around to face the person behind themselves. Nevertheless, mirrors do reverse things. This is clear if you look at one of your hands in a mirror. Without worrying about which side of the body the hand appears to be on, the fact is that the mirror image of a right hand looks like a left hand, and vice versa. Similarly, if you look at a screw in a mirror, the threads on the screw’s image are reversed (see the stripes on the candle in Figure 8.7). So, what do mirrors reverse? The answer is that mirrors reverse front and back. This can be seen in Figure 8.20 in which the number 5 is shown with its mirror image. The “front” of the object is the side closest to the mirror and it ends up closest to the mirror in the image as well. In the process, the image reversed the front and the back of the object. Figure 8.20 The number 5 and its mirror image.
5
5
8.7 Fermat’s Principle of Least Time*
201
The result that mirrors reverse the fronts and backs of objects does not just apply to flat mirrors or virtual images, but applies to all of the mirrors and images discussed here. Each image has the front and back reversed from those of the object for all of these mirrors, whether the mirror is flat, concave, or convex. This is still true when the image is inverted, as happens with a concave mirror in the “inverted image” configuration. In that case, the mirror reverses left and right, up and down, and front and back. Regardless of which axis is reversed, an important aspect of mirror reversal is that objects do look different. As before, your right hand looks like a left hand when seen in a mirror. In other words, the object and its image are not exactly the same shape, no matter how you try to rotate them. Objects of this type are called chiral, from the Greek word for hand. Chirality is an important property of many molecules, including biological ones in particular. For example, glucose is a simple sugar that is readily available in the United States as corn syrup; it is a right-handed molecule. The mirror image of this, left-handed glucose, has all of the same physical properties as normal glucose but does not arise in nature and cannot be digested. As another example, amino acids are the building blocks of proteins. All naturally occurring amino acids (except for glycine, which is not chiral) are left-handed (Figure 8.21). Presumably, biology could work just as well in a hypothetical mirror-image world, where all amino acids and sugars had the opposite chirality, but that’s not how life happened to evolve on Earth.
Figure 8.21 Natural and unnatural mirror images of the cysteine amino acid (L and D refer to the prefixes levro and dextro, from the Latin for left and right). The D-cysteine cannot be rotated to match the L-cysteine structure because that would put blue nitrogen atom in the front instead of the back.
Each time an object is reflected in a mirror, the image is reversed. Thus, one reflection produces a reversed image, two reflections produce a normal image, three mirrors produce a reversed image again, and so on.
8.7
Fermat’s Principle of Least Time*
The Law of Reflection (θr = θi ) is an elegantly simple and complete description of how rays reflect off mirror-like surfaces. However, it doesn’t address the question of
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why rays reflect in this way. That answer turns out to come from a different and much less intuitive observation, which is that light always follows the path of least time in going from its origin to its destination5 . This is called Fermat’s principle of least time after the French mathematician Pierre de Fermat who stated the principle in 1657, although the concept actually dated back at least as far as the Greek philosopher Hero of Alexandria (c. 60 C.E.). As a simple example, consider a ray of light that travels through empty space from point A to point B without being either reflected or refracted. One could imagine that the light might take any of many possible long and winding routes (Figure 8.22). However, according to Fermat’s principle, light actually takes only the one particular path that gets it to its destination as quickly as possible, which, of course, is a straight line. This agrees with the common observation that light rays propagate in straight lines. Figure 8.22 Different possible paths for light in empty space, of which the straight line path is the one that a ray of light actually takes.
A
B
As another example, consider light that goes from point A to point B and that also reflects off a mirror along the way, as shown in Figure 8.23. Here, one could imagine that the light might reflect off any of the different possible different points on the mirror, labeled in the figure as locations 0 to 10. The right figure panel shows the time that light would take to get to its destination by these different paths. It shows that the light gets to its destination most quickly if it reflects at position number 5, which is exactly in the center. Here, the incident and reflected angles are equal to each other, showing that Fermat’s principle agrees with the law of reflection.
Figure 8.23 (Left) Different possible paths for light to reflect off a mirror, of which only the solid one obeys the law of reflection. (Right) Time taken for light to reflect off different parts of the mirror.
5 More
accurately, light always follows the path for which slight variations in this path don’t significantly affect the time that it takes to traverse it.
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Fermat’s principle appears to suggest that light has supernatural powers. Not only does it travel extremely fast and seem to know where it’s going, but it can look ahead to figure out the most efficient way to get there. This isn’t true, obviously, but instead Fermat’s principle points toward much deeper connections. Fermat’s principle arises from constructive and destructive interference in a manner that is similar to Huygens’s principle (Section 3.4). The idea is that light does not actually follow only one path, but instead it follows all possible paths at once. Some of those paths are straight, some are curved, and yet others are quite wiggly; also, it really does reflect off all parts of the mirror at once. When the light then arrives at some destination, say at a light sensor, all of the light waves from these different paths add together. In this addition process, the group of paths that were very close to the fastest one all took very nearly the same amount of time, so their waves are in phase with each other and interfere constructively; they are sensed as light. In contrast, the infinite number of paths that were far away from this fastest path took a wide range of different times, so their waves are out of phase with each other and add destructively. Thus, the result is that a ray of light appears to only take the single path of least time6 . If multiple different light paths all take exactly the same amount of time, Fermat’s principle states that light rays choose all of the paths at once. For example, consider a concave mirror that creates a real image (the “inverted image” configuration); it’s possible to show that the light’s travel distance from object to image is identical for every possible reflection point off the mirror. This implies that light takes all of these paths, in agreement with our prior considerations of light rays that spread out from the object, reflect off different parts of the mirror, and then converge at the location of the real image (see Figure 8.13).
8.8
Summary
Waves reflect at boundaries in their media. These include locations where the medium ends, such as a rope tied to a doorknob, and locations where the wave speed changes, such as a thin rope joined to a thick rope. For two and three dimensional waves, boundaries need to be larger than a wavelength to reflect the waves substantially. Reflections are specular if the surface is smooth enough that adjacent rays get reflected in the same direction, and diffuse if the surface is rough. The law of reflection is that a light ray’s reflected angle is equal to its incident angle, where both angles are measured from the surface normal. An interesting example is that two or three mirrors that are perpendicular to each other, called a corner-cube retroreflector, redirects light rays back toward where they came from. When an object is placed in front of a plane mirror, its image is as far behind the mirror as the object is in front of the mirror. The image is upright, not magnified,
6 Fermat’s principle also applies to matter waves, where it forms a central part of modern theoretical
physics. See Richard Feynman’s popular book “QED: The Strange Theory of Light and Matter”.
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and virtual, where virtual means that the light appears to come from that location but doesn’t really. The image distance is observed using parallax, in which the image is observed from multiple locations. In a ray diagram of a plane mirror, multiple rays from the object reflect off the mirror and are then traced backward behind the mirror to find the image location. Concave mirrors converge incident light rays. Parabolic reflectors redirect a beam of parallel rays to the mirror’s focus, or redirect light from the focus to a parallel beam. For spherical mirrors, the focus is halfway between the mirror surface and the center of curvature. The image location and magnification can be determined using the values in Table 8.1, with ray diagrams, or with equations. The ray diagrams can be drawn with any rays from the object, but four principal rays are typically easiest to draw. The image location is where the rays cross, which is in front of the mirror for real images and behind the mirror for virtual images. The mirror equation relates the focal length to the object and image distances with 1 1 1 + . = f do di Also, the image magnification is M=
di hi =− . ho do
Spherical mirrors don’t focus light perfectly, but exhibit spherical aberration; parabolic mirrors are better but exhibit coma for objects that off the optical axis; and systems with multiple curved mirrors can essentially remove both aberrations. Convex spherical mirrors diverge light, typically providing a wide-angle view. Their images are always upright, reduced, virtual, and behind the mirror. They can be located with ray diagrams or the same equations as for concave mirrors, but using negative focal lengths. Systems with multiple mirrors can be analyzed by starting with the object and considering one mirror at a time, where each image is the object for the next mirror. Mirrors do not reverse either left and right or up and down, but do reverse front and back. They only appear to reverse left and right for us because we reverse our left and right sides when we turn around to see behind ourselves. An object that is not the same as its mirror image, such as a person’s right hand, is called chiral. Nearly all biological molecules are chiral. Each reflection off a mirror, regardless of the mirror shape, produces an image with the opposite handedness. Fermat’s principle of least time states that light rays take the path that takes the least time in going from origin to destination. This principle is consistent with the facts that light goes in straight lines in free space and reflects according to the law of reflection. It arises from the fact that light actually takes all possible paths, but only the light waves that are very close to the fastest path are able to interfere constructively with each other.
8.9 Exercises
8.9
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Exercises
Questions 8.1. As a candle is moved away from a plane mirror on a wall, what happens to its image? (a) gets smaller (b) gets larger (c) gets either smaller or larger, depending on the observer’s location (d) the magnification is negative and increases toward zero (e) the size does not change 8.2. For a flashlight that produces a beam of light, where is the light bulb compared to the reflector mirror? (a) at the center of curvature (b) at the mirror’s focus (c) halfway between the focus and the center of curvature (d) halfway between the mirror and the focus (e) up against the mirror 8.3. The focal length of a concave mirror has a magnitude of 20 cm. What is its radius of curvature? (a) 10 cm (b) 40 cm (c) -40 cm (d) 20 cm (e) -20 cm 8.4. What types of images can mirrors produce? Select all that are appropriate. (a) Magnified images (b) Reduced images (c) Upright images (d) Inverted images 8.5. (a) Give one example of a surface that produces a diffuse reflection. (b) Give one example of a surface that produces a specular reflection. 8.6. For each of the following, can you use this to see your own reflection? Briefly explain why or why not. (a) A white wall, (b) A silver spoon, (c) A single silver atom, (d) The water’s surface, (e) A white dinner plate. 8.7. What shapes of mirrors (flat, convex, or concave) may be used to (a) converge light rays, (b) diverge light rays (more than for no mirror at all), (c) neither
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converge nor diverge light rays, (d) form a real image, (e) form a virtual image, (f) produce an enlarged image, (g) produce a reduced image? 8.8. Several concrete “acoustic mirrors” were built in southeast England in the 1920s and 1930s that faced the English Channel. Explain how they could have been used to detect and locate incoming enemy aircraft.
8.9. A new skyscraper in London, called “20 Fenchurch Street” after its address, has a south-facing concave curved exterior. Before it was retrofitted with permanent awnings, it sometimes focused sunlight sufficiently well to melt parts of cars that were parked in the wrong places. The building’s facade is vertical at the ground and has a radius of curvature of about 540 m. (a) What is the focal length of this mirror? (b) Where is the sun’s image, relative to the base of the building, when the sun is to the south and on the horizon? (c) Is this a real image or a virtual image? (d) When the sun is 60◦ above the horizon (still due south), is the hottest spot on the ground closer to the building or farther away than when the sun was on the horizon?
8.10. Modern car headlights are complex optical instruments, designed to put the right amount of light in the right places. However, for the most part, they aim the light from a small bright halogen lamp filament to a collimated beam. Assume a simple design for this. (a) What shape is the reflector behind the light bulb (e.g. spherical, corner cube, parabolic, flat, convex)? (b) What is the name of the location where the light bulb filament is placed? (c) There is a separate filament for “high beam” output, which creates a beam of light that shines at a higher angle than the normal low beam. Is the high-beam filament above or below the low-beam filament (hint: draw a picture)? (d) The reflector usually extends out ahead of the light bulb. How much of the light emitted by the filament is directed into the collimated beam (choose from: 0%, 50%, or 100%)?
8.9 Exercises
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Problems 8.11. A “full-length mirror” is a flat mirror that is large enough so that you can see your entire body in it, from head to toe. Consider a woman who is 170 cm tall (5’6”) and whose eyes are 10 cm below the top of her head. (a) How high above the floor can the bottom of the mirror be, so she can still see her feet in the mirror? (b) How high above the floor does the top of the mirror need to be so she can still see the top of her head in the mirror? (c) From these answers, what is the minimum total length of the mirror? (d) Does you answer depend on how far she stands from the mirror? 8.12. Consider a 2-dimensional corner cube retroreflector in which two mirrors are perpendicular to each other (see Figure 8.5). For a ray that enters the reflector with incident angle θ, show that it will leave the reflector going toward the direction that it came from. 8.13. A concave mirror has focal length 10 cm and an object is 30 cm away from the mirror. (a) Draw a ray diagram for this system, showing the object, image, and any two rays. (b) Using the mirror equation, how far from the mirror is the image? (c) Using the magnification equation, what is the image magnification? (d) Do your diagram and quantitative results agree with each other? 8.14. A concave mirror has focal length 10 cm and an object is 3 cm away from the mirror. (a) Draw a ray diagram for this system, showing the object, image, and any two rays. (b) Using the mirror equation, how far from the mirror is the image? (c) Using the magnification equation, what is the image magnification? (d) Do your diagram and quantitative results agree with each other? 8.15. Suppose a room light is 2 m directly over your head and you are sitting 1 m away from a wall. Treating the wall as a diffuse reflector, calculate the time that the light takes to get to your eye, in ns, for its reflection off the wall at points (a) A, (b) B, and (c) C in the diagram. (d) Is there any point on the wall where the light takes less time than it does when it reflects at point B (use your intuition, guided by your prior results)? (e) Now suppose the wall is a specular reflector. For the light that reflects off the wall and goes to your eye, which point does it reflect off?
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8.16. A simple periscope has two flat mirrors that are parallel to each other (see Figure 8.18). Consider looking through a periscope that is 1 m tall, at an object that is 5 m in front of the top periscope mirror. (a) When you look at that object through the periscope, where is the object’s image relative to the lower mirror, including the image distance? (b) is this a real or virtual image? (c) Is the image right-side up or upside-down? (d) If there is text on the image, does it appear to be forward or backward? (e) Is the image larger, smaller, or the same size as the object? 8.17. A parabola is often defined with the math equation y = x 2 . However, it can also be defined geometrically as the set of points that are equidistant between a straight line and a point. Using this geometric definition and Fermat’s principle of least time, show that a parabola focuses light from a beam of parallel rays to its focus. 8.18. An ellipse can be drawn by attaching each end of a piece of string to two pins in a sheet of paper, pulling the middle of the string sideways as far as it goes with a pencil, and then drawing an arc with the pencil while keeping the string taught. The two pin locations are called the ellipse foci. (a) Using Fermat’s theorem, show that waves that are emitted from one focus get reflected to the other focus. (b) The Mormon Tabernacle, a religious building in Salt Lake City, Utah, is famous for its acoustics. It is built as an ellipse with the speaker’s pulpit at one focus. Where is the best place to sit to hear the speaker clearly?
8.19. In a “magic mirror illusion”, two concave mirrors face each other like a clamshell, an object is placed on top of the lower mirror, and a person sees inside through a small hole in the upper mirror. Assume each mirror has a focal length of 10 cm, the middles of the mirror faces are 10 cm apart, and the object is 1 cm above the lower mirror. Where is its image and what is its magnification? (Hint: consider the upper mirror first.)
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Puzzles 8.20. From underwater, the surface of the water can act as a mirror. Consider a sea turtle which is looking at a jellyfish that’s near the surface of the water. (a) Will the image that the turtle sees be a real image or virtual image? (b) Will the image of the jellyfish be left-right reversed, up-down reversed, both, or neither?
8.21. Consider the shadow of an object that is formed by direct sunlight. (a) Why is the edge of that shadow somewhat fuzzy, and not totally sharp? (b) Suppose sunlight reflects off a car windshield, which acts like a convex mirror. Will the virtual image of the sun appear larger or smaller than the actual sun? (c) After reflecting off the windshield, the sunlight forms a shadow of the same object. Will this shadow have a sharper or fuzzier edge? Why? 8.22. Consider looking into a corner cube retroreflector with one eye open. Where is the image of your eye? 8.23. Consider an object in a kaleidoscope that is made with 2 mirrors at a 60◦ angle to each other (left side of figure, below). (a) How many images will you see that arise from one reflection? (b) How many images from 2 reflections? (c) How many images from 3 reflections? (number of distinct images, not number of ways of getting them). Now consider an object in a kaleidoscope made with 3 mirrors (right side of figure). (d) How many total images will you see from this 3-mirror kaleidoscope?
8.24. The following figure shows two reflections of the Golden Gate Bridge in a soap bubble. Explain which surface of the bubble created each reflection.
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8.25. When the sun is low in the sky, its reflection off water with ripples appears as a long stripe of “sun glitter”, regardless of the wave direction. Explain.
8.26. If it is snowing extremely gently when it is quite cold, the snow crystals are sometimes tiny flat hexagonal plates that fall flat, with the plates nearly parallel to the Earth’s surface. When this happens, people sometimes observe “light pillars” above bright lights, such as street lights. (a) Draw a diagram that explains this phenomenon, showing both a person and a street light. (b) How far from the person, relative to the distance to the street light, are the snow crystals that reflect the light to that person’s eyes?
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Figure 9.1 Double rainbow near Vancouver, Canada.
Opening question What causes rainbows? (a) Light reflecting off raindrop surfaces (b) Light rays refracting as they enter and leave raindrops (c) The refractive index of water is different for different colors (d) White light is composed of all different colors (e) All of the above.
© Springer Nature Switzerland AG 2023 S. S. Andrews, Light and Waves, https://doi.org/10.1007/978-3-031-24097-3_9
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A glass of water is completely transparent, but not invisible. We are able to see it because both the water and the glass bend light rays through the process of refraction, producing optical effects that we’ve learned to associate with transparent objects. On the other hand, a glass that’s submerged in a dishpan is nearly invisible. In that case, light does not refract significantly, so there are few visual cues to see it with. Refraction also creates mirages over hot roads, makes water look less deep than it really is, and produces atmospheric phenomena such as rainbows. Optical lenses, whether in cameras or our own eyes, use refraction to focus light and create images. Refraction occurs because waves have different wavelengths when they speed up or slow down, which then forces a change in the wave direction. Although this differs from the physical causes of reflection, the two topics have many things in common. Both can be investigated with geometric optics, both produce real and virtual images, and, in many cases, both have similar ray diagrams and similar equations. Both obey Fermat’s principle of least time. In addition, reflection and refraction often occur together.
9.1
Refraction in General
9.1.1
Refractive Index
Recall from Section 2.3.4 that waves travel at different speeds in different media. For example, light travels at about 3 · 108 m/s in vacuum and 2.3 · 108 m/s in water. The ratio of the light speed in vacuum, c, to its speed in another medium, v, is the medium’s refractive index, n: c (9.1) n= . v (We saw this before as Eq. 2.5.) Using this equation, water’s refractive index is about 1.33. As other examples, the refractive index of glass is about 1.52 and that of air is about 1.0003. In practice, we typically use a value of 1.00 for air, treating it the same as vacuum, although the more accurate value is important on occasion. Refractive index values tend to fairly uniform throughout the visible wavelength range, but do vary some, as we’ll explore toward the end of this chapter.
9.1.2
Why Waves Refract
Light rays refract, meaning that they bend, when they transition between faster and slower media. The left side of Figure 9.2 shows an example. Here, three parallel light rays came in from the left side of the picture, refracted toward the right when they entered a plastic block, and then refracted back toward the left when they returned back to the air. This photograph also shows that the light rays reflected each time they changed speeds, as we saw in the last chapter.
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Figure 9.2 (Left) Refraction of light beams as they enter and leave a plastic block, along with reflected rays. (Middle) Diagram of waves slowing down and speeding up, refracting at each transition. (Right) Roller skating from a hardwood floor onto a carpet has the same bending effect.
Refraction is a direct consequence of waves changing speeds. The middle of Figure 9.2 shows wavefronts arriving from the left and encountering a new medium where they go slower. Their wavelength decreases when they slow down, and the wavefronts don’t break, which means that the waves have to bend. As a result, the propagation direction, shown in orange, also bends. Refracting waves always bend toward the direction of the slower medium. This can be seen in the image with the plastic block, where the rays turn to the right into the block, and then turn to the left as they return to the air, in both cases bending toward the slower medium. As an analogy, think about roller skating from a hardwood floor onto a carpet, as shown in the right panel of the figure. When skating at an angle to the edge of the carpet, one skate hits the carpet first, so it slows down and pulls the skater around toward the carpet. Waves bend in essentially the same way, as though they get “pulled” into the slower medium and get turned in the process.
9.1.3
Refraction From Density Gradients
Refraction can also occur when waves change speed gradually, rather than all at once at a transition. You have likely seen a mirage, which occurs when the ground is hotter than the surrounding air. Mirages are particularly common on hot roads, where they appear to create puddles on the road that reflect the sky, cars, or other objects1 . The left side of Figure 9.3 shows a typical example, showing a car on a hot road, and the right side shows a diagram of how it works. The observer’s regular view of the car is the same as it always is, with the light going straight from the car to the person’s eyes. However, rays of light also go from the car toward the road. Rather than being absorbed by the road as normal, they instead move into an area of hot air just above the hot road, where the air is less dense so the light travels faster, causing the rays to bend upward. Those bent rays also go to the observer’s eyes. To the observer, it
1 This description explains inferior mirages, in which the light bends upward over a hot surface. There are also superior mirages, in which the light bends downward, typically over cold water or polar ice sheets. This can cause islands and ships to appear higher up than their actual position, sometimes even making them visible when they would normally be beyond the horizon. See problem 9.6.
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Figure 9.3 (Left) Example of a mirage over a hot road. (Right) Diagram of how mirages work.
looks like the rays got reflected off the road, as would happen if there were a puddle of water there. A similar phenomenon is that sound is well known to carry especially well over water. For example, people can be having a quiet conversation in a fishing boat, which they think is private, but it turns out that people on shore can hear them just fine. This arises from sound wave refraction, in which the some of the sound waves propagate up and away from the boat, but then get refracted back downward, with the result that more sound gets to the shore than is normal for that distance. The sound refracts downward because the air is colder over the water and sound travels more slowly in colder air. Again, the waves refract toward the direction where they travel slower.
9.2
Refraction at Interfaces
9.2.1
Snell’s Law
The medieval Persian scientist Ibn Sahl gave the first quantitative description of refraction around the year 984. However, this result was largely forgotten by the time of the European scientific revolution, leading to several rediscoveries, first in 1602 by the English astronomer Thomas Harriot, then in 1621 by the Dutch astronomer Willebrord Snellius, and finally in 1637 by the French philosopher René Descartes. As it turned out, the description became named for Willebrord Snellius. All of these scientists considered a ray of light refracting, such as the one shown in Figure 9.4. Here, an incident light ray approaches from the upper left and then encounters a slower medium, such as glass or water. Some of the ray reflects, as we investigated in the last chapter, but most of it refracts into the new medium, shown with the arrow that goes down and right. They investigated how the angle of refraction, shown as θ2 in the figure, relates to the angle of incidence, shown as θ1 . The result, now called Snell’s law, is n 1 sin θ1 = n 2 sin θ2 ,
(9.2)
where n 1 and n 2 are the refractive indices of the two sides. Note that the angles of incidence and refraction are measured relative to the surface normal, like one does for reflection. Also, both the figure and this equation agree with the observation that the light ray always bends toward the slower medium. As an aside, it doesn’t matter which side is called side 1 and which side 2 because the light bends the same amount regardless of which way it propagates.
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Figure 9.4 Refraction of a light ray going into a slower medium. The reflected ray is shown with a dashed line.
Expressing Snell’s law as shown above in Eq. 9.2 emphasizes its symmetry and is easy to remember, but is rarely useful. Typically, you’ll want to rearrange it to solve for one of the parameters in terms of the other three. For example, if you want to to find the angle of refraction, meaning the angle of the refracted ray, then you would rearrange it to n1 (9.3) sin θ1 . θ2 = arcsin n2 The arcsine function is the inverse of the sine function (see Appendix D). Example. Light shines from air into water with an incident angle of 60.0◦ . What is the angle of refraction? Answer. We’ll define side 1 as the air side and side 2 as the water side. Use Eq. 9.3 and plug in n 1 as the refractive index of air, which is 1.00, n 2 as the refractive index of water, which is 1.33, and θ1 as the incident angle, which is 60.0◦ . 1.00 n1 θ2 = arcsin sin θ1 = arcsin sin 60◦ = 40.6◦ . n2 1.33 Thus, the refracted ray is 40.6◦ away from the surface normal. This makes sense because the angle is smaller in the water side than the air side, in agreement with the statement that light always bends toward the side where it travels slower.
9.2.2
Apparent Depth
A consequence of refraction is that things that are underwater appear to be closer than they really are. This creates the familiar illusion of a drinking straw appearing to bend upward at the water’s surface and, if you’re trying to pick something up that fell in water, you tend to get wetter than expected. Figure 9.5 shows why this is the case. It shows a woman looking at a fish that’s underwater. The light from the fish follows the bent path that is shown with a solid line. However, the woman interprets the view as though the light travelled along a straight line, so it appears to her that the fish is higher up than it really is, at the location of the dashed picture. In physics
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terms, this is the fish’s virtual image; it’s virtual because there’s no light from the fish actually at this location. Figure 9.5 Diagram of apparent depth, showing that objects appear to be less deep than they really are.
We can determine the fish’s apparent depth, meaning how deep it appears to be, with trigonometry. From the diagram, the angles and depths relate to each other as tan θ1 = x/d1 and tan θ2 = x/d2 . Dividing the second equation by the first and simplifying gives the ratio of the apparent depth to the actual depth as tan θ2 d1 = . d2 tan θ1 If the angles are reasonably small, then it’s a good approximation to say that tan θ1 ≈ sin θ1 , and the same for tan θ2 2 . Finally, Snell’s law can be rearranged to n 1 /n 2 = sin θ2 / sin θ1 . Together, these lead to the result d1 tan θ2 sin θ2 n1 = ≈ = . d2 tan θ1 sin θ1 n2
(9.4)
Thus, the ratio of apparent depth to actual depth is the same as the ratio of the refractive indices. It helps to plug in some numbers. When looking into water from air, n 1 = 1 and n 2 = 1.33, which means that d1 /d2 = 1/1.33 = 0.75. This means that objects in water appear to be about 3/4 as far away as they really are.
9.2.3
Refraction and Reflection at Different Angles
To build some intuition for Snell’s law, we consider how refraction varies as the incident angle is changed. We start from normal incidence, meaning that the light shines directly at the surface, and move toward grazing incidence, where the light just grazes over the surface. The left two panels Figure 9.6 shows these effects for a light ray that transitions from a faster medium to a slower one, such as from air to water. The ray always
2 This
is valid because tan θ = sin θ/ cos θ and cos θ is close to 1 when θ is small.
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splits into reflected and refracted rays, with their angles being described by the law of reflection (Eq. 8.1) and Snell’s law (Eq. 9.2), respectively. The arrow thicknesses show that most of the light is transmitted at normal incidence, while larger incident angles typically produce more reflection.
Figure 9.6 (Left and Middle) Reflection and refraction for a light source in air. (Right) A view of a diver from beneath. The bright region of the water surface represents the entire view of the world above the water.
Focusing on the refracted rays, the light ray that shines straight into the water isn’t bent at all. The bending then increases as the angle of incidence increases, in all cases with the light being “pulled” toward the water due to its slower propagation there. This trend continues up until the incident light ray just grazes the water’s surface, shown in the middle panel. Here, nearly all of the light gets reflected, while the tiny bit that gets refracted shines down into the water with the largest possible angle of refraction. This largest possible angle of refraction is called the critical angle and is denoted θc . We can compute its value by using Snell’s law and plugging in the the angle of incidence as θ1 = 90◦ . This gives θc = arcsin
n1 sin 90◦ n2
= arcsin
n1 . n2
(9.5)
Substituting in the values n 1 = 1.00 for air and n 2 = 1.33 for water shows that the critical angle for light shining from air into water is 48.8◦ . This means that if a diver is underwater and looks up at the surface, as shown on the right side of the figure, all light that comes from above is compressed into a cone that has an angle of 48.8◦ away from the middle. This view is sometimes called Snell’s window. Next, consider light shining in the other direction, from a higher refractive index medium to a lower index one, such as from water to air. This is shown in Figure 9.7. Again, the ray that shines straight at the surface isn’t bent, while refraction increases for larger incident angles, this time with the light bending away from the new medium. Also, as before, transmission is high for normal incidence and decreases for larger angles. The same critical angle as before now represents the angle of incidence that causes the refracted ray to just graze the water’s surface. That is, it would just graze the surface if there were a refracted ray at all, but what really happens is that all of the light becomes reflected once the critical angle is reached, which is called total
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Figure 9.7 Reflection and refraction for a light source in water.
internal reflection. All of the light is also reflected if the angle of incidence is larger than the critical angle. This is rather remarkable because it means that one can shine a light from water toward air, at a large angle of incidence, and absolutely all of it gets reflected back into the water.
9.2.4
Total Internal Reflection Examples*
Total internal reflection might seem exotic, but it is actually quite common and technologically useful. Figure 9.8 shows several examples along with ray diagrams that show how each one works. When you look down into a glass of water, you cannot see out of the sides of the glass, where your hand is, but instead you just see reflections of the bottom of the glass. These arise from total internal reflection.
Figure 9.8 Examples of total internal reflection, showing a glass of water, optical fibers, prisms inside a pair of binoculars, and a diamond. The bottom row shows a ray diagram for each example .
Optical fibers, such as those used to supply internet service to many homes, use total internal reflection to keep the light inside the fiber. They are typically manufactured with a high refractive index glass core that is surrounded by a low refractive index glass “cladding,” so the light reflects back and forth off this boundary repeatedly as it travels along the fiber.
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Binoculars use a folded light path in which the light zig-zags back and forth. This is accomplished by a pair of prisms, which are essentially just triangles of glass, that reflect the light through total internal reflection. Diamonds sparkle because their high refractive index (n = 2.42) gives them a small critical angle. When they are cut properly, this means that most rays of light that enter from the top facets totally internally reflect multiple times off the bottom facets and then leave again back through the top. Road signs, reflective paint, and most reflective fabrics offer yet another example of internal reflection. These cat’s eye retroreflectors, shown in the left panel of Figure 9.9, reflect light back toward where it came from using spherical glass beads. The middle panel shows that when a light ray hits a bead, it refracts into the bead, reflects off the back face, and then refracts back out into the air, getting fully turned around in the process. While incident rays that start sufficiently close to the bead edge are totally internally reflected off the back face, most other incident rays don’t hit the back face at a shallow enough angle, so they would typically get partially transmitted at that point. This is addressed by coating the back of the bead with a shiny material. Actual cats’ eyes (right panel), and those of most other nocturnal vertebrates, show up brightly in flashlights and headlights for the same reason. In their case, a shiny membrane called the tapetum lucidum reflects the light from just behind the retina; it improves night vision by sending any light that wasn’t absorbed by the photoreceptors on the first pass back across the photoreceptors a second time. Red-eye artifacts in photographs of people arise from the same mechanism, in their case with the red color arising from blood in people’s retinas.
Figure 9.9 Cat’s eye retroreflectors. (Left) A reflective road sign. (Middle) Ray diagram showing how cat’s eye retroreflection works. (Right) Retroreflection of a camera flash from a cat’s eyes.
9.2.5
How Much Light Gets Reflected*
While it’s true that light that shines directly at a surface is mostly transmitted and that reflection typically increases with larger angles of incidence, there’s more to the story. Augustin-Jean Fresnel, an early 19th century French scientist, investigated the relative amounts of light that get either reflected or transmitted. His results, called
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the Fresnel equations, are complicated in general3 but interesting at a few specific angles. First of all, if light rays point straight at the surface, then the reflected fraction is n1 − n2 2 , (9.6) R= n1 + n2 where n 1 and n 2 are the refractive indices on the two sides. Plugging in numbers for air and water shows that about 2% of light is reflected, while the rest is transmitted. Glass has a larger refractive index, so about 4% of light is reflected there. This isn’t much, but it clearly adds up if there are multiple surfaces. For example, a doublepane window has 4 glass-air interfaces, leading to about 16% reflection (or, to be mathematically proper, 96% is transmitted at the first interface, and then 96% of that is transmitted through the second interface, and so on, so the total transmission is 0.964 = 0.85; the total reflection is 1 − 0.85 = 0.15, so, really, 15% is reflected). For other angles of incidence, Fresnel found that the results depend on the light’s polarization relative to the surface. Light that is polarized parallel to the plane of the surface follows the standard trend of having greater reflection with larger incident angles. However, light with the opposite polarization (parallel to the plane of incidence, meaning the plane that includes the incident ray and the surface normal) can actually be completely transmitted, with no reflection at all. This occurs if the angle of incidence is equal to Brewster’s angle, which is about 53◦ for an air and water interface and about 57◦ for an air and glass interface4 . Setting the angle of incidence to Brewster’s angle can be useful for eliminating reflective losses, or for separating unpolarized light into its two polarizations.
9.2.6
Evanescent Waves*
When total internal reflection occurs, no light propagates beyond the surface but the light does actually extend beyond the surface by a tiny extent. These light waves that are outside of the surface, called evanescent waves, extend roughly one light wavelength beyond the surface before decaying away. Their effect can be seen by pressing your fingers firmly against the outside of a full glass of water. This makes your fingerprints visible on the inside of the glass, which is an example of frustrated total internal reflection. Light would totally internally
2 θi −n 2 cos θt Fresnel equations give the fraction of reflected light as Rs = nn 11 cos cos θi +n 2 cos θt and R p = n 1 cos θt −n 2 cos θi 2 n 1 cos θt +n 2 cos θi , where θi and θt are the angles of the incident and transmitted rays. Also, the s subscript is for s-polarized light, in which the electric field is parallel to the surface, and p is for p-polarized light, in which the electric field is parallel to the plane of incidence. 4 Sir David Brewster (1781–1868) was a Scottish physicist best known for his work on optics. As an equation, Brewster’s angle is θ B = arctan(n 2 /n 1 ). It represents the situation where the refracted and reflected rays are perpendicular to each other. This topic also arises in Section 11.4.5. 3 The
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reflect normally, but your fingers are within the evanescent wave distance, so the light is able to escape the glass in this case and get absorbed by your fingers. A particularly interesting application of evanescent waves is in TIRF microscopy, where TIRF stands for Total Internal Reflection Fluorescence (Figure 9.10). Here, some sort of sample, such as a biological cell, is placed on a prism and then a laser beam is totally internally reflected through the prism. The laser’s evanescent waves illuminate the sample, but only to about one wavelength in depth. These evanescent waves excite molecules that fluoresce and the fluorescence is then observed through the microscope. This method enables researchers to just look at proteins or other molecules that are very close to a cell surface, without having the signal overwhelmed by molecules that are deeper in the cell.
Figure 9.10 Total internal refraction fluorescence (TIRF) microscopy. (Left) Diagram of experiment. (Right) A TIRF image, showing cell structure that is very close to the cell surface.
9.3
Lenses
9.3.1
Types of Lenses
Lenses are optical devices that redirect light using refraction. They come in a wide range of shapes, as shown in Figure 9.11, but can generally be classified as either converging lenses or diverging lenses. Converging lenses are thicker in the middle and thinner on the edges, and they converge a beam of parallel rays down to a focus. Diverging lenses are the opposite. They are thinner in the middle and thicker on the edges, and they diverge parallel rays into an expanding beam. All of these qualify as thin lenses, in which lenses are much thinner than the radii of curvature of their two sides. Thin lenses are much easier to analyze than thick lenses, so they will be assumed throughout this section. As an example, the detailed shapes of the two lens sides, such as whether a lens is biconvex, plano-convex, or positive meniscus, doesn’t affect the behavior for thin lenses. Most lenses are spherical, meaning that the shape of each side is a portion of a sphere. There are also cylindrical lenses that only curve on one axis. Aspheric
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Figure 9.11 Common lens shapes.
lenses, which are cut in different shapes, are less common but are produced for some specialized applications such as high end cameras and telescopes.
9.3.2
Lens Coordinates
Recall from the discussion of curved mirrors (Section 8.3.2) that the optical axis is the imaginary line that runs through the mirror’s axis of symmetry, that distances are measured from the mirror surface, and that mirrors have two important points along this line: the focus, F, and the center, C, which is twice as far away. Lenses, shown in Figure 9.12, are similar but a little different. They also have an optical axis running through the axis of symmetry but in their case, distances are measured relative to the middle of the lens rather than one of the sides. Also, lenses don’t have just one focus but two, with one on each side; they are labeled here5 as −F on the side that the light comes from, and F on the side that the light goes to. Also, the point that’s twice as far as the focus isn’t called the center point, because it’s not the center of anything, but is named the 2F point, again with one on each side.
Figure 9.12 Ray diagrams and coordinates for converging and diverging lenses.
The left side of the figure shows that a converging lens focuses light from a beam of parallel rays to the focus at F. The distance to the focus is the lens’s focal length. The diverging lens, on the right side of the figure, spreads the light from a beam of parallel rays to an expanding beam. Tracing these rays backward with straight lines shows that they appear to come from the focus at −F. The focal length of this lens is
books label both foci as F, or one as F and the other as F . However, this book uses signs because this tends to be clearer and is also helpful when using lens equations.
5 Most
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still the distance from the middle of the lens to the focus, but is now given a negative value to show that it’s at the −F focal point.
9.3.3
Images From Lenses
If you look at an object through a lens, what do you see? Where is the image, how big is it, and is it real or virtual? As we did for curved mirrors, we can figure out the answers to these questions in several ways. You might just know the answers, such as from experience with a magnifying glass, or you can figure them out with ray diagrams, or you can calculate them with math equations. We’ll start with ray diagrams this time. Figure 9.13 shows a ray diagram for a converging lens for the situation where the object is between the −F and −2F points. It shows that the rays start at the object, get refracted by each surface of the lens, and then go straight from there. In this case, those refracted rays converge to a point, meaning that the light actually goes to that point and that the image is real. If you were to put a piece of paper at the location of the real image, then you’d see a picture of the object projected onto the paper. As the diagram shows, the image would be magnified and inverted. Alternatively, if you were to look at the object through the lens, you would see the image instead, and it would appear to be floating in the air in front of the lens.
Figure 9.13 Ray diagram for a converging lens, showing many rays between object and image.
This ray diagram is completely valid, but it is hard to draw because you need to use Snell’s law to calculate the angles of refraction for each ray at each lens surface. Figure 9.14 shows an equivalent ray diagram that is easier to draw. To draw a ray diagram for a lens, start by sketching out the optical axis and lens. Then, add the ±F and ±2F points and draw the object. Next, you need to draw at least two rays. There are four principal rays that are easy to draw, all of which are shown in this figure. For converging lenses, they are: (Red) A ray goes straight through the center of the lens. This occurs because the refraction when the ray enters the lens is exactly offset by the refraction when it leaves.
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(Green) A ray that goes from the −2F point goes to any point at the lens, and back through the 2F point; draw this ray through the object. (Blue) An incident ray that is parallel to the optical axis is refracted through the focus on the far side. This arises from the definition of the focus. (Purple) A ray that goes through the lens’s −F focus is refracted parallel to the lens’s axis. Again, this arises from the definition of the focus. When drawing these rays, don’t bother trying to draw them with refraction at each lens surface, but instead just draw straight lines that connect at the middle of the lens. Also, note that you’re allowed to extend the lens vertically as much as you need to make the figure work out, as is shown in the figure with the black dashed line. The rays clearly don’t actually refract when they miss the lens, but that’s unimportant because this diagram is only being used to find the image’s position and size.
Figure 9.14 Ray diagram for a converging lens, showing all four principal rays.
Figure 9.15 shows two more ray diagrams for converging lenses. The first one shows that if the object is between the lens and −F, then there’s a magnified virtual image on the same side of the lens as the object. This is the configuration for a standard magnifying glass, in which one looks at an object through a lens and it appears to be much larger than it really is. The second diagram shows the object moved back a little bit, to the −F point. Now, the object is at a focus, so the lens converts its point of light to a beam. Lighthouses use this lens arrangement to create beams of light. Moving the object past the −F point causes the rays in the beam to converge, as we saw in Figures 9.13 and 9.14. This creates a real image that is magnified and inverted and is the configuration that is used in cameras and our eyes, both of which we’ll explore more below. Continuing to move the object farther away causes the image to shrink and approach the lens. For example, an object at the −2F point produces an image of the same size at the 2F point. This trend continues until the object is infinitely far away and the image has shrunk down to a dot at the focus. This is now the beam-to-point configuration, and is what one uses when using a magnifying glass to focus the sun’s rays down to a point to start a fire.
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Figure 9.15 More ray diagram for a converging lens, this time for (Left) an object inside of the focus and (Right) an object at the focus.
Ray diagrams are more complicated to draw for diverging lenses, shown in Figure 9.16, because more of the rays have to be extended backward. However, fortunately, these pictures always lead to the same result, which is that the image is a virtual image, it is between the object and the lens, and it is reduced in size. For diverging lenses, the principal rays are: (Red) Straight through the center of the lens, exactly like for converging lenses, (Blue) straight to the lens and then refract as though the ray came from −F, and (Green) toward the 2F point and then refract at the lens as though the ray came from the −2F point.
Figure 9.16 Principal rays for a diverging lens.
Table 9.1 summarizes the image locations and magnification for lenses. It’s essentially identical to the corresponding table for curved mirrors (Table 8.1) but has a few minor modifications to account for the differences between mirrors and lenses.
9.3.4
Lens Equations
The other way to locate lens images is with the thin lens equation. It is6 1 1 1 = + , xi f xo
6 Most
(9.7)
introductory textbooks use a different version of the thin lens equation, called the Gaussian form, in which the object location has the opposite sign. That has the advantage of being identical to the curved mirror equation and seems simpler initially because the object position is typically a positive value. However, it gets confusing for virtual images and multi-lens systems, with the result that one has to memorize complicated sign rules. The version presented here, called the Cartesian form, avoids these problems.
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Table 9.1 Relationships between objects and images for lenses object position
image
image position
orientation and size
from 0 to −F at −F from −F to −2F at −2F from −2F to −∞ at −∞
virtual none real real real
Converging lenses from 0 to −∞ upright, magnified parallel beam from ∞ to 2F inverted, magnified at 2F inverted, same size from 2F to F inverted, reduced
magnifying glass point-to-beam inverted image inverted image inverted image
real
at F
beam-to-point
from 0 to −∞
virtual
inverted, zero Diverging lenses from 0 to −F upright, reduced
configuration name
where f is the lens’s focal length, xo is the object location, and xi is the image location. This equation uses the same coordinate system that we’ve been using throughout this section, including in all of the ray diagrams, in which the object is placed to the left of the lens and has a negative location. The image location also uses the same coordinate system, so it is negative if the image is on the left of the lens (and is virtual) and positive if it’s on the right (and is real). Finally, the focal length is positive for converging lenses and negative for diverging lenses. As a simple check on this equation, suppose we have a converging lens with focal length f and the object is at the −2F point, meaning that xo = −2 f . The equation evaluates to xi = 2 f . This matches our expectation, since we know that an object at −2F has an image at 2F. The equation for image magnification is M=
xi hi = , ho xo
(9.8)
where h i and h o are the heights of the image and object. The magnification is positive for upright images and negative for inverted images. This equation arises from the geometry of similar triangles, which states that if two triangles are similar (all the same angles), then the side lengths of the two triangles are directly proportional to each other; in Figure 9.14, the red line forms the hypotenuse of the two triangles while the object and image form the upright legs of the triangles. The magnification equation is easy to check as well. For the same situation as before, where the focal length is f , the object is at xo = −2 f , and the image is at 2 f , this equation shows that the magnification is −1. In other words, the image is the same size as the object but inverted, which again agrees with expectations. The only difference between these equations and their curved mirror counterparts (Eqs. 8.6 and 8.7) is that the object position (do for curved mirrors and xo here) has the opposite sign. This arises from the facts that light shines through a lens and reflects off a mirror.
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Example. A converging lens has a 20.0 cm focal length and the object is 10.0 cm away. Where is the image and what is its magnification? Answer. Use the thin lens equation, Eq. 9.7, and plug in numbers, remembering that the object location is negative: 1 1 1 1 1 1 = + = + =− . xi f xo 20.0 cm −10.0 cm 20 cm Then, take the reciprocal to get xi = −20.0 cm. The magnification equation, Eq. 9.8, gives M=
−20.0 cm xi = = 2.0. xo −10.0 cm
Thus, the image is 20 cm behind the lens (i.e. on the left side) and its magnification is 2, meaning that it’s upright and doubled in size. This lens is acting as a magnifying glass, as shown on the left side of Figure 9.15.
9.3.5
Cameras*
All cameras work in essentially the same way. As shown in Figure 9.17, a camera consists of a lens at the front, an aperture behind the lens that reduces the effective lens area, and a sensor at the back, which could be either film or a digital sensor.
Figure 9.17 Diagram of a camera taking a picture of some flowers.
Starting at the back, the sensor records the light. Traditionally, cameras used film that was 35 mm high because that’s what was available from early movie cameras; it became widely used in personal cameras in the 1920s, and quickly became the uni-
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versal standard7 . Film cameras are largely obsolete by now, but the 35 mm standard is still used for high end digital cameras and is also the basis for a substantial amount of camera terminology. Both film and digital sensors work by recording individual photons of light. Each photon starts a chemical reaction at a single molecule in film, or moves a single electron across a tiny barrier in digital sensors, and then these small effects get amplified in subsequent processing to create the final picture. Considering the lens next, its purpose is to create a real image of the object near the back of the camera. The lens (actually several lenses in a row in most cameras, but it acts like a single lens) has some given focal length, which determines the amount of magnification. Wide angle lenses have short focal lengths in order to provide low magnification and a wide field of view, normal lenses have intermediate focal lengths, and telephoto lenses have long focal lengths, which provide the strong magnification that is needed to photograph distant objects. For 35 mm cameras, these focal lengths typically vary from around 14 mm for wide-angle lenses to 100 mm or longer for telephoto lenses. In addition, zoom lenses (also called optical zoom), are special lenses that enable the photographer to adjust the focal length as desired. If the camera is aimed at an object that is very far away, such as a distant mountain, then the image is at the lens’s focal plane, which is the vertical plane that’s one focal length behind the lens (dashed line in the figure). In this case, the photographer focuses the camera by moving the sensor to the focal plane, where the image is, and then takes the picture (in practice, the lens is moved forward or back, not the sensor, but these are equivalent and it’s easier to think about the sensor being moved). On the other hand, if the photographer is taking a picture of something smaller and closer, like the flowers that are shown in the figure, then the object’s image is behind the lens’s focal plane. This is easy to address by moving the sensor back. However, there’s the problem that the sensor is flat but the image inside the camera is three-dimensional. As a result, there’s no single best place for the sensor. The best that can be done is to place the sensor at a particularly interesting part of the image, which will then yield a nice sharp picture for that portion, and to accept that closer and farther parts of the object will be fuzzy in the picture. This fuzziness arises from the fact that the rays are still converging toward the image, or are already diverging away from the image, at the sensor’s location. In the diagram, the flower with blue petal tips and shown with blue light rays is in focus, meaning that its image is at the sensor, whereas the flower at the back, shown with red rays, is out of focus because its image is in front of the sensor. While only an infinitely thin slice of the image is perfectly located at the sensor, there’s a range of distances that are acceptably in focus, which is called the camera’s depth of field. The depth of field can be modified by adjusting the size of the aperture. Shrinking the aperture doesn’t affect the 3-dimensional image inside the camera, but it means that the rays that create that image have more similar angles. As a result, they produce a less fuzzy picture on the sensor. However, the smaller aperture also
7 In
more detail, 35 mm film has a total width of 35 mm, but the image on the film is typically 24 mm high and 36 mm wide. These are the dimensions of a modern “full-frame” digital sensor.
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means that there’s less light to take the picture with. The extreme version of this is a pinhole camera, in which there is very little light, but essentially an infinite depth of field. Vice versa, a large aperture lets in more light but results in a shallow depth of field, as shown in Figure 9.18. Figure 9.18 Photograph of flowers with a shallow depth of field.
Aperture opening sizes are quantified as the f-number, which is the ratio of the focal length to the aperture diameter. In the camera diagram, the focal length is about twice as long as the aperture diameter, so it would have an f -number of 2. This is written as f /2, meaning that the aperture diameter is half of the focal length. Smartphone cameras, such as the ones shown in Figure 9.19, are essentially the same as 35 mm cameras, but dramatically scaled down8 . Their sensor sizes are around 3 to 5 mm across and their lens focal lengths vary from 1.5 to 7 mm. These focal lengths don’t mean much to most people, so they’re generally reported as the equivalent focal length for a 35 mm camera. As shown in the figure, these equivalent focal lengths are still fairly short, such that a phone’s “telephoto” lens would traditionally be considered a normal lens. Phone cameras also have tiny apertures, typically around 3 mm or less. Dividing the focal length by the aperture sizes yields f -numbers in the f /1 to f /2 range. These low f -numbers would produce a shallow
Figure 9.19 Cameras and related devices on the back of an Apple iPhone 12 Pro Max smartphone. Values given in parentheses are equivalents for a 35 mm camera.
8 The numbers shown in the figure are from “All Apple iPhone 13 and 13 Pro Camera upgrades: Explained” by Rishi Sanyal in Digital Photography Review, published on Sept. 22, 2021; and from https://www.kenrockwell.com/apple/iphone-12-pro-max.htm.
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depth of field if they were used on a 35 mm camera. However this doesn’t apply to phone cameras, because they have tiny images, because everything is smaller, and it turns out that small images have increased depths of field. Again, these effects are quantified by giving equivalent f -numbers for a 35 mm camera, typically yielding values in the f /8 to f /20 range, which correspond to a deep depth of field. Where smartphone cameras really excel is in their software and integration with additional sensors. They can capture images from multiple cameras on the back of a phone at once, each with a different fixed focal length, and combine them to create the effect of a zoom lens. They can also collect an image over time and then use those data to identify and remove blurriness from camera shaking, which is called electronic image stabilization. Some phones go further by detecting phone motion directly with motion sensors and then physically moving the lens or camera sensor to compensate, which is called optical image stabilization. Also, some cameras use infrared laser range finders (LIDAR, for Light Detection and Ranging) to improve their autofocus capabilities. In addition, some phones reduce the image noise that arises from insufficient light by combining multiple images in a row and then using software to correct for any motion that occurred during image collection. Phones aren’t as good as professional quality cameras yet, but are getting remarkably close.
9.3.6
Vision Correction*
Humans, like most vertebrates, have camera-type eyes, shown in Figure 9.20. Like cameras, our eyes have a lens in the front, an aperture next to the lens, and a photosensitive surface in the back. The lens is composed of three layers that are called the cornea, anterior chamber, and lens; the aperture is called the pupil and is created by the surrounding iris, which forms the color of people’s eyes; and the photosensitive surface is called the retina9 . Figure 9.20 Diagram of a human eye. The ciliary muscles are relaxed, so rays from a distant object come to a focus at the retina.
9 The cornea and retina terms, along with other parts of the eye such as vitreous humor and aqueous humor, are medieval Latin words. They were given those names by the medieval Arab Ibn AlHaytham, who was the first person to largely figure out how vision worked. See Zghal et al. Proc. of SPIE, 9665:966509, 2007.
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231
As with cameras, the eye’s lens (and cornea and anterior chamber) refracts light to create a three-dimensional image within the eye, of which the part that’s at the retina creates a sharp picture while the parts that are in front of or behind the retina appear fuzzy. To adjust the focus, our eyes don’t vary the distance between the lens and retina as cameras do, but they instead vary the focal length of the lens. When relaxed, the lens’s focal length is the same as the eye length, as shown in the figure, so distant objects are in focus. Compressing the lens with ciliary muscles then thickens the lens and reduces its focal length, which allows us to focus on closer objects; this is called accommodation. In bright daylight, our irises close down to let less light in, which also gives us a deep depth of field. In low light conditions, our irises open up, reducing our depth of field. Vision problems arise when the lens isn’t able to create a sharp focus at the retina. Two problems are particularly common. First, as children grow, their eyes grow, too. As they grow, they stay in focus through a feedback process in which the child’s brain senses whether the objects that the child is typically looking at, which are presumably distant objects, are in focus, and adjusts the growth rate accordingly. This process works well for kids who spend a lot of time outdoors, but fails for kids who spend too much time reading, playing video games, or doing other close-up work, because the brain then makes the eye grow longer so that it’s adapted to these close-up scenes (Figure 9.21). The child can still focus on even closer objects by shortening their lens’s focal lengths, but he or she can no longer focus on distant objects because there’s no way to flatten the eye lenses beyond what can be accomplished by relaxing the ciliary muscles. At that point, the child has near-sightedness or myopia and the only solution is to wear glasses or contact lenses10 . These are diverging lenses that correct for the excessive length of the eyes. Figure 9.21 Diagram of a human eye with myopia, in which rays from a distant object focus in front of the retina.
The best ways to prevent myopia are to limit time spent looking at close objects, to look at distant objects periodically, and to read with a bright light to increase
10 Genetic
factors can play a role in the development of myopia as well, but the feedback process described here, called emmetropization, appears to be the dominant cause in most cases. See L. Spillmann, “Stopping the rise of myopia in Asia” Graefe’s Archive for Clinical and Experimental Ophthalmology 258:943, 2020; J. Sivak, “The cause(s) of myopia and the efforts that have been made to prevent it” Clinical and Experimental Optometry 95:572, 2012.
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the depth of field. Also, increasing time spent outdoors appears to be beneficial. In addition, presumably, it’s best to take glasses off when reading because they force the eyes to continue their up-close focusing. The other common vision problem usually starts to be noticeable at around age 40. Over time, the eyes’ lenses becomes stiffer and the ciliary muscles become less effective, which combine to reduce the eye’s ability to change the focus. As a result, older people are still able to focus on distant objects, but they can’t shorten their focus for close objects, which is called presbyopia. The solution is to shorten the focal length externally when needed by wearing reading glasses, which are converging lenses. There’s no way to prevent presbyopia, and nor can it be treated, but it’s just an unavoidable part of aging. Vision correction, for eyeglasses or contact lenses, is described in terms of diopters. These are units of refractive power, given as the inverse of the focal length and measured in m−1 . For example, a converging lens with a 1 m focal length has a power of 1 diopter, and one with a 0.5 m focal length has a power of 2 diopters. Glasses for near-sighted people use diverging lenses, so they have negative focal lengths and are described with negative diopters. Figure 9.22 shows a typical eyeglasses prescription for a person with moderate myopia. The “OD” row is for the right eye and the “OS” row is for the left eye (latin oculus dexter and oculus sinister). In this case, the right eye needs a spherical lens correction of -4.00 diopters, meaning a diverging lens with a focal length of -25 cm, and also a cylindrical lens correction of -1.50 diopters. The axis of the cylindrical lens needs to be rotated 105◦ from the horizontal. The left eye is similar. If this person were to get reading glasses, then he would need 0.75 diopters to be added to each lens. He does not need special correction for intermediate distances (“int add”), and nor does he require a prismatic correction, in which the lens is made thicker on one side and thinner on the other, which shifts the view and is useful for eyes that don’t align properly.
Figure 9.22 A typical eyeglasses prescription.
9.4
Multiple Lens Systems*
9.4.1
Objects and Images*
Optical systems that use multiple lenses offer more control over image positions and magnification because they can be varied in many different ways. However, they almost always position the lenses sequentially along a single optical axis.
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233
One can find the final image by computing the image for the first lens, using that image as the object for the second lens, and so on for each lens. This can be done with either ray diagrams or math equations.
9.4.2
Microscopes*
Figure 9.23 shows a ray diagram for a simple 2-lens microscope. The object is the tiny insect at the left. The objective lens creates a magnified real image of the object, near the middle of the figure. Then, the eyepiece lens essentially acts as a magnifying glass to expand this real image into a much larger virtual image, at the far left, which is observed by the viewer. Figure 9.23 A ray diagram for a microscope. Fo and Fe are the objective and eyepiece lens focal points, respectively. Each dark or light band on the optical axis represents 1 cm.
To demonstrate how this same result could be determined mathematically, the diagram shows that the objective lens focal length is 1 cm, the eyepiece lens focal length is 1.6 cm, the object is 1.5 cm from the objective lens, and the lenses are 4.3 cm apart. The thin lens equation and lens magnification equations, Eqs. 9.7 and 9.8, give the first image results as 1 1 1 1 1 = + = − = 0.33 cm−1 , so xi = 3 cm xi f do 1 cm 1.5 cm M=
xi 3 cm = = −2. xo −1.5 cm
Thus, the first image is 3 cm to the right of the objective lens, is inverted, and is magnified 2-fold, all of which agrees with the figure. To find this image’s position relative to the eyepiece lens, we use this result that it’s 3 cm right of the objective lens, and the fact that the lenses are 4.3 cm apart, to give its position as 1.3 cm left of the eyepiece lens. Next, repeating the equations for the eyepiece lens gives 1 1 1 1 1 = + = − = −0.14 cm−1 , so xi = −6.9 cm xi f do 1.6 cm 1.3 cm
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xi −6.9 cm = = 5.3. xo −1.3 cm
This shows that the second image is 6.9 cm left of the eyepiece lens and is magnified 5.3-fold, which again agrees with the figure. The total magnification, from the initial object to the final image is the product of the two magnifications, which is −10.7 in this case. Thus, the final image is inverted and 10.7 times larger than the actual object. It’s impressive that only two lenses enable a dramatically enlarged image. Further, it seems that slight adjustments to the lens positions would allow essentially any degree of magnification. This is true to a point but, in practice, the magnification becomes limited by the wave nature of light. Light waves from the object need to go through the objective lens and, just as we saw for waves going through small holes, diffraction occurs at this lens that causes the waves to spread out. This makes the first image slightly fuzzy. Further magnification can enlarge that image but doesn’t remove the fuzziness. As a result, the best that can be achieved is diffraction-limited resolution, in which detail can be seen down to about half of a light wavelength. This limits visible light microscopes to a resolution of around 250 nm, which is small enough to see individual biological cells, but is not good enough to see individual proteins or most intracellular structure.
9.4.3
Telescopes*
Telescopes, shown in Figure 9.24 are remarkably similar to microscopes. Again, an objective lens creates a real inverted image and then an eyepiece lens acts as a magnifying glass to expand the first image into a larger virtual final image. However, the lens positioning and dimensions are different. Telescopes use objective lenses that have a large diameter, to gather as much light as possible, and a long focal length, to create a large first image. The object, meaning the star or galaxy that one is looking at, is effectively infinitely far away, so the first image is at the objective lens’s focus. The eyepiece is then positioned to put this image, which is now being treated as the object for the eyepiece lens, at its focus. This causes the final image, which is what the viewer sees, to be infinitely far away again.
Figure 9.24 A ray diagram for a telescope. Fo and Fe are the objective lens and eyepiece lens focal points, respectively.
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235
Another difference is that the magnification value that was used earlier isn’t meaningful for telescopes, when the actual objects are light-years away and often lightyears across. Instead, telescopes are described with angular magnification, meaning the amount of angular expansion in the image relative to that in the initial view. It is equal to Mangular = −
fo , fe
where f o and f e are the objective and eyepiece lens focal lengths. The negative sign shows that the image is inverted. It might seem that wave diffraction through a telescope’s objective lens would be utterly negligible due to the large lens diameter. Indeed, this is true for telescopes on Earth, where atmospheric distortion generally limits resolution, but it’s not true for space-based telescopes. There, diffraction provides essentially the only limit to resolution, so those telescopes are also described as having diffraction-limited resolution.
9.4.4
More Optical Systems*
The same principles have been applied to design a wide variety of other optical systems as well. For example, the telescope described here is fine for astronomical observations but inconvenient for terrestrial use because it produces an inverted image. The Galilean telescope design, largely developed by the Italian astronomer Galileo Galilei, solves this by replacing the converging eyepiece lens with a diverging eyepiece, thus creating a magnified upright image. Spyglasses solve this in a different way, by inserting one or two converging lenses, called erecting lenses, between the objective and eyepiece lenses. Binoculars solve it in yet another way, by putting a pair of prisms between the objective and eyepiece lenses, that again flip the image over. Peepholes in doors, such as one finds in hotel rooms, use a set of diverging lenses on the outside of the door to create a reduced virtual image, and then a converging lens on the inside for magnifying the virtual image. All of these systems can be analyzed through ray diagrams and the lens equations.
9.4.5
Electron Microscopes*
Electron microscopes are similar to normal visible light microscopes, but use electron waves rather than light waves or, equivalently, electrons rather than photons. Scanning electron microscopes produce shaded images of small objects, such as the blood cells shown on the left side of Figure 9.25. They focus a beam of high energy electrons onto the sample, which is typically coated in a thin layer of metal first, and then detect low energy “secondary” electrons that get knocked off the sample. The
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electron beam scans over the sample in a raster pattern, creating the image as it goes along. They provide relatively low resolution but realistic-looking images. Transmission electron microscopes produce higher resolution images of even smaller objects, such as the viruses shown on the right side of Figure 9.25. They send an electron beam through the sample and then focus the transmitted beam to produce an image onto an electron detector. Their optics are conceptually similar to those of light microscopes, but use precisely shaped magnetic fields to steer electrons rather than the glass lenses that are used for focusing light.
Figure 9.25 (Left) Scanning electron microscope image of several red blood cells and one lymphocyte (a type of white blood cell). Each cell is about 8 µm across. (Right) Transmission electron microscope image of coronaviruses. Each virus is about 0.1 µm across.
Electron microscopes offer vastly better resolution than visible light microscopes due to the picometer (10−12 m) scale wavelengths of electron waves.
9.5
Dispersion
9.5.1
Prisms
The refractive index of most media vary slightly over the visible spectrum. For example, the refractive index of window glass is about 1.53 for blue light and 1.51 for red light. Likewise, the refractive index of water is about 1.34 for blue light and 1.33 for red light. These differences in wave speeds for different wavelengths is called dispersion because they can be used to spread mixtures of waves out into their component wavelengths, such as white light into the colors of the rainbow (we first met dispersion in Section 6.5 in the context of water waves). Glass and water are called dispersive media, whereas the vacuum of empty space is nondispersive. Dispersion can be observed when white light refracts through a prism, as shown in Figure 9.26. Both the diagram and photograph show that red light is refracted less than blue light, which is due to to its smaller index of refraction. This trend, in
9.5 Dispersion
237
which longer waves have lower refractive indices than shorter waves, is typical for electromagnetic waves in transparent media11 .
Figure 9.26 A diagram and photograph of prisms dispersing white light into its component colors.
9.5.2
Achromatic Lenses*
Dispersion is often undesirable in optical systems because it can make the different colors in an image not align with each other, which is called chromatic aberration. Rays that are parallel to the optical axis lead to misalignment in which the different colors focus at different positions, as shown in the left side of Figure 9.27. In this case, one color ends up being in focus while others are fuzzy. Chromatic aberrations are even worse for incident rays that arrive at an angle to the optical axis. That still has the same focus problem, but also the colors get shifted sideways by different amounts.
Figure 9.27 (Left) A lens with chromatic aberration. (Right) An achromatic lens that corrects for chromatic aberration.
One solution to chromatic aberration is to replace lenses with mirrors, which are nondispersive. This is one reason why all modern large telescopes use mirrors rather than lenses (also, mirrors are lighter weight, easier to support, and only have one surface that needs polishing). Another solution is to replace some lenses in the system with achromatic lenses, also called achromats, that are clever combinations of lenses made of different
11 The typical case, in which longer waves have lower refractive indices, is called normal dispersion.
Anomalous dispersion, which tends to occur near absorption bands, represents the case where longer waves have larger refractive indices.
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materials that work together to be nearly nondispersive. A common type of achromat, shown on the right side of Figure 9.27, uses a converging (convex) lens made of crown glass, which has low dispersion, and a diverging (concave) lens made of flint glass, which has high dispersion. Like all kinds of glass, both crown and flint glass are mostly silicon dioxide, but also have various additives that affect the glass properties. Both crown and flint glass disperse light in the same direction, with a higher refractive index for shorter wavelengths, but the fact that they are used with opposite lens shapes means that the total dispersion cancels out. Cameras, binoculars, microscopes, and other optical devices make wide use of achromatic lenses in order to get the different colors in their images to line up correctly. Some also go a step further with apochromats, which are three lens combinations that are even better at reducing chromatic aberration.
9.5.3
Rainbows*
Rainbows offer another example of dispersion, albeit a somewhat complicated one. Their optics are similar to cat’s eye retroreflectors in that sunlight shines into many spherical raindrops and, in each one, the light refracts as it enters the drop, reflects off the back face, and then refracts again as it leaves the drop. These rays are shown on the left side of Figure 9.28, with several rays entering in the top half of a drop and leaving through the bottom half. The ray that enters in the middle of the drop is mostly transmitted through the drop, but some gets redirected straight back toward where it came from. Rays that enter closer to the top are reflected more strongly, and are redirected back at larger angles. The ones that hit close to the edge of the drop are reflected most strongly, and get redirected to an angle that’s about 42◦ away from the direction that the light came from. No rays are redirected to larger angles12 .
Figure 9.28 Rainbow diagrams. (Left) Rays entering a raindrop in the top and leaving in the bottom. (Right) A person’s view of a rainbow, along with white scattered light on the inside.
light that enters extremely close to the edge scatters to less than 42◦ . Thus, as the light moves from center to edge, the scattering angle increases to 42◦ , levels off, and then turns around to decrease again. This reversal creates even more intensity at 42◦ .
12 Additionally,
9.6 Fermat’s Principle of Least Time*
239
Now, suppose the sun is behind you and a lot of raindrops are in front of you, which is shown as a sheet of rain on the right side of the figure. The shadow of your head marks the antisolar point, which is the direction that’s directly opposite the sun’s location. Any raindrops near this location will redirect some of the sunlight that hits them back toward your eyes. Also, raindrops that are up to 42◦ away from this point will also redirect sunlight back toward your eyes, with the brightness increasing for larger angles. This light is especially bright at 42◦ , which is the rainbow, and then suddenly, there is no scattered light beyond that. Dispersion enters the story due to the larger refractive index for blue light than red light, and the fact that the light refracts as it enters and leaves the raindrop. This refractive index difference causes the red ring to appear at about 42◦ , the green ring at about 41.5◦ , the blue ring at about 41◦ , and the violet ring just inside of that. Inside of the violet ring, there is white scattered light because the raindrops also redirect all colors to smaller angles. On the other hand, there’s no scattered light outside of the rainbow because no rays get redirected to these larger angles. Figure 9.29 shows a photograph of a rainbow, with obvious white scattered light on the inside and no scattered light outside. Figure 9.29 Photograph of a rainbow showing white scattered light inside and no scattered light outside.
Other atmospheric optical phenomena have similar explanations, although with different sets of reflections and refractions. For example, a double rainbow includes a second rainbow that’s outside of the normal one, and that has the colors in reverse order. This double rainbow arises from the same basic process as a regular rainbow, but now the light reflects twice on the inside of the raindrop rather than just once.
9.6
Fermat’s Principle of Least Time*
We met Fermat’s principle of least time in Section 8.7. It states that light takes the fastest path between two points, which helps to explain why light rays travel in straight lines and why the law of reflection has equal angles of incidence and reflection. Fermat’s principle also applies to refraction (and, in fact, was initially developed to explain refraction). In an analogy for refraction, the left side of Figure 9.30 shows a lifeguard who wants to take the fastest possible path to rescue a struggling swimmer. The right side of the figure shows the times for the various paths, based on the assumption that the
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lifeguard runs twice as fast on sand as she can swim in the water. The best route is path C, in which she covers more distance on the sand than in the water, but still swims outward at an angle. This path, not coincidentally, obeys Snell’s law, in which n 1 sin θ1 = n 2 sin θ2 , where θ1 and θ2 are the angles shown in the figure and n 1 and n 2 are the “refractive indices” in the two media (in this case, n 2 /n 1 equals 2 because her speed is twice as fast on sand as in the water). This shows that Snell’s law and Fermat’s principle of least time give the same results, at least in this case.
Figure 9.30 (Left) Paths that a lifeguard can take to reach a swimmer. (Right) Times for the different paths, showing that path C, which obeys Snell’s law, is fastest.
More generally, paths that obey Snell’s law always represent the fastest path, whether for a lifeguard or a light ray. This is easy to show with particular examples, as in Figure 9.30, and also not hard to prove mathematically13 . If we take Fermat’s principle as a fundamental law of nature, then this result provides an explanation for why waves refract when they meet an interface with different wave speeds. How does light find this most efficient path? The answer, as we saw before, is that light actually takes all paths at once, in each case with the waves propagating forward much as in Huygens’s principle. Of these paths, the ones that are very close to the fastest one all interfere constructively, while those that follow slower paths have wider ranges of times and so interfere destructively. The result is that the light intensity is only bright along the path with the fastest time, and this is the same as the path that obey’s Snell’s law. Converging lenses that create real images, as in Figure 9.31, provide another interesting example of Fermat’s principle. Here, the light expands out from the object and is converged back together by the lens to create the real image, with all rays contributing brightness. In this case, all paths are equally fast. The light that goes through the middle of the lens has the shortest total path, but gets delayed the most by the lens thickness. Light rays that shine above or below the middle have longer paths, but correspondingly less glass that slows them down.
13 The
proof requires finding the travel time as a function of the position where the lifeguard enters the water and then taking the derivative with respect to this position. This derivative is set to zero to represent the minimum time situation. The resulting equation can then be rearranged into Snell’s law.
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Figure 9.31 Light rays going through a converging lens, all of which take exactly the same amount of time to go from object to image.
9.7
Summary
Light waves refract when they transition to a medium with a different refractive index. Their wavelengths change but the wavefronts don’t break, so the waves change direction, turning toward the slower medium. Mirages are examples of refraction that arises from density gradients. Refraction at interfaces is described by Snell’s law, n 1 sin θ1 = n 2 sin θ2 , where n 1 and n 2 are the refractive indices and θ1 and θ2 are the incident and refracted angles. It explains the phenomenon that underwater objects appear closer than they really are. If light transitions from a faster medium to a slower one, Snell’s law shows that the angle of refraction cannot exceed the critical angle, θc = arcsin
n1 . n2
If light transitions from a slower medium to a faster one, then incident light at or beyond the critical angle undergoes total internal reflection. This phenomenon is observed in glasses of water, fiber optics, diamonds, and elsewhere. Totally internally reflected waves do not propagate beyond the surface but nevertheless produce evanescent waves that extend about a wavelength beyond the surface; they can interact with objects that are within their range. Converging lenses have a convex shape and diverging lenses have a concave shape. For thin lenses, the optical axis has its origin at the lens center, its ±F points one focal length away, and its ±2F positions two focal lengths away. A converging lens focuses parallel rays to a real image at F, and a diverging lens expands parallel rays as though they had arisen from a virtual image at −F. To locate images for lenses, one can look up the answers (Table 9.1), draw ray diagrams using principal rays, or compute them from the equations 1 1 1 = + xi f xo M=
hi xi = . ho xo
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Here, xo and xi are the object and image positions, f is the lens focal length, and h o and h i are the object and image heights; xo is always negative for single lens systems. Cameras focus light from an object to a real three-dimensional image that is recorded by film or a digital sensor. Image portions at the sensor are in focus whereas other portions appear fuzzy. Reducing the camera’s aperture increases its depth of field. Smartphone cameras are very small, of which one consequence is that they have deep depths of field. Myopia occurs when people’s eyes grow to be too long, so images of distant objects are in front of the retina. Accommodation, meaning the ability to focus, can only reduce the focal length and not increase it, so myopia is addressed with eyeglasses or contact lenses with diverging lenses. Presbyopia is the inability to focus the eyes, primarily due to lens stiffening, and is essentially universal among people over age 40. It is corrected by wearing reading glasses with converging lenses. For multiple lens systems, the image from one lens is the object for the next lens. For both microscopes and telescopes, an objective lens creates a real image and then an eyepiece lens expands it to a magnified virtual image. Both are limited by wave diffraction through the objective lens, leading to diffraction-limited resolution. Variation of refractive index with wavelength leads to dispersion. Light shining through a prism disperses into its component colors, with red bending least due to its lower refractive index. Achromatic lenses correct for chromatic aberration, where different colors focus at different positions, typically by using a pair of lenses made of different kinds of glass. Dispersion creates color separation in rainbows. Fermat’s principle of least time states that light takes the fastest path; it helps explain refraction because the path that obey’s Snell’s law is the fastest one.
9.8
Exercises
Questions 9.1. Why does a straw in a glass of water appear to bend at the water’s surface? (a) straws become floppy when they get wet (b) our eyes don’t work correctly with light rays that come through water (c) light rays from the straw bend when they leave the water, so the straw appears to be where it isn’t (d) light rays illuminating the underwater part of the straw bend as they enter the water (e) light rays interact differently with wet and dry parts of the straw 9.2. Suppose you want to focus sunlight to start a fire, using a lens. (a) Do you want a concave or convex lens? (b) Will you be forming a real or virtual image? (c) Do you want a large diameter or small diameter lens? 9.3. Consider the sea turtle that is underwater in the following figure. (a) Can it see everything that is in the air above the water (i.e. the birds shown in the picture)?
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(b) At what angle away from the vertical does the turtle look to see the feet of the duck that is about to land on the water? (c) Does the turtle see reflections of anything that is above the water? (d) Can the turtle see everything that is in the water (i.e. the fish and jellyfish)? (e) Does the turtle see strong reflections of things that are underwater; if so, which ones?
9.4. (a) Does light go faster in air or in water? (b) Does red light or blue light travel faster in water? 9.5. If you look at an object that’s on the other side of a fire, candle, toaster, engine exhaust or some other heat source, it often appears to shimmer, which is called schlieren. Explain how the regions of hot and cold air could cause this. 9.6. The following image shows a superior mirage (also called a Fata Morgana), in which light bends downward due to cold air near the Earth’s surface. Draw a diagram that shows why the boat appears to floating in the air.
Problems 9.7. (a) Write down Snell’s law. (b) Draw a picture that illustrates Snell’s law, with the angles of incidence and transmission labeled. 9.8. Consider a laser, with a 640 nm wavelength, which is pointed straight down into water. The refractive index of water is 1.33. (a) What color is the laser? (b) What is the laser frequency in air? (c) What is the laser frequency in water? (d) What is the laser light speed in water? (e) What is the laser wavelength in water? (f) What color does the laser appear to be in the water? 9.9. A laser beam shines from air into a sheet of glass (n = 1.52) with an incident angle of 35◦ . (a) What is the angle of refraction? (b) Then, the same beam shines from this glass into water (n = 1.33), now with an angle of incidence equal to the previous angle of refraction. What is the new angle of refraction into the water? (c) A different laser beam shines directly from air to water, with an angle of incidence of 35◦ ; compute the angle of refraction. 9.10. Dominic is an enologist (a wine maker) and he needs the sugar concentration of his grapes to be at least 20 brix (1 brix is 1 gram of sugar per 100 grams of
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juice). He lost his refractometer, so he shines yellow light into some grape juice and observes that the angle of incidence is 65.0◦ and the angle of refraction is 41.8◦ . (a) What is the refractive index of the grape juice? (b) Using the equation B = 650(n − 1.333), where B is the sugar concentration in brix and n is the refractive index, what is the sugar concentration in brix? (c) Is the juice sweet enough for Dominic to use it for wine? 9.11. An aquarium has a vertical front and lights above it, as shown in the figure below. (a) Is it possible to look in the front of the tank and to see out the top (i.e. is the red ray that is shown actually possible or will it undergo total internal reflection)? (b) If the previous answer is yes, then what is the minimum angle of refraction for the light that shines in the top and leaves the aquarium through the front?
9.12. (a) Draw a diagram that illustrates total internal reflection. (b) Write down Snell’s law for the case of total internal reflection (e.g. light going from glass toward air, where air is n 1 and glass is n 2 ). (c) What is the critical angle for light going from glass to air (the index of refraction for glass is 1.5)? (d) What is the critical angle for light going from cubic zirconia to air (this is fake diamond, index of refraction is 2.16)? 9.13. Consider water waves in 10 m deep water that have a 1 m wavelength. (a) What is the wave speed? (b) What is the wave frequency? These waves come to a region of water that is 5 cm deep, with a 60◦ angle of incidence (see the figure). (c) What is the wave speed in the shallow water? (d) Defining the refractive index as 1 in deep water, what is the refractive index for the shallow water region (i.e. what is the ratio of wave speeds)? (e) What is the wavelength in shallow water? (f) What is the angle of refraction into the shallow water?
9.14. A converging lens has a focal length of 8 cm. A 3 cm tall flower is 20 cm from the lens. (a) Draw a ray diagram of this situation, showing the lens, optical axis, object, image, and at least two principal rays. (b) Compute the image location and height using math equations. (c) Do your two answers agree with each other? If not, go fix them. 9.15. A diverging lens has a focal length of 5 cm. A 4 cm tall object is 5 cm from the lens. (a) (a) Draw a ray diagram of this situation, showing the lens, optical
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axis, object, image, and at least two principal rays. (b) Compute the image location and height using math equations. (c) Do your two answers agree with each other? If not, go fix them. 9.16. Two converging lenses are lined up sequentially. Each has a focal length of 10 cm, and they are 5 cm apart. An object is 20 cm from the first lens. (a) Compute the image location and magnification for the first lens. (b) Compute the final image location and total magnification for the pair of lenses. 9.17. Consider light going from an object through a convex lens and then to an image. The lens has a refractive index of 1.5, a radius of 5 cm, a maximum thickness of 1 cm, and a focal length of 26 cm. Suppose the object is 52 cm in front of the lens. In this problem, compute all times to 3 decimal places. (a) Where is the image of the object? (b) Is this a real or virtual image? (c) How long does it take light to go from object to image when it goes along the optical axis (path A in the figure)? (d) How long does it take light to go from object to image when it goes via the lens edge (path B)? (e) Which is faster, or are these the same?
Puzzles 9.18. The physical sunset time is the time when the top edge of the sun actually goes below the horizon. However, atmospheric refraction causes the sun to appear to set at a slightly different time. In other words, when we look at the setting sun, it appears to be in a slightly different place than where it really is. (a) By drawing rays on the figure shown below, does the setting sun appear to be higher or lower than it really is? (b) Is the apparent sunset time before or after the physical sunset time? (c) The atmosphere has normal dispersion, in which blue light has a higher refractive index than red light. Which image sets first, red or blue?
9.19. A trout that’s swimming in a river sees a mayfly that is flying above the water, and it decides that it wants to eat the mayfly. To the fish, the mayfly appears to be 12 cm above the water. How high above the water is it really?
Part III Light
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Figure 10.1 Painting titled “Last Painter on Earth” by James Doolin (1932–2002), which depicts Last Chance Canyon, California. Painted 1983, 72×120 inches, oil on linen.
Opening question A painter mixes together equal amounts of magenta and cyan paints. What color paint does he create? (a) Red (b) Black (c) Blue (d) Green (e) Violet
© Springer Nature Switzerland AG 2023 S. S. Andrews, Light and Waves, https://doi.org/10.1007/978-3-031-24097-3_10
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Think about a bright red strawberry. It’s undoubtedly red, but what does it mean to be red? Would it still be red if you closed your eyes? What if it were illuminated with only blue light? Might it be a red strawberry to you, but an orange strawberry to someone else, a green strawberry to your pet dog, and a color that we can’t even imagine to a parrot? Fundamentally, color is about the perception of light by our eyes and brains. It arises from the different frequencies of light waves that enter our eyes, but doesn’t truly attain the concept of color until those light waves are absorbed in our eyes and the resulting signals are transformed into mental images in our brains. Perhaps surprisingly, most people seem to transform these signals in nearly the same way, so colors appear about the same for everyone. However, there are exceptions. Color perception is different at night than during the day, it depends to some extent on the culture that people grew up in, and, of course, it is different for people who are color blind. Animals are different yet, often perceiving either more or fewer total colors than people can. The biology of color perception also leads to interesting effects when different colors are combined. For example, the red strawberry would appear neither red nor blue when illuminated with blue light, but black. Color mixing, which is now highly scientific, creates all of the colors that we see on printed materials, phone displays, and television screens.
10.1
Color Vision
10.1.1 How Vision Works The left panel of Figure 10.2 shows a ray of light entering an eye. It is focused by the cornea, anterior chamber, and lens to create an image of what we’re looking at on the eye’s retina, which is at the back of the eye. At the retina, shown in the middle panel, the light shines through several layers of signaling cells, through the majority of several photoreceptor cells, and finally gets absorbed by bent retinal pigment molecules that are toward the back side of the photoreceptor cells1 . When a retinal molecule absorbs a photon of light (right panel), the light’s energy momentarily breaks a weak chemical bond. This allows the retinal molecule to snap into a lower-energy straight shape. The shape change then distorts nearby proteins, which initiates a chemical signal within the cell. That signal gets passed back through the photoreceptor cell and into the signaling cells, which do some image processing such as to detect motion, object edges, and colors. These processed signals then continue on to the brain where they get combined with many other signals to form a mental image.
1 Retinal is
a form of Vitamin A, which the body synthesizes from carotene, which we ingest when eating carrots and other vegetables. Hence the common advice to eat carrots to improve your vision.
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Figure 10.2 (Left) A diagram of a yellow light ray entering a human eye and being absorbed at the retina. (Middle) A small portion of the retina, showing signaling, rod, and cone cells. Cone cells colors show whether a particular cell is sensitive to red, green, or blue light. The dark patch inside each cell is its nucleus. The dashed line shows a signal going from a triggered cell to the brain. (Right) Retinal molecules that are within rod and cone cells. Light absorption temporarily breaks one of the chemical bonds, enabling the molecule to snap into a straight shape.
There are two types of photoreceptor cells, rods and cones. Rods are extremely sensitive to light, even able to be triggered by a single photon. They are also very sensitive for observing motion, due to the ways that they are connected together by signaling cells. Rods are abundant over almost the entire retina, but not in the spot at the center of our vision, which is called the fovea. The fact that the fovea has very few rods means that the center of our vision isn’t very good in low light conditions, such as at night. On the other hand, the rods over the rest of the retina give us sensitive peripheral vision, meaning vision of things that we are not looking at directly. For example, you may have noticed that a flashing light at night, such as from an airplane that’s flying overhead, appears bright when you see it “out of the corner of your eye”, meaning with your peripheral vision; however, it doesn’t appear as bright when you look at it directly. Cones are the dominant photoreceptor cells in the fovea. They are much less sensitive to light than rods, but they enable color vision by coming in three separate types; one is primarily sensitive to red light, another to green light, and the third to blue light. Because there are many cones in the fovea, we are good at seeing the colors of things when we look directly at them. Both rods and cones detect light using the same retinal molecules, but vary what wavelengths these molecules are sensitive to by surrounding them with different proteins2 . Figure 10.3 shows the resulting spectral sensitivities. It shows that “red” cone cells have at least some sensitivity from red to violet light, but are most sensitive to yellow light, “green” cones are sensitive to slightly shorter wavelengths, and “blue” cones are only sensitive over the range from cyan to violet. These blue cones are sensitive into the ultraviolet too, but we can’t see that light because it is blocked by the lenses in our eyes.
2 The proteins around the retinal molecules, called opsins, have electrically charged atoms in them. These atoms exert electric fields on the retinal molecules, and the different electric fields, arising from slightly different protein structures, create different retinal absorption spectra through a mechanism
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Figure 10.3 Spectral sensitivity of human eye photoreceptors. The black dashed line is for rods and colored lines are for cones. These graphs have been normalized so that each one has a peak value of 1.
The fact that the sensitivity curves overlap implies that most light wavelengths are able to excite multiple photoreceptor cell types. These individual cells can’t detect what color of light they receive, but simply report the amount of light that they detect. As a result, it is up to the neighboring signaling cells and then the brain to determine the color of light. They do this by combining the information that they receive from many different cone cells, along with their knowledge of which photoreceptor cell is which3 . For example, suppose you look at a street light that emits yellow light at 589 nm. From Figure 10.3, your red cone cells would be fully excited, your green cone cells would be about 50% excited, and your blue cone cells would not be excited at all. Your brain has learned that this pattern of excitation corresponds to yellow light, so the light appears yellow to you.
10.1.2 Light and Dark Adaptation When we walk from bright sunlight into a dark room, we see only blackness initially. Soon though, our vision returns as our eyes transition from being light adapted to dark adapted. A quick but relatively minor part of this adaptation process is that the irises of our eyes expand to let more light through our pupils (when looking at someone’s eye, the pupil is the central black region and the iris is the brown, blue, or green ring that surrounds it). Further dark adaption occurs as the photoreceptor cells gradually convert straight retinal, which was made straight by the large quantities of sunlight that had been entering our eyes before, into bent retinal. Having more bent retinal means that more of the light entering our eyes gets absorbed, thus making it easier to see when there isn’t much light. If the room is dark enough, such as a dark movie theater, it becomes too dark for our cones to be of much use, leading us to see almost exclusively with our rods.
called the Stark effect. See Kochendoerfer, Gerd G., et al. “How color visual pigments are tuned.” Trends in biochemical sciences 24.8 (1999): 300–305. 3 While it might seem reasonable that the rod cells would contribute to color perception, using the fact that their sensitivity spectra are different from those of the cone cells, they do not actually appear to do so.
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This has several interesting consequences. One is that we lose sensitivity to red light because rods aren’t sensitive to red light. For example, a bright red scarf is very obvious during the day, but appears almost black at night. Another consequence of seeing primarily with our rods is that we can use a red flashlight as much as we want without ruining our dark adaptation. This is because the red light is only absorbed by our red cone cells, which aren’t useful for seeing in the dark anyhow, and not by the rods, thus preserving the rods’ dark adaptation. Yet another consequence is that we perceive dim light as being more blue and green than it really is. For example, people often think that moonlight illuminates things with a bluish tint, in contrast to the white color of sunlight. In truth though, it’s the opposite; moonlight is actually redder than sunlight, but our brain assumes that it’s greenish-blue because we see it with our rods and that’s where rod cells are most sensitive. Staring at a particular picture for a while and then at a white background, or seeing a sudden flash of bright light, causes an afterimage (Figure 10.4). This arises from essentially the same processes as light and dark adaptation. Here, the part of the retina that was looking at a bright part of the picture uses up its bent retinal and so becomes light-adapted. Vice versa, the part that was looking at a dark part of the picture gains bent retinal through the cell regenerating it, and so becomes dark-adapted. When one’s gaze is then shifted to a uniform background, the different sensitivities of the two sets of cells causes them to report the same incoming light with different results. The dark-adapted cells report more light and the light-adapted cells report less light, thus producing the reverse of the original image. Figure 10.4 Images for black-and-white and color afterimages. To see an afterimage, focus on a spot at center of one of the pictures for about 30 seconds without moving your eyes. Then look at a white background. The right image is of Michael Jackson.
10.1.3 Different People see Different Colors* Between 4 and 10% of males, depending on ethnic background, are at least partially color blind. Up to about 1% of females are at least partially color blind as well, where this lower prevalence arises from the fact that the most common types of color blindness arise from errors in the X chromosome, and females have two X chromosome copies while males only have one. While these percentages may seem small, they imply that a room with 15 men in it probably has at least one color blind person.
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In red-green color blindness, the most common type, the affected individual typically has the same three cone types as usual, but the spectral sensitivity of the green cone is shifted to nearly overlap that of the red cone. As a result, the brain isn’t able to differentiate between red and green colors. Figure 10.5 shows an image that is used to test for red-green color blindness; people with normal vision, meaning not color blind, see the number 3 in the image whereas those who are color blind cannot see it. Remarkably, this type of red-green color blindness can be essentially solved with “color blind glasses,” which absorb those light wavelengths in which there is the greatest spectral overlap between the red and green receptors. This accentuates the differences between the two types of cones, leading to restored color vision. Less common types of color blindness arise from cone cells that have other spectral shifts or from one of the cone types being entirely missing from a person’s retina. Figure 10.5 An image used to test for color blindness. This is an Ishihara chart, developed by the Japanese ophthalmologist Shinobu Ishihara around 1917.
All people with normal vision have essentially identical visual systems, from their retinal molecules to their neural processing systems. This seems to result in fairly similar color perception. Nevertheless, there are subtle differences. In one of the better studied examples, the Russian language has one word for light blue, “goluboy,” and another for dark blue, “siniy,” but no word that encompasses all shades of blue. As a result, Russian speakers tend to perceive light blue and dark blue as being more different from each other than English speakers do4 . Other cultural differences in color perception have been reported as well, but again are relatively minor. People’s eyes change substantially as they get older, in part because proteins in the eye’s lens break down over time and with exposure to UV light. This progressively scatters more and more of the light entering the eye, and blue light in particular. Also, older people’s pupils get smaller due to weakened eye muscles. Together, these mean that elderly people have much less light to see with than young people do, as shown in Figure 10.6. While one might expect that the reduced blue and violet light would change color perception with age, it actually has very little effect. Undoubtedly this is because people’s brains compensate for the changing input information, much as everyone adapts when watching a television set that has a slightly redder or bluer
4 Winawer,
Jonathan, et al. “Russian blues reveal effects of language on color discrimination.” Proceedings of the National Academy of Sciences 104.19 (2007): 7780–7785.
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color balance. Eventually, so much light is scattered that people have a hard time seeing and are said to have cataracts. The solution to cataracts is to replace an eye’s natural lens with an artificial plastic lens, thus restoring transparency. People who have had cataract surgery report substantial color changes initially, with extra vivid violet hues in particular, but that the perceived colors return to normal after a few days as the brain adjusts to the new inputs. Figure 10.6 Amount of light entering the eye at different wavelengths as a function of age. From Turner and Mainster, British Journal of Ophthalmology, 2008.
10.1.4 Color Vision in Animals* All vertebrates, including mammals, birds, reptiles, amphibians, and fish, have eyes that have the same basic design as ours. In all cases, light shines through a lens and is focused on rod and cone photoreceptor cells on the retina, which is at the back of the eye. However, the ratio between rod and cone cells differs widely. Diurnal animals, such as parrots, chameleons, and ourselves, are active during the daytime when the world is well-lit. As a result, they generally have mostly cone cells in their foveas, which gives them good color vision, but at the cost of poor vision at night. In contrast, nocturnal and crepuscular (active at dawn and dusk) animals, such as cats, deer, and owls, generally have many more rod cells in their foveas. This gives them vision that is more sensitive to low light, but at the cost of poor color perception. In addition, many animals don’t have three types of cone cells like we do, but have either more or fewer types. For example, dogs have only yellow and blue cones. This allows them to differentiate yellow from blue, but they cannot distinguish between red and green objects (Figure 10.7), making their vision similar to that for people who are red-green color blind. On the other hand, dogs’ eyes are sensitive to some UV light that is invisible to us, which they undoubtedly see as blue. Dolphins are even worse off, having only a single type of cone cell, thus giving them minimal color perception, if any.
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Figure 10.7 Comparison of human and dog views of the world. (Left) Color appearance for different light wavelengths, (Middle) A person’s view of a scene, (Right) A dog’s view of the same scene.
In contrast, birds often have four colors of cone cells, adding one in the near ultraviolet portion of the spectrum. In addition, birds have tiny colored oil drops within each cone cell that filters out some light colors in order to narrow each cone’s spectral sensitivity range. As a result, birds can see colors that we cannot see, or really even conceive of, because our color concepts are generally limited to those that we can see. For example, many flowers have ultraviolet markings on them, which are undoubtedly visible to birds but are invisible to us (Figure 10.8).
Figure 10.8 Comparison of human and bird views of the world. (Left) A person’s view of a flower (creeping cinquefoil). (Right) A photograph of the same type of flower in which the UV light is shown and the visible light was blocked. A bird sees all the colors that people do, plus these UV colors.
Invertebrates generally have very different eye designs than ours. For example, most insects have compound eyes, which are composed of hundreds or thousands of tiny eye units that each point in a different direction. Nevertheless, they still detect color using a collection of photoreceptor cells that are sensitive to particular colors, much like our cone cells. Many insects, including house flies and fruit flies, have two types of photoreceptor cells so their color vision is probably similar to that of dogs. Honeybees have three types of photoreceptor cells so their color vision is probably similar to ours; however, their spectral sensitivities are shifted to shorter wavelengths, so they see less red but add some ultraviolet. Remarkably, one type of Australian butterfly has 15 different types of photoreceptor cells, potentially giving it extraordinary color vision. Or maybe not; mantis shrimp have 12 types of photoreceptors,
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which also suggested extraordinary color vision, but further research showed that its color vision wasn’t actually particularly good5 .
10.2
Color Models
10.2.1 Color Wheel Isaac Newton showed that white sunlight could be split into the colors of the rainbow using a prism. He arranged these colors into a circle6 , creating an early version of the color wheel with which we are familiar today, both of which are shown in Figure 10.9. Figure 10.9 Newton’s color circle and a modern color wheel.
He named the colors on his circle as red, orange, yellow, green, blue, indigo, and violet, which are sometimes shortened to the abbreviation ROY G BIV. It is somewhat dubious whether indigo is actually distinct from blue and violet or is better described as a gradation between the two. However, Newton included it because he thought that the spectrum should have seven colors, much like the seven musical pitches in an octave in Western music (also, Aristotle said that there were seven colors, which he gave as crimson, violet, green, blue, black, white, and yellow, so Newton may have been following this tradition). Arranging the colors into a circle makes perceptual sense, in that we perceive a continuous gradation of colors from red to violet and then continuing on through magenta and back to red again. However, it’s important to point out that the color circle does not arise from the physics of light. Instead, the physical reality is that the
5 Invertebrate color vision is more complicated than ours is, more diverse, and less well understood. For fruit fly color vision, see: Kelber and Henze “Colour vision: parallel pathways intersect in Drosophila” Current Biology 23: R1043 (2013). For the referenced butterfly color vision, see Chen et al. “Extreme spectral richness in the eye of the common bluebottle butterfly, Graphium sarpedon” Frontiers in Ecology and Evolution 4:18 (2016). For honeybee color vision, see: De Ibarra, et al. “Mechanisms, functions and ecology of colour vision in the honeybee” Journal of Comparative Physiology A 200:411 (2014). For mantis shrimp color vision, see Thoen, et al. “A different form of color vision in mantis shrimp” Science 343:411 (2014). 6 Newton, Isaac, Opticks Book I, Part II, Plate III, Figure 11 (1704).
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colors of the rainbow lie in a linear spectrum that starts at red and goes to violet; beyond these two endpoint colors are infrared and ultraviolet respectively, without magenta and not arranged in a closed circle. It turns out that the color wheel makes perceptual sense because of the three types of cone cells in our eyes, for red, green, and blue. It makes sense to arrange the red, green, and blue colors in a triangle, and to then join these limiting points by the colors that we see when we mix them together. For example, pure yellow light (600 nm) excites both our red and green cone cells, so it appears halfway between red and green on the color circle. Similarly, an equal mixture of red and green lights also excites our red and green cone cells, so we see this mixture as the same shade of yellow. The fact that our eyes cannot perceive the difference between pure colors (e.g. yellow) and mixtures of different colors (e.g. a mixture of red and green) means that technology for printing and displaying colors does not need to work with pure colors but can use mixtures of only a few primary colors instead. When these primary colors are mixed in roughly equal proportions, they are called secondary colors. Secondary colors can then be mixed with primary colors to create tertiary colors and so on. More importantly, primary colors can be mixed in variable amounts to create essentially any other color. The subsequent sections will discuss several color models, each of which provides a way to describe different colors quantitatively.
10.2.2 Light Addition with the RGB Color Model Computer, phone, and television displays produce colors by adding together red, green, and blue lights, which is the RGB color model and is an example of additive color mixing. Figure 10.10 illustrates this approach by showing the situation in which red, green, and blue spotlights are aimed at a white screen to produce partially overlapping circles of colored light. The outside of the figure is black because no light reaches that region. The red, green, and blue primary colors appear where the surface is illuminated by a single light, the cyan, magenta, and yellow secondary colors appear where two lights overFigure 10.10 Red, green, and blue (RGB) color mixing using lights.
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lap, and white appears where all three lights overlap. Note the similarity between this figure and the color wheels shown above (Figure 10.9). Colors that are opposite each other, whether in this diagram of the RGB color model or on a color wheel, are called complementary colors. These are: red and cyan, green and magenta, and blue and yellow. Also, black and white can be considered complementary as well. Complementary colors create a high visual contrast. They are also the colors that we see in afterimages because the eye adapts to one color when staring at a point and then reports the “opposite” color when it is redirected to a neutral background. Additive color mixing can also be expressed as equations. For example, red + green = yellow red + green + blue = white yellow + blue = white. The last equation can be seen to be correct by the fact that yellow is the same thing as red + green, and red + green + blue add to white. More generally, adding any pair of complementary colors produces white light. Light addition becomes much more complicated if we allow the brightnesses of the individual lights to vary. For example, a lot of red plus a medium amount of green add to create orange. Or, if we mix a little red, a lot of green, and a lot of blue, then the result is turquoise. These combinations are all part of the RGB color model but are hard to predict without experimentation, so they are largely beyond the scope of this book. Figure 10.11 shows a microscope image of a modern phone display, showing the separate red, green, and blue pixels that compose it. These pixels are too small to see normally, so our eyes add their brightnesses together to produce white in this case, or other colors if the primary colors have different brightnesses. One way to see these pixels is to smear a little water on the display to produce droplets that then work like tiny magnifying glasses. Figure 10.11 Close-up of an Apple iPhone 5s display.
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Examples. Solve the following color equations. (1) blue light + red light = ? (2) cyan light + red light = ? Answers. (1) Magenta light. Refer to Figure 10.10. (2) White light. Cyan is green + blue; then adding red yields all colors, which is white.
10.2.3 Light Subtraction with the CMYK Color Model Colored objects, whether a yellow lemon or a blue shirt, typically exhibit their colors using pigments that remove specific colors of light. This is called subtractive color mixing. Here, white light is typically incident on the object, some wavelengths of the white light are absorbed by pigments in the object, and the rest of the light wavelengths are reflected. The color that we perceive arises from the mixture of the reflected light wavelengths. Essentially the same thing happens for partially transparent objects, such as for rose-colored sunglasses. Here, some colors of light are absorbed by pigments in the glass while the rest are transmitted. As with light addition, subtractive color mixing can be performed by mixing only three different primary colors, each of which is a pigment in this case. Modern printing uses the CMYK color model, in which these three primary color pigments are cyan, magenta, and yellow (the ‘K’ stands for black, described below). Figure 10.12 illustrates the mixing of these pigments. It represents the situation in which cyan, magenta, and yellow circles of colored glass are placed on a white screen and illuminated from behind with white light. Each piece of colored glass absorbs its complementary color. For example, the cyan glass absorbs red light and transmits green and blue; as a result, the light that gets through is a mixture of green and blue light, which appears cyan. Likewise, the magenta glass absorbs green light and transmits red and blue which together appear magenta, and the yellow glass absorbs blue light and transmits red and green which together appear yellow. Figure 10.12 Cyan, magenta, yellow, and black (CMYK) color mixing using pigments.
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Figure 10.13 Diagrams for determining the colors that result from subtractive color mixing.
Determining the colors that result from subtractive color mixing is more challenging than with additive color mixing, but can be deduced by drawing simple diagrams. In these diagrams, start with the color of incident light, subtract off all light that is removed by pigments, and then see what remains. The left side of Figure 10.13 illustrates the solution to the following problem: white light shines through a magenta filter and then a yellow filter; what remains? The diagram shows red, green, and blue arrows on the left because they add to the white incident light. Then, each filter removes its complementary color, as shown with X’s in the appropriate rows of the filter boxes (the rows are red, green, and blue, in that order). This diagram shows that only red remains, thus showing that magenta and yellow pigments combine to produce red light. This also implies that mixing magenta and yellow paints would produce red paint. As an equation, this problem and solution is white light + magenta pigment + yellow pigment = red. The diagram on the right side of Figure 10.13 is similar, but now the incident white light is sent through filters with yellow and blue pigments. This time, all of the light colors are absorbed, so no light emerges. As an equation, white light + yellow pigment + blue pigment = black. Here, the yellow pigment absorbs the blue light and then the blue pigment absorbs the red and green light, leaving nothing, so the result is black. More generally, combining pigments with complementary colors always produces black. Note that this diagrammatic approach is the same whether the light is transmitted through colored filters or reflected off colored surfaces. As with additive color mixing, this scheme is only approximate. It assumes that each pigment either fully transmits or fully absorbs each color of light, whereas real pigments have variable absorption amounts. For example, Figure 10.14 shows a collection of gumballs, each colored with food coloring pigments. The middle panel illustrates the color of red and yellow gumballs according to scheme explained above, where red pigment transmits only red light, and yellow pigment transmits red and green light while absorbing all blue light. The right panel shows actual reflectance spectra of red, yellow, orange, and magenta gumballs. The red and yellow spectra clearly agree qualitatively with the middle panel, but have some differences too. The pigments don’t actually have 0% reflectance at some wavelengths and 100% reflectance at others, and they don’t have perfectly sharp transitions between these regions. Because real pigments do not absorb or transmit completely, using more
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Figure 10.14 (Left) A collection of gumballs, each colored with food color pigments. (Middle) An idealized reflection spectrum for gumballs with red and yellow pigments. (Right) Actual reflection spectra for different colored gumballs.
pigment generally leads to greater light absorption, particularly at the wavelengths that are already highly absorbed. In other words, using more pigment leads to a darker appearance and more saturated colors7 . The same limitations with real pigments makes it difficult in practice to produce black by mixing cyan, magenta, and yellow. In principle, this combination should produce perfect blackness, as shown at the center of Figure 10.12. In practice though, mixing complementary pigment colors generally produces brown, and not black. For printing technology, the solution is to use black pigment as a fourth primary color, leading to the CMYK color model. Despite the fact that the CMYK model is used by most modern photocopiers, laser printers, and paint mixing machines, it is far from universal. It was preceded by the RYB color model, which is based on red, yellow, and blue pigments. The RYB model continues to be widely used by artists and is taught in most art classes (including, most likely, art classes that you took). Its versions of the red, yellow, and blue colors are shifted slightly from those described above. For example, yellow and blue mix to form black in the CMYK color model, but they mix to form green in the RYB model; this is because the RYB blue is between the blue and cyan colors of the CMYK model. Except where noted, this book assumes that color names refer to those of the CMYK model.
7 Subtractive color mixing can be performed quantitatively by multiplying the incident light intensity spectrum by the transmission or reflection spectra for each of the included pigments. This yields the final light spectrum. Alternatively, the pigment absorption spectra can be added together and then converted to a transmission spectrum. See Chapter 4.
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Examples. Solve the following color equations. (1) magenta light filtered by cyan filter = ? (2) white light + yellow pigment + green pigment = ? Answers. (1) Blue light. See the diagram. (2) Green pigment. See the diagram.
10.2.4 Leaf Colors in Summer and Fall* In the fall, trees often create spectacular displays of yellow and red leaves (Figure 10.15). How do they do this? First, it turns out that trees that turn yellow, such as aspens, poplars, sugar maples, and ginkos, have yellow pigment in them at all times. This yellow color is masked during the summer by green chlorophyll, which allows the leaves to photosynthesize, but is then revealed when the chlorophyll is removed during the fall. In agreement with this, subtractive color mixing shows that yellow pigment + green pigment = green, meaning that we can’t see the yellow during the summer even though it’s there.
Figure 10.15 Quaking aspen (Populous tremuloides), Japanese maple (Acer Palmatum ‘Bloodgood’) and red-osier dogwood (Cornus sericea ssp. stolonifera) leaves with and without chlorophyll.
Some trees that turn red, such as Japanese maples and flowering plum trees, work the same way. They always have red pigments, which makes the leaves bright red in the fall when the chlorophyll is removed. During the summer, subtractive color mixing predicts that red pigment + green pigment = black. In practice, this isn’t quite right but is close. These trees actually have dark purple leaves in the summer, arising from the combination of the two pigments. Finally, most trees that turn red, such as dogwood and cherry, gain red pigment during the fall as the chlorophyll is removed. We can see this because, again, red
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pigment + green pigment = black, and this isn’t at all close to the actual summer leaf color. Instead, these leaves are green during the summer, which implies that the red pigment must not be there.
10.2.5 HSV Color Model The RGB and CMYK color models are ideally suited for lighted display and color printing technologies, respectively. However, they are not intuitive. Computer graphics researchers addressed this in the 1970s by developing the HSV model or HSB model, which are the same thing; the letters stand for hue, saturation, and either value or brightness. This model is essentially an extended and formalized version of the basic color wheel. The hue describes the color of the light, such as red, orange, yellow, green, blue, violet, or magenta. The saturation describes how colorful the light is, from being a dull shade at the low extreme, to being a vivid color at the high extreme. Finally, the value or brightness describes how bright or dark the light is, ranging from black to white. Figure 10.16 shows the HSV model using a cylinder, where the angle around the cylinder is the color wheel and other axes show the saturation and value. Figure 10.16 Cylinder showing the hue, saturation, and value coordinates.
It is helpful to consider how the hue, saturation, and brightness components of a color correspond to light transmission in the visible spectrum: roughly, the hue gives the wavelength of the maximum emission, the saturation describes the width of the peak, and the brightness gives the height of the peak. All of these color models, meaning the RGB, CMYK, and HSB models, can be easily explored using a variety of modern drawing software. For example, if you draw an object using Microsoft PowerPoint, you can then color the object using a color picker. The color picker options include the RGB, CMYK, and HSB models.
10.2.6 Color Spaces* Each of the models presented describes color using three variables. The RGB model uses the amounts of red, green, and blue light, the CMYK model uses the amounts
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of cyan, magenta, and yellow pigments (plus black, which is technically redundant), and the HSV model uses the hue, saturation, and value parameters. These give rise to the concept of a color space, where the position in color space is described by the values of the three variables, much like a 3-dimensional position in a room might be described by the three coordinates of Cartesian space. That there are always three color space variables is not coincidence, but arises from the fact that we have three types of cones in our eyes.
Figure 10.17 Excitation of cone cells by light, where L is for long-wavelength or red, M is for medium wavelength or green, and S is for short-wavelength or blue. (Left) Single 530 nm light input. (Right) Mixture of 530 nm and 630 nm light inputs.
There are two problems with all of these color spaces. First, they are all somewhat arbitrary, depending on the precise light wavelengths or pigments chosen. This makes them inadequate for precise work, at least without adding some sort of standardization. Second and more importantly, it turns out that they can’t actually represent all of the colors that people are able to see. That is, there are some colors that people can see but that can’t be created by mixing red, green, and blue lights together. These problems were solved by designing yet more color spaces, now developed around human perception rather than around display and printing technologies. The LMS color space follows the most intuitive approach. Here, the three color space parameters are the excitation levels of the three types of cones, with L being the excitation level of the long-wavelength (red) cones, M being the excitation level of the medium-wavelength (green) cones, and S being the excitation level of the short-wavelength (blue) cones. By construction, every color of light that people can perceive can be mapped to this color space. For example, consider pure green light that has a wavelength of 530 nm. The left panel of Figure 10.17 shows that this light would excite the red cones to level 0.80, the green cones to level 0.98, and the blue cones to level 0.01, putting this color at point (0.80, 0.98, 0.01) in the LMS color space. Scaling all of these values up or down together would lead to a light with the same apparent color (hue) but would be brighter or dimmer. Suppose we then added a second light, now at 630 nm (right panel of Figure 10.17), which would add red excitation of 0.42 and green excitation of 0.06. Adding these to the prior excitation vales leads to a total excitation of (1.22, 1.15, 0.01). This color would appear yellow
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to us (note that a very similar cone excitation could be created with pure yellow light of about 570 nm). These two examples show that any possible light input, whether a single wavelength or a mixture of wavelengths, can be mapped to a single point in LMS color space. However, it’s not reversible; there are perfectly valid points in LMS space that cannot be produced with any possible light inputs. For example, the point (0, 1, 0) represents excitation of the green cone exclusively. However, looking at the cone sensitivity spectra shows that any light wavelength that can excite green cones also excites either red or blue cones as well, making it impossible to excite the green cone by itself. Figure 10.18 Response spectra for the fictitious “CIE standard observer” with cones labeled X, Y, and Z.
In practice, the LMS color space is rarely used because the cone sensitivity spectra were only well established relatively recently. Instead, theCIE 1931 XYZ color space has become the established standard (CIE stands for Commission internationale de l’éclairage, which translates to International Commission on Illumination; the color space was developed in 1931). It is based on experiments conducted in the 1920s in which subjects matched colors from mixtures of red, green, and blue lights to those from single wavelengths of light. After some mathematical manipulations, those experimental results led to the definition of the “CIE standard observer,” a fictitious person who has the cone sensitivity spectra shown in Figure 10.18. These cones are labeled X, Y, and Z and correspond roughly to the actual red, green, and blue cones. Beyond this change of sensitivity spectra, the XYZ color space is essentially the same as the LMS color space described above. Importantly, the XYZ color space still represents human vision completely correctly, despite the fact that its parameters don’t actually correspond to anything physical. The three-dimensional structure of the XYZ color space makes it difficult to visualize, so a single slice of it is typically shown (Figure 10.19). This slice is taken on the diagonal in which the three color coordinates add to 1 and is then called either the CIE xy color space or the chromaticity diagram. Because the three color coordinates add to 1, all points in this diagram represent the same brightness, or luminosity. One could conceivably show other slices with higher or lower luminosities, but that isn’t conventional. As part of the rescaling to constant luminosity, the X , Y ,
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and Z parameters are replaced by x, y, and z, which then have the constraint that x + y + z = 1. Figure 10.19 Chromaticity diagram, or CIE xy color space. See the text for details.
Several things are notable about this diagram: • Moving toward the right implies a larger x value, which is more red light. Moving up implies a larger y value, which is more green light. And moving down and left implies a larger z value, which is more blue light. • The colored portion represents all colors of light that can be perceived by people and is called the gamut of human color perception. Its outer edge shows the cone excitations that result from pure wavelengths of light, which are labeled on the figure, while its inside shows cone excitations that result from mixtures of light wavelengths. Its bottom edge is called the “line of purples”; these colors can only arise from mixtures of primarily red and blue lights. The black dot near the middle has nearly equal excitation of all cones so it appears gray. • The gray region in the upper right portion of the figure does not have colors because this is outside of the XYZ color space. Here, x + y > 1 so z (blue cone excitation) would have to be less than 0, which doesn’t make sense. • The white region outside of the colored diagram represents cone excitations that cannot be produced by any wavelengths of light, due to the overlapping cone sensitivity spectra. In principle, a person could perceive these colors if there were some way to excite some cones but not others, but this is impossible using light. For example, the point (0, 1, 0) represents excitation of the Y cone only, but that that’s not possible because any wavelength of light that excites Y also excites either X or Z as well.
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• The triangle in the diagram shows all of the colors that are possible with a standard RGB electronic display, which is called the display’s gamut. The corners of the triangle represent the display’s primary colors (red, green, and blue), while the inside of the triangle represents colors that can be created from mixtures of these primaries. The small circle near the center is the display’s output when each primary color is turned on to equal brightness. Despite its complications, the XYZ color space is almost universally used for quantitative work on color. It is used by marketers who want to ensure that their products look correct in advertisements, film studios who want to ensure that colors appear accurately in movies, and computer graphics researchers who want to develop better displays. The XYZ color space also helped with the development of the modern RGB and CMYK color models by showing which primaries would lead to a large gamut.
10.3
Summary
In human vision, light is focused through the eye and onto the retina, where it converts bent retinal to straight retinal in photoreceptor cells. This change initiates biochemical signals that are processed in nearby cells and then sent to the brain for further processing into images. Rod cells are very sensitive to light and motion but cannot detect color; we use them primarily in our peripheral vision and in dark conditions. Cone cells, which are abundant in our foveas, are less sensitive but can detect color. People have red, green, and blue cones, each of which is sensitive to different light wavelengths. Our eyes become dark adapted when the irises expand and the rod and cone cells have converted most of their straight retinal to bent retinal. In very dark places, we see using our rods primarily, leading to several shifts in color perception, such as red becoming almost black. Different portions of our retinas become light or dark adapted when we stare at a stationary image, leading to afterimages. Most people seem to see colors reasonably similarly, although color blind people have reduced color perception. Many animals have different color perceptions. Nocturnal animals generally have more rod cells and fewer cone cells, dogs have only two types of cone cells, and birds have four types of cone cells. Also, many animals are sensitive to ultraviolet light, enabling them to see flower markings and other patterns that are invisible to people. Colors are often arranged in a color wheel, traditionally including red, orange, yellow, green, blue, indigo, and violet, and sometimes magenta, before returning to red. The wheel is not based on the physics of light, but represents the way colors are perceived. For example, people cannot distinguish between a pure yellow light and a mixture of red and green lights, and people can perceive magenta, but magenta is not in the electromagnetic spectrum. The color wheel naturally arises from the three types of cones.
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The RGB (red-green-blue) model is used for additive light mixing, as in digital displays. These primary colors mix in pairs to produce secondary colors, and all three mix to form white. Mixing colors that are opposite each other in the RGB diagram, called complementary colors, produces white light. The CMYK (cyan-magenta-yellow-black) model is used for subtractive mixing with colored pigments, where each pigment removes its complementary color. In concept, pigment spectra have sharp transitions between complete absorption and complete transmission, but real pigments have smooth transitions between incomplete absorption and incomplete transmission, so use of more pigment generally produces a darker and more saturated color. The red-yellow-blue (RYB) model is a subtractive mixing model that continues to be widely used by artists. Leaf colors in the summer and fall illustrate subtractive color mixing. The HSV (hue-saturation-value) model is a more intuitive method for picking colors. Hue represents the color from the color wheel, saturation represents the intensity of that color, and value represents the lightness. Each of these color models represents a three-dimensional color space, but their gamuts are smaller than that of human perception. The LMS (long-, medium-, and short-wavelength) color space uses its parameters as the excitation levels of the red, green, and blue cone cells, representing all colors that people could possibly perceive, and some that people cannot perceive. The XYZ color space is similar, but for a “standard observer” who has three sensitivity spectra that are different from the actual cones. The XYZ color space is often depicted in a chromaticity plot, which shows the gamut of human perception, the gamut of the RGB or other color models, and the colors that arise from mixtures of light.
10.4
Exercises
Questions 10.1. Tuan has a bucket of magenta paint. He wants it to be blue. Which pigment should he add? (a) cyan (b) green (c) yellow (d) black (e) it’s impossible 10.2. Suppose you’re sitting in a red tent, where all of the light is red, and you’re searching for your green socks. What color will they appear to be once you find them? (a) red (b) yellow (c) green
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(d) blue (e) black 10.3. If a parrot, which has four types of cone cells including ultraviolet sensitivity, were to observe a standard television, what would it perceive? (a) it wouldn’t see a picture at all because it’s shown with the wrong colors (b) the image would appear in black, white, and shades of gray (c) all of the colors would be completely scrambled (d) the image would look roughly correct, but missing some colors (e) the image would be much more vibrant than it is used to seeing 10.4. At night time, if you look directly at the flashing light of a distant airplane, it’s often quite dim. However, if you see the same flashing light with your peripheral vision, it’s usually much brighter. What causes this effect? 10.5. Fire engines used to be bright red because people thought they would be easy to see. However, many cities have replaced them with bright yellow fire engines because those are easier to see at night. Why is yellow easier to see at night? 10.6. Deer hunters often wear bright orange clothing with camouflage patterns. The clothing is obvious to people but not to deer. Deer, like dogs, have only two types of cones. Why can’t deer see the orange clothing? 10.7. List at least two traffic safety signals that a person with red-green color blindness would have difficulty seeing. 10.8. What do the letters stand for in the following abbreviations: (a) ROYGBIV, (b) RGB, (c) CMYK, (d) HSV? 10.9. Describe the colors of the following objects when illuminated with red light: (a) a yellow tennis ball, (b) a red shirt, (c) a green leaf, (d) a white sheet of paper. Problems 10.10. Complete the following color “equations.” (a) red light + blue light = light, (b) yellow light + blue light = light, (c) blue light + light = cyan light. 10.11. Complete the following color “equations,” assuming illumination by white light in each case. (a) cyan paint + magenta paint = paint, (b) cyan paint + paint, (c) magenta paint + paint = black paint. red paint = 10.12. Using the color subtraction scheme presented here, (a) what color results when white light is passed through a cyan filter (e.g. a cyan sheet of plastic)? (b) what color results when white light is passed through 4 cyan filters in a
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row? (c) In reality, would the transmitted light in these two cases be exactly the same or would it appear somewhat different? Explain. 10.13. Explain why the sun is white when overhead and red during sunrise and sunset, using the fact that air acts like a filter that removes cyan light. 10.14. Using the color picker in a computer drawing program, such as Microsoft Powerpoint, choose olive green. What are its values in (a) RGB, (b) CMYK, (c) HSB? Puzzles 10.15. In 3D movies, viewers typically wear glasses with one red lens and one blue lens, with which they then view red and blue movie images separately. Could this work equally well if the colors were red and yellow, instead? Explain. 10.16. Sketch the color space for a dog’s vision in which the x-axis is the excitation level for the yellow cones and the y-axis is the excitation level for the blue cones. Label the regions of the space that correspond to light of different wavelengths, including rough wavelength numbers. Indicate what regions of the space are inside and outside the gamut of dog color perception.
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Electromagnetic Waves
Figure 11.1 Lasers in a light show.
Opening question Why is the sky blue? (a) We see the water vapor in the air, and water is blue (b) It’s from the ozone layer, which is a layer of blue air in the upper atmosphere (c) The sky reflects the color of the oceans (d) Air molecules scatter the bluer portions of sunlight more than the redder portions (e) It’s not really blue, but just appears that way
© Springer Nature Switzerland AG 2023 S. S. Andrews, Light and Waves, https://doi.org/10.1007/978-3-031-24097-3_11
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For a long time, people thought that electricity was a mysterious force that caused glass rods that were rubbed with silk handkerchiefs to pick up little balls of fluff, and that magnetism was a completely different mysterious force that caused certain “lodestone” rocks to point toward the north pole. Making sense of electricity and magnetism, and furthermore showing these two topics are intimately connected to each other, remains one of the crowning achievements of science. Even more remarkable was the finding that electricity and magnetism could combine to become visible light. The notion that rubbing a balloon on your hair and then sticking it to the ceiling has anything to do with starlight that shines across the galaxy is truly amazing. This chapter investigates what it means for light to be an electromagnetic wave. It introduces electricity and magnetism, looks at the electromagnetic spectrum in more detail than we’ve seen in prior chapters, and investigates the polarization and scattering properties of electromagnetic waves.
11.1
Light Waves as Electric and Magnetic Fields
11.1.1 Scalars, Vectors, and Fields It’s helpful to start with a few definitions. A scalar is just a fancy name for a number. For example, a scalar could represent the temperature, the air pressure, the elevation above sea level, someone’s age, or anything else that can be quantified with just a number. The number is allowed to have units but it doesn’t have to. A vector is an arrow, which generally includes both a direction and a length. For example, an airplane’s velocity is a vector, where the vector direction shows the airplane’s direction and the vector length is the airplane’s speed. Also, the slope of a hillside is a vector, where the vector direction is the direction that is most uphill and the vector length is the steepness. A field is a physical quantity that has a value for each point in space. There are both scalar fields and vector fields. The left panel of Figure 11.2 shows a temperature map of the world’s land areas; it is a scalar field because it shows the temperature, which is a scalar value, at each point in space. The right panel of Figure 11.2 shows a map of the world’s ocean currents. This is a vector field because it shows the current speed and direction, which is a vector, at each point in space.
Figure 11.2 Temperature and ocean current maps, illustrating scalar and vector fields.
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There are often close correspondences between scalar fields and vector fields. For example, a map that shows the elevation at different points on a hill represents a scalar field, while a different map that shows the slope at different points on the same hill, perhaps to show the direction in which water would flow, represents a vector field. These two maps show essentially the same information but in different ways.
11.1.2 Electric Fields Suppose you are wearing socks and shuffling your feet across a rug on a dry day (Figure 11.3). This gives your body an electric charge because some of the rug’s electrons rub off it and go into you. These extra electrons, each of which is negatively charged, spread out within you to make your whole body negatively charged. Now, suppose there’s a bit of dust nearby that happens to have a positive charge, perhaps because you scuffed it off the rug. This dust is attracted toward you because your negative charge attracts its positive charge. The attraction between the dust and you can be interpreted in either of two ways. First, it can be interpreted as “action at a distance,” meaning that your negatively charged body exerts a force on the positively charged dust, over the distance between you and it. The force then causes the dust to move toward you. This is relatively simple to think about, but is still a little inconvenient because it means that the dust is moving because of things that are some distance away from it. The second interpretation is that the negative charge of your body creates an electric field in the space around you, shown by the red arrows in Figure 11.3. The electric field, which is a vector field, points away from any positive charges and toward any negative charges. There’s no way to see or feel this electric field, but we can tell that it’s there because of the motion of the dust particle; the dust is a tiny positive charge and it’s being pushed toward you, so the electric field at its location must point toward you. The dust only responds to the electric field at its exact location, thus removing the necessity for action at a distance. This electric field interpretation turns out to be quite useful for understanding electromagnetic waves. Figure 11.3 A person scuffing his feet across a rug. Red arrows show the electric field, the black ball is a little bit of positively charged dust, and the green arrow shows the force exerted on the dust by the electric field. This force reveals the electric field.
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As an aside, shuffling your feet across the rug makes your head and hair negatively charged, like the rest of you. Those negative charges all repel each other, causing your hair to go out in all directions.
11.1.3 Magnetic Fields Magnetic fields are conceptually similar, but with north and south magnetic poles instead of positive and negative electric charges. Again, opposites attract each other, so north poles attract south poles and vice versa. However, a key difference is that north and south poles cannot be isolated, but always occur together1 . For example, every bar magnet has a north pole at one end and a south pole at the other end. The Earth’s iron core acts like very large magnet, again with north and south poles (Figure 11.4). Confusingly, the Earth’s magnetic south pole is close to the geographic north pole (in the Arctic) and vice versa. This is because early scientists named magnetic poles so that the north pole of a magnet would point toward the north pole of the Earth, without realizing that it is the opposite poles that attract each other. In any case, the Earth’s magnet creates a magnetic field that surrounds the Earth, shown with blue arrows in Figure 11.4. It is a vector field that points away from north poles and toward south poles. As with the electric field, the magnetic field can’t be seen or felt by humans but we know it’s there because it exerts forces on magnets, such as magnetic compass needles. Figure 11.4 The Earth’s magnetic field, shown with blue arrows. The field exerts forces on the compass needle, shown in the middle, causing it to point north.
1 Isolated
magnetic poles, called magnetic monopoles, could exist in principle without violating any physical laws. However, no good evidence for their existence has ever been found.
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11.1.4 Changing Electric and Magnetic Fields Electric and magnetic effects are reasonably independent of each other, so long as nothing moves or changes. However, electricity and magnetism become tightly linked when things move. For example, if electrons move along a copper wire, then they create a magnetic field that surrounds the wire. This magnetic field is the same as that created from a bar magnet or other permanent magnet, so a device that does this is called an electromagnet. Electromagnets are the central elements of electric motors, electric generators, headphone speakers, and magnetic door locks. As another example, if electrons move through an existing magnetic field, then the magnetic field pushes the electrons sideways. This effect was used in most television sets up until the early 2000s; an electron gun at the back of the television set aimed a beam of electrons toward the inside of the screen, and then controllable electromagnets steered those electrons to the correct locations on the screen. The same effect creates auroras, also called Northern or Southern Lights, as shown in Figure 11.5. These are produced by large numbers of fast electrically charged particles, including electrons, protons, and helium atom nuclei, that are emitted by the sun in what’s called the solar wind. As these high energy particles encounter the Earth’s magnetic field, their electric charges cause them to get deflected sideways, which funnels them down Earth’s magnetic field lines toward the north and south poles. There, they excite nitrogen and oxygen molecules that re-emit the particles’ energy as auroras. Figure 11.5 An aurora seen above a house in northern Alaska.
11.1.5 Electromagnetic Waves Most of these electric and magnetic behaviors were reasonably well understood by the mid-1800s, but they were thought of as separate phenomena. In a series of papers published from the 1850s to 1870s, the Scottish scientist James Clerk Maxwell managed to combine all of these separate behaviors, plus one more that he
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discovered, into a set of four relatively simple mathematical equations (Figure 11.6). The result, called Maxwell’s equations, describe electric fields, magnetic fields, and all of the interactions among them. Figure 11.6 Maxwell’s equations. This is a photograph of a plaque that is on a statue in Edinburgh, Scotland, Maxwell’s birthplace.
Maxwell noticed that his equations expressed an important property: changing electric fields create changes in magnetic fields and changing magnetic fields create changes in electric fields. Amazingly, no physical charges or magnets were required, implying that these fields could keep on generating each other over and over again, without ever ending. Maxwell computed the speed at which these fields would propagate across space, using experimental values that had been found from studies of electricity and magnetism, and found that they would travel at about 3 · 108 m/s. This matched the speed of light, which had already been measured quite accurately (see Section 2.3.3), from which Maxwell famously concluded “We can scarcely avoid the conclusion that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena”2 . In other words, he claimed that light is an electromagnetic wave. Maxwell’s proposal that light was an electromagnetic wave seemed promising, but there remained no experimental evidence for it. This evidence was provided by Heinrich Hertz in the 1880s. He built one piece of equipment that produced radio waves using sparks and another piece of equipment that produced sparks when in an oscillating electric field. He separated these by several meters and observed that sparks in the first led to sparks in the second, in agreement with Maxwell’s predictions. Further evidence since then has thoroughly confirmed Maxwell’s explanation for light. Figure 11.7 shows a diagram of a light wave, which is based on Maxwell’s prediction and agrees with the current understanding. The light, which is shining from left to right through the figure, propagates in a straight line along the green arrow3 (it does not move in a wiggly path). Red up-and-down arrows show the electric field at
2 From a lecture at Kings College (1862) as quoted by F. V. Jones, “The Man Who Paved the Way for Wireless” New Scientist (Nov 1, 1979) p. 348. 3 More precisely, this figure shows a plane wave, in which there is a wide beam of light and all of it is propagating in the direction of the green arrow.
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different points along the light’s path at some instant of time. Likewise, blue arrows show the magnetic field at different points along the light’s path at the same instant. If we could watch the light propagate, the waves in the electric and magnetic fields would move along with the light, moving in the direction of the green arrow. Figure 11.7 Electromagnetic wave.
Note that the electric and magnetic field arrows are perpendicular to the direction of light propagation. Having these “displacements” perpendicular to the wave propagation direction means that light is a transverse wave, making it similar to a string wave but different from a sound wave. Because the electric field arrows are shown pointing up and down, this figure shows light with vertical polarization, using the standard convention that the polarization is defined by the electric field.
11.1.6 How Electromagnetic Waves Work* To create radio waves, a radio transmitter pushes electrons up and down a metal antenna at the desired frequency. Each time the electrons get pushed toward one end of the antenna, and correspondingly get depleted from the other end, this create an electric field in the space around the antenna, shown in red on the left side of Figure 11.8. This field reverses every time the electrons switch ends. In addition, the electron motion creates a magnetic field around the antenna, shown in blue, which again reverses each time the electrons change direction. These electric and magnetic fields are out of phase with each other, with the electric field strongest when the electrons are at an end, and the magnetic field strongest when the electrons are flowing through the antenna. This is not electromagnetic radiation. Instead, this is called the near field because the electric and magnetic field strengths diminish rapidly with increasing distance away from the antenna, essentially vanishing by a few wavelengths away (the wavelength is λ = c/ f , where f is the electron oscillation frequency).
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Figure 11.8 (Left) Near field radiation from an antenna, in black, where the electric field arises from charge displacement, and the magnetic field arises from steady charge motion. (Right) Far field radiation, which is only well well-established several wavelengths away, where the electric and magnetic fields arise from charge acceleration.
However, the electrons in the antenna don’t just move at a fixed speed but are constantly accelerating either up or down, which causes their electric fields to change over time. Likewise, the fact that the electrons reverse direction frequently causes the magnetic field to change over time, too. Changing electric fields create changes in magnetic fields, and changing magnetic fields create changes in electric fields, which can continue in an endless cycle, so these changing fields do propagate away from the antenna, as shown on the right side of Figure 11.8. These are electromagnetic waves, and are called the far field or radiation field. The fact that the electric and magnetic fields create each other often leads to the misconception that the wave energy goes back and forth between the electric and magnetic fields, alternating between them. To see that it doesn’t, observe that Figure 11.7 shows that the electric and magnetic fields are always in phase with each other (unlike the near field situation). The energy is stored in these fields, so both fields have high energy at once, then both have no energy at all, then both have high energy again, and so on. Thus, the energy actually rises and falls in both fields at once, rather than going back and forth between the two fields. This behavior arises from the way the fields create each other, which is that spatial variation in one field causes the other field to change over time, and vice versa,4 , shown in Figure 11.9. When the electric field decreases over space, the magnetic field must increase over time, which then moves the wave forward by a little bit. Likewise, when the magnetic field increases over space, the electric field must decrease over time, again moving the wave forward. Because the electric and magnetic fields change together, they stay in phase with each other.
4 This
description in terms of causality, in which spatial variation in one field causes the other field to change over time, is conventional and valid, but not the only explanation. An alternate view is that the two fields are simply two components of the same underlying entity, in which spatial variation of one component always occurs with temporal variation of the other component, but neither variation causes the other (See Wikipedia “Jefimenko’s equations”).
11.2 The Electromagnetic Spectrum Figure 11.9 Electromagnetic wave, showing factors that cause it to propagate.
281 electric field decreasing over space causes magnetic field to increase over time
wave propagation
magnetic field increasing over space causes electric field to decrease over time
An important aspect of the far field is that the wave energy is actually stored in the fields themselves. That is, the electromagnetic interactions have fully left the antenna and have become the propagating fields. This means that if the waves are absorbed, say by a second antenna, then the waves will be diminished but this absorption won’t feed back to have an effect on the electron motions in the first antenna. Your local radio station works in the far field, so it doesn’t need to increase its broadcasting power if lots of people are tuned into it; likewise, light bulbs emit the same amount of light regardless of how much of it gets absorbed. The near field situation is different. Here, the electric and magnetic fields are often better seen as intermediaries between interacting objects. Placing a second antenna within the near field range does impact the electron motions in the first one, giving it more load. This effect is used in smartphone touch screens which use near field feedback from a person’s finger to sensors that are inside the screen to determine where that finger is. Another interesting question is why light propagates more slowly in glass, water, or other media than in vacuum. The answer is that the electric field of a light wave pushes and pulls on electrons in the matter, causing them to oscillate. Those oscillating electrons then emit their own light waves, which is at the same frequency as the incident light, but delayed by about a quarter of a cycle. The superposition of the delayed waves with the incident waves, which have now been attenuated a little due to their interaction with the electrons, is a light wave that moves more slowly than the original one did. It still has perpendicular electric and magnetic fields, and they are still in phase with each other, but they propagate a little more slowly than before.
11.2
The Electromagnetic Spectrum
We have already met the electromagnetic spectrum several times, but this is a good place to explore it in more detail. The relevant portion of the spectrum varies continuously over about 20 orders of magnitude, going from long radio waves at one extreme to gamma rays at the other. Different regions of the spectrum interact with matter in different ways, and are used by people in different ways, leading to its division into the bands shown in Figure 11.10. These include both the main bands (radio,
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Figure 11.10 Electromagnetic spectrum. Radio frequency abbreviations: LF, MF, and HF are low, medium and high frequency, with prefixes V for very, U for ultra, S for super, E for extremely, and T for terribly. Ultraviolet abbreviations: UV is ultraviolet, UV-A, B, and C are three UV bands, and VUV is vacuum ultraviolet.
microwave, infrared, etc.) and some more specific bands5 (VHF, UHF, SHF, etc.). Different organizations tend to define the boundaries between the bands slightly differently so there is no consensus on exactly where each band starts and stops, although they are consistent about the general ranges. We’ll go through the main bands in order from long to short wavelength. Long radio waves have wavelengths of kilometers, thousands of kilometers, or longer. These very long wavelengths enable them to diffract around mountains and even the curvature of the Earth, allowing them to be detected thousands of miles away. Their long wavelengths also enable them to penetrate deep into seawater, even to the bottom of the ocean for the longest waves, making them useful for communicating with submarines. However, a downside to the long wavelengths is that very long antennas are required to transmit and receive them, which are typically inconvenient and inefficient. Also, their low frequencies mean that they can only transmit information very slowly. As a result, long radio waves are not used for much besides submarine communication. Radio waves with shorter wavelengths, normally just called radio waves, are easier to transmit and receive, and can transmit information faster than the long radio waves because of their higher frequencies. They are widely used for communica-
5 These are the International Telecommunication Union (ITU) designations, which are widely used in radio communication. Other designations are often used in other fields, such as by atmospheric and earth scientists.
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tion, with essentially all frequencies tightly regulated by the Federal Communications Commission (FCC) in the United States and other agencies elsewhere. AM radio6 uses frequencies around 1 MHz and has wavelengths of several hundred meters, while shortwave radio (used primarily for amateur radio and international broadcasting) uses higher frequencies and shorter wavelengths. Both types of waves are long enough to diffract around buildings and hills. Also, their low frequencies enable them to reflect off both the ground and the ionosphere, which is an electrically conductive upper layer in the Earth’s atmosphere, enabling them to propagate for thousands of miles while “skipping” back and forth between the ground and ionosphere. The situation changes for the higher frequency FM and VHF radio waves. Their shorter wavelengths reduce their diffraction around obstacles and their higher frequencies reduce their ionosphere and ground reflections, so they follow line-ofsight propagation. This means that they can only be detected in places where one can see the antenna that they are being broadcast from (not accounting for trees and building walls, which are typically transparent to radio waves). Line-of-sight propagation applies to all electromagnetic waves with shorter wavelengths, too. Microwaves have wavelengths in the centimeter range. Their high frequencies enable them to transfer information very rapidly, leading them to be used for a wide range of modern communications. This includes television, cell phones, Wi-Fi, and Bluetooth. They are also used for high speed data links between communication towers and satellites, often using dish-shaped antennas (Figure 11.11). Figure 11.11 Radio and microwave frequency antennas. The thin vertical antennas broadcast in or near the VHF frequency range (e.g. FM radio), the dish shape antennas, which have rain covers on them, send or receive microwave transmissions that are transmitted as focused beams, and the thick vertical cylinders are cell phone antennas.
In a much less sophisticated application, microwave ovens use high intensity microwaves that are about 12 cm long. Microwaves heat food by rotating water molecules or parts of sugar molecules, which naturally spin at roughly the same
6 AM stands for amplitude modulation, meaning that the information is transmitted by varying the radio wave amplitude. FM stands for frequency modulation, in which information is transmitted by varying the radio wave frequency.
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frequency as the microwaves (∼ 1010 Hz); the spinning molecules then transfer their energy to neighboring molecules through collisions. This microwave absorption is sufficiently weak that the waves can penetrate several centimeters into the food before being fully absorbed, causing microwave ovens to heat well into the food, rather than just at the surface like a frying pan does. On the other hand, plastic and ceramic dishes don’t have any molecules that can rotate, so they absorb very little microwave energy and thus stay cool. Yet another microwave application is in airport security “millimeter wave scanners,” which are the body scanners where you put your hands over your head. These use the fact that millimeter length microwaves penetrate clothing but reflect off people and other solid objects, enabling the scanner to effectively see your whole body, along with anything you’re carrying. The wavelengths are too long to provide detailed views of private parts, but still reveal more than some people would like. Infrared radiation frequencies are too fast for modern electronics to synchronize with the individual waves, and so are not used for communication in the same way that radio and microwaves are. Furthermore, far infrared radiation, meaning the portion of infrared that is farthest from visible light, is difficult to detect with cameras or other sensors that detect photons, because these photons have very low energies. As a result, this spectral band is sometimes called the terahertz gap (its frequencies are around ∼ 1012 Hz) due to its lack of technological application. Near infrared light, by contrast, can be detected with cameras and sensors that detect photons, allowing it to be viewed with night vision goggles, transmitted by TV remote controls, and used in fiber optic cables. Infrared radiation is sometimes called heat radiation, giving the false impression that it’s somehow different from other types of electromagnetic radiation. It isn’t, of course, but it is true that all warm objects radiate strongly in the infrared through thermal radiation (see Section 12.1). This makes infrared images good for observing how hot objects are. Infrared radiation has the same frequency range as most molecule vibrations, leading to its strong absorption by water, gases in our atmosphere, and many other substances. Visible light represents the narrow region of the electromagnetic spectrum that our eyes can see. It ranges from 400 nm, which is violet, to 700 nm, which is red. The sun, like other very hot objects, has strong thermal radiation in the visible portion of the spectrum, making Earth well-lit with visible light. Visible light is invariably observed by detecting photons, regardless of whether the detector is a human eye, a film camera, or a digital camera. Figure 11.12 shows a person’s portrait in near infrared, visible, and ultraviolet light, showing that these different bands reveal different details. Visible light frequencies are too fast for molecular vibrations and also too slow for many electron motions in atoms and molecules, so many molecules don’t absorb it. This makes air, water, glass, and most plastics transparent to visible light. However, visible light is absorbed by the relatively slow electron motions in exceptionally large molecules, such as carotene, as well as by the slow electron modes in many metal atoms, including the iron atoms that make our blood red, the magnesium atoms that make plants green, and the metal atoms that are used in most synthetic pigments.
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Figure 11.12 Portraits of a woman using infrared on the left (720–850 nm), visible in the middle (440–640 nm), and ultraviolet on the right (335–365 nm).
Ultraviolet light has shorter wavelengths than visible light. It also has higher energy photons, which causes most of its interesting behaviors. For example, “black lights” are ultraviolet lights; they emit ultraviolet photons, which are invisible to us, but that excite fluorescent molecules in our clothing or other objects. These excited molecules then emit visible light. Photons of UV-A, which is the portion of the ultraviolet that is closest to visible light, are rarely powerful enough to knock electrons completely off molecules to produce charged molecules called ions, but are still powerful enough to start many chemical reactions. However, the higher energy UV-B and UV-C photons are more likely to be ionizing radiation because they often have enough energy to create ions. This enables them to start more chemical reactions, including ones that can damage skin cells and the DNA within those skin cells. This can then lead to sunburn and skin cancer. Most sunscreen is basically just a dye that absorbs ultraviolet light. Fortunately for us, the Earth’s air absorbs most of the sun’s ultraviolet light. The sun’s highest energy photons are absorbed high in the upper atmosphere when they knock electrons off molecules to create ions, which then form the ionosphere. Lower energy ultraviolet photons are absorbed when they break apart normal oxygen molecules that have two atoms; these free atoms then recombine with other oxygen molecules to form oxygen molecules with three atoms, calledozone, which then form the atmosphere’s ozone layer. Yet lower energy ultraviolet photons get absorbed by those ozone molecules. In addition to the sun, ultraviolet light is emitted by extremely hot objects, such as lightning bolts and welding arcs, and also by specialized ultraviolet lamps. X-rays are even higher energy ionizing radiation. The lower energy “soft Xrays” are rapidly absorbed by air or pretty much anything else, but the higher energy “hard X-rays” can shine through many solid objects relatively easily. This makes them useful for X-ray imaging, such as for looking at people’s bones with medical radiography (Figure 11.13) and for looking for weapons with airport security scanners. X-ray wavelengths are around the sizes of individual atoms. This causes them to diffract strongly off regular lattices of atoms, after which they add together through superposition to produce complex diffraction patterns. Determining the lattice struc-
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Figure 11.13 X-ray image of a person’s hand.
ture from the diffraction pattern is called X-ray crystallography, which is an important scientific method for determining the atomic structures of minerals, DNA, proteins, and many other substances. Gamma rays often have higher photon energies than X-rays, but not necessarily. Instead, they differ from X-rays in that, by definition, X-rays are emitted by high energy electrons, whereas gamma rays are emitted by radioactive decay of atomic nuclei. These nuclear reactions include the natural decay of radium, a naturally occurring element on Earth, the reactions that occur in nuclear reactors, nuclear explosions, and the nuclear reactions that power the sun and other stars. This nuclear origin doesn’t affect the actual radiation, of course, so gamma rays behave the same as X-rays of similar wavelengths. As with X-rays, gamma rays are a form of ionizing radiation and they can penetrate solid objects relatively easily.
11.3
Scattering
When light hits an object, it can be absorbed, reflected, or transmitted. It can also be scattered, the focus of this section, meaning that the light gets redirected toward some new direction that’s effectively random. In general, waves interact strongly with large objects, meaning objects that are the size of a wavelength or larger, and weakly with small objects, often just passing by with no effect at all. These trends hold true for light waves as well, shown with the red and blue curves in Figure 11.14, which turns out to be responsible for most light scattering effects. As a result, scattering can be divided into three categories depending on whether the objects that scatter the light are larger, the same size as, or smaller than the light wavelengths.
11.3.1 Scattering Off Large Objects Compared to a visible light wave, a large object is anything that is about 1 µm or larger (corresponding to the right portion of Figure 11.14). This size range includes the bumps on a piece of frosted glass, such as one might find in a bathroom window,
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scattered color scattering type
Rayleigh
red Tyndall
Mie
white scattering geometric
scattering amount
1 10-2 10-4 10-6 10-8 10-10
light wavelengths blue: 450 nm red: 650 nm
1 nm
10 nm
100 nm
1 μm
10 μm
particle diameter
Figure 11.14 Scattering efficiency from particles of different sizes for red light (650 nm) and blue light (450 nm), where curves represent Mie theory. The top row shows the colors that are scattered most strongly.
the individual droplets within a cloud7 , and the crystals that make up porcelain, whether a bathroom sink or fine china. All of these objects appear white because these “large” objects scatter all colors of light equally effectively. Large objects scatter light through geometric scattering, in which the shape of the object directs the scattering. With a piece of frosted glass, for example, the glass itself is clear, but the incident light rays hit the rough glass surface at different locations, where they encounter different surface orientations, and then reflect to different directions (left panel of Figure 11.15). In reverse, when you look at a piece of frosted glass, the rays that enter your eyes were reflected from many different directions, so you don’t see distinct images in those reflections. Instead, you see many different sources of light at once, which then mix together into an average of their colors. Because the frosted glass surface scatters all wavelengths of light equally efficiently, the surface itself appears to be white. The light rays that go through the frosted glass are also scattered, in this case because they refract by different amounts, again due to the different surface orientations. In other words, frosted glass is translucent, meaning that it transmits light but one cannot see images through it.
Figure 11.15 Geometric scattering examples. (Left) Scattering off and through a rough surface, such as a frosted glass window. (Middle) Scattering from spherical particles, such as cloud droplets. (Right) Scattering from internal crystal faces, such as porcelain.
7 Cloud droplet sizes range from about 1 to 100 µm in diameter, and are about 10 to 15 µm in diameter on average.
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Clouds scatter light in a similar manner. Here, each light ray that hits a cloud droplet is both reflected and refracted, redirecting the ray to many different directions (middle panel of Figure 11.15). Some of those new rays hit more cloud droplets and get scattered again. Again, the droplets scatter all colors of light, making clouds appear white. In white porcelain, the surface is often very smooth, with bumps that are much smaller than a light wavelength, but the object still appears white. What happens is that most of the light penetrates through the surface, but then encounters many crystal faces within the porcelain, each with a different orientation. Then, as before, the light scatters repeatedly off these faces, causing the directions of the light rays to be completely scrambled by the time they finally emerge back into the air.
11.3.2 Scattering Off Medium Size Objects Scattering gets more complicated when object sizes are comparable to light wavelengths (center portion of Figure 11.14) because resonance can occur between the light wave electric fields and electric fields in the objects. These effects were explained for spherical objects by the German physicist Gustav Mie (pronounced “me”) in the early 1900s, leading to its being called Mie scattering. Typically, but not always, Mie scattering appears white. Haze in the air, typically from dust, pollutants, or pollen, causes Mie scattering (left panel of Figure 11.16). This creates the whitish color of the sky near the horizon on a clear day, and it obscures distant mountains with whitish haze. The white color of whole milk also arises from Mie scattering, in its case because the fat droplets from the cream are somewhat larger than the wavelengths of visible light (blue line in Figure 11.16).
Figure 11.16 Mie scattering examples. (Left) Haze in mountains scattering sunlight. (Right) Particle size distribution in milk with different fat contents; fat droplets make whole milk appear white through Mie scattering.
An intriguing example of Mie scattering, which does not create a white result, occurs on Mars. The Martian atmosphere has very little air but a lot of dust particles
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that range in size from about 2 to 4 µm across, making them slightly larger than light wavelengths. This size range leads to complicated Mie scattering effects that, overall, tend to scatter more red light at large scattering angles and more blue light at small scattering angles8 . The increased red scattering at large angles means that Mars rovers that look up at the sky observe a pink sky, unlike the blue sky that we have here on Earth. Meanwhile, the increased blue scattering at small angles means that the sky appears blue in the direction of the sun.
11.3.3 Scattering Off Small Objects As objects are made smaller than the light wavelength, they rapidly become much less effective at scattering light. This is shown in the left portion of Figure 11.14, which shows that the scattering efficiency decreases by a factor of 1010 over the particle size range shown, which is an enormous decrease. Any given particle is even smaller when compared to red light waves then blue ones, so it scatters even less red light. In other words, small particles scatter more blue light and thus appear blue. This scattering is called Tyndall scattering for the larger of these very small particles and Rayleigh scattering for the smallest particles, including individual air and other gas molecules, but the mechanisms are essentially the same and all create a blue appearance9 . As an example of Tyndall scattering, skim milk doesn’t have the relatively large fat droplets of whole milk, but instead has a large number of small particles called casein micelles that are roughly 100 nm in diameter (see Figure 11.16). These particles Figure 11.17 Peyto Lake, in Banff National Park, Canada. The turquoise color of the lake arises from Tyndall scattering by fine particles of rock called rock flour.
8 See: Ehlers, Chakrabarty, and Moosmüller. “Blue moons and Martian sunsets.” Applied Optics 53:1808-1819 (2014); Guzewich, Smith, and Wolff. “The vertical distribution of Martian aerosol particle size.” Journal of Geophysical Research: Planets 119:2694-2708 (2014). 9 John Tyndall was a prominent 19th century Irish scientist who was a pioneering mountaineer and glaciologist, proved the existence of the greenhouse effect, and discovered bacterial spores, among other accomplishments. Lord Rayleigh, with given name John William Strutt, was an equally prominent 19th century English physicist who figured out how we localize sound using two ears, described surface waves that are now called Rayleigh waves, and investigated thermal radiation.
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scatter more blue light than red light, because they are less small when compared to blue wavelengths, which gives skim milk a slightly bluish hue. Other examples of Tyndall scattering include the blue color of some people’s eyes and the blue color that is sometimes observed for smoke (particularly two-cycle engine smoke, such as from lawnmowers, snowmobiles, and chainsaws). Also, some glacial lakes and streams are a brilliant turquoise color due to Tyndall scattering by very fine particles of rock that are suspended in the water (Figure 11.17). Rayleigh scattering produces the blue color of the sky. In this case, the nitrogen and oxygen molecules in the air are each roughly spherical and about 0.35 nm in diameter, which is vastly smaller than a wavelength of visible light but still large enough to scatter light very slightly. We will focus on nitrogen here because it is more dominant in the air, but oxygen works in exactly the same way. When visible sunlight passes by a nitrogen molecule, the light’s electric field induces a slight polarization of the molecule, pushing the electrons toward one end and then the other (left panel of Figure 11.18). This doesn’t excite any normal modes in the nitrogen molecule because all of its resonant frequencies are much faster than the frequencies of visible light. However, the electrons’ back and forth motion causes them to emit light at the same frequency as the light wave but in a direction that’s perpendicular to the molecule axis, much as a radio antenna emits radio waves. This is the scattered light.
Figure 11.18 Rayleigh scattering. (Left) Scattering of blue light by a nitrogen molecule, but red light is unaffected. (Middle) The sun emits white light, of which the blue scatters away and is seen as the blue sky, leaving the red behind to be seen as the color of the sunset. (Right) A photograph of the sun setting over Phoenix, Arizona, showing blue sky overhead and a reddish sun.
When sunlight shines through the air above our heads, the blue components of the sunlight get scattered most strongly, some of which goes down toward our eyes (middle panel of Figure 11.18). Thus, when we look up, we see the blue light that was scattered away from its original direction and down toward us instead. After the sunlight has gone through many miles of atmosphere, which occurs whenever the sun is low in the sky, most of the blue and green components of the light have already been scattered away. This leaves just the red, orange, and yellow components, with the result that the sun appears reddish at sunrise and sunset. Thus, the red sun that we see at sunrise and sunset is a direct result of the loss of blue light to Rayleigh scattering. Sometimes, we also observe red clouds at sunrise and sunset as well. There is no new physics here. Instead, they are simply illuminated by the same red sunlight that we see at sunrise and sunset, and they then scatter that red light with geometric scattering.
11.4 Polarization
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Polarization
11.4.1 Electromagnetic Wave Polarization As with other transverse waves, electromagnetic waves can have multiple possible polarizations, including vertical, horizontal, and diagonal, as shown in Figure 11.19. By convention, this polarization is defined by the orientation of the light wave’s electric field vector.
Figure 11.19 Vertically, horizontally, and diagonally polarized waves.
Electromagnetic waves can also be unpolarized, meaning that they include a mixture of all different polarizations. Incandescent and fluorescent light bulbs, the sun, and most other light sources emit unpolarized light.
11.4.2 Polarizers A polarizer is a device that transmits waves with one polarization and blocks those with the perpendicular polarization. Figure 11.20 shows a picket fence acting as a polarizer for waves on a rope. In the left panel, the fence transmits vertically polarized waves because the rope goes up and down in a gap between the boards without interference. However, the fence blocks horizontally polarized waves, shown in the middle panel, leading to wave absorption. The fence combines both behaviors for a diagonally polarized wave, in the right panel, by transmitting the vertical component and blocking the horizontal component. Note that the resulting waves have a smaller amplitude than the incident ones and are polarized parallel to the fence boards.
Figure 11.20 Vertically and horizontally polarized waves on ropes that encounter picket fences.
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Electromagnetic wave polarizers are similar, but are made from parallel conducting wires instead of wooden boards. As always, the size scale is set by the radiation wavelength, so, for example, the wire lengths and separations need to be on the order of centimeters for microwaves and on the order of hundreds of nanometers for visible light waves. Electromagnetic waves do not actually slip between the parallel wires, as the rope does with the picket fence, but are absorbed if they drive electrons back and forth along the wires. This means that waves polarized parallel to the wires are absorbed, and waves polarized perpendicular to the wires are transmitted, exactly opposite to the situation for rope waves. Despite this difference, illustrations of electromagnetic polarizers, both here and elsewhere, invariably use lines to show the polarizer’s transmission axis (i.e. the direction of the boards in the rope and fence model). In the 1930s, an American scientist named Edwin Land figured out that he could make the tiny parallel wires that are required to polarize visible light by dissolving long conducting molecules in plastic and then stretching the plastic to orient the molecules into long parallel chains. He called the resulting material Polaroid 10 , which continues to be widely used for applications such as polarized sunglasses and LCD displays.
11.4.3 Multiple Polarizers Interesting effects can be achieved with two or more polarizers in a row11 . For convenience, assume that the first one is oriented to transmit vertically polarized light, as shown in all panels of Figure 11.21. If unpolarized light shines into this first polarizer, then it transmits the vertically polarized components of the light while blocking the remaining horizontally polarized components. If this vertically polarized light beam then encounters a second polarizer that’s also aligned vertically, shown in the leftmost portion of the figure, then the light is already aligned with the polarizer, so it just continues on through it with no additional absorption. Rotating the second polarizer a small amount, shown in the next column of the figure, reduces the polarizer’s alignment with the light, so less light gets through. Rotating the polarizer by 90◦ , shown next, creates crossed polarizers, in which the two polarizers block all of the light. The first blocks all of the horizontal components and the second blocks all of the vertical components, leaving nothing left. Remarkably, adding a third polarizer in between crossed polarizers, shown in the rightmost column, lets light through again if the middle one is at a diagonal to the other two. This makes sense by following the light polarization; it starts vertical after
10 The
Polaroid term is a trademark of the Polaroid Corporation, founded by Edwin Land, but has come into common use. 11 Technically, the first polarizer is called the “polarizer” and the second is called the “analyzer”, but this terminology can be confusing because it gives the incorrect impression that these are different devices.
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Figure 11.21 Two or three polarizers in a row, shown as diagrams in the top row and as what one sees in the bottom row. Red arrows show light polarization, and lines and white arrows show polarizer transmission axes.
the first polarizer, is at a diagonal after the middle polarizer, and then hits the last polarizer while still at a diagonal, so some light gets through the last one as well. These discussions of multiple polarizers can be made quantitative through the use of Malus’s Law, named for the early 19th century French scientist Étienne-Louis Malus. It states that the intensity of linearly polarized light that is transmitted through a polarizer is equal to I = I0 cos2 θ,
(11.1)
where I0 is the incident light intensity and θ is the angle between the light’s initial polarization and the polarizer’s transmission axis. The transmitted light’s polarization is parallel to the final polarizer’s transmission axis. For example, the two polarizers are parallel in the left column of Figure 11.21, so θ = 0◦ . This means that cos θ = 1, cos2 θ = 1, and I = I0 , implying that all of the light goes through the second polarizer, in agreement with what we saw before. In contrast, the crossed polarizer situation is represented by θ = 90◦ . In this case, cos θ = 0 and cos2 θ = 0, showing that no light goes through the second polarizer, again in agreement with prior results. It doesn’t have a name or an equation, but it’s also worth knowing that incident unpolarized light loses half of its power (and becomes polarized) when it goes through a polarizer. At least, this is true in principle and for high quality polarizers that are used in optics labs. In practice, Polaroid film actually removes substantially more than half of the light, although we will usually ignore this when doing calculations.
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Example. Unpolarized light shines through two sequential polarizers, where the second one is rotated 30◦ from the first one. What fraction of the incident light remains after the second polarizer? Answer. Half of the light goes through the first polarizer. Then, use Malus’s law for the second polarizer, using the fact that its polarization axis is rotated 30◦ relative to the light’s current polarization, I 3 = cos2 θ = cos2 30◦ = . I0 4 Combine these two fractions to get the final result: 21 of the incident light goes through the first polarizer, and 43 of the remaining light goes through the second polarizer, which multiply to give the result that 38 , or 37.5%, of the incident light remains at the end.
11.4.4 Liquid Crystal Displays Liquid crystal displays, commonly known as LCDs and shown in Figure 11.22, are widely used in modern electronic displays, including those on most calculators, laptop computers, and overhead projectors12 . In essence, they are based on the example of three polarizers in a row, but have better light transmission and use an electronically controllable middle layer. The middle layer, the liquid crystal, is a gooey solution of long electrically conducting molecules that act as a polarizer, as in Polaroid plastic. Importantly, they preferentially arrange themselves so that they are all lined up, parallel to each other. OFF state: molecules align to ends Light rotates and is transmitted
ON state: molecules align to electric field Light does not rotate and is absorbed
+
+
+
-
+ +
+
-
-
-
Figure 11.22 Diagram of an LCD display.
12 This
figure continues with the convention of showing polarization that aligns with linear molecules, whether in polarizers or the LCD crystal. However, in reality, all polarization arrows should be rotated by 90◦ because a linear molecule absorbs light that is polarized along the molecule’s axis, and transmits the perpendicular polarization.
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In the OFF state, meaning that no electric field is applied, these molecules align themselves to tiny grooves that are etched onto the inside faces of the crossed polarizers. The grooves on the front polarizer are rotated by 90◦ from those on the back polarizer, so the liquid crystal molecules rotate smoothly from one polarizer to the other. When light goes through this from back to front, it is polarized by the back polarizer, and then rotated gradually as it goes through the liquid crystal, getting to 90◦ rotation by the end; this rotation works the same as the middle polarizer in the three-polarizer example, except that this one loses much less light because it rotates the light continuously rather than in just a few steps. At this point, the light is aligned with the transmission axis of the front polarizer, so the light goes on through, and the viewer observes brightness. In the ON state, an electric field is applied across the two polarizers, which re-aligns the liquid crystal molecules to point toward the front or back of the display instead of crosswise. In this orientation, they don’t rotate the light polarization, so light that enters the back polarizer is then blocked by front polarizer. No light gets through, so the viewer observes darkness. Dividing the LCD up into tiny independently-controllable rectangular pixels, and adding red, green, and blue color filters on the pixels, completes the components of a color display.
11.4.5 Sources of Polarized Light Shining unpolarized light onto a polarizer, as described above, produces polarized light through a process called selective absorption, because one polarization is transmitted and the perpendicular one is absorbed. Polarized light can also arise through other mechanisms. First, spatially oriented wave sources often produce polarized waves. For example, vertical radio antennas emit vertically polarized radio waves. This is because the upand-down electron motion in the antenna creates creates up-and-down electric fields in the surrounding space, which then propagate away as vertically polarized waves. These waves can then be received by other vertical antennas, because the waves’ vertical electric fields are aligned correctly to drive electrons up and down in those antennas as well. Individual molecules emit and absorb light in essentially the same manner, again with the polarization parallel to the direction of the molecule’s electron motion. However, this is rarely observable because each molecule in a sample tends to have a random orientation, so the effects of the different molecules usually add up to remove any overall polarization effect. For example, the hot molecules in a candle’s flame have all different orientations, so each individual photon is polarized along the axis of the molecule that emitted it, but the total light is unpolarized13 .
13 The mathematics of this orientational averaging is complicated, but described in Andrews, “Using
rotational averaging to calculate the bulk response of isotropic and anisotropic samples from molecular parameters.” Journal of Chemical Education 81:877 (2004).
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On the other hand, there are also cases where randomly oriented molecules can emit polarized light, of which the Rayleigh scattering that produces the blue sky is a notable example. Figure 11.23 shows how this occurs when the sun is close to the horizon, focusing on the three principal molecule orientations. Sunlight, which is Figure 11.23 Blue sky is polarized due to Rayleigh scattering. Nitrogen molecules with the three basic orientations are shown, of which only one scatters sunlight toward the ground, so this orientation sets the light polarization direction.
unpolarized, is not absorbed by molecules that are oriented parallel to the sun’s rays, because the light’s electric field is perpendicular to the rays. However, vertically oriented air molecules are able to absorb sunlight. These molecules don’t contribute to the blue sky that is seen by an observer on the ground though because they scatter the sunlight in directions that are perpendicular to their own orientations, which is not toward the ground. Finally, horizontal molecules that are perpendicular to the sun’s rays are also able to absorb sunlight, and they also scatter some of that light toward the ground, so they are the ones that are primarily responsible for the blue sky. This scattered light is polarized parallel to the axes of these molecules, making the blue sky polarized. This effect occurs over much of the sky, but is strongest at 90◦ away from the sun. People can only detect this polarization when wearing polarized sunglasses, but bees, wasps, migratory songbirds, mantis shrimp, and many other animals are able to see it directly and use it for navigation14 . Another way that light becomes polarized is through low angle reflection off smooth non-metallic surfaces. For example, sunlight glare off a wet road, reflections from windows, or reflections from the surface of a calm pond are all substantially polarized (Figure 11.24). Here, some of the incident light is transmitted into the new medium, meaning the glass or water, and some is reflected off it. It turns out that light that is polarized parallel to the surface has greater reflection, while light polarized perpendicular to the surface has greater transmission, which means that the reflected light tends to polarized parallel to the surface. Water surfaces are
14 With
training, many people actually can see polarization through a faint vision effect called Haidinger’s brush. Animals that see polarization can usually only detect it for blue light, and using parts of their eyes that are oriented upwards. In cases where polarization can be sensed, it is generally detected with oriented light receptors within the eyes.
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Figure 11.24 Two photographs of the same rock, taken with a polarizer filter. (Left) Filter transmits horizontally polarized light. (Right) Filter transmits vertically polarized light.
typically horizontal, so polarized sunglasses are designed to block glare by absorbing horizontally polarized light. Remarkably, there is a particular angle of incidence, called Brewster’s angle, where all of the light that is polarized perpendicular to the surface is transmitted (it’s about 53◦ for the interface between air and water15 ). At this angle, the reflected light is perfectly polarized. This can be seen near the middle of the right panel of Figure 11.24, where there is essentially no glare at all.
11.4.6 Polarization as Superposition* The superposition principle for waves, which we met in Section 3.1.1, says the displacements from multiple separate waves add together to give the total displacement for all of the waves at once. When applied to polarized waves, it implies that a vertically polarized wave can be added to a horizontally polarized wave, where these two waves have the same wavelength and phase, to create a diagonally polarized wave. This is shown on the left side of Figure 11.25. By extension, it is possible to describe waves with any possible polarization angle as a weighted sum of the vertical and
+ = Figure 11.25 (Left) Vertically and horizontally polarized waves adding to create diagonal polarization. (Right) Circularly polarized light wave.
15 Brewster’s
angle, named for the British scientist Sir David Brewster (1781-1868), is given by θ B = arctan(n 2 /n 1 ), where n 1 is the refractive index for the incident ray and n 2 is the refractive index for the refracted ray. See Section 9.2.3.
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horizontal “basis polarizations”. All of these waves are linearly polarized because the electric field vectors lie in a single plane, whether vertical, horizontal, or diagonal. Now suppose that there is a 90◦ phase shift between the vertical and horizontal waves that are added together. In this case, the total electric field goes up, then forward, then down, then backward, then up, and so on, around in a circle, which is called circular polarization. The electric field strength is always the same, but its direction rotates in a circle around the light’s direction of propagation, as shown on the right side of Figure 11.25. More precisely, the figure shows a right circularly polarized wave, because the electric field rotates clockwise over time as seen by an observer looking back along the ray toward the source16 . Vice versa, counterclockwise rotating waves are left circularly polarized. Although not shown here, it is possible to add left and right circularly polarized waves to create vertical, horizontal, or other linearly polarized waves. Thus, the two circular polarizations can be used as the basis polarizations, rather than the vertical and horizontal polarizations. Which set is better depends on the problem that needs to be solved.
11.4.7 Birefringence and Optical Activity* Some materials have different refractive indices for vertically and horizontally polarized light, which causes the two light polarizations to travel at different speeds. These so-called birefringent materials are typically crystals with asymmetric structures, of which calcite (calcium carbonate) is particularly common. The left side of Figure 11.26 shows a calcite crystal on a sheet of graph paper, showing that the background appears twice through the crystal, with one image for each light polarization. Birefringent materials are widely used for advanced optics applications, such as to separate the two polarizations of light more efficiently than is possible with Polaroid. Plastic is often birefringent as well, in its case because it is composed of very long molecules that tend to get stretched out in parallel directions during manufacturing or when the plastic is stressed later on, and the oriented molecules slow the two light polarizations by different amounts. This birefringence is observable by placing the plastic between crossed polarizers. Suppose the first polarizer transmits diagonally polarized light, which can also be described as equal portions of vertically and horizontally polarized light. Then, these two polarizations go through the plastic at different speeds and add back together again at the end, but with a phase shift due to the different transmission speeds. This phase shift changes the total polarization (toward being circular or even the perpendicular linear polarization, depending on the amount of phase shift), which then allows some of the light to go through the
16 This convention of defining the circular polarization direction as that seen by an observer looking
back along the light path is standard in most fields, including optics and chemistry, but is not universal. In particular, quantum physicists typically use the opposite convention.
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second polarizer. Often, the amount of phase shift depends on the light wavelength, creating bright colors in the final result, as shown on the right side of Figure 11.26.
Figure 11.26 (Left) A crystal of calcite showing two images of the background. (Right) Protractor and stencil in crossed polarizers.
Yet other materials exhibit circular birefringence, meaning that they have different refractive index values for right and left circularly polarized light, which is often called either optical activity or optical rotation. These materials rotate the plane of linearly polarized light; linearly polarized light is the sum of left and right circularly polarized light, and these two polarizations propagate through the material at different speeds, so they add back together at the end with a phase shift that is observed as a rotation of the linear polarization. As with the birefringent crystals described above, circular birefringence also arises from an asymmetry in the material. In this case, it arises from molecules that don’t have reflection symmetry, meaning those that are chiral (see Section 8.6). Most biological molecules are chiral, including sugars, proteins, DNA and others. For example, glucose (corn syrup) only exists naturally in its right-handed form; as a result, it’s optically active, happening to rotate linearly polarized light in a clockwise direction17 . Placing an optically active material between crossed polarizers enables light to get through, this time because the chiral molecules rotate the angle of the light polarization so that it’s no longer blocked by the second polarizer. Circular birefringence is often used in chemistry labs to test the purity of chiral molecules.
17 The
physical causes of optical activity are complicated, arising from the interactions of both the light’s electric and magnetic fields on the chiral molecules. It’s related to circular dichroism, which is the differential absorption of left and right circularly polarized light and is described in Andrews and Tretton, “Physical principles of circular dichroism.” Journal of Chemical Education 97: 4370-4376 (2020).
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Summary
Electromagnetic waves are propagating electric and magnetic fields, each of which is a vector field. The electric field represents the force that would be exerted on a small positively charged object, and the magnetic field represents the rotational force that would be exerted on a small compass needle. Electric fields can be created from electron displacements, and magnetic fields from steady electron motion, but these represent the near field, which diminishes rapidly with distance, and are not electromagnetic radiation. On the other hand, electron acceleration leads to changes in each field, which then creates changes in the other field; these changes continue indefinitely and are electromagnetic radiation. The electric and magnetic fields in electromagnetic waves are transverse to the propagation direction and are in phase with each other. The electromagnetic spectrum includes: long radio waves, radio waves, microwaves, infrared light, visible light, ultraviolet light, X-rays, and gamma rays, listed in order of both increasing frequency and decreasing wavelength. All of these bands are electromagnetic waves, with identical physics, but they interact with matter in different ways and have different applications. Radio and microwaves are widely used for communication, with radio better for long distances and microwaves better for high data rates; both are detected with electronics that synchronize with the waves. Near infrared is also used for communication but is too high frequency for synchronous detection, so is detected as photons instead. High energy ultraviolet light, X-rays, and gamma rays are ionizing radiation, meaning that they can knock electrons off atoms or molecules. This enables them to damage biological molecules, such as DNA, potentially leading to cancer. Light scatters efficiently off objects that are as large or larger than the wavelength, and less efficiently for smaller objects. Thus, large objects, such as the bumps on ground glass, cloud droplets, and porcelain crystals, scatter all wavelengths of light and appear white, which is called geometric scattering. Objects that are close to the light wavelength, such as haze in the air, often appears white as well but can also be blue or pink depending on the precise size; this is called Mie scattering. Objects that are smaller than a light wavelength scatter less efficiently, but scatter blue more than red due to the wavelength difference, leading to a blue appearance. This is called Tyndall scattering for larger objects and Rayleigh scattering for smaller ones, leading to the blue color of smoke, glacial lakes, and the sky. Most light sources emit unpolarized light. A polarizer, such as a sheet of Polaroid, can then polarize the light by blocking one polarization and transmitting the perpendicular polarization. With two polarizers, the second one transmits all of the light incident on it when it is aligned with the first one, transmits less when rotated some, and blocks all light when rotated by 90◦ . This is quantified by Malus’s law, I = I0 cos2 θ.
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Adding a third polarizer between crossed polarizers lets light through again because the light gets a new polarization after the middle polarizer. LCD displays are based on this latter result, using a controllable middle polarizer. Polarized light can also arise from polarized emission (e.g. a vertical radio antenna), Rayleigh scattering, or low angle reflection off non-metallic surfaces. Linearly polarized waves can be described as a sum of the vertical and horizontal basis polarizations. These basis polarizations can also add to create left or right circularly polarized light, in which the electric field goes around in a circle. Alternatively, the circular polarizations can be the basis polarizations, which then add to create other polarizations. Birefringent materials have different refractive indices for different light polarizations.
11.6
Exercises
Questions 11.1. An airport security officer discovers that someone’s suitcase is emitting substantial amounts of gamma radiation. What might be inside? (a) a chemical explosive (b) a computer that’s overheating (c) something that’s radioactive (d) a live animal (e) a two-way radio that’s turned on 11.2. What would you be able to see if you had soft X-ray vision (Assume there’s enough X-ray light to see) (a) just the air (b) pretty much the same as with visible light (c) you could see through walls, but people would still appear fully dressed (d) everyone would appear to be naked (e) all you’d see of people would be their bones 11.3. What would you be able to see if you had hard X-ray vision? (Assume there’s enough X-ray light to see) (a) just the air (b) pretty much the same as with visible light (c) you could see through walls, but people would still appear fully dressed (d) everyone would appear to be naked (e) all you’d see of people would be their bones
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11.4. Many kitchen appliances have a “brushed metal” surface, in which the metal is first polished to make it shiny and then roughened to create grooves and ridges that are about 0.5 to 1.5 µm high. What type of light scattering occurs here? (a) Rayleigh scattering (b) Mie scattering (c) Metal scattering (d) Tyndall scattering (c) No light is scattered 11.5. Old fashioned television sets had “rabbit ear” antennas that stuck up out of the top of the TV. Sometimes, the TV would receive its signal better if these antennas were tilted to new angles. Why did this help? (a) It aligned the antennas with the wave polarization (b) It reduced wave scattering (c) It improved resonance between the antenna and the waves (d) It decreased wave emission from the antennas (e) All of the above 11.6. Which of the following waves are substantially polarized? Select all that apply. (a) Sunlight (b) Sunlight after passing through a polarizer (c) Sunlight reflected off a mirror (d) Sunlight reflected off a water surface (e) The blue sky 11.7. Which of the following waves can be made polarized? Select all that apply. (a) Red laser light (b) White sunlight (c) Long radio waves (d) Infrared light from a hot stove (e) A “black light” (ultraviolet light) 11.8. Aluminum foil has a shiny side and a dull side. The surface roughnesses18 of the two sides are about 184 nm and 28 nm. (a) Which of these values corresponds to the shiny side? (b) Explain how you can tell. 11.9. Draw a diagram of an electromagnetic wave which shows: (a) electric field vectors, (b) magnetic field vectors, (c) an arrow that shows the direction of wave propagation, and (d) the wavelength.
18 These values were measured parallel to the rolling direction, and are from C. Sinagra, F. Bravac-
cino, C. Velotti, “Aluminium foil: focus on the surfaces”, packmedia.net webmagazine, 11/10/2016.
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11.10. Snow blindness is essentially a sunburned cornea, which is painful but recovers with time. Why would this be more likely to occcur when wearing cheap sunglasses that reduce visible light but don’t block UV light, rather than not wearing sunglasses at all? 11.11. Would it be safe to look directly at the sun while wearing a pair of sunglasses that blocked all UV light? Why or why not? 11.12. Why are you told to wear a lead apron when your dentist takes X-rays of your teeth? 11.13. Give examples of (a) a scalar quantity, (b) a vector quantity, (c) a scalar field, (d) a vector field. 11.14. (a) Why is the sky blue? (b) Why are sunsets red? (c) Why are clouds white? 11.15. If the Earth’s atmosphere were 50 times denser than it is, what color would the sun appear when it’s overhead? 11.16. Suppose you travel to high altitude, such as to the top of a high mountain or flying in an airplane. Will the sky overhead appear lighter blue or darker blue than normal? Problems 11.17. List the following electromagnetic bands in order of (a) increasing frequency, (b) increasing wavelength, (c) increasing photon energy: FM radio, microwave, shortwave, gamma ray, visible. 11.18. Which band is wider, measured as the ratio of the highest frequency to the lowest frequency: visible light (390 to 700 nm) or AM radio (530 to 1700 kHz)? 11.19. When light reflects off a horizontal water surface (e.g. a wet road), its electric field is mostly parallel to the water’s surface. (a) To reduce glare, should your polarized sunglasses transmit light with a vertical or horizontal electric field? (b) Would wearing a second pair of polarized sunglasses at the same time reduce the glare even more, have essentially no further effect, or be completely opaque? 11.20. A store has a rack of sunglasses for sale, all of which it claims are polarized. Your friend doesn’t believe that they are actually polarized. Describe one way in which you can test whether the sunglasses are polarized or not.
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11.21. What percent of unpolarized light is transmitted through (a) two parallel polarizers? (b) two crossed polarizers? (c) Three polarizers, of which the second is rotated 45◦ from the first and the third is rotated another 45◦ (i.e. the first and third are crossed polarizers and the second is at a diagonal between them)? Puzzles 11.22. The following two photographs (of Colorado’s Yampa River) differ in that they were taken with filters that transmitted different light polarizations. (a) Which used a filter that transmitted horizontally polarized light, A or B? (b) Is the predominant polarization of the sky’s Rayleigh scattering horizontal or vertical? (c) Approximately where is the sun (e.g. close to horizon behind the photographer, close to the horizon to the left, nearly overhead, etc.)?
11.23. Recalling that a standing wave in a cavity is the same thing as two traveling waves propagating in opposite directions, (a) draw at least 3 diagrams of the electric and magnetic fields of a standing electromagnetic wave, showing them at different times in the cycle. (b) Are the electric and magnetic fields in phase or out of phase? (c) Where is the energy when the electric field has its minimum values? (d) Where is the energy when the magnetic field has its minimum values? 11.24. Mars’s atmosphere has very little air, but a lot of dust particles that range from 2 to 4 µm across. These dust particles are made of the same red minerals that form the surface of Mars. The mid-day sky is pink and the setting sun is blue, shown in the photograph below. The text says that these observations arise from Mie scattering, but could the same results arise from light absorption by the red colored dust particles instead? Explain.
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11.25. Aristotle claimed that looking at the daytime sky from the bottom of a very deep well blocks out the blue scattered light, making stars visible just as they are at night time. Does this make sense or not? Explain.
Thermal Radiation
12
Figure 12.1 The sun, imaged in the extreme ultraviolet at 30.4 nm.
Opening question Which is hottest? (a) Red-hot wires in a toaster (b) A bluish-white welding arc (c) A yellow candle flame (d) The surface of the sun (e) The surface of Jupiter © Springer Nature Switzerland AG 2023 S. S. Andrews, Light and Waves, https://doi.org/10.1007/978-3-031-24097-3_12
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Thermal radiation is electromagnetic radiation that is emitted by all objects, such as the sun, the walls of a room, and even an ice cube. Warmer objects emit more light but cooler objects emit light, too. The emission spectrum of thermal radiation is remarkably simple, depending only on the object’s color when it is cold and the object’s temperature, but not the object’s composition. This spectral simplicity is fascinating because it hints at the presence of underlying unifying principles in physics, one of which turns out to be the quantization of radiation. However, we leave that to the next chapter, focusing here on the phenomenology of thermal radiation. Thermal radiation is essential to life on Earth. Nearly all of the energy on Earth arises from sunlight; the sun warms the Earth, enables plants to grow, and, eons ago, grew the plants that turned into today’s fossil fuels. Meanwhile, the Earth also cools off by emitting radiation to space. The balance of these radiation fluxes helps determine the Earth’s temperature, whether on the short-term, such as how cold it gets on a clear night, or on the long-term, such as the coming and going of ice ages.
12.1
Thermal Radiation
12.1.1 Qualitative Trends Anyone who has watched the glowing embers of a fire, such as those in Figure 12.2, is well aware of two correlations among heat, color, and brightness. First, the color of the embers shows their temperature, where cold ones are black, warm ones are red, hot ones are orange, and very hot ones are nearly white. Second, the hotter embers glow more brightly than the cooler ones. We see this brightness difference visually as bright and dim embers. We also feel this brightness difference directly, by feeling the primarily infrared radiation as more or less radiative heat. Figure 12.2 Burning embers in a fire, showing that hotter embers are both whiter in color and brighter overall.
These same trends occur for thermal radiation from all objects, which represents the conversion of some heat energy in an object to emitted electromagnetic waves.
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12.1.2 Blackbodies Thermal radiation is simplest when it’s emitted from objects that are black when cold, which are called blackbodies. To be a perfect blackbody, an object needs to be totally black, meaning that it absorbs all incident light, of all wavelengths. This ideal is hard to reach, but most dark colored objects, including everything from black T-shirts to cast iron frying pans, behave as reasonably good blackbodies. Even water, which is transparent in visible wavelengths, acts as a good blackbody in the infrared, where it does absorb essentially all light. A piece of charred wood isn’t quite perfect, but is a good image of a blackbody to have in mind. Although not exactly an object, a hole that leads to an enclosed cavity can also behave as an excellent blackbody. Consider, for example, looking into the spout of an empty teapot (with the lid on). The hole looks black, regardless of the teapot’s internal color, because all light that shines into the hole gets absorbed somewhere on the inside of the teapot and so doesn’t return. As a result, the hole in the spout absorbs all incident light and hence acts as a blackbody that’s just as good as any black object. Figure 12.3 shows the spectra of light emitted by blackbodies, called blackbody radiation, at different temperatures. It shows the two effects described above. The peaks of the emission spectra move toward shorter wavelengths as the temperature increases, showing the trend toward whiter emitted light for hotter objects. Also, the area under each curve increases with higher temperatures, showing greater emission as objects get hotter. Figure 12.3 Blackbody radiation spectra at three different temperatures.
12.1.3 Wien’s Displacement Law The color of blackbody radiation can be expressed by focusing on how the peak of the emission spectrum varies with temperature. This relationship is described by the equation λmax =
b T
b = 2.898 · 10−3 m K.
(12.1)
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where λmax is the peak emission wavelength, b is a constant with the above value, and T is the absolute temperature in Kelvin1 . The b constant is expressed in SI units, as usual, which means that λmax gives the light wavelength in meters. This equation is called is called Wien’s displacement law, after Wilhelm Wien, a German physicist, who figured it out in 1893. Wien’s displacement law shows that the wavelength of peak emission moves to shorter wavelengths with higher temperatures, in agreement with common experience. Vice versa, the wavelength of peak emission gets longer for cooler objects, often moving into the infrared, or even to microwaves or radio waves for very cold objects. Most hot objects that we are familiar with, from candles to light bulbs, have their emission peaks in the infrared region. Wien’s displacement law can also be inverted to give the temperature from the peak emission wavelength, giving T = b/λmax . This form is useful for calculating the temperature of an object from its spectrum. For example, (pronounced Beetlejuice) is a red giant star that forms a shoulder of the Orion constellation. Spectral measurements show that its emission spectrum peaks in the near infrared at around 805 nm; plugging this into Wien’s displacement law shows that its surface temperature is about 3600 K. It’s impressive that we can determine the temperatures of stars that are billions of miles away by simply seeing what color they are. Example. A person’s skin temperature is about 91F. What is the wavelength of peak emission? Answer. First, convert the temperature from Fahrenheit to Celsius using the relationship ◦ C = 59 (◦ F − 32◦ F), and then from Celsius to Kelvin with T (K) =◦ C + 273.15, ◦
C=
5 ◦ (91 F − 32◦ F) = 33◦ C 9
T (K) = 33◦ C + 273.15 = 306 K. Thus, a person’s skin temperature is 33◦ C, or about 306 K. Next, use Eq. 12.1 to get the peak emission wavelength, λmax =
b 2.898 · 10−3 m K = = 9.5 · 10−6 µm. T 306 K
1 The absolute temperature scale, measured in Kelvin units, was introduced in Section 6.3.2. The b constant relates to the fundamental physical constants as b ≈ 0.201405 khcB , where h is Planck’s constant, c is the speed of light, and k B is Boltzmann’s constant.
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This shows that the peak wavelength for a person’s thermal radiation is about 9.5 μm, which is in the mid-infrared region of the spectrum. All people have “black” skin in the infrared, meaning that they absorb essentially all infrared light and they emit as blackbodies.
12.1.4 Color Temperature The correlation between temperature and color leads to the concept of a color temperature, which is the temperature of a blackbody as a function of its visual appearance. Figure 12.4 shows the color temperature scale for temperatures from 0 K to 10,000 K, showing that cold blackbodies are black but that they then transition through red, orange, yellow, white, and eventually blue as they are made hotter.
Figure 12.4 Color temperature scale, along with the approximate effective temperatures of some objects. Objects in upright text are thermal emitters, while those in italics emit through other mechanisms.
Many familiar light sources emit light through thermal radiation, some of which are listed on the diagram. For example, incandescent light bulbs emit light from hot metal filaments that have temperatures around 2700 K, so they emit yellowish orange light. Halogen light bulbs use hotter filaments, so they emit whiter light. The surface of the sun is even hotter, at around 5788 K, and emits white light. The color temperature scale can also be applied to light that is emitted through nonthermal mechanisms. For example, fluorescent and LED light bulbs produce light through quantum effects rather than thermal radiation, so their light color doesn’t correspond to their actual temperatures (also, their spectra differ from the blackbody spectra shown in Figure 12.3). However, their colors are still typically described as particular color temperatures. As examples, LED light bulbs that emit about the same color light as traditional incandescent light bulbs are typically labeled as having a 2700 K color temperature, while those that emit a whiter light are often labeled as having a 4200 K color temperature. Even the blue sky can be assigned a color temperature, typically between 10,000 to 20,000 K depending on the amount of haze. This implies that its color is the same as what would be emitted by a blackbody emitter with that temperature, despite the fact that the sky’s blue color arises from Rayleigh scattering, not thermal radiation.
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Confusingly, people often describe indoor lighting with terms that trend in the opposite direction as color temperature, describing redder lights as “warmer” and bluer lights as “cooler”. For example, a “warm white” light bulb emits yellow light and has a color temperature of about 3000 K whereas a “cool white” light bulb has a hotter color temperature, emitting white light with a color temperature of around 4000 K.
12.1.5 Stefan-Boltzmann Law The other important correlation for thermal radiation is that hotter objects emit more total radiation than cooler ones. The Austrian physicists Josef Stefan and Ludwig Boltzmann quantified this relationship in 1884, about a decade before Wien’s work, finding that the total radiative power emitted by a blackbody is P = AσT 4
σ = 5.67 · 10−8 W m−2 K−4 .
(12.2)
In this equation, called the Stefan-Boltzmann law, P is the total emitted power in units of watts, A is the surface area of the object, σ is the Stefan-Boltzmann constant, and T is the absolute temperature2 . The units on σ don’t make much intuitive sense, but work out correctly when σ is multiplied by the other terms in the equation. Looking at the Stefan-Boltzmann law, the emitted power is seen to be proportional to the surface area. This makes sense because the atoms on the surface produce the radiation, so it makes sense that more surface atoms would create more radiation. The other important factor in the Stefan-Boltzmann law is its T 4 dependency. This shows that the amount of emitted light depends very strongly on the temperature, with hot objects emitting quite a lot more radiation than cool objects. For example, suppose you start with a hot cup of coffee. Initially, its high temperature causes it to radiate a lot of power, so it cools off quickly. Later on, when it’s only lukewarm, it radiates much less power, so it cools more slowly and hence stays lukewarm for a long time. As another example of the Stefan-Boltzmann law, Alnilam is a blue supergiant star that forms the middle of Orion’s belt. Its surface area is about 1000 times larger than the sun’s and its temperature is about 4.8 times hotter. The Stefan-Boltzmann law has area to the first power and temperature to the fourth power, so Alnilam’s total emission is 1000 × 4.84 times greater than the sun’s, which is a factor of about 530,000. Thus, Alnilam is about a half million times brighter than the sun (however, it doesn’t appear exceptionally bright in the sky because it’s a long way away).
2 See
σ=
Section 4.1 to review energy and power. Also, σ relates to fundamental physical constants as
2π 5 k 4B 15h 3 c2
where k B is Boltzmann’s constant, h is Planck’s constant, and c is the speed of light.
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Example. A cast iron pot is removed from a 450◦ F oven. Its surface area is 0.25 m2 and it is an excellent blackbody. How much power does it radiate? Answer. Converting from Fahrenheit to Celsius and then to Kelvin (see the prior example) shows that 450◦ F = 232◦ C = 505 K. Plugging the pot surface area and this temperature into the Stefan-Boltzmann law shows that the emitted power is P = AσT 4 = (0.25 m2 )(5.67 · 10−8 W m−2 K−4 )(505 K)4 = 922 W. This is about the same amount of power as is emitted by a standard toaster.
12.1.6 Remote Temperature Measurement* A pyrometer is a device for measuring an object’s temperature remotely, meaning that the instrument doesn’t contact the object. There are several varieties, but all estimate temperature by assuming blackbody radiation and then measuring either the emitted color or the total radiative power. A spectral pyrometer, widely used for determining the temperatures of objects in furnaces and pottery kilns, measures the spectrum of the emitted light and then matches it with known blackbody spectra for different temperatures. A simple version of this is the “disappearing filament pyrometer” (Figure 12.5) in which a wire filament is heated up until it seems to disappear because its color matches that of the object in the furnace. At this point, the filament and object have the same temperature, so knowledge of the filament’s temperature yields the temperature of the object. Figure 12.5 Illustration of a “disappearing filament pyrometer”, in which the filament temperature is adjusted until its color matches the background color.
Infrared thermometers, such as medical ones that are used to scan a person’s forehead to see if they have a fever or not, use a different approach. They measure the total amount of infrared radiation that is emitted from a known portion of the surface area. Then, using the Stefan-Boltzmann law, they compute the surface’s temperature from the radiation power. These are better than spectral pyrometers for relatively cool objects, such as people, because it’s easier to measure total amounts of infrared light than infrared spectra.
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Thermal Radiation Interactions
12.2.1 Radiation Coupling and Emissivity Thomas Wedgwood, a young man from an illustrious British family3 , published an important scientific paper about thermal radiation in 1792. He held black and silver objects in a hot crucible and found that the black ones heated up to red-hot temperatures faster than the silver ones. Then, he removed the objects from the crucible and found that the black ones cooled off more quickly. These results didn’t depend on the material, but only on the objects’ colors. Although he didn’t understand it at the time, these results arose from the facts that (1) black objects absorb more radiation than silver objects, which is probably familiar to most of us, and (2) black objects emit more radiation than silver objects, which is probably less familiar. As a modern example of the same phenomenon, consider a black car and a silver car. When left in the sun, shown on the left side of Figure 12.6, the black car absorbs all wavelengths of visible light and, presumably, most wavelengths of infrared and ultraviolet light. Light is energy, so it gets very hot inside. The silver car, by contrast, reflects most of the sun’s visible light and some of the infrared light, so doesn’t get as hot. However, the situation reverses during a clear night, shown on the right side of the figure. Then, there is no sunlight to heat the cars up, so the energy transfer goes the other way, with the cars radiating heat away from themselves as infrared light and cooling off in the process. The black car emits more radiation, so it cools off faster. Meanwhile, the silver car emits less radiation and cools off more slowly4 .
Figure 12.6 Black and silver cars during the day and at night. The black car absorbs more sunlight during the day, so it gets hotter. It also emits more radiation at night, making it colder.
3 Thomas Wedgwood was the son of the famous potter Josiah Wedgwood and uncle to the naturalist Charles Darwin; Thomas later went on to be a pioneer in photography. The paper referred to here is Wedgwood, T. “Continuation of a paper on the production of light and heat from different bodies” Philosophical Transactions of the Royal Society of London 82 (1792): 270–282. 4 While instructive and largely valid, this description is also simplistic. At night, the emission differences are smaller than one might expect because most cars have similar finishing coatings, giving them similar infrared emissivity. Also, this description ignores conductive and convective heat transfer with the surrounding air, which are not affected by paint color.
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We can connect these effects to material that we covered in Section 4.3.1 by saying that black objects couple strongly to radiation, with high energy transfer, whereas silver objects couple poorly to radiation, with much less energy transfer. Coupling always acts equally for energy transfer in both directions. Here, this means that strong coupling, meaning a black color, creates both high absorption and high emission, while weak coupling, meaning a silver color, leads to both low absorption and low emission. The emissivity, given by (Greek letter epsilon), quantifies the coupling between an object and radiation. It ranges from 0 for a perfect reflector, or a perfectly transparent object, to 1 for a perfect blackbody. If light is shining on an object, the amount of radiation that gets absorbed is a factor of less than the incident light, Pabsor b = Pincident .
(12.3)
The remainder of the incident light, which has fraction 1 − , is either transmitted through the material or reflected off it. Due to the reversibility of coupling, the same relationship applies to emission. Here, an object emits a factor of less light than it would if it were a perfect blackbody, Pemit = Pblackbody = AσT 4 .
(12.4)
The first part of this equation shows how coupling affects emission, while the second part substitutes in the Stefan-Boltzmann law for blackbody radiation. In fact, this latter equation can be considered as the complete Stefan-Boltzmann law, leaving Eq. 12.2 as the special case for perfect blackbodies. Although not shown here, the emissivity typically depends on wavelength, so one needs to use the emissivity for the correct wavelength range. For example, snow is white in visible wavelengths, giving it a visible emissivity that’s close to 0, but is black over most of the infrared, giving it an infrared emissivity that’s close to 1.
12.2.2 Two-Way Thermal Radiation Objects don’t just absorb radiation when they’re cold and emit radiation when they’re hot, but they both absorb and emit radiation all the time. For example, consider a blackboard on a classroom wall. It constantly emits radiation, so one might expect it to lose energy and get cold. However, it doesn’t get cold because it also absorbs radiation from the classroom walls, ceiling, floor, furniture, and people. It absorbs and emits the same amount of power, so it stays at a constant temperature. The same principle applies to everything else in the room too. This insight leads to a useful technique, which is that we can determine how much radiation an object absorbs from its environment by calculating how much power it would emit if it were at the environment’s temperature. These two values are always equal to each other. This technique even works if the object has a different temperature from the environment, because its temperature doesn’t affect how much energy it absorbs.
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Back to the blackboard example, we can also go a step farther by completely removing the blackboard and the wall behind it to leave a giant hole in the wall. In that case, the amount of radiation that leaves the room through the hole in the wall would be the same as the radiation that the blackboard would emit if it were there. Two-way thermal radiation explains the phenomenon that cloudy nights tend to be warmer than clear nights. The Earth’s surface radiates about the same amount of heat away in both cases. If there are clouds, then the clouds absorb this radiation, warm up, and radiate the energy back to the Earth’s surface, which keeps the surface warm. However, on clear nights, more of the radiation emitted by the Earth’s surface simply continues away into space, so the surface cools more. As a practical matter, these considerations of two-way radiation enable us to use the Stefan-Boltzmann law to calculate the power that is emitted from a hole in an object, such as the spout of a teapot. The equation is P = AσT 4 , as always, but now A represents the cross-sectional area of the spout and T represents the temperature of the inside of the pot. Example. A room with a temperature of 68◦ F has an open window that has an area of 1 m2 . How much radiation goes out the window? Answer. Compute the radiation that would be emitted by this 1 m2 if it were a blackbody rather than an open window. Converting temperature units shows that 68◦ F = 20◦ C = 293 K. Substituting numbers into the Stefan-Boltzmann law gives the emitted power as P = AσT 4 = (1.0 m2 )(5.67 · 10−8 W m−2 K−4 )(293 K)4 = 418 W. The same amount of power would have to be absorbed by the blackbody for its temperature to stay constant, implying that 418 watts hits this surface. But it’s actually an open window, so 418 watts of radiation goes out the window.
12.3
Earth’s Climate*
12.3.1 Earth’s Energy Budget* Temperatures on Earth, or indeed on any of the planets, can be computed remarkably accurately from little more than Wien’s displacement law (Eq. 12.1) and the StefanBoltzmann law (Eq. 12.2). This is worked out in detail in some problems at the end of this chapter. In brief, Wien’s displacement law can be used to show that the sun’s surface temperature is about 5778 K. Substituting this into the Stefan-Boltzmann law, along with values for the radius of the sun, Earth, and the Earth’s orbit, gives the result that 341 watts of sunlight reaches each square meter of the Earth’s surface, on average. See Figure 12.7, which labels this energy input as “Incoming solar radiation.” About 30% of this, which is called the Earth’s albedo, gets reflected back
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317
into space by white clouds, snow, light-colored desert sand, and reflections off the oceans5 . The remaining 239 W/m2 gets absorbed by the Earth’s atmosphere and surface, warming the Earth in the process. Global Energy Flows W m-2 102 Reflected Solar Radiation 101.9 W m-2 Reflected by Clouds and Atmosphere 79
Incoming Solar Radiation 341.3 W m-2
341
239
79 Emitted by Atmosphere 187
Outgoing Longwave Radiation 238.5 W m-2
Atmospheric 22 Window 30 Greenhouse Gases
Absorbed by 78 Atmosphere 17
Latent 80 Heat
Reflected by Surface 23
374
22
333
Downwelling
Radiation 161
Absorbed by Surface
17
80
Thermals Evapotranspiration
396 Surface Radiation
333 Absorbed by Surface
Net absorbed
0.9 W m-2
Figure 12.7 Earth’s global energy flows. This shows the global annual mean Earth’s energy budget for the March 2000 to November 2005 period, with all values in W/m2 . Republished from Trenberth and Fasullo, Surveys in Geophysics, 33:413–426, 2012.
The Earth then emits the same amount of energy back to space through thermal radiation; it has to be very nearly the same amount because the planet as a whole is staying roughly the same temperature from day to day, without getting dramatically hotter or colder over time. From the Stefan-Boltzmann law, emission of 239 W/m2 implies that the Earth’s average temperature must be about 255 K, which is about −18◦ C. This prediction is consistent with temperatures in the atmosphere, where most of the outgoing radiation comes from. However, the Earth’s surface is substantially warmer than this, with an average temperature of about 15◦ C. This warming is produced by greenhouse gases in the atmosphere, including primarily water vapor, carbon dioxide, and methane, which act as a blanket. These greenhouse gases absorb most of the outgoing radiation from the Earth’s surface and then re-radiate it, with some going out to space and the rest returning back to the surface. The returned radiation, labeled as “Downwelling radiation” in Figure 12.7 and often called back radiation, warms the Earth up, before being emitted yet again. This warming of the Earth through the blocking of infrared
5 The
“albedo” and “emissivity” terms are typically used in different contexts, but represent essentially the same thing; the albedo equals 1 − .
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radiation by the atmosphere is called the greenhouse effect 6 ; it warms the Earth by about 33◦ C above what the temperature would be if we didn’t have an atmosphere. The greenhouse effect is a very good thing, at least from the perspective of people and most of Earth’s other current inhabitants, because it makes the Earth habitable for us. Without it, most of Earth’s water would be frozen and life as we know it could not survive.
12.3.2 Greenhouse Effects on Mars and Venus* Mars and Venus have greenhouse effects as well. The Martian atmosphere is composed primarily of carbon dioxide so, like on Earth, its atmosphere is transparent to visible light but mostly opaque to infrared light. Nevertheless, Mars’s greenhouse effect7 is weak because its atmosphere is very thin, with only about 1% as much total air as we have here on Earth. Without the greenhouse effect, Mars would be about −76◦ C, which is colder than the Earth due to Mars’s increased distance from the sun. The greenhouse effect warms Mars only about 5◦ C, up to a chilly −71◦ C. This is about the temperature of Earth’s South Pole in winter during a cold spell. Venus, in contrast, has a very strong greenhouse effect. Current research suggests that Venus may have been similar to Earth billions of years ago, with moderate temperatures, oceans of liquid water, continents of dry land, and possibly even snow in places8 . It’s conceivable that there was life on Venus’s surface at that time, although no evidence has been found yet to support that. However, at some point within the last billion years, volcanos likely released large quantities of greenhouse gases into the atmosphere, which warmed the planet. This initial warming was then amplified through positive feedbacks. First, the warmer oceans evaporated to water vapor, which both thickened the atmosphere and added more greenhouse gases. This heated up the rocks, which released carbon that turned into carbon dioxide; this again thickened the atmosphere and added greenhouse gases. Together, these processes led to a runaway greenhouse effect, which now heats the planet by about 500◦ C above what it would be without an atmosphere. This creates an average surface temperature of about 464◦ C, making it the hottest planet in the solar system9 . Venus’s surface is hot enough to evaporate water instantly, decompose any biological molecules, and melt
6 The
greenhouse effect term comes from the notion that greenhouses are warm because their glass walls transmit visible light but block infrared light. However, research has shown that actual greenhouses primarily stay warm by trapping air, which keeps it from moving away and getting replaced by cooler air. 7 Haberle, Robert M. “Estimating the power of Mars’ greenhouse effect” Icarus 223:619–620. 8 Ernst, Richard, “Venus was once more Earth-like, but climate change made it uninhabitable” The Conversation, Dec. 13, 2020. 9 Venus is closer to the sun than the Earth but actually absorbs less sunlight because it has a very high albedo, of 75%. As a result, Venus would be colder than the Earth if not for its greenhouse effect.
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lead; even carefully designed spacecraft haven’t survived for more than a couple of hours on Venus’s surface. It is tempting to be judgmental about these greenhouse effects, calling them either good or bad. However, they arose from natural physical processes, making them neither good nor bad but just facts of the solar system. If there is life on Mars or Venus, which seems extremely unlikely but not completely impossible, then that life has undoubtedly adapted to the local greenhouse conditions and so relies on those conditions for its survival. On the other hand, the greenhouse conditions in both places largely preclude the possibility that people could ever live on either Mars or Venus. Mars is far too cold for humans or any of the plants and animals that we rely upon, while Venus is far too hot. There has been some discussion of terraforming Mars so that it would be habitable by people, part of which would require giving it an atmosphere with a greenhouse effect. However, this is almost certainly impossible because Mars wouldn’t retain an atmosphere, even if it were given one. This is both because its gravity is too weak to hold it for long and because the planet doesn’t have a magnetic field that would shield the atmosphere from high energy solar particles.
12.3.3 Global Warming* Global warming is an increase of the average Earth’s surface temperature that has happened primarily over the last 100 years. It arises from a slight enhancement of the greenhouse effect due to human-caused emissions of greenhouse gases, largely arising from effects of the industrial revolution. In particular, carbon dioxide is a greenhouse gas produced by the burning of fossil fuels (coal, oil, gasoline, natural gas, etc.), deforestation, and cement production. Methane is another major greenhouse gas. It is produced in natural and artificial wetlands (artificial ones include rice paddies and rivers that have been dammed to create reservoirs), by oil wells, and by livestock. Methane is generally described as a stronger greenhouse gas than carbon dioxide because it blocks more of the infrared light that would otherwise escape the Earth. On the other hand, carbon dioxide is being produced faster and has a longer lifetime in the Earth’s atmosphere, making it more important for climate change. Research on ancient climatic conditions has shown a remarkable correspondence between atmospheric carbon dioxide and temperature over the past 800,000 years, a time frame that encompasses the entirety of human existence and then some, as shown in Figure 12.8. The carbon dioxide concentration data were collected by measuring the gas composition of bubbles in Antarctic ice that fell as snow thousands of years ago. The temperature data were collected by measuring the ratio of hydrogen isotopes in the same ice samples, where an isotope refers to the mass of an atom. In this case, water with heavier hydrogen atoms requires higher temperatures to evaporate and so precipitation that includes heavier hydrogen atoms is indicative of a warmer climate. The rises and falls in the graphs show several ice ages and warming periods. The current carbon dioxide concentration, at the right side of the figure, is much higher than it has been at any time during the past 800,000 years,
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suggesting that the temperature will rise to levels that have been unprecedented over the past 800,000 years as well. Of course, this is the first time that carbon dioxide has been released through industrial production, so one might imagine that the close correlation between carbon dioxide and temperature might not continue. However, scientific research doesn’t support this possibility, showing instead that global temperatures are rising rapidly and will continue to rise for several centuries10 .
Figure 12.8 Carbon dioxide concentrations (blue), global average temperature (red), and ice ages (gray) over the past 800,000 years. Human evolution dates are rough estimates, but included to give a sense of scale. Carbon dioxide data are from Lüthi et al., Nature 453:379, 2008 and temperature data are from Jouzel et al., Science 317:793, 2007. See the figure credits for details.
As with Venus’s greenhouse effect, Earth’s global warming is accompanied by positive feedback mechanisms. For example, as the climate warms, more snow melts in polar regions, which increases the amount of sunlight that gets absorbed and warms the Earth more. Also, what was permanently frozen ground in the arctic is starting to thaw, which is releasing methane to the atmosphere and adding greenhouse gases that create more warming. In addition, many forest regions are becoming drier, leading to more forest fires and lower carbon uptake by trees. Nevertheless, it is extremely unlikely that anthropogenic carbon emissions will create a runaway greenhouse effect on Earth, as occurred on Venus, because there simply aren’t enough readily available fossil fuels to start a runaway condition. Also, the Earth has been much warmer in the distant past, reaching up to 15◦ C warmer than the present 50 million years ago11 , and it obviously cooled off successfully since then.
10 These
are central results of the Intergovernmental Panel on Climate Change (IPCC), which publishes reports every few years, starting in 1990. These reports present a comprehensive review of climate research and are based exclusively on peer-reviewed scientific literature. They are available at https://www.ipcc.ch. 11 Westerhold et al. “An astronomically dated record of Earth’s climate and its predictability over the last 66 million years” Science 369:1383–1387, 2020.
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In addition to the Earth generally being warmer, climate change has many other impacts. It is reducing ice and snow cover in both polar regions and mountains. Melting polar ice from Antarctica and Greenland is raising sea levels, while the snow loss in mountains reduces an important drinking water supply in many parts of the world (e.g. the American West and South Asia). Sea levels are also rising because the water in the oceans expands as it warms up. Climate change is also changing regional climates, leading to more intense tropical storms, hotter heat waves, and, overall, wetter regions getting wetter and drier regions getting drier. The warmer Earth is also changing ocean chemistry because warmer water can dissolve less oxygen, thus enabling the ocean to support fewer fish. Although not arising from global warming directly but from increased atmospheric carbon dioxide, the ocean is becoming more acidic because carbon dioxide dissolves into seawater where some of it turns into carbonic acid. The Earth has warmed and cooled many times in the past, leading to a succession of ice ages and interglacial periods. Those temperature changes are not fully understood but are known to have been caused in part by variations in the Earth’s atmosphere by volcanic and biological influences, periodic changes in the Earth’s orbit around the sun due to the influences of other planets, which changed the incoming solar radiation, and the motion of Earth’s tectonic plates, which changed global air and water flow patterns. Life on Earth survived those changes and will undoubtedly survive the current anthropogenic climate change as well. However, the three prior greenhouse phases in the Earth’s atmosphere coincided with mass extinctions12 , so it is likely that the current one will as well. In fact, we are already in a mass extinction phase, called the Holocene extinction event, but, so far, it has been caused primarily by habitat loss, invasive species introductions, pollution, and over-harvesting rather than climate change. It is yet to be seen how climate change will contribute to this extinction event and how humans will fare.
12.4
Summary
All objects emit thermal radiation. This radiation depends on temperature in two important ways. First, hotter objects emit light at shorter wavelengths. This is quantified for blackbodies, which are objects that absorb all incident light, according to Wien’s displacement law, b . T It also gives rise to the concept of a color temperature scale, which represents the temperature of a blackbody that has some particular color and transitions from black λmax =
12 Mayhew, Peter J., Gareth B. Jenkens, and Timothy G. Benton, “A long-term association between global temperature and biodiversity, origination and extinction in the fossil record”, Proc. R. Soc. B 275:47–53, 2008.
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to red, orange, yellow, white, and blue, for increasing temperature. Second, hotter objects emit more total radiation. This is described by the Stefan-Boltzmann law, P = AσT 4 . Pyrometers are devices that measure objects’ temperatures remotely, using either the color of the emitted light or the total amount of radiation. Black objects couple more strongly to radiation than silver objects, both absorbing and emitting more light. The emissivity, , describes this coupling by representing the fraction of radiation that an object absorbs or emits, relative to a perfect blackbody. Black cars couple strongly to radiation, so they both heat up and cool off faster than silver cars. An object that is the same temperature as its surroundings must absorb the same amount of radiation as it emits, which is helpful for calculating how much radiation an object absorbs from its surroundings. Planetary temperatures can be calculated reasonably accurately using little more than Wien’s displacement law and the Stefan-Boltzmann law. Differences between these results and actual average surface temperatures arise primarily from the greenhouse effect, in which particular gases in the atmosphere transmit visible sunlight to the surface but absorb out-going infrared radiation. The greenhouse effect warms the Earth’s surface by about 33◦ C, which is essential for the survival of life as we know it. Mars has a very weak greenhouse effect and Venus has a very strong greenhouse effect, in both cases making those planets uninhabitable by humans. Global warming is an enhancement of the Earth’s greenhouse effect that is caused by anthropogenic emissions of greenhouse gases, including primarily carbon dioxide and methane. It is causing many changes, including polar ice melting and sea level rise.
12.5
Exercises
Questions 12.1. List the following in order from coldest to hottest, according to their actual temperature: (a) the surface of a red giant star (b) a bluish-white welder’s arc (c) the surface of the sun (d) a blue light-emitting diode (LED) (e) a black car sitting in the sun 12.2. List the following in order from coldest to hottest in terms of color temperature, not actual temperature: (a) a “warm white” LED light bulb (b) a “cool white” fluorescent light bulb (c) a red traffic light (d) the blue sky (e) the surface of the sun
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12.3. The Lockheed SR-71, named “Blackbird” for its black color, was the fastest airplane ever made, with typical speeds over Mach 3. Air friction at these speeds caused the plane’s surface to reach temperatures over 500◦ C. What would have been some good reasons for painting the plane black? (Choose all that are appropriate.) (a) To keep the pilot warm (b) To keep the pilot cool (c) For camouflage at night (d) Black paint reflects radar (e) It makes the plane quieter 12.4. By comparing the colors of burning wood in Figure 12.2 with the color temperature scale in Figure 12.4, estimate the temperature range for the burning embers. 12.5. (a) Which will get hotter when sitting in the sun: a black cat or a white cat? (b) Which will get colder when sitting outside on a clear night: a black cat or a white cat? 12.6. (a) What color roof will keep a house cooler on a hot summer day, black or white? (d) What color roof will lose less heat on a cold winter night, black or white? 12.7. The United States Department of Energy launched a “Cool roof initiative” in 2010 that required new and re-roofed Department buildings to use highly reflective roofing materials. Because of this policy, the United States National Nuclear Security Administration switched to cool roofs and then found that the switch reduced its heating and cooling costs by about 70%, enabling them to save about $500,000 per year13 . Explain how using highly reflective roofing materials can reduce both heating and cooling costs. 12.8. New houses are often built with styrofoam insulation in the walls that’s covered on one side with shiny aluminum foil. Explain why this foil is used, given that this insulation is on the inside of the wall, where it’s never seen? 12.9. Two hikers get lost in the desert and have to spend the night. One wraps himself in a large black garbage bag (with a hole for his head) and the other wraps herself in a “space blanket,” which is a large sheet of shiny mylar film. Who will stay warmer? 12.10. Europa is a rocky moon that orbits Jupiter. Would you expect it to be hotter or colder than our Moon? (Neither moon has any significant atmosphere.) Explain.
13 https://energy.gov/articles/secretary-chu-announces-steps-implement-cool-roofs-doe-and-
across-federal-government
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12.11. The military sometimes uses infrared cameras to locate enemy soldiers at night. Why is it hard to camouflage a person’s infrared emission? Discuss some camouflage approaches that might work, at least partially. 12.12. You pour yourself a hot cup of coffee and are about the add some cream when the phone rings. Will the coffee be hotter after the phone call is over if you add the cream now, or if you wait until after the phone call to add it? Problems 12.13. Consider a piece of pottery in a kiln that is at 1200◦ C. (a) What is the temperature of the pottery in Kelvin? (b) What is the wavelength for peak emission by the pottery? (c) What type of light is this (e.g. infrared, visible, or ultraviolet)? (d) Does the peak emission wavelength get shorter or longer as the kiln is made hotter? 12.14. Consider the same piece of pottery, still in a kiln at 1200◦ C, and assume that it has a surface area of 100 cm2 . (a) What is the temperature of the pottery in Kelvin? (b) How much total power does it radiate (units should be in W). (c) Does the emitted radiant power increase or decrease as the kiln is made hotter? 12.15. Consider a 100 W incandescent light bulb which has a temperature of 2700 K. What is the surface area of the filament in mm2 ? (Hint: solve the StefanBoltzmann equation for the area.) 12.16. Sirius is a bluish-white star in the Canis Major constellation (close to Orion). The peak of its emission spectrum is at about 292 nm. What is the temperature of the surface of the star? 12.17. Consider 1 square meter of ground that is covered by snow, on a 0◦ C day. Assume the snow’s visible emissivity is 0.05 and infrared emissivity is 0.95, and that the incident sun intensity (assumed to be all visible light) on this snow is about 100 W. (a) How much power does the snow absorb? (b) How much power does the snow emit? (c) Will these radiative influences warm or cool the snow overall? 12.18. Suppose a man, who is dressed in just a swimsuit, has a skin temperature of 91◦ F, a surface area of 2 m2 , an emissivity of 1, and is in a 68◦ F room. (a) How much power does he radiate? (b) How much radiant power does he absorb? (c) How much more radiation does he emit than absorb? (d) How many food calories does he burn in an hour, simply by radiating infrared light? 12.19. An ice-cold glass of lemonade, which is at 0◦ C and has a surface area of 170 cm2 , is sitting in a room that’s 30◦ C. How much more radiation power does the lemonade absorb from the room than it emits?
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12.20. Hypersonic weapons are unmanned military planes that fly at speeds over Mach 5. They are hard to detect by ground-based radar because they fly only tens of kilometers above the Earth’s surface, in contrast to ballistic missiles, which fly hundreds of kilometers high. This question investigates whether hypersonic weapons can be detected by spy satellite. The nose cone of a hypersonic weapon, which we’ll assume has a surface area of 1 m2 , gets heated to about 2000 K due to friction with air molecules. (a) What is the wavelength of peak emission? (b) How much power does the nose cone radiate? (c) Assuming that a spy satellite’s resolution is around 100 m2 , meaning that each pixel in its photograph represents this much area on the Earth’s surface, how much power does the Earth, at 300 K, radiate from 100 m2 ? (d) How many times brighter or dimmer is the hypersonic weapon (assume the spy satellite detects all light wavelengths equally well)? (e) Would this suggest that a spy satellite could see the weapon or not? Explain. 12.21. Black holes are collapsed stars. Gravity is so strong around them that nothing can escape the edge of the black hole, called the “event horizon,” including even light. However, whenever a pair of particles forms just outside of the event horizon, it’s possible for one to fall in and the other to fly outward, with the net result that black holes actually radiate energy. This energy turns out to 23 be identical to a blackbody of temperature T = 1.2·10 M , where M is the black hole mass in kg and T is the black hole temperature in kelvin. (a) Compute the temperature of a black hole that has one solar mass (M = 1.99 · 1030 kg), giving the answer in kelvin. (b) What is the wavelength of peak blackbody emission? (c) Calculate the radius of a 1 solar mass black hole’s event horizon, using the relationship R = 1.48 · 10−27 M, where R is in meters and M in kg. (d) How much total power does a 1 solar mass black hole radiate? 12.22. The universe began in the Big Bang about 14 billion years ago and immediately started expanding and cooling. About 378,000 years later, it had cooled to about 3000 K, at which point electrons and protons combined to form neutral atoms, and light could then travel long distances without constantly getting absorbed and re-emitted. This light, which had the spectrum of a 3000 K blackbody, still exists but has been redshifted dramatically due to continued universe expansion so that it now has the spectrum of a 2.73 K blackbody. It is called the cosmic microwave background, fills all space, and is seen uniformly in all directions. (a) What is the peak wavelength of this radiation? (b) Suppose a satellite has a surface area of 100 m2 ; how much energy does it absorb from microwave radiation? 12.23. Procyon B is a white dwarf star in the Canis Minor constellation. Its mass is 1.20 · 1030 kg, its temperature is 7740 K, and it emits 1.88 · 1023 W of power. (a) What color is it? (b) What is its radius? (c) What is its density in kg/m3 ? (d) What is the mass of 1 cm3 of this star?
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The Earth’s energy budget 12.24. The solar constant. (a) The surface of the sun is about 5778 K and its radius is about 696,000 km. How much total power does the sun emit? (b) Imagine an entire sphere placed around the sun that has the radius of the Earth’s orbit (149,500,000 km). What is the surface area of this sphere, in m2 ? (c) How much of the sun’s power goes through each square meter? This value is called the solar constant. 12.25. Projection effect. The total amount of sunlight that hits the Earth is the solar constant times the area of the Earth that is visible to the sun. However, we want to average this over the entire Earth, which is done here. (a) Defining the Earth’s radius as R E , give an equation for the area of the Earth that is visible to the sun, Avisible (hint: it’s just the area of a circle). (b) What is the surface area of the entire Earth, Atotal as a function of R E ? (c) What is the ratio of Avisible to A Ear th , simplifying as possible? This is the fraction of the solar constant that hits each square meter of the Earth’s surface, on average. (d) Multiply this ratio by the solar constant from the last problem to get the average incoming sunlight on each square meter of the Earth. 12.26. Earth albedo. From the last problem, you should have found that average incoming sunlight is about 342 W/m2 . About 30% of this light reflects off clouds, the oceans, snow, and light colored sand without being absorbed, called Earth’s albedo. (a) What is the power of the reflected light for each square meter? (b) What is the power of the absorbed light for each square meter? 12.27. Earth infrared emission. In the last problem, you should have found that the Earth absorbs about 239 W of sunlight per square meter. The Earth is extremely close to being at steady state, meaning that it is neither gaining nor losing energy. Thus, each square meter of the Earth’s surface must also emit about 239 W of energy. (a) What is the temperature, in K, of a square meter that emits about 239 W? (b) What is this in Celsius? (c) What is wavelength of maximum emission for light from the Earth? 12.28. The greenhouse effect. In the last problem, you should have found that the Earth’s temperature would be about −18◦ C if it didn’t have a greenhouse effect. However, the actual average temperature is actually about 15◦ C. (a) How many degrees warmer is the Earth than what it would be if it didn’t have an atmosphere? (b) How much power is radiated from 1 square meter of the Earth’s surface, using the actual Earth surface temperature? (c) How much power does 1 square meter of the atmosphere send back to the Earth’s surface? To calculate this, assume that the atmosphere isn’t gaining or losing energy; if it’s losing 239 W to space, from above, and it’s gaining the amount of energy that you calculated in part b of this problem, then the difference must be the amount of energy that it sends back to Earth’s surface. (d) What fraction of the energy emitted from the Earth’s surface gets reflected back to Earth (i.e. the ratio of the two prior numbers)?
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12.29. The following figure is a diagram of the numbers that you computed in the last several problems. (a) Copy it and write your numbers next to the arrows as appropriate. (b) Compare your results to Figure 12.7; which numbers there are essentially identical to the ones that you calculated? (c) What processes are included in 12.7, which you ignored?
12.30. Global warming. You should have found above that about 39% of the radiation that the Earth emits is sent back to the Earth, meaning that about 61% of the energy escapes to space. Suppose the atmospheric reflectivity increased by 2%, so now 41% of the light returns to Earth and 59% goes to space. (a) Now, how much radiation goes from Earth to the atmosphere (the outgoing radiation is still 239 W from above, so divide this by 59%)? (b) What Earth temperature would produce this much outgoing radiation, in ◦ C? (c) By how many degrees did this atmosphere change warm the Earth? Puzzles 12.31. How much total radiation energy is in an empty teapot? Give the answer in terms of its temperature T , volume V , the Stefan-Boltzmann constant σ, and the speed of light c. The teapot’s emissivity is 1. For convenience, assume it is a spherical cavity with radius r and assume that all light that is emitted from one side travels distance 2r before being absorbed again at the far side. 12.32. The Moon is about as far from the Sun as the Earth is, so the average temperature on the Moon should be about the same as the average temperature on the Earth but without the greenhouse effect. However, the Moon rotates very slowly, so there is a large difference between day and night sides. Calculate the temperature on the Moon on the equator at mid-day in ◦ C. Assume that visible radiation comes in from the sun and infrared radiation gets emitted to a hemisphere of sky (ignore any radiation arriving from the sky). The Moon’s albedo is about 12%, assume that it emits as a perfect blackbody, and use a solar constant value of 1362 W/m2 . 12.33. Model the Venusian atmosphere as some number of layers, where each layer absorbs all of the IR light from below and then emits half of the energy upward and half of the energy downward. Consider 1 square meter of the surface. (a) Define the emission of the top layer to space as P1 , where this is layer number 1; how much power does one layer below it, layer 2, emit upward as a function of P1 ? (b) How much power does the third layer emit upward? (c) Write an
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equation for Pn . (d) The surface is at 462 ◦ C, so how much power does it emit upward? (e) Combine these last two results to solve for n, which is the number of layers; use the fact that P1 = 791 W.
Part IV Modern Physics
Photons
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Figure 13.1 A nanodiamond is trapped by a beam of light. (J. Adam Fenster).
Opening question What is a photon? (a) A particle of light (b) A packet of energy (c) A particle that moves at the speed of light (d) A massless object that has momentum (e) All of the above © Springer Nature Switzerland AG 2023 S. S. Andrews, Light and Waves, https://doi.org/10.1007/978-3-031-24097-3_13
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By the late 1800s, the wave description of light successfully explained diffraction and interference, agreed with research on electricity and magnetism, and had been thoroughly confirmed with radio wave experiments. However, it couldn’t explain a few seemingly minor problems, one of which was the shape of the blackbody radiation spectrum. As it turned out, solving this problem led to a completely new concept of light, and then of everything else, too. The new concept was that light is both waves and particles at the same time. This makes little intuitive sense but, if one is willing to accept a little suspension of disbelief, it explains many things. It explains why ultraviolet light produces sunburns, how light can change frequencies in some situations, and how light can exert forces. This chapter and the following one explore quantum mechanics, which is the physics of photons, atoms, and molecules. It is a bizarre topic but almost all aspects of it have been figured out by now. Remarkably, experiments that test quantum mechanics invariably show it is correct, even when it makes predictions that seem utterly impossible.
13.1
The Quantum Revolution
13.1.1 Explaining Blackbody Radiation In 1874, a teenage student in Germany named Max Planck was discouraged from studying physics by his advisor, who told him that theoretical physics was essentially fully solved, so there weren’t many interesting problems left1 . Indeed, many important things had already been figured out, including Newton’s laws of motion and Maxwell’s unification of electricity, magnetism, and light propagation. Nevertheless, a few puzzles still remained. For example, why don’t electrons fall into atoms? Why do atoms only absorb light with particular frequencies? Why can some chemical reactions be started with ultraviolet light but not visible light? And, of a more technical nature, what determines the shape of the blackbody radiation spectrum? Planck ignored his advisor’s advice and studied physics anyhow. He was successful, gradually rising through the ranks of German academia, but was not particularly notable for many years. Eventually, the blackbody radiation spectrum became a popular topic to investigate, with important work by Stefan, Boltzmann, Wien and others (see Chapter 12). Planck investigated it as well. He invested several years of work
1 From: Max Planck: Wege zur Physikalischen Erkenntnis. Reden und Vorträge, Band 1. Leipzig 1943, page 128, quoting his advisor, Philip von Jolly. In the original, “die theoretische Physik nähere sich merklich demjenigen Grade der Vollendung, wie ihn etwa die Geometrie schon seit Jahrhunderten besitzt,” which translates to “theoretical physics is noticeably approaching that degree of perfection that geometry has had for centuries”; accessed in German language Wikipedia “Philip von Jolly”.
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on it, with many failures, but finally achieved major success by finding an equation that fit the blackbody spectrum perfectly, shown in Figure 13.2 with a black line2 . Figure 13.2 Blackbody radiation spectrum for 5000 K. Shading represents the experimental result. The solid line represents Planck’s result, which agrees with experiment perfectly, and the dashed line represents the Rayleigh-Jeans law.
However, this success was marred by the fact that neither Planck nor anyone else found his result satisfying because it was based on what seemed like an absurd assumption. He had assumed that light energy could only be absorbed or emitted in discrete amounts; for example, an object could absorb or emit 1, 2, 3, or any other integer units of light, but it could not absorb or emit non-integer amounts, such as 1.5 or 2.3 units of light. Nothing in Maxwell’s equations of electricity and magnetism required integers at all, so this assumption didn’t seem reasonable. Because the problem still wasn’t solved adequately, more physicists continued to join the fray. In 1905, five years after Planck’s theory, the British physicists Lord Rayleigh and James Jeans came up with a theory that was particularly well-founded on the understanding of light at the time and which, naturally, did not include Planck’s assumption. Their work, called the Rayleigh-Jeans law, agreed with experiment at low light frequencies but turned out to be grossly in error at high frequencies (dashed line in Figure 13.2). This error, later dubbed the ultraviolet catastrophe because of the large errors for ultraviolet frequencies, showed that Maxwell’s equations were fundamentally incomplete. Thus, some sort of new physics was required and Planck’s assumption of discrete units of light, absurd as it seemed, was the only known solution. Albert Einstein was one of the first people to suggest that light discreteness was real, rather than something to be explained away. He also went further, claiming that discreteness wasn’t just something that happened when light was absorbed or
3
equation is B = 2hc2f (eh f /k B T − 1)−1 , where B is the light intensity, h is Planck’s constant, f is the light frequency, c is the speed of light, k B is Boltzmann’s constant, and T is the temperature. The spectrum in Figure 13.2 is the same as the 5000 K spectrum shown in Figure 12.3 but looks different for several reasons: the x-axis shows frequency rather than wavelength, both axes use log scaling, and the brightness is scaled to be proportional to Hz−1 here rather than nm−1 as in the prior figure. 2 Planck’s
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emitted, but that light itself is actually a stream of discrete particles. His view was that each particle of light is a separate little packet of energy. As is usual for Einstein’s work, more recent results have confirmed his understanding. His proposed particles of light are now called photons.
13.1.2 Particles and Waves Light is now well understood to be both particles and waves at the same time, which is called particle-wave duality. This isn’t just a political compromise to satisfy both particle and wave believers, and nor are there different kinds of light, but instead all electromagnetic radiation is always both particles and waves at the same time. The wave aspects of light are exactly as we saw them before (Section 11.1.5). To summarize, light is electromagnetic waves, with perpendicular electric and magnetic fields. Each field changes over time due to spatial variation in the other field, causing the fields to oscillate back and forth over time. These changing fields oscillate with a particular frequency ( f ), have a specific wavelength (λ), propagate at the speed of light (c), and carry energy. The particle aspects of light are the focus of this chapter.
13.1.3 What is a Photon? Simply put, a photon is a small packet of pure energy. It is also a single particle of light. Photons travel at the speed of light, because they are light. Also, each photon has a color, wavelength, and frequency that matches the corresponding light waves. Additionally, photons have polarizations, including vertical and horizontal, or rightand left-handed, just like their wave counterparts3 . On the other hand, photons are not matter and have no mass. To be more precise, photons have no rest mass, meaning that if a photon could be stopped, which is impossible because they always travel at the speed of light, but if one could be stopped, then it would be found to have no mass. However, despite having no mass, photons do have momentum. For example, when you stand in the sunlight, the light actually pushes on you, extremely gently. A final attribute of photons is their physical size, which is surprisingly complicated and discussed below. Table 13.1 summarizes these properties. There’s no good way to draw a picture of a photon, although Figure 13.3 makes an attempt and provides a useful mental image. It illustrates a green photon as a short wave packet, which is a short burst of waves.
3 Individual photons can be polarized along the horizontal or vertical axis, just like light waves. They can also be in a superposition of these states, giving them right or left handed circular polarizations, again like light waves. In the latter cases, the photons are said to have a spin, which is similar to the spinning of a conventional particle, but is really a purely quantum phenomenon.
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Table 13.1 Properties of photons. Physical property
Do photons have it?
Photon value
Energy Speed Color Wavelength Frequency Polarization Mass Momentum Size and shape
Yes Yes Yes Yes Yes Yes No Yes Undefined
E = h f = hc λ Speed of light is c Same as light waves Same as light waves = λ Same as light waves = f Same as light waves Photons don’t have rest mass p = λh See Section 13.4.2
Figure 13.3 Illustration of a vertically polarized green photon as a wave packet that moves at the speed of light. The vertical axis shows the electric field, the axis going into the page shows the magnetic field, and the shaded region indicates either the photon size or the probability of where it might be found, depending on the interpretation.
13.2
Photon Energy
13.2.1 Planck-Einstein Relation A photon’s energy is typically its most important attribute. It is given by the PlanckEinstein relation, which states that the energy of a single photon is given by E = hf
h = 6.626 · 10−34 J s.
(13.1)
The h variable is Planck’s constant, which has the value shown, and f is the frequency of the light waves. This equation shows that a photon’s energy is proportional to the light frequency. This means that low frequency light, like radio waves, have low energy photons, whereas high frequency light, like X-rays, have high energy photons. Planck’s constant, h, is one of the fundamental physical constants of our universe. It is also extremely tiny, reflecting the fact that quantum behaviors are typically only relevant for tiny systems, such as individual atoms and molecules. Recall that wave speed, frequency, and wavelength relate to each other, for all types of waves, through the equation v = λ f (Eq. 2.7). Combining this with Eq. 13.1, and writing the speed of light as c, gives an alternate version of the PlanckEinstein relation,
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hc . (13.2) λ In agreement with the prior trends, light with short wavelengths, such as X-rays, have high photon energies, whereas light with long wavelengths, such as radio waves, have low photon energies. E=
Example. Fireflies emit light of around 550 nm. Supposing one firefly flash has about 1 mJ of energy, how many photons is this? Answer. Each 550 nm photon has an energy of E photon =
hc (6.626 · 10−34 J s)(3 · 108 m/s) = = 3.6 · 10−19 J. λ 550 · 10−9 m
This is the energy of one photon, so the number of photons is n photon =
E total 10−3 J = = 2.8 · 1015 photons. E photon 3.6 · 10−19 J/photon
Each photon comes from a chemical reaction of one molecule of a chemical called luciferin, so this means that about 2.8 · 1015 luciferin molecules react to create each firefly flash.
13.2.2 Photoelectric Effect Einstein’s insight that photons are tiny packets of energy immediately solved another of the puzzles at the time. It was well known that light was able to knock electrons off some materials, which is called the photoelectric effect, but the puzzle was that this only happened if the light had a short enough wavelength. Figure 13.4 shows the typical experimental setup. A battery applies a voltage across two metal plates that are separated by a small gap and light can shine onto one of the two plates. The light energy bumps electrons off one metal plate that then cross the gap to complete the electrical circuit. Running this experiment shows that electric current only flows around the system when the light has a sufficiently short wavelength, typically with ultraviolet light, and not otherwise. Even using very intense light makes no difference if the wavelength is too long. It made sense that light could bump electrons off metal surfaces, since it was known that light waves carried energy, but the need for short wavelengths couldn’t be explained. Einstein realized that the particle explanation for light solved the puzzle because individual ultraviolet photons have a lot of energy, so they can eject electrons off a metal plate, but individual visible light photons are too weak to have an effect, so they don’t eject electrons. As an analogy, consider trying to break a window by throwing balls at it. Ping-pong balls, which are analogous to visible light photons, don’t break
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Figure 13.4 Equipment for observing the photoelectric effect.
the window, no matter how many you throw. On the other hand, throwing just one golf ball, which is analogous to an ultraviolet photon, does break the window. Additional experiments showed that the speed of the ejected electrons depended on the light wavelength, with shorter wavelengths giving the ejected electrons higher speeds. It was also found that increasing the light intensity, assuming it had short enough wavelength, ejected more electrons. Both of these results make sense with the photon description.
13.2.3 Photoelectric Effect Examples* Photomultiplier tubes are extremely sensitive light sensors that can detect individual photons and work through the photoelectric effect. When a photon collides into its sensor end, it knocks off a single electron. That electron is electrically attracted to a second surface, where it knocks off more electrons, which is repeated to eventually create an avalanche of electrons that can be easily measured with standard electronics (Figure 13.5).
Figure 13.5 (Left) A photomultiplier tube and its socket, used for single photon detection. (Right) A diagram showing how photomultiplier tubes work.
Night vision goggles work in a similar fashion. Again, photons knock electrons off a metal surface that then start avalanches of more electrons. In this case, those latter electrons strike a “phosphor” screen that glows, thus giving the overall result of converting a dim light into a much brighter light.
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Digital cameras form yet another example of the photoelectric effect. In both major types of digital sensors, which are CCD (charge-coupled device) and CMOS (complementary metal oxide semiconductor), light sensing starts when a photon ejects an electron from a piece of silicon. These photoelectrons get stored briefly. Then, when the camera is done taking the picture, the ejected electrons get quantified, which is where the sensor types differ. A very different example occurs in space, where there’s no atmosphere to carry away extra electric charges. Sunlight knocks electrons off the parts of satellites that are in the sun, which gives them a net positive charge. Some of those electrons rebind to satellite portions that are in the shade, giving them a net negative charge. The resulting static electricity differences across the satellite can then interfere with delicate spacecraft electronics. An intriguing effect arises on the Moon. Sunlight knocks electrons off dust particles, which gives them positive charges. These dust particles then repel each other, so they rise up off the ground and form a thin haze that rises several kilometers above the moon’s surface.
13.2.4 Photochemistry* Yet another puzzle in the late 1800s was the discovery that many chemical reactions can be started by ultraviolet light but not visible light. For example, nothing happens if hydrogen and chlorine gases are combined in a balloon, even if it is exposed to visible light. However, it explodes if it is exposed to a small amount of ultraviolet light. Once again, the photon description of light solves this puzzle. The explosion happens because ultraviolet photons have enough energy to break chlorine bonds, which starts the chemical reaction, whereas visible photons don’t have enough energy. Similarly, we can get sunburned by ultraviolet light but not by infrared or visible light, which again is due to the difference in photon energies. Furthermore, ultraviolet light carries enough energy to damage the DNA in our skin cells, which can lead to skin cancer. The threshold at which photons have enough energy to start chemical reactions is often in the near ultraviolet range, but can also be at higher or lower energies. For example, vision and photosynthesis are examples of chemical reactions that are initiated by visible light photons.
13.2.5 Compton Scattering Despite the strong evidence in favor of Einstein’s particle theory of light, many physicists were not convinced initially. That changed in 1923 when the American physicist Arthur Compton discovered what is now called Compton scattering. He aimed X-ray light of some given wavelength at a material, which happened to be graphite in his original experiment, and found that the X-rays that bounced off had
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a longer wavelength than the initial light. This is shown in Figure 13.6. This could not be explained with the wave theory of light because that theory didn’t offer any mechanisms for light wavelengths to change4 . However, it could be explained with the particle theory of light. What happens is that an incoming X-ray photon bounces off an atom, while knocking an electron loose in the process. This means that some of the photon’s energy goes to the electron, leaving the photon with less energy than it had originally. As a result, the outgoing photon has less energy, meaning that it has a longer wavelength. This explanation finally lead to widespread acceptance of particle-wave duality by the late 1920s. Figure 13.6 Compton scattering. An incident high-energy photon (purple) ejects an electron and is also scattered as a lower energy photon (blue).
13.3
Photon Momentum
13.3.1 Classical Momentum Recall from Section 6.1.2 that momentum is a measure of how hard it is to stop a moving object. For all normal objects, meaning nearly everything except photons, the momentum is p = mv,
(13.3)
where m is the object’s mass and v is the object’s velocity. For example, a cement truck on a freeway has high momentum because it has high mass and high speed, whereas a flying butterfly has low momentum because it has low mass and low speed. Momentum is important because it is conserved during collisions. For example, when the cement truck hits the butterfly, some of the momentum is transferred from one to the other, but the total momentum of the two objects together does not change.
4 Light
wavelengths can only change through non-linear effects, which are rare, and none were known at that time.
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13.3.2 Photon Momentum Photons have momentum, too, meaning that they push on things when they bump into them. However, their momentum cannot be computed from Eq. 13.3 because that equation only applies for objects that travel substantially slower than the speed of light, whereas photons travel at the speed of light. Instead, photon momentum is given by h . (13.4) λ This shows that long wavelength photons, such as from infrared light, have low momentum while short wavelength photons, such as from ultraviolet light, have high momentum. This follows the same trend as the photon energy, showing that short wavelength photons are more powerful overall, carrying both more energy and more momentum, while long wavelength photons are less powerful. Photon momentum was first observed in the same Compton scattering experiment described above, where it was found that if the electron was ejected to the left, then the photon was scattered to the right, and vice versa. This occurred because the total momentum of the photon, electron, and remainder of the atom needs to be conserved throughout the interaction, which is only possible if the photon and electron scatter in opposite directions. p=
Example. Which has more momentum? 1016 photons of 650 nm laser light (about 1 second of laser pointer light), or a 1 mg gnat walking at 0.1 m/s? Answer. The photon momentum is plaser = 1016
h 6.626 · 10−34 J s = 1016 = 1 · 10−11 kg m/s. λ 650 · 10−9 m
The gnat’s momentum is p = mv = (1 · 10−6 kg)(0.1 m/s) = 1 · 10−7 kg m/s. Thus, the gnat has 10,000 times more momentum than the light.
13.3.3 Radiometers* Science gift shops often sell radiometers, like the one shown in the left panel of Figure 13.7, which spin when placed in sunlight. Offhand, it seems that they might work through photon momentum, although it turns out that they actually don’t. Nevertheless, they are instructive to think about. To understand how the light’s momentum could make a radiometer spin (middle panel), suppose the light is shining at the vanes in a horizontal beam. This figure
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Figure 13.7 A radiometer, which spins in sunlight. The middle diagram shows how the momentum of light would cause it to spin with the white side moving away from the light. The right diagram shows how differential heating of air molecules actually causes it to spin the other direction.
shows a top view of a radiometer, with equal numbers of photons hitting the black side of one vane and the white side of the opposite vane. The photons that hit the black side of a vane, shown at the bottom of the figure, are absorbed by the black paint, so they transfer their momentum to the vane and push on that side of the radiometer. The same number of photons hit the white side of the opposite vane, so they push the same amount when they hit. However, those photons then reflect off and go back toward where they came from, effectively pushing themselves off the radiometer as they reflect, creating an additional force. Thus, the photons that hit the white side exert a greater force than those that hit the black side, which means that the radiometer should spin with the white side moving away from the light. In practice, science store radiometers spin the other direction, rotating with the black side moving away from the light. This turns out to arise from the air in the radiometer (right panel). What happens is that the black sides of the vanes absorb more light, which makes them warmer. This heats up the air that is next to them, and hot air molecules move faster than cold air molecules (see Chapter 6), so these hot air molecules have more momentum than those on the white sides, which causes them to push harder. Air molecules have a lot more momentum than photons, with the net result that radiometers spin with the black side moving away from the light5 .
13.3.4 Solar Sails* Much as conventional sailing ships are driven by air molecules pushing against fabric sails, spaceships can be driven by photons pushing against solar sails. They sail using sunlight, which is available everywhere in the solar system but, obviously, is brighter closer to the sun. Solar sails are more economical than burning rocket fuel but have the disadvantage of offering low power due to the low momentum of photons.
5 This example discusses a Crooke’s radiometer, which is the commercially available type. If the air is removed to leave a high vacuum around the radiometer vanes, then it’s called a Nichols radiometer, and it really does spin with the white side moving away from the light due to the photon momentum.
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IKAROS, a Japanese spacecraft whose name is an acronym for Interplanetary Kite-craft Accelerated by Radiation of the Sun6 (Figure 13.8), is the only spacecraft that has sailed through interplanetary space using a solar sail, so far. It was launched in 2010 and successfully sailed past Venus six months later, thereby proving that solar sails can work successfully. IKAROS has a 200 m2 square sail that is about the thickness of the thinnest commercially available plastic food wrap, and has LCD panels (see Chapter 11) to control reflectivity and thus allow for some sailing control. Figure 13.8 Artist’s conception of IKAROS.
Several other spacecraft have been built with solar sails since then, almost exclusively to test the practicality of using solar sails to control satellites that are in Earth orbit. So far, it appears that solar sails are practical for re-orienting spacecraft and creating minor trajectory shifts, but provide too little power to be practical for most other applications.
13.3.5 Laser Tweezers* The opening picture for this chapter showed an example of laser tweezers, more formally called optical trapping, which is another application of photon momentum. This is a clever technique for holding a tiny diamond, glass bead, or other transparent object at the center of a laser beam. Figure 13.9 shows how it works. The middle panel illustrates a glass bead that is already centered in the laser beam, shown with the red arrow, so the beam shines through the bead without bending. These photons simply pass through the bead without changing direction, so there is no net force on the bead.
6 The name also refers to Icarus, a mythological Greek man who attempted to escape from Crete by flying with wings made of feathers and wax.
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Figure 13.9 Function of laser tweezers. The laser light is shown in red and the force on the bead is shown in blue.
However, if the bead drifts off to the left, as shown in the left panel, then the bead acts as a lens and bends the laser light further to the left. Because the photons’ momentum gets sent to the left, the bead gets pushed toward the right, which is back into the middle of the laser beam. Vice versa, if the bead is too far to the right, as shown in the right panel, then the bead bends the laser light further right, which pushes the bead back to the left. Either way, the bead is always pushed toward the center, causing it to get trapped there by the laser light. This technique allows researchers to hold, move, or generally manipulate individual tiny particles, including beads, cells, or bacteria. Laser tweezers can also be used to measure tiny forces. For example, if some external force is constantly pulling the bead to the left, then the light that refracts through the bead will also get aimed to the left, which is easy to observe. With this method, biophysicists have been able to measure the elasticity and breaking strengths of individual DNA molecules and proteins7 . Further development of optical tweezers, most notably by the American physicist Steven Chu, enables the trapping and manipulation of single atoms.
13.3.6 Doppler Cooling* Yet another application of photon momentum arises in Doppler cooling, shown in Figure 13.10, in which laser light is used to slow down individual atoms until they are moving so slowly that their effective temperature is near absolute zero. To see how it works, consider an atom that only absorbs light of some specific frequency (blue in the figure), and imagine aiming laser light that has a slightly lower frequency at it from all sides (green). If the atom is stationary, as in the left panel of Figure 13.10, then the laser frequency doesn’t match the atom’s absorption frequency, so the atom doesn’t absorb any photons and nothing happens. However, if the atom moves, then it perceives the light beam that it’s moving toward as having an increased frequency due to the Doppler shift (see Chapter 4). This shifted laser frequency matches the atom’s absorption frequency so the atom absorbs photons, and the momentum of these photons pushes
7 See Bustamante et al., “Single-molecule studies of DNA mechanics”, Current Opinion in Structural Biology 10:279, 2000; and Kellermayer et al., “Folding-Unfolding Transitions in Single Titin Molecules Characterized with Laser Tweezers” Science 276:1112, 1997.
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Figure 13.10 Doppler cooling. (Left) The atom is stationary so all laser beams appear the same color, which does not match the absorption color, and photons are not absorbed. (Right) The atom is moving to the right so the laser beam from the right is Doppler shifted to higher frequency, causing these photons to be absorbed.
the atom backward, which slows it down. The atom re-emits the same amount of light energy as new photons, but they are emitted in random directions, so they don’t affect the atom’s average speed. Doppler cooling is remarkably effective, enabling researchers to cool atoms from oven temperatures to only a few millionths of a degree above absolute zero in less than a millisecond8 .
13.4
Particle-Wave Duality
13.4.1 Quantum Interpretation of the Double-Slit Experiment Young’s double slit experiment (Chapter 3) showed beyond any possible doubt that light is a wave, since only waves are able to exhibit interference patterns. Meanwhile, quantum mechanics results, including Planck’s blackbody distribution theory, Einstein’s explanation for the photoelectric effect, and Compton’s demonstration of photon energy and momentum, showed beyond any possible doubt that light is a particle. How can light be both waves and particles at the same time? To help address this question, the double-slit experiment can be performed in a way that explicitly tests whether light is a particle or wave. It is run with a light source that is so dim that only one photon is in the air at a time. The idea is that if light is a particle, then each particle would have to go through only one slit and wouldn’t interfere with anything, so it would just produce a spot behind the slit that it went through. However, if light is a wave, then it would go through both slits at once and would produce an interference pattern. Figure 13.11 shows a simulated version of the experimental result (which is consistent with actual experiments). Remarkably, it shows both particle and wave behaviors at once. Each photon creates an individual dot on the film, in agreement with the
8 See
Metcalf and van der Straten, “Laser cooling and trapping of neutral atoms,” The Optics Encyclopedia: Basic Foundations and Practical Applications, 2007.
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particle explanation, but these dots cluster together to form the interference pattern that is expected from the wave explanation. Furthermore, because only one photon is in the air at a time, this result shows that individual photons must interfere with themselves. Figure 13.11 Simulated result of double slit experiment carried out with individual photons.
The Copenhagen interpretation of quantum mechanics, named for its development in Copenhagen during the 1920s by the German physicist Werner Heisenberg and the Danish physicist Niels Bohr, helps explain this. It states that the light wave does not just represent the electric and magnetic fields, but also represents the probability of finding the photon at any particular location. In this role, it is called a wave function. In other words, one is likely to find a photon at locations where the light wave displacements are large and unlikely to find it where displacements are small9 . When a very dim light is shone at the slits, the wave function goes through both slits at once and forms an interference pattern at the film. However, the film can only record the location of photons at individual points, so the wave must collapse into a single point at the film. This point has a random location that is chosen based on the wave function displacement values. The Copenhagen Interpretation correctly predicts not only the double-slit experiment with light, but all other quantum mechanics experiments that have been tested as well. Nevertheless, it is still a mystery how the wave function actually collapses, which Einstein called “spooky action at a distance”10 . We’ll investigate this topic more in Section 14.6.
9 More precisely, the probability of finding a photon at some particular location is proportional to the square of the wave function value at that location. 10 This quotation is typically attributed to Einstein’s attitude about the EPR paradox (Section 14.6) but the surrounding context suggests that he was actually talking about wave function collapse in general. The full sentence, translated from a letter that Einstein wrote to Max Born in March 1947, is “But I can’t seriously believe in it [quantum mechanics] because the theory is incompatible with the principle that physics should represent a reality in time and space, without spooky long-distance effects.” See a video by Sabine Hossenfelder (May 8, 2021) at http://backreaction.blogspot.com/ 2021/05/what-did-einstein-mean-by-spooky-action.html
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13.4.2 Photon Size* A photon is typically described as a point particle, with no size whatsoever. This makes sense when considering that there is no limit to the number of photons that can exist in a fixed volume of space, such as a hot blackbody cavity. Also, individual photons can interact with tiny objects, such as individual hydrogen atoms that have a diameter of around 10−10 m, or even a single atom nucleus, with a diameter of around 10−15 m. In both of these cases, photons are absorbed in their entirety and all at once, which makes sense for photons that are comparably tiny. In this point particle explanation, the photon is just a point but its position is some random location that has a probability that is given by the corresponding wave function. On the other hand, it’s often helpful to think of photons as having larger sizes. For example, when a carotene molecule, which is about 2 nm long, absorbs a photon, the photon is not absorbed at a tiny spot within the molecule, but by the entire molecule at once. Likewise, when a radio antenna absorbs a radio wave photon, it gets absorbed over the entire length of the antenna. Vice versa, molecules and radio antennas that emit photons emit them from their entire size. Thus, it’s reasonable to say that when individual photons interact with entire objects, then the size of the photon must be the same as the size of the object. Yet larger sizes arise when considering diffraction and interference experiments. First of all, recall that all types of waves diffract when they encounter a hole in a barrier (Section 3.4). As we saw before, they go through the hole without much power loss if the hole is much larger than a wavelength, they diffract substantially if the hole size is similar to a wavelength, and they essentially ignore the hole and just reflect off the barrier if the hole is much smaller than a wavelength. Considering light waves, this can be interpreted as saying that photons fit through holes that are larger than about a wavelength but not through smaller holes. Thus, in these diffraction and interference situations, photons behave as though their sizes are about one light wavelength. In two-slit interference, we showed that single photons are able to interfere with themselves, implying that each individual photon goes through both slits at once. These slits are often a millimeter or more apart from each other, which is over a thousand times longer than a light wavelength (assuming visible light), implying that a single photon can be a millimeter or more across. Then, it gets bigger. After going through the slits, the photon spreads out to an even larger size, typically on the order of several millimeters, as it approaches the photographic film behind the slits, where it then collapses back into a single dot on the film. An even more extreme version of the same thing is an interferometer (Section 3.3.3), where a single photon gets split into two pieces that might be meters, or even kilometers, apart from each other. Thus, there are cases where a single photon’s size could be reasonably considered to be multiple kilometers across. Yet another answer for a photon’s size is appropriate for light that comes from some light source, which might be a light bulb, a laser, a star, or a hole in an otherwise opaque lampshade. If it were possible to measure the electromagnetic field in the light beam, one would see a chaotic collection of waves that have regular peaks and
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troughs over short distances but that fade into different sets of waves over longer distances. Figure 13.12 illustrates this with waves on the ocean surface. The length scale over which the waves are reasonably regular is called the coherence length of the light source. Lasers have a large coherence length, typically in the centimeters to meters range, meaning that the waves across the entire laser beam are all in phase with each other, all moving together in lockstep. Light that seems to come from point sources, such as a tiny pinprick hole in a lampshade or a distant star, also has high coherence. On the other hand, light from large sources, such as the sun or standard light bulbs, have short coherence lengths that are in the micron range11 . Coherence lengths set the limit on the sizes of interference phenomena; for example, two-slit interference experiments don’t exhibit fringes if the slits are farther apart than the light’s coherence length. Thus, another useful measure for the size of a photon is the light’s coherence length. Figure 13.12 Photograph of the ocean surface. The coherence length, shown with red arrows, represents the distance over which the waves are reasonably regular, before fading into new wave patterns.
Taking all of these arguments together, there is no good single answer for the size of a photon. Instead, there are situations where photons are best considered as point particles, and other situations where larger sizes, even up to kilometers or longer, are more meaningful.
13.4.3 Spectral Broadening of Pulses The beginning of this chapter stated that photons have specific frequencies and also that they can be represented by wave packets. However, there’s a problem here. It is that wave packets are never single frequencies, due to the fact that they start and stop, but are always the sums of many different frequencies. The left side of Figure 13.13 shows how waves with many slightly different frequencies add together to create a wave packet.
11 See Divitt and Novotny, “Spatial coherence of sunlight and its implications for light management
in photovoltaics”, Optica, 2:2334, 2015.
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Figure 13.13 (Left) A wave packet as the sum of many component waves, according to the superposition principle. (Right) Electric fields of short light pulses, on the left, and the corresponding energy spectra for those pulses on the right. These pulse frequencies are centered at 0.5 · 1015 Hz, which corresponds to 600 nm, and the pulse standard deviations are 8 fs in the top row and 2 fs in the bottom row.
Another way of looking at this is to consider the spectrum of a wave packet, shown on the right side of the figure. Long pulses (top row) are relatively close to single frequencies, so they have narrow spectral peaks, while short pulses (bottom row) include more frequencies, so they have broad spectral peaks. This relationship between pulse duration and frequency range is quantified by defining σt as the pulse duration and σ f as the width of the the spectral peak. Carrying out the mathematics12 shows that σf ≥
1 . 4πσt
(13.5)
This is an inequality because it’s always possible to add more frequencies to a brief pulse. However, there is no way to have a narrower range of frequencies than this. This is not really a physics result, but just a mathematical consequence of the superposition of waves. All it says is that if you want to add waves together to create a wave packet, then you need a wide range of frequencies to produce short wave packets and a narrow range to produce long wave packets. It also applies to all types of waves. As an example with sound waves, short bursts of sound don’t have well-defined tones but instead sound like clicks, snaps, taps, or other percussive noises; they sound this way because they are composed of a very wide range of frequencies. Back to photons, this relationship implies that if a photon has a short wave packet, meaning that you know almost exactly when it was emitted, then it must have a broad spectrum. Vice versa, if you know that a photon was emitted at some point but you don’t know when, then it can have a very narrow spectrum. A typical example arises with atom or molecule relaxation. Suppose an atom is excited to a high energy level using an electric current, such as a neon atom in a neon
12 We assume a Gaussian “envelope,” meaning overall shape, of the electric field pulse, and we define σt as the standard deviation of the squared electric field over time. Then, take the Fourier transform of the electric field pulse and square that to get the energy spectrum. The result is another Gaussian, now with standard deviation σ f .
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sign. That atom is unstable so, at some point, it emits a photon and returns back to its ground state, which typically takes around 10−8 seconds13 . From Eq. 13.5, this corresponds to a spectral width, or natural line width, of 8 · 106 Hz. In terms of wavelength, this width is around 10−5 nm. Admittedly, this is a very narrow range of frequencies, so it’s typically legitimate to speak of a photon having a specific frequency, as we did at the beginning of the chapter, but, technically, there is actually a range of frequencies. Example. An ultrafast laser emits 750 nm light in a single 20 fs pulse (1 fs = 10−15 s). What is the minimum width of the spectral peak in nanometers? Answer. Compute the spectral peak width from Eq. 13.5, σf =
1 = 4 · 1012 Hz. 4πt
This is the answer in frequency units. Convert to wavelength by first computing the light frequency as c f = = 4 · 1014 Hz. λ Taking the ratio σ f / f shows that the peak width is 4 · 1012 /4 · 1014 = 1% of the light frequency. Multiplying this same 1% by the 750 nm wavelength shows that the peak has a minimum width of 7.5 nm.
13.4.4 Energy-Time Uncertainty Combining the mathematical relationship between wave packets and their spectra, Eq. 13.5, with the Planck-Einstein relation, E = h f , leads to another inequality, σ E σt ≥
h . 4π
(13.6)
This is the uncertainty principle for energy and time (it’s closely related to the better-known uncertainty principle for position and momentum, which we’ll meet in the next chapter). In a sense, there’s nothing new here. It’s just a combination of the mathematical result that wave packets are necessarily composed from a range of frequencies and the physics result that frequencies correspond to energies. More specifically, it shows that it’s impossible to know both when a photon was emitted and
13 There
are many ways to measure wave packet and spectrum widths. Previously, we used the standard deviations, σt and σ f . However, relaxation times are typically given as exponential decay constants, τ and spectral line widths are often given as the full line width at half of the maximum height (FWHM). Different conventions change the precise relationship some but it is always the case that f ≈ 1/2πt where t and f are measures of the wave function and spectrum widths.
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exactly how much energy it has. Instead, there is minimum combined uncertainty, where better knowledge of the photon’s energy (σ E ) implies less knowledge about the time when it was emitted (σt ), and vice versa. Thinking about this a little deeper shows that this is also a profound statement about the nature of reality. It describes an intrinsic fuzziness of our universe, showing that neither the energy of an event nor the time that it happened can ever be perfectly well defined. Instead, there is always some tradeoff between the two pieces of information. This uncertainty principle can also be seen to be a direct consequence of particlewave duality. The wave nature of light means that a photon’s duration corresponds with its frequency range, and the particle nature means that frequencies correspond to energies.
13.4.5 Particle-Wave Duality for Other Wave Types* The finding of particle-wave duality for electromagnetic waves raises the question of whether other types of waves correspond to particles as well. For the most part, the answer is yes. However, it’s more complicated than this. Matter waves are the wave natures of electrons, atoms, and other physical particles. As such, they clearly have particle counterparts, which is the focus of the next chapter. Gravitational waves, the gravitational ripples that propagate through space-time, are widely expected to be quantized as particles called gravitons. If this is correct, then those gravitons would probably be similar to photons in that they would travel at the speed of light and have no rest mass. They would mediate the gravitational force, much as photons mediate electromagnetic forces. However, all of this is speculative because no current theories connect gravitation with quantum mechanics and also there is no direct evidence for gravitons. The only real experimental information about them is that the rest mass of a graviton must be less than 10−67 kg, which is about 37 orders of magnitude lighter than an electron and is consistent with their having no rest mass at all. Because gravity is so much weaker than other forces, a detector that is sensitive enough to measure a single graviton would need to be built from masses equal to the size of Jupiter14 , which is clearly impractical. Finally, mechanical waves also exhibit particle-wave duality, although their particles are typically described as quasiparticles or collective excitations rather than true particles. Conventional work on the quantum behaviors of mechanical waves focuses on lattices of atoms or molecules, as in a crystal. These lattices have a set of normal vibrational modes, each with its own frequency, and it turns out that one can only put quantized amounts of energy into each normal mode. These quantized amounts follow the Planck-Einstein relation, E = h f , exactly as for photons. A quantum of vibration is called a phonon, where each phonon is a massless particle that travels at
14 See
Rothman and Boughn, “Can gravitons be detected?” Foundations of Physics 36 (2006): 1801–1825.
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the speed of sound. Although unconventional, these same ideas also apply to more complex systems, such as waves on a string, sound waves in a room, or water waves on a pond. Again, these waves can be represented as a stream of discrete phonons, each of which has energy h f and travels at the wave speed (which is the group velocity for dispersive media). Example. Someone plucks a guitar G string (196 Hz) very gently, giving it 0.01 J of energy. How many phonons are in the string wave? Answer. Compute the energy of one phonon with the Planck-Einstein relation, E = h f = (6.626 · 10−34 J s)(196 s−1 ) = 1.3 · 10−31 J. Divide the string energy, 0.01 J, by this phonon energy to find that there are 7.7 · 1028 phonons in the string wave.
13.5
Summary
Physics seemed to be well understood at the end of the 19th century but there were still some problems that it could not explain. Many of them were solved over the next few decades by the development of quantum mechanics, of which the first major discovery was that light is both particles and waves at the same time, called particle-wave duality. Photons, which are particles of light, are small packets of energy. They also have wavelike properties such as wavelengths, frequencies, and polarizations. A photon’s energy relates to the light wave frequency through the Planck-Einstein relation, E = hf . This shows that high frequency (short wavelength) photons contain more energy than low frequency (long wavelength) photons. It explains the blackbody radiation spectrum, the photoelectric effect, photochemistry, and Compton scattering. Photons don’t have rest mass but do have momentum, which is equal to p=
h . λ
This differs from the momentum equation for objects that do have rest mass, which is p = mv. Photon momentum follows the same trend as energy in which short wavelengths correspond to more powerful photons. Photon momentum can be used for solar sails, to optically trap glass beads, and to cool molecules. Particle-wave duality is typically understood through the Copenhagen interpretation, in which a wave function represents the probability of a particle’s location. This wave function propagates, diffracts, and interferes just like other waves, but
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then collapses into a particle at the moment it is observed. This explains the pattern of dots that appear when the double slit experiment is run with low light levels. Photons are usually considered to be point particles but various measures of the wave function size are often more meaningful, such as the wave packet size, wavelength, or coherence length. By the superposition principle, short wavepackets necessarily correspond to broad spectra and vice versa. Converting from frequencies to energies with the PlanckEinstein equation leads to the uncertainty relation for energy and time, σ E σt ≥
h . 4π
This shows that there is a fundamental limit to how well the combination of photon energies and their emission times can be known. Particle-wave duality applies to other types of waves as well, but with differences. For matter waves, the particles have masses; for gravitational waves, the gravitons are extraordinarily weak and also unproven; and for mechanical waves, the phonons are better described as collective excitations rather than true particles.
13.6
Exercises
Questions 13.1. Which one of the following is true? (a) X-ray photons are faster than infrared photons (b) radio wave photons have more energy than infrared photons (c) red photons have higher frequencies than blue photons (d) X-ray photons have more momentum than radio photons (e) none of the above; all photons are the same 13.2. Which properties do photons have (choose all that are appropriate)? (a) rest mass (b) energy (c) momentum (d) speed (e) wavelength 13.3. Which has the most energy? (a) one blue photon (b) one red photon (c) two blue photons (d) two red photons (e) they are all the same
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13.4. Which has the most momentum? (a) one blue photon (b) one red photon (c) two blue photons (d) two red photons (e) they are all the same 13.5. A beam of white light reflects off a mirror that’s attached to a wall. Which way does the light push on the mirror? (a) away from the wall (b) into the wall (c) parallel to the wall (d) there is no force on the mirror (e) it depends on the light wavelength Problems 13.6. Considering a red photon and a blue photon, which has the greater value for each of the following properties (for each part, answer “red”, “blue”, or “same”): (a) wavelength, (b) frequency, (c) speed, (d) energy, (e) momentum, (f) rest mass. 13.7. Consider green light with a wavelength of 550 nm. (a) What is the light frequency? (b) What is the energy of a single photon? (c) Does a red photon have a higher or lower frequency? (d) Does a red photon have a higher or lower energy? 13.8. A red laser pointer emits 2 mW of 650 nm laser light. (a) What is the energy of each photon? (b) How many photons per second is this? 13.9. (a) What is the photoelectric effect? (b) How did its explanation change the understanding of the physics of light? (c) Give an example of how the photoelectric effect is used in modern technology. 13.10. Hydrogen and chlorine gases are combined in a balloon. Both are dimeric, meaning that each hydrogen molecule is two hydrogen atoms and each chlorine molecule is two chlorine atoms. The hydrogen bond strength is 432 kJ/mole and the chlorine bond strength is 240 kJ/mole. One mole of molecules is 6.02 · 1023 individual molecules. (a) What is the bond strength of a single hydrogen molecule, measured in J? (b) What is the bond strength of a single chlorine molecule, measured in J? (c) What is the maximum wavelength photon that has enough energy to break one of these two bonds, which will then start a chemical reaction? 13.11. “Breakthrough starshot” is a research program that has the goal of sending tiny spacecraft to nearby stars. In their plan, these spacecraft will have shiny light sails and will be accelerated to about 20% the speed of light by shining a ground-based laser at them for about 10 minutes each. Supposing each
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spacecraft weighs 2 g and the laser wavelength is 10.6 μm (a CO2 laser), how much laser power will be required? 13.12. Suppose an electron is not moving and it gets hit with one green photon with 500 nm wavelength, and assume the electron absorbs the photon. An electron’s mass is 9.11 · 10−31 kg. (a) Use the conservation of momentum to compute the electron’s final speed. (b) Use the conservation of energy to compute the electron’s final speed (the kinetic energy of the electron is 1 2 2 mv , where m is the electron’s mass and v is its velocity). (c) Based on these results, is it possible for a lone electron to absorb a photon? 13.13. Consider a sidewalk section that has an area of 1 m2 and assume the sun is directly overhead. Also, assume the sun is emitting primarily green light, which has a wavelength of 500 nm. (a) What is the energy of one sunlight photon? (b) Using the fact that the power of the sunlight on this square meter is about 1367 W, how many photons hit the sidewalk section in one second? (c) What is the momentum of one sunlight photon? (d) What is the momentum of all photons that hit this sidewalk section in one second? (e) Now suppose an ant falls on the sidewalk. A typical ant weighs 4 · 10−6 kg and falls at 1.8 m/s. What is the ant’s momentum? (f) Which exerts a greater momentum, the ant or 1 second of sunlight? (g) Which would exert a greater momentum if the sidewalk were silvered, like a mirror? 13.14. When an excited hydrogen atom relaxes from its lowest energy excited state to its ground state, it releases a photon with an energy of 1.64 · 10−18 J. (a) What is the photon wavelength? (b) What is the photon momentum? (c) What is the recoil speed of the hydrogen atom? (A hydrogen atom mass is 1.67 · 10−27 kg; the total momentum is zero, so the atom momentum’s is equal and opposite the photon’s momentum.) 13.15. A hydrogen atom takes an average of 1.6 · 10−9 s to relax from its first excited state to its ground state, releasing a photon with energy 1.64 · 10−18 J. (a) What is the photon frequency? (b) What is the spectral width of the photon in nm? (c) What is the spectral width as a percent of the photon frequency? 13.16. Suppose an atom takes 10−8 seconds to relax from an excited state to its ground state, emitting a photon in the process. (a) Taking this as the wave packet duration, how long is the wave packet in meters? (b) Discuss whether this is a reasonable measure of the photon’s length. 13.17. The Very Long Baseline Array is a system of 10 radio telescopes, which are up to 8,611 km away from each other, that measure high frequency radio waves (in the microwave region) from distant stars. Every telescope measures the same radio wave peaks and troughs, which are then combined through a method called digital interferometry. (a) Do these radio waves have a large or small coherence length, compared to their wavelengths? (b) Would it be reasonable to say that a single radio wave photon is at least 8,611 km across? Explain why or why not.
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13.18. The Voyager 1 spacecraft is at the edge of the solar system, about 11 billion km away. (a) How long does a radio signal from Voyager 1 take to get to the Earth? (b) Its radio works at 8 GHz. What is the wavelength? (c) What type of light is this? (d) What is the energy of 1 photon? (e) It has a 23 watt radio wave transmitter. How many photons per second is this? (f) This radio wave is aimed toward the Earth by a dish; by the time the signal gets to the Earth, the area of the radio beam is 4 · 1015 km2 . What is the photon flux density, in photons per second per square km? (g) How about in photons per second per square meter? Puzzles 13.19. Consider an empty cubical black cardboard box that has side length L and is at temperature T (most answers will be equations). (a) How much total power do the sides of the box radiate into the box through blackbody radiation? (b) Using the fact that an average photon travels distance 2L/3 between when it is emitted and absorbed, what is an average photon’s lifespan? (c) How much total radiation energy is inside the box? (d) What is the energy of one of these photons, assuming that all photons have the wavelength that corresponds to the peak of the blackbody distribution spectrum? (e) What is the density of photons in the box? (f) Evaluate the photon density for T = 300 K. (g) How many times more air molecules are there than photons, using the fact that the air molecule density at 1 atm pressure and 300 K is 2.45 · 1025 molecules/m3 . (h) How far apart are the photons, on average, at 300 K (the cube root of the volume per photon)? 13.20. Consider a spherical teapot of temperature T and radius r . The inside of the teapot is painted black and radiates blackbody radiation. (a) How much pressure does this radiation exert against the sides of the teapot, as an equation? Note that pressure is force per unit area, P = F/A, and force is change of momentum per time unit, F = p/t. Assume that all emitted light is at the peak of the blackbody distribution spectrum and that photon emission and absorption occur with photons moving perpendicular to the surface. (b) Evaluate this pressure in atmospheres for T = 2000 K, using the facts that the SI unit for pressure is Pa and 1 atm = 101, 325 Pa. 13.21. Zoro is flying toward the star Betelgeuse so fast that the red starlight appears blue to Zoro due to the Doppler shift. When she measures the energy of one of these starlight photons, will it have the energy of a red photon or a blue photon? Explain.
Matter Waves
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Figure 14.1 Neon dragon, from Museum of Neon Art, Glendale, CA.
Opening question What causes atoms to have distinct energy levels? (a) Standing electron waves (b) Quantization of light into photons (c) Electrons bumping into each other (d) Interactions between neighboring atoms (e) It’s still unknown
© Springer Nature Switzerland AG 2023 S. S. Andrews, Light and Waves, https://doi.org/10.1007/978-3-031-24097-3_14
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Look around and observe your surroundings. You probably see furniture, people, buildings, trees, and other familiar objects, all with their respective shapes and colors. These are large objects, in contrast to the truly tiny world of electrons and protons, where quantum mechanics becomes important. Also, these objects behave normally, unlike the bizarre world of quantum mechanics where waves are particles, and particles are waves, and nothing can be known exactly. And yet, it could also be said that essentially everything about your surroundings is the way it is because of quantum mechanics. The wave-like properties of electrons create most of the colors that you see. They also explain how electrons, protons, and neutrons make atoms, how atoms bind together to form molecules, and how molecules bind together to form liquids and solids. Quantum mechanics explains why air is transparent, blood is red, and security lights are yellow. It is also the theory behind much modern technology, from computers to lasers, neon signs, and even fabric whiteners. This chapter introduces the basic principles of quantum mechanics. Remarkably, most of them are just the same physics of waves that we’ve already seen.
14.1
Matter Waves
14.1.1 De Broglie Relations Louis de Broglie was a French aristocrat who majored in history and intended to pursue a career in humanities. These plans were derailed by World War I, in which he served as a radio engineer; then, his brother inspired him to turn his attention to physics, where he ended up writing one of the world’s more famous PhD dissertations. In it, he extended Einstein’s discovery that light is made of particles to propose that matter is made of waves. These matter waves needed to have wavelengths and frequencies, so de Broglie simply rearranged the two most important equations for photons — the photon momentum equation, which gives a photon’s momentum as p = h/λ, and the PlanckEinstein relation, which gives a photon’s energy as E = h f (see Chapter 13). This led to the de Broglie relations for matter waves, λ=
h p
and
f =
E . h
(14.1)
The former equation is the de Broglie wavelength and the latter is the de Broglie frequency. As usual, h is Planck’s constant, equal to 6.626 · 10−34 J s. In contrast to the situation for photons, de Broglie set the momentum and energy values in these equations to their standard values from classical physics. The classical momentum of an object is p = mv par t. ,
(14.2)
where m is its mass and v par t. is its velocity; the “part.” subscript clarifies that this is the velocity of the particle, as opposed to its matter wave, which will turn out to
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be relevant later on. The classical energy of an object is E=
1 2 + U (x). mv 2 par t.
(14.3)
The first term is the kinetic energy, meaning the energy of the particle’s motion. The second term, U (x), represents the object’s potential energy as a function of its position, which we’ll return to in the next section. These equations show that heavier and/or faster objects have more of pretty much everything. They have more energy, more momentum, and faster frequencies. On the other hand, they have shorter wavelengths (like light waves, where shorter wavelengths correspond to higher energy photons). Quantum effects are most important when objects have long wavelengths, so electrons and other lightweight particles tend to exhibit the strongest quantum effects. These four equations form the foundation ofnon-relativistic quantum mechanics. This is the physics of systems that are small enough that particle-wave duality is important and that move slowly enough that Einstein’s theory of relativity can be safely ignored1 . Most of the mathematical results in the rest of this chapter follow directly from these equations. Example. Find de Broglie wavelengths for (a) a helium atom at a temperature of 2 K, which moves at 100 m/s (mass is 6.65 · 10−27 kg), (b) a 0.059 kg golf ball moving at 60 m/s. Answer. (a) Use the de Broglie wavelength equation for helium: λ=
6.626 · 10−34 J s h h = = = 1.0 · 10−9 m. p mv par t. (6.646 · 10−27 kg)(100 m/s)
(b) For the golf ball: λ=
h 6.626 · 10−34 J s h = = = 1.9 · 10−34 m. p mv par t. (0.059 kg)(60 m/s)
The helium atom has a “large” wavelength of about 1 nm, which is similar to the atom’s size, so quantum effects are likely to be important. In contrast, the golf ball’s wavelength is so tiny that quantum effects are completely irrelevant.
1 The de Broglie relations are still valid with relativity, but the momentum and energy equations need to be replaced by their relativistic versions. These are: p = γm 0 v and E = γm 0 c2 , where m 0
is the rest mass and γ = 1/ 1 − vc2 . This relativistic energy includes the mass energy along with the kinetic energy, which increases the de Broglie frequency and wave phase velocity, but doesn’t affect any observable results. 2
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14.1.2 Wave Functions The news of de Broglie’s work spread quickly and generally received a positive reception. For example, Einstein wrote that de Broglie had “lifted a corner of the great veil”2 . However, de Broglie’s wave theory was incomplete because it didn’t explain how the matter waves moved or even what they really were. Erwin Schrödinger, an Austrian physicist, made the next major advance. While on a Christmas ski vacation with his mistress3 in the Swiss Alps in 1925, he came up with the idea that each particle was represented by a wave function, given by ψ(x). This function was a wave that propagated through space, much like electromagnetic waves, but was for matter instead of light. He also figured out an equation, now called the Schrödinger equation, that described how these waves propagated4 . One of Schrödinger’s insights was that the mathematics became simpler if the wave function displacements were complex numbers. Complex numbers have two parts to them, called the real and imaginary components; for example, 2 + 3i is a complex number in which √ the 2 part is real and the 3i part is imaginary. The i factor represents the value of −1, which has no physical meaning but allows for some clever math such as the facts that i 2 = −1 and i · (−i) = 1. The top of Figure 14.2 shows a wave function for a particle as a function of the position along the x axis, with its real component on the vertical axis and its imaginary component on the axis that comes out of the page. As with photons, matter waves are best understood as probability waves, where large displacements imply high probability of finding the particle at that location and small displacements imply low probabilities. More precisely, the actual probability of the particle being at some position along the x-axis is given by the absolute square of the wave function at that location; the absolute square is essentially the same as the square of a number, but is a purely real number without any imaginary component5 . The bottom of Figure 14.2 shows this probability; it shows the likelihood of the particle being at each position along the x-axis. The area under this probability function is equal to 1, meaning that the electron has 100% probability of being somewhere.
2 APS News, “This Month in Physics History. October 18, 1933: Louis de Broglie elected to Academy” vol. 19, no. 9, p. 2, October 2010. 3 In addition to his physics work, Schrödinger is well known for his womanizing, such as that he had three daughters by three different mistresses, and he effectively had two wives for a while. There doesn’t seem to be a record of which mistress accompanied him during the Christmas 1925 trip. See “The lone ranger of quantum mechanics” by Dick Teresi, New York Times Section 7, page 14, Jan. 7, 1990. 4 The Schrödinger equation for wave propagation in one dimension is the differential equation 2 2 ∂ ∂ h i ∂t ψ(x, t) = − 2m + U (x, t) ψ(x, t), where = 2π and U (x, t) is the potential energy as ∂x 2 a function of position and time. 5 To take the absolute square, one multiplies a number by its complex conjugate, where the complex conjugate is the same as the original number except that the signs of the imaginary components are reversed. For example, the complex conjugate of 2 + 3i is 2 − 3i and the absolute square of 2 + 3i is (2 + 3i)(2 − 3i) = 4 + 6i − 6i − 9i 2 = 4 + 9 = 13. Complex conjugates are denoted by asterisks, so the absolute square of ψ(x) is ψ(x)ψ ∗ (x).
14.2 Traveling Matter Waves Figure 14.2 (Top) A matter wave function, showing a wave packet that is propagating along the x-axis. (Bottom) The absolute square of the wave function, which is the probability of the particle being at any particular position along the x-axis.
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14.1.3 What is Waving? While de Broglie and Schrödinger showed that matter waves were clearly important, a persistent question was what was actually waving. This was expressed by the German scientist Erich Hückel (and translated into English by the Swiss physicist Felix Bloch) in the poem Erwin with his psi can do Calculations quite a few. But one thing has not been seen: Just what does psi really mean.
The answer to this mystery is that there simply is no physical reality for matter waves. Instead, matter waves are only mathematical entities; they are figments of our imagination6 . That said though, matter waves behave like other waves and lead to physically sensible predictions, such as the result that squaring the wave displacements produces the probability of finding the corresponding particle at any given position. They also enable accurate, elegant, and sometimes even intuitive explanations of the physical world. Thus, as most physicists do, we will treat them here as though they are actual waves.
14.2
Traveling Matter Waves
14.2.1 Free Particles Consider a particle, such as an electron, proton, or atom, that’s just moving along through space, with nothing in its way. This free particle has some kinetic energy, because it’s moving, but it doesn’t have any potential energy because it’s not inter-
6 Even Niels Bohr, one of the founders of quantum mechanics, questioned their reality; he wrote “There is no quantum world. There is only an abstract quantum mechanical description. It is wrong to think that the task of physics is to find out how Nature is. Physics concerns what we can say about Nature.” From A. Peterson, Bulletin of the Atomic Scientist 19:12, 1963.
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acting with anything. This means that we can set the potential energy to zero, U (x) = 0.
(14.4)
If we define the particle’s mass as m and its velocity as v par t. , we can then compute its momentum, energy, de Broglie wavelength, and de Broglie frequency: p = mv par t.
E=
1 2 mv 2 par t.
λ=
h mv par t.
f =
mv 2par t. 2h
.
(14.5) We want to learn more about the particle’s matter waves, so we compute their velocity from the frequency-wavelength relation, v phase = λ f , which is true for all types of waves. The “phase” subscript reminds us that this is the wave phase velocity, meaning how fast the wave peaks move. Substituting in from above and cleaning up lead to v phase = λ f =
mv 2par t. v par t. h · = . mv par t. 2h 2
(14.6)
This is a surprising result. It says that the wave peaks don’t move at the same speed as the particle. Instead, they only move half as fast. This is reminiscent of the situation with water waves, in which a group of waves travelled with a “group velocity” that was different from the waves’ “phase velocity”. In that case, the difference arose from dispersion, meaning that the phase velocity depended on the wavelength. To see if the same thing occurs here, we rearrange the de Broglie wavelength to v par t. = h/mλ and then substitute that into Eq. 14.6 to give h . (14.7) 2mλ Indeed, this shows that the matter wave phase velocity depends on the wavelength, implying that they are dispersive. The fact that matter waves are dispersive has a couple of important consequences. First, as just discussed, they have different phase and group velocities. We won’t derive it here, but it can be shown that the group velocity is the same as the particle’s velocity. This makes sense because the group of waves is the particle, so it should move at the same speed as the particle. On the other hand, the phase velocity is largely meaningless; it’s half as fast as the group velocity in the example shown, but can be different in other situations (the phase velocity would have been different if we had set the potential energy to some value other than zero in Eq. 14.4). Second, it means that matter waves spread out as they propagate. Figure 14.3 illustrates this by showing the matter waves at three time points for a single particle that moves from left to right. This spreading is the same as that seen in water waves produced from a rock that’s dropped in a pond, where a single initial wave spreads out to a broad group of waves. In terms of the matter waves, it means that the particle starts with a narrow range of possible positions initially but then quickly spreads out to having a wide range of possible positions. v phase =
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Figure 14.3 The same matter wave at three points in time, showing that it spreads out. The time is measured in femtoseconds (10−15 s).
14.2.2 Classical and Quantum Roller Coasters For particles that aren’t free, we need to consider how the potential energy varies over space. The left panel of Figure 14.4 illustrates this by showing a normal roller coaster. The cars start at the top of the hill at the left end, where they have high potential energy, and roll down the hill, speeding up as they go. In the process, they convert potential energy to kinetic energy, which reaches its greatest value when the cars reach the bottom of the first dip (blue cars). The energy conversion process then reverses as they coast up the next hill, where they slow down and convert their kinetic energy back to potential energy, eventually going slowly over the top of the hill (orange cars). Then they go down and speed up again.
Figure 14.4 (Left) A classical roller coaster. (Right) A quantum roller coaster. Gray shading represents a potential energy surface.
These hills can be seen as a potential energy surface, meaning that they represent the cars’ potential energy as a function of their position. As a math expression, it is the U (x) function introduced above. The cars’ total energy, shown by the dashed black line at the top of the figure, stays constant over the whole journey due to the conservation of energy (we ignore friction). Subtracting the potential energy from this total energy yields the kinetic energy, which can be visualized in the figure as the distance between the potential energy surface and the dashed line (upper red arrow). For example, we can immediately see that the cars’ kinetic energy is low at the hill tops and high at the hill bottoms. The right panel shows a quantum roller coaster in which two particles move from left to right, each represented by a wave function. The gray region still represents a potential energy surface, now created by electrical forces rather than gravity but that doesn’t really change anything, and the dashed line still represents the total energy. The distance between these heights still represents the kinetic energy. The only changes are stylistic: (1) this diagram shows wave functions instead of cars, and
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(2) it uses the convention that wave functions are drawn on the line that represents the total energy rather than directly on the potential energy surface. Despite this change in drawing position, it’s important to remember that the particles aren’t flying over the surface like airplanes, but, like the roller coaster, are moving along the surface and responding to its height and shape as they go along. If you could watch the particles move, they would move just like the cars; they would speed up as they went downhill, slow down as they went up the next hill, and then speed up again on the far side. As before, we’ll calculate some of the wave parameters. First, the frequency is given by the de Broglie frequency relation, f =
E , h
where E is the total energy. This total energy doesn’t change as a particle moves up and down the hills, so the frequency doesn’t change either. In retrospect, this is a familiar result. Recall that light waves don’t change frequency when they change speeds by transitioning from air to glass. We also saw this behavior with a row of cars that slowed down where the road got bumpy (see Section 2.4.4). The same thing happens here, where the matter waves maintain their frequency even as they propagate faster or slower. A complication in this case is that we are allowed to set the zero value of potential energy anywhere we want. In Figure 14.4, it’s set to the bottom of the figure, but it could have been set higher or lower. This essentially arbitrary definition affects the value of the matter wave frequency, so these absolute values don’t actually mean much. On the other hand, it is important that matter wave frequencies are constant so long as the total energy doesn’t change, and they change when the total energy does change7 . The wavelength is next. We combine the kinetic energy (E kinetic = 21 mv 2 ) and momentum ( p = mv) equations to give E kinetic =
p2 . 2m
(14.8)
√ Then, solving for p gives p = 2m E kinetic . This is substituted into the de Broglie wavelength equation to find that the wavelength is h . λ= √ 2m E kinetic
(14.9)
This shows that matter wavelengths depend on the kinetic energy, and that particles with high kinetic energy (blue waves in the figure) have short wavelengths. Vice
7 Another
complication is that the wave speeds are not what one would expect. The group velocity increases as the particle goes downhill and decreases as it goes uphill, as expected, but the phase velocity behaves in exactly the opposite manner, increasing as the particle moves slower.
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versa, those with low kinetic energy (orange waves) have long wavelengths. Again, this is a familiar result, this time from photons, where long wavelength photons have low energies and short wavelength photons have high energies. We’ll see the same trends repeatedly throughout the following sections. In all cases, wave frequencies depend on total energies. Also, wavelengths depend on the kinetic energy, with long waves for low kinetic energy and short waves for high kinetic energy.
14.2.3 Barriers and Tunneling What happens when a quantum particle hits a barrier? For example, suppose an electron is moving through a piece of metal and there’s a tiny gap in the metal that the electron would have to jump across. Can it cross the gap or not? The classical version of this problem is a ball rolling up to a ridge in the carpet. If the ball has a lot of kinetic energy, then it just rolls up and over the ridge, and continues along on the far side. On the other hand, if it doesn’t have enough kinetic energy, then it goes part way up the ridge, stops, and rolls back toward where it came from. Even having just slightly too little energy is not good enough; it still gets reflected back toward where it came from. Figure 14.5 illustrates the quantum version with three particles, one on each horizontal line. The one on top has more total energy than the barrier height, the one in the middle has slightly less energy than the barrier, and the one on the bottom has much less energy. All three particles start with essentially the same wave packets, shown in green, which are moving toward the barrier from the left (they have different wavelengths, due to their different kinetic energies). When they hit the barrier, shown in blue, the top particle mostly goes through, the middle particle mostly reflects, and the bottom particle almost completely reflects. These same results are also seen after the particles have interacted with the barrier, shown in red, where the top particle is mostly transmitted and the others are mostly reflected. Figure 14.5 Three particles hitting a barrier. Each row represents a different particle, where the top one has high energy, the middle one has medium energy, and the bottom one has low energy. Matter waves for each particle are shown in green at the start, blue as the particle hits the barrier, and red after hitting the barrier.
These results mostly agree with classical version, but not perfectly. Considering the top particle, it split into two pieces with most of it making it through the barrier
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and a small fraction getting reflected back. This arises from the wave behavior of the particles, occurring because all waves partially reflect when they hit a boundary where they change speed. This includes string waves transitioning from a lighter to a heavier string, and light waves going from air to glass. Matter waves act the same way. In this case, the waves for the top particle change speed twice, once for each side of the barrier, which then leads to the same thin film interference behaviors that we met in Section 3.3.1, but now with matter waves. As it turns out, this thin film transmits most of these waves and reflects a small fraction. Changing either the barrier thickness or particle energy would modify the relative amounts of constructive and destructive interference, thereby changing the precise amount of reflection. More remarkably, part of the medium energy particle, which does not have enough energy to make it over the barrier, gets transmitted through the barrier anyhow. This is called quantum tunneling. It is the matter wave equivalent of frustrated total internal reflection, which we saw earlier for light waves (Section 9.2.6). What happens is that the wave packet hits the barrier, in this case from the left side. As it interacts, the waves extend into the barrier, as evanescent waves, by up to a few wavelengths in depth. If those evanescent waves can reach through to the far side, then they become real waves again, which propagate along just like normal. Meanwhile, the rest of the incident waves get reflected back toward where they came from. Tunneling is impossible for classical particles, but occurs in quantum mechanics because of the particles’ wave natures (tunneling is not just a quantum behavior, but occurs for all types of waves; this includes sound waves, which are completely classical8 ). Finally, the low energy particle appears to be completely reflected by the barrier, as one would expect. Presumably, there is some tiny tunneling probability for it, too, but that probability is too small to be observed in the figure and almost certainly too small to be relevant. In general, quantum tunneling is a fairly rare phenomenon because it requires narrow barriers (relative to the particle’s de Broglie wavelength) and particle energies that are quite close to the barrier height.
14.2.4 Tunneling Examples* Despite the rarity of quantum tunneling, there are several real-world examples of it. Nuclear reactions, whether in nuclear power plants, nuclear bombs, or the interior of the sun, only occur because of quantum tunneling. There are two types of nuclear reactions, nuclear fusion in which two light atom nuclei fuse together to form a heavier nucleus, and nuclear fission in which one heavy atom nucleus decays by splitting apart into two lighter ones. Either way, there is a high potential energy barrier that usually prevents the reaction from occurring, as shown in the left panel of Figure 14.6. Even in the center of the sun, where the temperature is about 15 million
8 See Yamamoto, Sakiyama, and Izumiya, “Visualization of acoustic evanescent waves by the stroboscopic photoelastic method” Physics Procedia 70 (2015): 716–720; and Peng, et al. “Acoustic tunneling through artificial structures: From phononic crystals to acoustic metamaterials” Solid State Communications 151 (2011): 400–403.
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kelvin, the hydrogen atom nuclei, which are just protons, don’t have enough energy to cross over this potential energy barrier. However, the protons can occasionally tunnel through the barrier. After tunneling, they “relax” to a lower energy fused state and release a great deal of energy in the process9 .
Figure 14.6 (Left) Nuclear fusion tunneling, in which a proton, shown with a blue wave packet, tunnels through an energy barrier to fuse with another proton. (Middle) Ammonia molecule potential energy landscape. Dashed lines show the energy of room temperature molecules, which undergo umbrella inversion by tunneling through the energy barrier. (Right) Light spots are individual silicon atoms in a silicon carbide crystal, imaged with a scanning tunneling microscope.
A less exotic tunneling example occurs with ammonia molecules, shown in the middle of Figure 14.6. Ammonia has the chemical formula NH3 and is widely used in household cleaners. The molecule has the shape of a pyramid, with a nitrogen atom at the point and three hydrogen atoms forming a triangular base. This is a reasonably stable pyramid, but it can also undergo “umbrella inversion,” in which the nitrogen atom goes through the hydrogen atom base and out the other side, much like an umbrella turning inside-out in a strong wind. Room temperature ammonia molecules don’t have enough energy to invert according to classical physics, but they can with quantum mechanics because the nitrogen atom can tunnel through the potential energy barrier that is created by the triangle of hydrogen atoms. This tunneling is quite rapid due to the low potential energy barrier, with rates of around 3 · 1010 inversions per second. A third example of quantum tunneling is the scanning tunneling microscope. This instrument connects a probe and a material of interest to an electric power supply and then lowers the probe to within less than a nanometer from the material’s surface. Electrons then tunnel across the gap at a rate that depends very strongly on the separation. As a result, the amount of electric current indicates the surface height very precisely, to within less than the thickness of a single atom. The probe is scanned across the surface to map out the locations of each atom, of which an example is shown in the right side of Figure 14.6.
9 When two protons fuse together to form a deuterium nucleus, they release a positron and a neutrino;
this is shown as the nuclear reaction p + p → 2 H + e+ + ν. The positron and neutrino carry energy away from the newly fused nucleus, enabling it to relax to a lower energy state.
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14.3
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Diffraction and Interference
14.3.1 Electron Diffraction As de Broglie was developing his theory of matter waves in Europe, two American physicists, Clinton Davisson and Lester Germer, observed strange patterns from electrons that they had bounced off a nickel surface. They didn’t know what caused them but Max Born, a German physicist who was deeply involved in quantum mechanics, realized that they arose from electron interference. In other words, this experiment showed that the wave nature of electrons caused their waves to interfere, just like light waves interfere in the double-slit experiment. After learning about what they were seeing from Born, Davisson and Germer then improved their experiment to show even stronger evidence of electron interference, thus providing a strong piece of evidence in favor of the wave theory of matter. Figure 14.7 shows a modern example of electron diffraction (i.e. interference), this time off an iron crystal from a piece of steel. It was produced by aiming a beam of electrons at the metal surface and then imaging the reflected electrons with a photographic plate. The surface acts like a diffraction grating (Section 3.5.3) by reflecting the electron waves from a periodic array of atoms, each one acting as a separate reflector. Those reflected waves interfere constructively with each other in some places, which creates the white spots on the film, and destructively elsewhere, which creates the dark regions. The spot in the center of the pattern is much brighter than the rest, so the experimenter blocked this spot with a stick to prevent it from overwhelming the rest of the image. Figure 14.7 Pattern produced by electrons diffracting off an iron crystal in a piece of steel. The 6-fold symmetry arises from the iron atom symmetry in the crystal.
As with other types of interference and diffraction, electron diffraction effects are strongest if their wavelengths, meaning their de Broglie wavelengths, are similar to the lattice spacing. As an example, iron atoms in a crystal have a spacing of about 0.25 nm, so electrons need to have de Broglie wavelengths of around 0.25 nm to exhibit strong diffraction effects. From Eq. 14.9, this length corresponds to an electron kinetic energy of around 3.9 · 10−18 J, which is a speed of about 2.9 · 106 m/s. Electron energies are often quantified in electron-volts, abbreviated eV, where 1 eV = 1.602 · 10−19 J. Thus, an electron with an 0.25 nm wavelength has 24 eV of energy, which is generally considered to be fairly low energy.
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14.3.2 Diffraction for Research* Electrons are relatively easy to work with because they can be emitted from traditional incandescent light bulb filaments, accelerated with electric fields, aimed with magnetic fields, and detected with many sensors, including simple photographic film. As a result, electron diffraction patterns are widely used for investigating a wide variety of crystals, ranging from simple inorganic crystals such as iron and silicon, to complicated organic crystals such as protein crystals. Electrons interact with atoms through their attractions to positively charged nuclei and repulsions from the surrounding electrons, thereby creating diffraction patterns that represent the electric charge distributions in the crystal. Neutrons, which are particles that are like protons but uncharged, diffract as well. Because they don’t have electric charges, they aren’t affected by the electric fields within a crystal’s atoms. Instead, they only reflect off atomic nuclei, through nuclear forces10 , enabling them to penetrate deep into a material. It also means that their diffraction patterns depend on the masses of the nuclei, such as whether a hydrogen atom is normal or “heavy hydrogen,” meaning that its nucleus includes a neutron as well as a proton (also called deuterium). As a result, neutron diffraction is particularly good for showing which types of atoms are where in a crystal, which is particularly useful for protein structure determination11 . On the other hand, this requires low energy neutron beams, and these are hard to produce, hard to aim, and hard to detect, so neutron diffraction is only studied at a relatively few facilities. Helium atom diffraction is also useful. Its primary benefit is that helium atoms are big enough that they can’t go through other atoms, as electrons and neutrons do, but they instead bounce off the very top layer of atoms on a surface. This is useful for imaging surfaces and surface phenomena. For example, helium diffraction can be used to image mechanical waves that travel across the surface of a crystal12 . Neutrons and helium atoms are thousands of times heavier than electrons. This means that they need to travel at relatively slow speeds to have de Broglie wavelengths that are on the order of crystal spacings.
14.3.3 Interference to Test Quantum Mechanics* A persistent goal in physics has been to create interference patterns with the largest possible particles. In addition to just the sheer challenge of it, this tests quantum mechanics theory to see if it disagrees with experiment at some point. So far, the theory has never failed.
10 Neutrons
are scattered primarily by the strong nuclear force, which is one of the four known fundamental forces and only has a range of about 10−15 m. They are also scattered by magnetic fields. 11 See, for example, Fukuda et al., “High-resolution neutron crystallography visualizes an OHbound resting state of a copper-containing nitrite reductase”, Proc. Natl. Acad. Sci. USA 117:4071, 2020. 12 See, for example, Glebov et al. “A helium atom scattering study of the structure and phonon dynamics of the ice surface”, J. Chem. Phys. 112:11011, 2000.
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One notable large particle is a molecule called buckminsterfullerene (buckyballs for short, and named for Buckminster Fuller, an American architect who popularized geodesic domes). Each molecule is composed of 60 carbon atoms, is about 0.7 nm across, and has the shape of a soccer ball (Figure 14.8). Buckyballs can form interference patterns13 , which is remarkable because this shows that even large molecules, which are so large that they behave classically in almost every way, exhibit particlewave duality. This means that a single buckyball can be aimed at two slits, as in Young’s two-slit interference experiment, and it goes through both of them at once. Figure 14.8 Molecular structure of buckminsterfullerene.
The current record-holder is a molecule that is about 30 times larger yet, with over 2000 atoms14 , the size of a small protein. Again, it exhibits particle-wave duality, exactly as quantum mechanics theory predicts. The de Broglie wavelength often forms a convenient test of whether quantum mechanics theory is likely to be relevant. In these experiments, the buckyball wavelength was 2.5 pm (2.5 · 10−12 m), which was hundreds of times smaller than the molecule itself. The larger molecule was even more extreme, with a wavelength of only 53 fm (53 · 10−15 m), which is only a few times larger than some atomic nuclei and again vastly smaller than the molecule. In both cases, one wouldn’t normally expect to see quantum effects due to the tiny wavelengths, but these experiments show that they still can be found with sufficient effort.
14.4
Standing Matter Waves
Matter waves, like other types of waves, exhibit standing waves when they are in confined spaces. Standing waves represent the normal modes of a system, whether of a jumprope, guitar string, air vibrating in a flute, or ocean water sloshing back and forth as tides. In the case of matter waves, electrons form standing waves when they are confined by electric fields, atoms form standing waves when they are confined by chemical bonds, and protons and neutrons form standing waves when they are confined by the nuclear force.
13 Arndt
et al. “Wave-particle duality of C60 molecules”, Nature 401:680, 1999.
14 Fein et al. “Quantum superposition of molecules beyond 25 kDa” Nature Physics, 15:1242, 2019.
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14.4.1 Electrons in a Cavity We start by considering standing waves in a cavity, as we did when we first met standing waves in Section 3.2.3. As an example, an electron in a carotene molecule, shown in Figure 14.9, can move freely over most of the molecule but cannot leave, so the molecule acts as a cavity and the electron’s wave function forms standing waves across its length.
Figure 14.9 Chemical structure of a β-carotene molecule. The red region represents the electron cavity, within which many electrons can travel reasonably freely.
To define this system more precisely, which is often called the particle in a box problem, we assume a potential energy well in which the electron has zero potential energy within the molecule and infinite potential energy outside of it, as shown on both sides of Figure 14.10 with gray shading. The infinite energy outside the molecule is just another way of saying that the electron cannot leave.
Figure 14.10 Potential energy surface, energy levels, wave functions (left), and probability distributions (right) for matter waves in a cavity.
Because the electron cannot be outside the cavity, its wave function must have a value of zero everywhere outside. This includes the cavity edges, which means that the wave function inside the cavity must also go to zero at each edge. In many ways this is similar to a guitar string, where the string’s position is fixed at each of the endpoints but the rest of it can vibrate15 . These boundary conditions enable us to find the wave function’s normal modes. As we’ve seen before with other standing waves in cavities, the only waves that have
15 This
paragraph defines the boundary conditions for the wave function. More generally, wave function boundary conditions are that (1) the wave function must be continuous, and (2) its first derivative must be continuous except where the potential energy becomes infinite.
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zero displacement at each end are those where the wavelengths are even fractions of the cavity length. This means that the possible standing waves have L = λ/2, L = λ, L = 3λ/2, and so on, where L is the cavity length. The left side of Figure 14.10 shows these standing waves with blue lines; each wave function oscillates over time between the solid lines and dashed lines that are shown in the figure. These possible standing waves can be generalized to λ=
2L , n
(14.10)
where n is some integer, such as 1, 2, 3, etc. (this is a repeat of Eq. 3.1, where we first met standing waves in cavities). Squaring the wave functions gives the probability for the electron’s location, which is shown on the right side of the figure. It shows that electrons in the n = 1 state are most likely to be near the middle of the molecule, whereas those in higher states are more evenly spread out over the entire molecule. It also shows that there are multiple points along the molecule where the probability goes to zero, meaning that the electron cannot ever be there. This seems impossible when thinking about a particle moving around but remember that the electron isn’t just a particle; it also is a wave. We saw previously that matter waves with greater kinetic energy have shorter wavelengths, which was quantified in Eq. 14.9. We can now use that relationship to compute the total energies for these matter waves (the total and kinetic energies are the same in this case because the potential energy was set to zero within the cavity). Setting the wavelength from that equation equal to the normal mode wavelengths that we just found in Eq. 14.10 leads to 2L h =λ= √ n 2m E n nh 2m E n = 2L n2h2 . En = 8m L 2
(14.11)
These energies for the matter wave normal modes, called energy levels are shown in Figure 14.10 with black dashed lines, on which the wave functions are drawn. As expected, note that matter waves with shorter wavelengths have higher energies. These energy levels illustrate two important points about matter waves in confined spaces. First, the normal modes only have specific energies. This is one of the most important results of quantum mechanics. And second, even the lowest energy level always has some kinetic energy. This arises from the relationship between wavelength and energy, and the fact that there is a maximum possible wavelength in confined spaces. This lowest possible kinetic energy, called the zero-point energy, also implies
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that quantum particles are always in motion. In this particular case, the zero-point energy is the energy of the n = 1 state, which is E1 =
h2 . 8m L 2
14.4.2 Filling in Electrons Up to this point, the de Broglie relations and their refinement into wave functions by Schrödinger have been the only quantum mechanics theory that we have used. This is adequate for any system with one electron, or any other single particle, but another rule is needed for systems that have multiple electrons. This rule is the Pauli exclusion principle, named for the Austrian physicist Wolfgang Pauli, which states that no two electrons can share the same quantum state. As it turns out, electrons have a property called spin, which behaves somewhat like the classical spinning of tops and balls that we’re used to, but not quite. In particular, electrons can’t have variable amounts of spin but have only two quantum spin states, which are conventionally labeled as “up” and “down.” As a practical matter, this means that we can draw electrons wherever we like on a diagram of energy levels, but we can only put up to two electrons on any particular level and those two must have opposite spins. Back to carotene, each molecule has 22 electrons that are reasonably free to move16 , so we need to fill these electrons into the energy levels that we just derived. Figure 14.11 shows the electron arrangement that has the lowest amount of energy, called the ground state, in which there are two electrons in each of the first 11 levels and none above that. Thus, the image that we should have in mind of these 22 electrons is that all of their matter waves extend over the entire molecule, of which two standing waves are in the n = 1 normal mode, two in the n = 2 normal mode, and so on, up to the final two in the n = 11 normal mode. The molecule also has higher normal modes, but they aren’t occupied by electrons. Figure 14.11 Energy levels for a carotene molecule, showing the quantum number on the left. These levels are filled with 22 electrons. A transition is shown in which an electron in n = 11 absorbs a photon and moves up to n = 12.
16 Most
of the carotene electrons are tightly bound to either a single atom or to a chemical bond between two atoms. However, the chain of alternating single and double bonds creates what’s called a conjugated π system, in which one electron from each carbon atom is free to move over the entire system.
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It would be nice to test this theory by observing the actual electron wave functions or their energies, but this isn’t possible. Instead, what can be done is to investigate transitions between energy levels, which is typically accomplished by adding or removing energy with a photon. Figure 14.11 shows the lowest energy transition that is possible from the ground state, in which a photon excites an electron from the n = 11 level to the n = 12 level. It uses the convention that photons are depicted with wiggly lines, to remind us of electromagnetic waves. The energy of this transition, meaning the energy of the photon that gets absorbed in the process, can be found by plugging numbers into Eq. 14.11 for the n = 11 and n = 12 energy levels. Other values are that m is the electron mass, which is 9.11 · 10−31 kg, L is the carotene molecule length, which is about 2.4 nm, and h is Planck’s constant, which is 6.64 · 10−34 J s. From these, E 12 − E 11 = 1.51 · 10−18 J − 1.27 · 10−18 J = 2.42 · 10−19 J. This energy corresponds to a photon with a wavelength of 825 nm (from Eq. 13.2). For comparison, the experimental answer (from Figure 4.12) is around 450 nm, meaning that our prediction is off by almost a factor of 2. This agreement isn’t very good, but is still remarkably close given the crudeness of the model. More importantly, this analysis shows how one can go from abstract ideas about standing matter waves in molecules to actual measurable properties. More thorough calculations that are based on the same principles can yield extremely accurate spectral predictions.
14.4.3 Molecular Vibrations It’s a reasonably good approximation to think of chemical bonds between atoms as springs that can be compressed, stretched, or bent, but that rebound back afterward. These springs also allow atoms to bounce back and forth repeatedly, which are molecular vibrations. Formally, this is called a harmonic oscillator, which includes molecular vibrations as explored here, but also clock pendulums, marbles rolling back and forth in bowls, and many other things that oscillate back and forth. For concreteness, consider the hydrogen chloride molecule, HCl, shown in Figure 14.12 (hydrogen chloride makes hydrochloric acid when dissolved in water, which is widely used for industrial applications). While the two atoms actually bounce back and forth against each other, the chlorine atoms barely move because they are much heavier than hydrogen atoms, so we’ll treat these vibrations as though only the hydrogen atom moves. With this approximation, the classical view is that the hydrogen atom acts as a ball that oscillates back and forth at some natural frequency17 , given as f . This classical oscillation can have any amplitude, including zero. Also,
1 k natural frequency is f = 2π m where k is the force constant of the spring and m is the oscillator mass, which in this case is essentially just the mass of the hydrogen atom.
17 This
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like a kid on a swing set, it can be driven to higher or lower amplitude through resonance with a driver that oscillates at the same frequency; a typical driver would be an electromagnetic wave that alternately pushes and pulls on the hydrogen atom using the fact that it has a slight positive electric charge.
Figure 14.12 Hydrogen chloride molecule, showing the chemical bond as a spring.
To investigate the quantum description, Figure 14.13 shows the potential energy well for the hydrogen atom as a function of its distance from the chlorine atom. The potential energy is zero at the bottom of the well, when the spring is fully relaxed, and increases when the hydrogen atom is moved either closer to or farther from the chlorine, because it takes energy to either compress or stretch the spring.
Figure 14.13 Potential energy surface, energy levels, wave functions (left), and probability distributions (right) for the hydrogen atom in a HCl molecule.
More mathematics are required here than for the particle-in-a-box problem presented above, so I’ll just present the answers. The potential energy well turns out to be represented fairly accurately by a quadratic function, and the normal modes for the hydrogen atom matter waves are shown on the left side of Figure 14.13. These matter waves don’t stop abruptly at the sides of the well, because the sides are sloped, but extend a short distance into them with evanescent waves. As usual, waves with more kinetic energy have shorter wavelengths, which applies both to those at higher energy levels and to those that are closer to the middle of the well. From these wave functions, the energy levels are found to be 1 , (14.12) En = h f n + 2 where f is the classical oscillation frequency and n can be any integer starting from 0, meaning n can be 0, 1, 2, etc. The lowest energy normal mode, n = 0, has a zero-point energy, implying that the hydrogen atom cannot be stopped at the relaxed position, but is always moving.
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These classical and quantum descriptions are very different. The classical hydrogen atom bounces back and forth; also, it has frequency f , regardless of its amplitude. In contrast, the quantum one just sits in a particular normal mode, with its probability spread out in one or more stationary bumps along the position axis (right side of Figure 14.13). Its frequency is the de Broglie frequency, f = E/h, which depends on the energy level. How can these descriptions be compatible with each other? Regarding the position, it’s true that the quantum version can be in a single normal mode, but it doesn’t have to be. Just like when you hold a rope that’s tied to a doorknob, you can put it in a single normal mode by swinging it like a jumprope, but you can also shake it up and down briefly to put a pulse in the rope that travels back and forth; this traveling pulse is a superposition of normal modes. Similarly, the matter wave can be in a superposition of multiple normal modes that add up to create a pulse that bounces back and forth. This pulse represents the hydrogen atom, showing that the hydrogen atom’s classical bouncing motion is consistent with the quantum description. The frequency also has a connection. From Eq. 14.12, the spacing between adjacent energy levels, e.g. n and n + 1, is always h f . This is the energy of a photon that has frequency f , meaning that a photon with frequency f can add energy to (or remove energy from) the vibration. This same frequency, f , is also the difference between the de Broglie frequencies for adjacent energy levels. Also, this is the same frequency that one would need for transferring energy through classical resonance. Thus, the classical frequency is still the essential frequency of the system even in the quantum description. This also shows that although the actual values of the de Broglie frequencies are meaningless, their differences represent both the classical resonant frequency and the quantum photon frequency that is required for transitions18 . As an aside, note that the matter waves described here don’t represent just a single particle but an entire hydrogen atom. Furthermore, if we consider the hydrogen and chlorine atoms bouncing against each other with both atoms moving, as we really should, then the matter waves represent the combined motions of both atoms. This shows that a single wave function can represent the motions of multiple particles at once. Additionally, recall from Chapter 4 that larger molecules have many vibrational modes, such as the three modes for water molecules that are shown in Figure 14.14. Each of them is quantized in the same manner, yielding a separate vibrational wave function for each vibrational mode.
Figure 14.14 Vibrational modes of water molecules, repeated from Figure 4.17.
18 These
frequency correlations are only strictly true for harmonic oscillators, meaning systems with quadratic potential wells, but are also reasonably close for other systems as well. In addition, all single-particle systems appear to approach these correlations as the particle energy is increased.
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14.4.4 Structures of Atoms One of the bigger unsolved problems in the early 1900s was how atoms were stable, without the electrons falling into the nuclei. The understanding at the time was that atoms were composed of a small positively charged nucleus (correct), and that they had one or more electrons that orbited around this nucleus like planets around the sun (incorrect). Physicists reasoned that the orbiting electrons should emit electromagnetic radiation, as electrons do in radio antennas, so the electrons should gradually lose energy and spiral into the nucleus. However, this wasn’t observed. Schrödinger’s matter wave description of electrons solved the problem. Figure 14.15 shows a slice through the potential energy well for an electron around a hydrogen atom’s nucleus with gray shading. It arises from the attraction of opposite electric charges, in this case between the negatively charged electron and the positively charged proton that is the nucleus. This is only one slice because it’s really a threedimensional system, which is hard to draw. There’s no well-defined bottom to this potential energy well, so the convention is to define the top of the surface as the zero energy level, which means that the potential energy is negative everywhere below that. The kinetic energy is still represented by the height difference between the potential energy surface and the total energy, which is still a positive value. Figure 14.15 Potential energy surface, energy levels, and the first few wave functions for an electron in a hydrogen atom. Blue wave functions are for the 1s and 2s orbitals, and the magenta wave function is for 2p orbitals.
The matter wave normal modes for this surface are called atomic orbitals, of which slices through the first few are shown on the figure. These wave functions show that the electron is likely to be close to the nucleus, and sometimes even right at the nucleus, but can also be farther away. As before, the system has discrete energy levels and n is the index. In this case, n is a non-negative integer, such as 1, 2, 3, etc., and is called the principal quantum number. One consequence of this being a 3D system is that there are multiple different orbitals at the n = 2 and higher levels that have the exact same energy (two are shown in the figure for n = 2). Despite being called orbitals, these wave functions are better described as electron clouds because the electron doesn’t orbit the nucleus. Figure 14.16 shows these clouds as density plots of where the electron is most likely to be found, with darker regions indicating higher probability. It also shows some of the orbital names, which include 1s, 2s, 2px , 2p y , 2pz , 3s, etc. The number part of the name is the principal
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quantum number, while the letters and subscripts describe the orbital shape and orientation. These letters follow the sequence s, p, d, and f for historical reasons19 .
Figure 14.16 Some hydrogen atom electron orbitals, where darker regions represent higher probability of finding the electron at that position. Red dots represent the nucleus. All images use the same scale, the z-axis goes up-down, and the x-axis goes left-right.
While atomic orbitals are closely associated with atoms, they are really just the normal modes for spherical systems, called spherical harmonics. In the same way, the wave functions that we saw for the particle-in-a-box problem (Figure 14.10) are the normal modes for cavity systems, and are the same as the normal modes for guitar strings, jump ropes, and water waves in dishpans. Likewise, if one created a spherical gong and whacked it, its vibrations would be best described as a sum of spherical harmonics. Atomic energy levels are reasonably simple for atoms that only have one electron, such as hydrogen (symbol H) or a helium atom with a positive charge (He+ ), and are given by En = −
m Z 2 e4 1 · . 8h 2 20 n 2
(14.13)
The constants are: m is the mass of an electron, which is 9.11 · 10−31 kg, Z is the number of protons in the nucleus (1 for hydrogen, 2 for helium, etc.), e is the electric charge of an electron, which is 1.60 · 10−19 C, h is Planck’s constant, and 0 is called the electrical permittivity of space, which is 8.85 · 10−12 C2 N−1 m2 . This equation returns negative numbers, which is due to the convention of setting the zero energy level at the top of the potential energy surface. As expected, the lowest energy state,
19 Before
atomic orbitals were understood, spectroscopists observed that atomic absorption and emission lines for alkali metals (e.g. sodium) had different characteristics. They named these lines as sharp, principal, diffuse, and fundamental, which later turned into the s, p, d, and f orbitals.
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meaning the one with the most negative value, is the one with n = 1. Energies then increase toward zero as n increases toward infinity, which can be expressed as E ∞ = 0. Just because Eq. 14.13 only returns negative values doesn’t mean that an electron can only have negative energies. To the contrary, electrons can also have positive energies. These electrons are no longer bound to the atom, and can’t be described by a quantum number, but instead have escaped the nucleus altogether, making them more like the traveling waves that we investigated first. An atom that’s missing an electron is called an ion, so atoms with escaped electrons are said to be in an ionized state (ions also include atoms with extra electrons). By extension, the ionization energy is the minimum energy that is required to remove an electron from an atom. For example, the ionization energy for a hydrogen atom that starts in the n = 2 state is E ∞ − E 2 , which simplifies to −E 2 . The left side of Figure 14.17 shows the same hydrogen energy levels but in a different format. It also labels several transitions between the different energy levels, labeled by their initial discoverer. The Balmer series, first observed by the Swiss mathematician Johann Balmer, involves relaxations down to n = 2, many of which are in the visible region of the spectrum. In contrast, the Lyman series, which goes down to n = 1, have much larger energies and so are in the ultraviolet region while the Paschen series, which goes down to n = 3, have smaller energies and are in the infrared region20 . All of these spectral lines were observed long before matter waves were even conceived of, and thus long before the emission was understood. The right side of the figure shows a hydrogen discharge tube, in which an electric current excites hydrogen atoms to high energy levels and the atoms then emit light as they relax to lower energy states. It is essentially a neon light, but with hydrogen
Figure 14.17 (Left) Hydrogen atom energy levels and transitions between them (not drawn to scale). Energies are given in electron-volts, where 1 eV= 1.602 · 10−9 J. (Right) A hydrogen discharge tube, showing the magenta color that arises from the Balmer series transitions.
20 Theodore Lyman IV was an American physicist and Friedrich Paschen was a German physicist. Also, the Rydberg formula, introduced below, was figured out in 1888 by the Swedish physicist Johannes Rydberg.
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instead of neon. The characteristic magenta color for hydrogen arises from the red, blue, and violet transitions of the Balmer series. While emitted light wavelengths can be computed from Eq. 14.13, an easier method is to use the Rydberg formula, 1 1 1 2 − 2 (14.14) R = 1.097 · 107 m−1 . = RZ λ n 21 n2 This is essentially the same as Eq. 14.13 but it collects the constants into the single Rydberg constant, R, with the value given above, and it focuses on the difference between the two energy levels n 1 and n 2 . Also, it gives the result as a light wavelength rather than an energy. As a practical matter, note that it doesn’t matter which energy level gets assigned to n 1 and which to n 2 because it only affects the sign of the answer and it’s easy to change the sign at the end. Other atoms are substantially more complicated than hydrogen due to interactions between multiple electrons, but are conceptually similar and have the same orbitals. Their orbital energies can’t be computed from Eq. 14.13, partly because they depend on the orbital shape in addition to the principal quantum number. Nevertheless, the energy levels have a fairly consistent order, which is: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, ... They get filled up in order, as described above, with up and down spins in each energy level. This sequence creates the structure of the periodic table, shown in Figure 14.18, and determines the chemical reactivity of each element.
Figure 14.18 Periodic table of the elements, showing the atomic orbitals of the outermost electrons.
Back to the question that we started with, these orbitals and energy levels explain why electrons don’t fall into the nucleus. The answer is that electrons are waves, and the lowest energy standing wave that’s possible in an atom is the 1s orbital. This wave cannot be crammed into the nucleus, but instead extends outward from it by a substantial amount. The only way to make this wave function smaller would be to shorten its wavelength. But, doing so would give it a higher energy and higher momentum, which would then make it bigger again. Thus, it is the wave nature of electrons that gives them a minimum size and prevents them from falling into the nucleus.
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Example. What color is the n = 4 to n = 3 transition for He+ ? Answer. This atom has one electron, so we use the Rydberg formula. Its nucleus has two protons, so Z = 2. Also, set n 1 = 3, and n 2 = 4, 1 = RZ2 λ
1 1 − 2 2 n1 n2
−1
= (1.097 · 10 m 7
)2
2
1 1 − 2 2 3 4
= 2.13 · 106 m−1 .
Invert this to get λ4→3 = 468.8 nm. This is between blue and violet. In practice, it’s not observed in the helium emission spectrum because the vast majority of helium atoms are neutral, not ionized.
14.4.5 Chemical Bonds* Our final example of standing matter waves will be two atoms that are side-byside. For convenience, we’ll start with the simplest possible molecule, which is H+ 2, meaning that there are two protons and one electron. The protons repel each other, of course, while the electron is attracted to both of them. There are two ways to find the wave functions for this system. First, we follow the same approach as above by drawing out the potential energy surface and finding the normal modes for the electron matter waves, as shown in Figure 14.19. We ignore the proton motions because each one is nearly 2000 times heavier than the electron, so they move very slowly by comparison and the electron can adapt as needed. These electron wave functions are called molecular orbitals, of which the lowest one is labeled 1σ and the next is 1σ ∗ . As we’ve seen many times, the higher energy mode has a shorter wavelength, which is seen in this case by the node in the middle of the upper wave function.
Figure 14.19 Potential energy surface, energy levels, and wave functions for an electron around two protons (i.e. the H+ 2 molecule). Red lines indicate the proton positions. The right side zooms in on the wave functions.
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Although not shown in the figure, the 1σ energy level is below the 1s energy level of an isolated hydrogen atom. Again, this agrees with prior results, this time from the particle-in-a-box result that the zero-point energy is lower if electrons can travel farther. More importantly, the 1σ energy is sufficiently below the 1s energy level that the atoms have less total energy when they are together than apart, despite the repulsion between the two protons. As a result, the atoms stay together, which is a chemical bond. The 1σ orbital is called a bonding orbital while the 1σ ∗ orbital, which has more energy than the isolated 1s orbitals, is an antibonding orbital. The other approach gets to the same normal modes but in a different way. It starts by considering the electron as being in the 1s orbital around just one of the two protons. The two possibilities for this are shown at the two sides of Figure 14.20, and labeled “1s,left” and “1s,right”, describing which proton the electron is moving around. These two states aren’t normal modes of the entire system, but that’s okay. They still have de Broglie frequencies and energies; the energies are shown as the vertical position in the figure, as normal.
Figure 14.20 The left and right wave functions represent 1s atomic orbitals around the separate nuclei (shown in red). They combine to create 1σ and 1σ ∗ molecular orbitals around both nuclei, which have lower and higher energies, respectively.
These two 1s states have the exact same frequencies as each other, so they resonate. As it turns out, if two states resonate with each other, then they can be combined to find the two normal modes, of which one has a lower frequency than the resonant frequency and is found by adding the two motions together, and the other has a higher frequency than the resonant frequency and is found by taking the difference of the two motions. For example, suppose two kids are on adjacent swings on a swing-set, and the swingset is sufficiently wobbly that there’s coupling between the two swings. Either kid can swing alone, which is equivalent to the electron being in either 1s state, but this always shakes the other kid due to resonance. Thus, the normal modes are found by “adding” the two motions, meaning that both kids swing exactly in phase with each other, and by “subtracting” the motions, meaning that both kids swing exactly out of phase with each other; the former normal mode has a lower frequency than either kid swinging alone and the latter has a higher frequency. Back to the matter waves, the energy levels shown at the center of Figure 14.20 represent normal modes of the entire system, of which the lower one is the sum of the two 1s waves, and the upper one is the difference of the two 1s waves. This brings us back to the same 1σ and 1σ ∗ molecular orbitals that we saw above, but now with the realization that these necessarily have lower and higher frequencies,
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and thus lower and higher energies, than the isolated 1s orbitals. Also, the amount that the energy shifts arises from the amount of coupling between the isolated 1s states. This coupling, in turn, depends on how close the two protons are to each other. There’s very little coupling and very little energy shift if they are far apart and more when they are close together. Again, the decreased energy of the 1σ molecular orbital relative to the 1s orbitals explains why molecules can be more energetically favorable than separate atoms. Figure 14.21 presents yet another view of chemical bonds. In these pictures of the 1σ and 1σ ∗ molecular orbitals as electron clouds, notice that the electron is more likely to be found between the nuclei than outside of them in the 1σ orbital, and vice versa for the 1σ ∗ orbital. As a result, the electron’s attraction pulls the nuclei together in the former case and pulls them apart in the latter case. This shows, again, that bonding orbitals create bonds while antibonding orbitals weaken them. Figure 14.21 Bonding and antibonding molecular orbitals as electron clouds. Red dots represent the nuclei.
Using these molecular orbitals, we can consider what would happen with more electrons. One electron would go into the bonding orbital (1σ), because it has the lower energy, which creates a net attraction between the nuclei. A second electron would also go into the bonding orbital, leading to a stronger attraction (Figure 14.22). This is the chemical bond in a neutral hydrogen molecule, H2 . By the Pauli exclusion principle, a third electron would not be allowed to go into the bonding orbital, but would have to go to the antibonding orbital instead which would weaken the bond. Finally, a fourth electron would also go into the antibonding orbital, now completely negating the influence of the electrons in the bonding orbital. This represents the situation with helium atoms, in which there are two electrons per atom. The four electrons in the two atoms would fill both the bonding and antibonding orbitals, so there isn’t a net bond and the atoms would fall apart into isolated atoms. Indeed, helium exists naturally as isolated atoms and not as dimers. Figure 14.22 Molecular orbital diagram for a hydrogen molecule, H2 . Each atom contributes one electron, shown on the sides, both of which move to the 1σ (bonding) molecular orbital, shown at the bottom.
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The same basic approach applies for larger atoms and larger molecules. For example, nitrogen, oxygen, and fluorine molecules (N2 , O2 , and F2 ) bond together due to similar molecular orbitals. Those orbitals arise from resonance between their 2p orbitals, which create molecular orbitals called 2π (bonding) and 2π ∗ (antibonding) molecular orbitals. Likewise, the carotene molecule at the beginning of this section has resonance all along the chain of carbon atoms, again with their 2p orbitals. This leads to an even longer molecular orbital that is called a conjugated π system.
14.5
Energy Level Transitions
14.5.1 Light Absorption When an atom (or molecule) absorbs a photon, its energy increases by the photon’s energy. However, this is only possible if the atom has an energy level that is at exactly the right amount of energy above its initial state; otherwise, the photon goes right on by, and isn’t absorbed. The left side of Figure 14.23 illustrates this process by showing the energy diagram for an atom that starts in its ground state and has an excited state that is higher by the energy of a green photon. This atom can absorb a green photon, which moves it from its ground to excited state in the process, but red and blue photons pass on by without interacting at all.
Figure 14.23 (Left) Energy level diagram, showing that absorption occurs if the photon energy matches the energy level difference. (Right) The sun’s spectrum, showing the background blackbody radiation and the absorption lines, along with which atom or molecule creates them.
The right side of the figure shows an example of this selective absorption. The background rainbow colors represent the light that’s emitted from the surface of the sun, which follows the blackbody radiation spectrum and is completely continuous. The black lines appear in this spectrum because the sun’s atmosphere, which is outside the sun’s surface, has atoms and molecules in it that absorb photons with specific energies. These absorbed photons don’t make it to Earth, so they create black lines in the sun’s spectrum which we see. Many molecules, including water, nitrogen, oxygen, and most plastics, simply don’t have energy levels that are a visible photon’s energy above the ground state, so they don’t absorb visible light and appear transparent. On the other hand, large π-conjugated molecules, such as carotene and most fabric dyes, have more closely spaced energy levels, so they do have transitions in the visible energy region and
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appear colored. Also, many metal atoms have closely spaced energy levels that enable them to absorb visible photons. Examples include oxidized iron (Fe3+ ), which produces the red color of rust, blood, and brown beer bottles. This and other metal atoms produce the pigments in most paints.
14.5.2 Light Emission Emission is the opposite process, in which an electron transitions from a higher to a lower energy level and gives off the energy difference as a photon. This is shown in the energy diagram on the left side of Figure 14.24.
Figure 14.24 (Left) Energy level diagram, showing that emission occurs from atomic or molecular relaxation from an excited state to a lower state. (Right) Emission spectra of several elements.
There are many ways to excite atoms up to their excited states in the first place. These include making them very hot, as occurs on the sun, running an electric current through them, as in neon signs, and, of course, exciting them with the correct energy photons. The right side of the figure shows emission spectra from several types of atoms. In each case, the spectral lines represent energy differences between higher and lower energy levels. The hydrogen spectrum, shown on top, can be computed from the Rydberg formula (Eq. 14.14) because hydrogen atoms only have one electron. The spectrum shown here represents part of the Balmer series, in which atoms relax from higher energy levels down to the n = 2 energy level. Hydrogen’s other possible relaxations occur either in the ultraviolet or infrared ranges, so don’t appear here. The other spectra are more complicated because those atoms have more electrons, but nevertheless provide convenient fingerprints for identifying which atoms are present. For example, the next spectrum, helium, was discovered from the spectral lines that it creates in the sun’s spectrum. Those lines could not be assigned to any known element, so scientists deduced that it must come from some new element, which they then named helium. Subsequent work revealed a gas on Earth that had
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the same spectrum, showing that it was the same element21 . Helium’s spectrum also appears in most stellar spectra, from which we now know that most stars in the universe are composed of primarily hydrogen and helium. The next two emission spectra, sodium and neon, are likely to be more familiar from everyday life. Sodium atoms emit a pair of extremely bright yellow lines, at 589.0 and 589.6 nm, which are observed in the yellow lights that are often used for street lights, parking lot lights, and security lights. These emission lines can also be observed by sprinkling salt in the flame of a candle or gas stove. Finally, neon atoms emit a large number of bright red lines, which together create the red color of neon signs. These sharp emission lines raise the question of why the sun exhibits a continuous blackbody spectrum when it’s composed of mostly hydrogen and helium atoms. It turns out that there are several reasons. First, the sun isn’t just hydrogen and helium atoms, but also includes hydrogen molecules, hydrogen ions, and helium ions, along with oxygen, carbon, neon, iron, and other atoms, each with its own emission lines. Also, these atoms bump into each other, exist in strong magnetic fields, and are moving rapidly toward or away from us, all of which spread out the spectral lines. Finally, and most importantly, the sun is 860,000 miles thick, so even if it emits weakly in some part of the spectrum, this is sufficient, over enough of the pathlength in the sun, to glow the same amount in that spectral region as well (vice versa, if the sun were cold, it would absorb all incident light and appear black, without any of the incident light getting through to the far side).
14.5.3 Fluorescence Adding additional excited states allows for fluorescence, in which a molecule (or atom) executes an absorption and emission cycle. This is shown in Figure 14.25, where the orange balls represent individual molecules, most of which are in the ground state22 . The cycle starts with a ground-state molecule absorbing a high energy photon, shown in blue, which raises it to an excited state. Before it has a chance to emit a photon of the same energy and return to the ground state, which would be called scattering, it instead loses some of its energy to heat through nonradiative transitions, such as by adjusting the shape of the molecule or by bumping
21 Many people played a role in discovering helium. Jules Janssen, a French astronomer, discovered
a previous unassigned emission line in the sun’s spectrum at 587.49 nm. Normal Lockyer, an English astronomer, observed the same line the same year and, together with the English chemist Edward Frankland, realized that it was a new element, which they named helium. In 1881, Italian physicist Luigi Palmieri detected helium on Earth by observing the same spectral line in gas emitted from Mt. Vesuvius lava. Finally, the Scottish chemist Sir William Ramsay isolated helium from radioactive rocks, again observing its presence through the same spectral line. 22 Systems generally relax to have more atoms and molecules in low energy states than high energy states. This is particularly true for electronic states because their excited states typically have sufficiently high energies to be beyond the range of thermal excitation.
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into other molecules. Then, it emits its remaining energy as a low energy photon, shown in green. The molecule might then undergo more non-radiative transitions before returning to the ground state (not shown). The time between absorption and emission is called the fluorescence lifetime, which is typically between 0.5 and 20 nanoseconds. Figure 14.25 Energy level diagram of fluorescence. Balls represent individual molecules.
Figure 14.26 shows several examples of fluorescence. First, a fluorescent light bulb is constructed from a tube that contains a mercury-vapor discharge lamp, like a neon sign but with mercury instead of neon, which happens to have several strong ultraviolet emission lines. These UV photons get absorbed by white powder that coats the inside of the tube, called phosphor. The phosphor undergoes some internal conversions and fluoresces by emitting the remaining energy as visible photons.
Figure 14.26 Examples of fluorescence. (A) A compact fluorescent light bulb. (B) A woman with fluorescent paint and clothing. (C) Structure of green fluorescent protein (GFP). (D) Green fluorescent proteins fluorescing in a worm, showing the different cell types (the head is at the top). (E) A pet fish that has been genetically modified by adding blue fluorescent proteins.
Next, fabric washed in laundry brightener glows spectacularly under “black lights,” which are really ultraviolet lights. This is because laundry brightener isn’t actually a soap that gets clothes extra clean, but is fluorescent dye that converts invisible UV photons to visible photons, which makes the fabric appear extra white. Fluorescent body paint obviously fluoresces as well. Green fluorescent protein (GFP) is a barrel-shaped protein that has a little segment in the middle that fluoresces green. It was first isolated from jellyfish in the 1960s and then developed into a biological tool in the 1990s. GFP has revolutionized
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biology research because its DNA, or DNA for similar fluorescent proteins that have been developed more recently, can be added into worms, flies, mice, or other organisms which then causes the organism to express the protein as fluorescent markers. Researchers can then illuminate the organism with UV or other high energy light to see where the fluorescent proteins ended up. This research has led to rapid advances in cell biology research, including for cancer and other applications. On a more whimsical note, GFP and related proteins have also been added to some pets, most notably fish, to make them more colorful.
14.5.4 Phosphorescence* Phosphorescence is essentially the same thing as fluorescence, but has a much longer excited state lifetime of minutes to hours. Examples include “glow-in-the-dark” things, such as toys, star stickers for bedroom ceilings, and the hands on some clocks. Fireflies, marine plankton, and other light-producing organisms are often incorrectly called phosphorescent but are actually bioluminescent. This means that they excite biological molecules called luciferin through chemical means, and those molecules then relax and emit light.
14.5.5 Lasers Laser function is surprisingly similar to fluorescence, with the principal difference being that the emission doesn’t happen by itself but is stimulated by other photons. Figure 14.27 illustrates this process. As with fluorescence, the atoms or molecules of the laser medium go through a cycle in which they start in the ground state, get excited by absorbing a high energy photon, which is called pumping, and then relax through non-radiative transitions. This relaxation takes the atoms or molecules to a so-called metastable state, where they wait briefly, and then they get pushed down to a lower energy level by a photon and emit another photon in the process, which is called stimulated emission. Finally, they may go through additional nonradiative relaxations before getting pumped back up again in the next cycle. The central process is succinctly described in the word “laser,” which is an acronym for “light amplification through stimulated emission of radiation”. The stimulated emission portion of this cycle only works because there is a population inversion, in which most of the laser medium is in the metastable state, rather than the ground state. If this weren’t true, then any photons would simply excite the medium up to the metastable state and would get absorbed in the process, which would make the light dimmer. As it is though, most of the medium is in the metastable state, so any photons push the medium down to a lower energy level, with another photon getting emitted in the process, so the light gets brighter. The right side of the figure shows a diagram of a laser. It includes a flash tube, which produces the high-energy photons for pumping, and the laser medium that
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Figure 14.27 (Left) Energy levels and their occupation in a laser medium. (Right) Diagram of a laser.
actually performs the lasing. In addition, the laser medium is located in an optical cavity that has mirrors on both ends, in order to create a strong laser light that shines back and forth within the medium. One of the mirrors lets a small fraction of the light through, which is the emitted laser beam. A cooling system is not shown, but is essential for large lasers, because the vast majority of the flash tube light just gets turned into heat. An important aspect of the lasing process is that the waves of the emitted photons are in phase with those of the stimulating photons (because the energy transfer from laser medium to electromagnetic wave occurs through resonance). With amplification back and forth in the laser cavity, all of the waves end up in phase with each other, thus creating a light source with high spatial coherence (see Section 13.4.2), in which each light wave is highly uniform across the width of the beam. This high coherence allows laser beams to stay narrow over long distances, as is familiar with laser pointers.
14.6
Quantum Weirdness
14.6.1 Heisenberg Uncertainty Principle We met the energy-time uncertainty in the last chapter (Section 13.4.4), which showed that it’s impossible to know both the exact energy of a photon and the exact time when it was emitted. Instead, better knowledge of one quantity necessarily implies worse knowledge of the other, which was quantified with the equation σ E σt ≥
h . 4π
A similar principle applies to particle positions and momenta. Consider the wave packet in Figure 14.28. This wave packet represents a particle, and that particle could be found anywhere along the length of the wave packet. Thus, there is no way of knowing exactly where the particle is. However, we can say that it’s most likely to be somewhere near the middle of the wave packet, because that’s where the amplitude is largest, which is made more precise by saying that its location has some uncertainty, given as σx . The same issue arises with the wavelength. Again, there is no way to give a specific wavelength for the wave packet because all wave packets are necessarily
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composed of a range of wavelengths. This is because wave packets start and stop, and this can only be achieved by adding waves with many different wavelengths together. Each of these different wavelengths corresponds to a different momentum through the de Broglie wavelength equation, λ = h/ p, which means that the wave packet represents a range of momenta. Thus, the momentum is some “best guess” value from the central wavelength, plus or minus some uncertainty σ p . Figure 14.28 A matter wave packet propagating to the right that has standard deviation σx .
A longer wave packet is more vague about the particle’s position, but has a better defined wavelength, and hence more precise momentum. Vice versa, a shorter wave packet is more precise about the particle’s position but less precise about its momentum. Thus, there is an inescapable trade-off between knowledge of a particle’s position and knowledge of its momentum. This is quantified in the Heisenberg uncertainty principle, named for Werner Heisenberg (also just called the uncertainty principle), h . (14.15) 4π As with energy-time uncertainty, this relationship arises from particle-wave duality. The wave nature contributes the mathematical trade-off between uncertainties in wave packet location and wavelength, while the particle nature relates the wavelength to particle momentum through the de Broglie wavelength relation. The uncertainty principle also applies to measurements. Even if an electron had a very uncertain position initially, it would still be possible to measure a position for it very precisely (chosen randomly from the absolute square of the wave function). But, the act of doing so would perturb the electron’s momentum, making that very uncertain. Likewise, a precise determination of the electron’s momentum would perturb its position. No experiment can measure position and momentum precisely at the same time, and nor can an experiment measure one property without perturbing the other one. The uncertainty principle connects to nearly all of the topics that we’ve met so far in this chapter. (1) We started by considering wave packet spreading, explaining that it arose from the fact that matter waves are dispersive, with shorter waves traveling faster than longer waves. A different explanation is that wave packets necessarily have a range of momenta, from the uncertainty principle, so they spread out because of these different speeds (see Figure 14.29). (2) We also considered diffraction effects. Recall in single-slit diffraction that waves spread out much more after going through narrow slits than wide slits. With matter waves, if a particle goes through a narrow slit, then we know its position (perpendicular to its direction of travel) very precisely. This implies that its momentum is poorly defined from the uncertainty principle, σx σ p ≥
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Figure 14.29 Diagram of a particle’s position uncertainty increasing over time. In this “phasespace” graph, the x-axis shows position and the y-axis shows momentum. The particle starts with a minimum uncertainty wave packet at the left (σx σ p = h/4π), and its position uncertainty increases over time due to the range of possible momentum values. This figure uses the same values as Fig. 14.3.
which is consistent with it spreading out behind the slits. (3) For standing matter waves, there is always a zero-point energy that arises from the finite wavelength. This can also be seen as arising from the fact that the position is partially known, so the uncertainty principle states that we can’t know the momentum exactly; in particular, the momentum can’t be exactly zero, so the energy can’t be zero. Finally, (4) The uncertainty principle shows that electrons can’t fall into atomic nuclei. This is because an electron that stayed in the nucleus would have zero uncertainty in both position and momentum, which would violate the uncertainty principle. On a deeper philosophical note, Heisenberg’s uncertainty principle places a limit on the amount of detail that exists in the physical world. In a sense, reality has an inherent graininess, rather like an impressionist painting. This graininess isn’t quite so simple as saying that physical space is divided up into very small brush strokes, or tiny pixels as in a digital photograph, but isn’t far from it. Importantly though, this lack of detail arises from limitations to reality itself, and not from limitations of our measurement equipment.
14.6.2 Schrödinger’s Cat Experiment* Quantum mechanics wave functions are fully predictable, meaning that if we know the initial wave function for some quantum system, then we can use Schrödinger’s wave equation to compute the wave function for any time in the future. This computation might be very difficult, but it’s possible in principle. From this final wave function, we can compute the probability of observing various properties of the system, such as where the particles are or what their momenta are. There’s nothing random here at all. However, the act of observing the system does introduce randomness because it causes the wave function to collapse into exactly one of its possible states. How this collapse occurs, which is called the measurement problem, is completely mysterious. Schrödinger didn’t understand wave function collapse either, so he came up with a thought experiment that showed the problems with the concept. In this experiment, which was only imagined and never carried out, a live cat is placed in a closed box, along with a radioactive atom, a hammer, and a vial of poison. If the atom decays,
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which is a quantum event, then it will release the hammer, which will cause the vial of poison to open, and the cat will die. If the atom does not decay, then the vial stays closed, the hammer stays put, and the cat survives. See Figure 14.30. Figure 14.30 Diagram of Schrödinger’s cat thought experiment.
According to the Copenhagen interpretation of quantum mechanics, the radioactive atom will follow both of its possibilities until it is observed, at which point it will collapse to being either intact or decayed. This act of following both possibilities until it collapses is very much like a photon in a double-slit experiment that goes to all parts of the screen until it is observed, at which point it collapses to a single point. There is nothing controversial about the atom being in both states at once, which is more formally described as a superposition of the wave functions for the decayed and non-decayed states. However, if the atom is in both states, then the hammer must also be both unchanged and released, and the poison vial both intact and broken, and the cat both alive and dead. Everything in the box, including the cat, remains in this superposition state until the moment that a person opens the box and observes the result, at which point the whole works collapses to one of the two states. Schrödinger knew perfectly well that a cat being both alive and dead at the same time was absurd, which was the whole point of his thought experiment. It raised the question of what constitutes an observer. Does the sensor for the radioactive atom qualify as the observer, or the cat, or only the person who opens the box? Or, in contrast to the Copenhagen interpretation, does the quantum state collapse all by itself at some point, without requiring an observer at all? Alternatively, perhaps the quantum state never collapses, but instead there are different copies of reality for every possible outcome. In this many worlds interpretation, there are two parallel universes, one of which has a live cat and a happy owner and the other has a dead cat and a sad owner, and both universes carry on indefinitely. By extension, this means that every random outcome that arises from quantum events leads to a separate splitting of the universe into yet more parallel universes. This explanation is the most consistent with our current understanding of quantum mechanics, but is clearly absurd. The answer to this problem remains unknown.
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14.6.3 The EPR Paradox* A central mystery of the Schrödinger’s cat experiment is the issue of entanglement, meaning that multiple objects share a single quantum state. The radioactive atom, hammer, vial of poison, and the cat in Schrödinger’s cat experiment are all entangled together because they are not independent, but share the atom’s quantum state, which happens to be in a superposition of being both intact and decayed. Observing any one of these objects causes it collapse to a single quantum state, so the other objects must collapse to the corresponding state at the same time. For example, peeking at the radioactive atom forces it to collapse to either being intact or decayed, and the cat must then collapse to being either alive or dead at the exact same moment. In 1935,Albert Einstein,Boris Podolsky, andNathan Rosen, all physicists working in the United States at the time, pointed out the logical problems with entanglement. They argued that either (1) quantum mechanics makes absurd predictions or (2) the theory of quantum mechanics is somehow incomplete23 . This became known as the the EPR paradox, after the first letters of the authors’ last names. The examples used to illustrate the paradox have evolved over time, but have always focused on the fact that it doesn’t make sense for an observation on one object to instantly affect some other object that might be very far away. Figure 14.31 shows what is now the typical example. Here, there is an atom at the center which repeatedly gives off pairs of photons that go in opposite directions, and these two photons always have opposite polarizations. However, their polarizations are unknown before they are observed, which means, according to quantum mechanics, that each photon is actually unpolarized at this point. At the moment that one photon of a pair is observed, say by Alice, it collapses to being either horizontally or vertically polarized due to Alice’s polarizer; this causes the other photon, being observed by Bob, to immediately collapse to the other state24 . Alice’s and Bob’s polarizers are perpendicular to each other, and they only detect photons that go through their polarizers. This means that
Figure 14.31 Experiment to test the EPR paradox. The photon source emits pairs of unpolarized photons that are entangled. When Alice observes that a photon has horizontal polarization, its entangled partner will instantly collapse to have vertical polarization, measured by Bob, and vice versa.
23 Their paper is: Einstein, Podolsky, and Rosen, “Can quantum-mechanical description of physical
reality be considered complete?” Phys. Rev. 47:777, 1935. Alice and Bob names are traditional for the EPR paradox, dating at least to Bennett et al. “Teleporting an Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels” Phys. Rev. Lett. 70:1895, 1993.
24 The
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both Alice and Bob detect photons if they collapse to one state, and neither of them detects photons if they collapse to the opposite state, but it’s never the case that just one of them detects a photon. Because the photons are going in opposite directions, each at the speed of light, they can’t send signals to each other unless those signals travel faster than light, which is impossible. Thus, there are only two possibilities. Either (1) the photons magically collapse to opposite polarizations at the exact instant that one of them is measured, without any signal being transmitted between them, which is called quantum teleportation, or (2) the photons had “hidden” polarizations all along, with one vertical and the other horizontal, and were never entangled in the first place. The second option is the only sensible answer and in fact is essentially what Einstein, Podolsky, and Rosen concluded had to be the case. However, it wasn’t clear how these hidden states were stored. This remained an unsolved problem for nearly 30 years. Then, in 1964, the Northern Irish physicist John Bell discovered that one can test for photon entanglement simply by rotating one of the polarizers by 45◦. Now, it’s no longer the case that Alice and Bob’s results always match each other, due to the randomness of the observations. However, if Alice and Bob collect many results on when they do and don’t detect photons, Bell showed that the predicted statistical results are different if the photons are entangled versus if they have hidden polarizations. This has been tested experimentally many times. Remarkably, results conclusively show that the photons are in fact entangled, meaning that quantum teleportation is real, and that Einstein, Podolsky, and Rosen were wrong. Recent tests have investigated quantum entanglement over increasingly long distances, of which one notable experiment separated the detectors by 143 km, between the Canary Islands La Palma and Tenerife25 . These experiments always show that quantum mechanics is correct, and there are no hidden polarizations, despite it seeming utterly impossible. Again, this shows that wave function collapse is simply not understood; it remains as one of the deepest mysteries in physics.
14.6.4 Quantum Decoherence* Yet another puzzle, but one which has been largely figured out by now, asks why quantum mechanics only applies to very tiny things. For example, a pendulum on a grandfather clock is a harmonic oscillator, very much like the molecular vibrations discussed above, but it behaves differently. It swings back and forth rather than sitting in a single quantum energy level with the pendulum position spread out over its entire trajectory. Likewise, large objects don’t really go through two slits at once, but go through either one or the other. Also, cats can’t actually be both alive and dead at the same time. Why not?
25 Ma et al. “Quantum teleportation over 143 kilometers using active feed-forward” Nature 489:269,
2012.
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In truth, quantum mechanics applies to everything, of all sizes. However, large objects are different in a couple of important ways. First, they have extremely small de Broglie wavelengths; for example, a tiny dust particle that’s floating around in a room has a wavelength of less than 10−23 m, which is orders of magnitude smaller than a single proton. Larger objects, like cats, have even smaller wavelengths. Second, large objects tend to interact more with the surrounding environment, whether by bumping into air molecules or by absorbing or emitting photons. Every one of these interactions entangles the object with the environment, with the result that the initial nice clean quantum state of the object rapidly gets spread out over the air molecules, photons, and surfaces of the environment. And, vice versa, the quantum state of the environment gets mixed into that of the object. If these environmental interactions are different for different parts of the object’s wave function, which is inevitable, then this creates a phase shift in part of the wave function that is known as quantum decoherence. Even a tiny phase shift completely ruins the wave coherence for large objects, due to their small de Broglie wavelengths, with the result that the wave nature is no longer apparent and the objects behave strictly classically26 . For example, we learned about matter wave diffraction experiments with buckyballs, of which a simple version would involve buckyballs going through two slits and then interfering with themselves to create interference fringes on a detector. For this experiment to show interference fringes, shown on the left side of Figure 14.32, the buckyball matter waves that go through one slit need to be almost exactly in phase with those going through the other slit, where this tight tolerance is necessary because their de Broglie wavelengths are about 10−12 m. The researchers who performed this experiment succeeded in observing interference fringes because they worked extremely hard to remove all possible environmental interactions27 . However, if they hadn’t worked quite so hard and the buckyballs had bumped into air molecules or absorbed or emitted photons during their flights through the two slits,
Figure 14.32 (Left) Matter wave interference without environmental interactions, which produces interference fringes. (Right) The same interference experiment but with environmental interactions, with blue dots representing air molecules that scatter the matter waves, which now produces the classical result of one spot behind each slit.
26 See:
Ball, Philip “The universe is always looking” The Atlantic Oct. 20, 2018; Zurek, Wojciech H. “Decoherence and the transition from quantum to classical” Physics Today 44:36, 1991; Schlosshauer, Maximilian “Quantum decoherence” Physics Reports 831:1, 2019. 27 Arndt et al. “Wave-particle duality of C60 molecules”, Nature 401:680, 1999.
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shown on the right side of Figure 14.32, then this would have ruined the nice clean matter waves and ruined the interference fringes. In this case, the molecules would have been observed directly behind the slits, as expected for classical particles.
14.6.5 Macroscopic Quantum Systems* Despite the challenges in observing quantum effects in macroscopic systems, there are a few intriguing examples. All of them are extremely cold, so that their de Broglie wavelengths are as large as possible, so thermal energy has as little effect as possible, and to minimize interactions with the environment. In addition, when atoms have low enough energies, they all tend to settle into the same lowest possible energy level, which leads to its own peculiar quantum mechanical behaviors28 . Helium is a gas at room temperature, but condenses to a liquid at about 4 K. Cooling it even further, down to 2.17 K, produces another phase transition, this time to a very bizarre state. Here, it becomes a superfluid, meaning that it flows without any friction at all. As a superfluid, helium can flow through tiny pores without any resistance, can form vortices that spin forever, and can even flow up and over the sides of whatever container it is in (left side of Figure 14.33). Superfluidity arises from most helium atoms being in their lowest energy state, which then limits the interactions that they can have with each other.
Figure 14.33 (Left) Superfluid helium. (Right) A magnet levitating above a superconductor.
Cooling atoms off even more, now down to the range of 170 nK, leads to BoseEinstein condensates, named for the Indian physicist Satyendra Bose and Albert
28 This
is not a violation of the Pauli exclusion principle, introduced above. That principle applies to electrons and other so-called fermions, which are particles with half-integer spin. The atoms considered here are bosons, meaning that they have integer spins, and so aren’t subject to the Pauli exclusion principle. Also, remarkably, pairs of electrons that are called Cooper pairs can also act as bosons.
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Einstein29 . Here, the de Broglie wavelengths of the atoms are so large that they overlap each other, with all of the atoms of the gas spread out over the same space. Bose-Einstein condensates are sufficiently wavelike that condensed atoms have been shown to exhibit interference with each other30 . They have even been used to create atom lasers, in which they produce beams of coherent atoms31 . Essentially the same process as superfluidity also applies to electrons, or more accurately pairs of electrons, to produce superconductivity. In this state, electrons can flow through metal or other materials with zero electrical resistance. In addition to being physics curiosities (right side of Figure 14.33), superconductors also have important engineering applications. Most significantly, they allow large electric currents to flow around a loop of wire without any resistance, which helps create the strong magnets that are required for medical MRI machines (magnetic resonance imaging) and some laboratory equipment (e.g. nuclear magnetic resonance, or NMR, spectrometers).
14.7
Summary
Physical matter is described by particle-wave duality, much like light. The wave nature is quantified by the de Broglie relations, which give the wavelength and frequency of matter waves as λ=
h p
f =
E , h
where h is Planck’s constant, E is the particle’s total energy (E = 21 mv 2 + U (x)), and p is its momentum ( p = mv). Schrödinger described these matter waves as wave functions, ψ(x), in which the displacements are complex numbers and the absolute squares of these displacements, which are real numbers, represent the probability of where a particle is likely to be found. Measuring the particle’s position leads to wave function collapse and causes the particle to be at its measured position. Potential energy landscapes represent the potential energy of a particle as a function of position. The difference between a particle’s total and potential energy is its kinetic energy, and matter waves with higher kinetic energies have shorter wavelengths. Matter waves are dispersive, so they spread out as they propagate and have different phase and group velocities. Phase velocities are largely meaningless, while the group velocity represents the velocity of the actual particle. Traveling matter
29 See
Townsend, Christopher, Wolfgang Ketterle, and Sandro Stringari, “Bose-Einstein condensation” Physics World 10:29, 1997; Ketterle, Wolfgang. “Nobel lecture: When atoms behave as waves: Bose-Einstein condensation and the atom laser” Reviews of Modern Physics 74:1131, 2002. 30 Andrews, M. R., C. G. Townsend, H-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Observation of interference between two Bose condensates” Science 275:637, 1997. 31 Mewes, M-O., M. R. Andrews, D. M. Kurn, D. S. Durfee, C. G. Townsend, and W. Ketterle, “Output coupler for Bose-Einstein condensed atoms” Physical Review Letters 78:582, 1997.
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waves that encounter a potential energy barrier are mostly reflected if they have less energy than the barrier height and mostly transmitted if they have more energy. However, those with less energy can penetrate into the barrier with evanescent waves and, if the barrier is low and thin enough, then tunnel through the barrier to continue propagating on the far side. Nuclear reactions, ammonia inversion, and scanning tunneling microscopes are tunneling examples. Matter waves diffract and interfere like other waves, which was important initially for proving their existence. Electric, neutron, and helium atom diffraction are all used in research for investigating the atomic structures of crystals, where the different particles interact with the atoms in different ways. Matter wave diffraction patterns are strongest when matter wave wavelengths are similar to the atomic spacings in the crystals. Large molecules, such as buckyballs, have also been shown to exhibit matter wave diffraction despite their exceedingly short de Broglie wavelengths of 10−12 m or less. Particles in potential energy wells exhibit standing matter waves that are characterized by multiple normal modes. They are labeled by one or more quantum numbers, where higher number modes have shorter wavelengths, faster frequencies, and higher energies. Systems with standing matter waves include: waves in a cavity (the particle-in-a-box problem), of which electrons in carotene molecules are good examples; harmonic oscillators, exemplified by molecular vibrations; electrons in atoms, such as hydrogen atoms; and electrons among multiple atomic nuclei, such as hydrogen molecules. The energy levels for the first two of these systems are: En =
h2n2 8m L2
En = h f
n+
1 2
n = 1, 2, 3, ... n = 0, 1, 2, ...
The lowest energy levels always have positive kinetic energy, called the zero-point energy. For matter waves that represent electrons, the Pauli exclusion principle states that only two electrons are allowed in each normal mode. Matter waves can also represent atoms, collections of atoms, or even the normal modes of molecular vibrations. Atomic orbitals, better described as electron clouds, have names such as 1s, 2s, 2px , etc., where the number part is called the principal quantum number and the letters describe the orbital shape and orientation. For hydrogen atoms, the energy levels are En = −
m Z 2 e4 1 · 8h 2 20 n 2
n = 1, 2, 3, ...
where n is the principal quantum number; it can range from 1 to ∞, where ∞ represents ionization. The wavelengths for transitions between energy levels are given by the Rydberg formula, 1 = RZ2 λ
1 1 − 2 2 n1 n2
.
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Molecular orbitals are conceptually similar, but are for electrons around multiple nuclei. Their names include 1σ, 1σ ∗ , 2σ, 2σ ∗ , 2π, etc., where orbitals without asterisks are bonding orbitals, which create chemical bonds, while those with asterisks are antibonding orbitals. Atoms and molecules can change energy levels by absorbing or emitting a photon, where the photon energy must match the energy level spacing. In fluorescence, a molecule absorbs a high energy photon to transition from its ground state to an excited state, undergoes some non-radiative relaxation down to a lower excited state, and then emits a lower energy photon as it returns to its ground state. Lasers follow essentially the same process, but now molecules are pumped to excited states and relax to a metastable state. There are more molecules here than in the ground state, a situation called population inversion, so laser photons stimulate further emission by driving molecules down to their ground states. The wave natures of particles, along with the fact that quantum mechanics allows superpositions of these waves, has several strange consequences. The Heisenberg uncertainty principle, h , 4π shows that it’s impossible to determine a particle’s exact position and momentum at the same time. Instead, better knowledge of one necessarily implies worse knowledge of the other, where this limitation arises from an intrinsic fuzziness to reality. The Schrödinger cat thought experiment illustrates the fact that quantum superposition implies that macroscopic objects can be in a superposition of states, such as cat being both alive and dead at the same time. This is clearly absurd, leading to the yet unsolved question of how wave function collapse actually occurs. The EPR paradox addresses the related question of how wave function collapse occurs for entangled particles. Measuring one particle causes instantaneous collapse of the entangled particles, even if a signal between them would need to travel faster than light. This makes no sense but has been confirmed by experiments. Macroscopic systems do not generally show quantum behaviors due to quantum decoherence, in which interactions with the environment perturb the matter waves; combined with the tiny de Broglie wavelengths of large objects, this decoherence destroys wave interference, which then leads to classical particle-like behaviors. However, quantum behaviors are seen in some exceptionally cold systems, including superfluids, BoseEinstein condensates, and superconductors. σx σ p ≥
14.8
Exercises
Questions 14.1. All hydrocarbons, which are molecules that are made of hydrogen and carbon (e.g. methane, propane, butane, and gasoline) burn in the presence of oxygen with a blue flame that has a temperature of about 1950◦ C. Which one of the following could explain this blue flame?
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There are additives in fuels that make the flame blue for safety Blackbody radiation from the burning gases All hot gases are blue Excited carbon dioxide emits blue light Rayleigh scattering
14.2. When hydrocarbons burn without sufficient oxygen, they create a flame that is white, yellow, or orange, depending on the temperature. Which one of the following could explain this flame color? (a) Emission from sodium atoms in the fuel (b) Emission from hydrogen atoms in the fuel (c) Emission from hot air molecules (primarily nitrogen, oxygen, and argon) (d) Fluorescence from carbon dioxide (e) Blackbody radiation from particles of unburned fuel 14.3. Which one is true about Heisenberg’s uncertainty principle? (a) It reflects an underlying limit to the physical world (b) It reflects the limitations of current technology (c) It only applies to electrons (d) It is the central assumption of quantum mechanics (e) It was an interesting hypothesis but later disproven 14.4. Compare the de Broglie wavelengths of a charging elephant and a sleeping mosquito. Which one is true? (a) The elephant’s wavelength is longer (b) The mosquito’s wavelength is longer (c) They have the same wavelength (d) De Broglie wavelengths don’t apply to animals (e) Which wavelength is longer depends on the animals’ temperatures 14.5. Many animals can see ultraviolet light, so some hunters are concerned that wearing clothing with fabric brighteners in them, which are essentially fluorescent dyes, will make their clothes glow in the ultraviolet and thus be visible to animals. Is this a valid concern? Why or why not? 14.6. (a) Draw an energy level diagram for fluorescence, with arrows for excitation, non-radiative relaxation, and emission. (b) In which state are most of the atoms? (c) Is fluorescent emission at a longer, shorter, or the same wavelength than the excitation? 14.7. (a) Draw an energy level diagram for a simple laser showing pumping (excitation), non-radiative relaxation, and stimulated emission. (b) In which state
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are most of the atoms? (c) Is laser emission at a longer, shorter, or the same wavelength than the excitation? 14.8. Match the scientist on the left with the discovery on the right. (a) Werner Heisenberg (1) Only one electron can be in any quantum state (b) Louis de Broglie (2) Cold atoms can overlap each other (c) Wolfgang Pauli (3) Matter waves are complex wave functions (d) Erwin Schrödinger (4) Position and momentum can’t be known simultaneously (e) Satyendra Bose (5) Particles have wavelengths and frequencies Problems 14.9. Electrons in a piece of metal move at a range of speeds, of which the fastest move at the so-called Fermi velocity, which is about 1570 km/s for copper. (a) What is the de Broglie wavelength of these electrons? (b) How does this compare to the separation between copper atoms, which is about 0.26 nm? (c) Would it be more accurate to say that each electron is bound to an individual atom, or is spread out over several atoms? 14.10. Consider two electrons in an atom, one in the 1s orbital and the other in the 2s orbital. (a) Which electron has the longer de Broglie wavelength, or are they the same? (b) Which electron has the higher de Broglie frequency, or are they the same? 14.11. Consider vibrations of a HCl molecule that are represented by a quantum harmonic oscillator. Suppose the molecule is in a superposition of the two lowest energy states, n = 0 and n = 1. (a) Sketch the wave function at time 0, by adding the two component wave functions (the appropriate solid blue lines in Figure 14.13). (b) At time τ /2, where τ is the harmonic oscillator period and is equal to 1/ f , it’s reasonable to consider the n = 0 wave function as unchanged, but the n = 1 wave function as having rotated around through half of its period, shown with the dashed lines in Figure 14.13. Sketch the wave function at time τ /2. (c) At time τ , the n = 0 wave function is still unchanged, but the n = 1 wave function has now completed one full rotation. Sketch the wave function again. (d) Describe the motion that these wave functions are showing. 14.12. Arrange the following molecules in order from weakest bond (or no bond at all) to strongest bond. Use an “=” symbol if two molecules have the same bond strength. Each molecule is composed of one helium atom and one hydrogen atom. HeH3+ , HeH2+ , HeH+ , HeH, HeH− . 14.13. Use the Rydberg formula for hydrogen atom emission. (a) Compute the wavelength for the case where n 1 = 2 and n 2 = 3. (b) What color is this? (c) Compute the wavelength for the case where n 1 = 1 and n 2 = 2. (d) Compute the wavelength for the case where n 1 = 1 and n 2 = ∞. (e) What color is this? (f) Are there any n 1 and n 2 combinations that can give a shorter wavelength? (g) What is the energy of one photon of this wavelength (i.e. the ionization energy)?
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14.14. Electron microscopes achieve higher resolution than light microscopes because electrons can have much shorter wavelengths. (a) For an electron traveling at 1.9 · 107 m/s, what is its wavelength? (b) How many times shorter is this than the 350 nm wavelength of a violet photon? 14.15. Consider a sample of hydrogen atoms that are in their ground states (hard to achieve on Earth, but common in interstellar space). (a) Draw an energy level diagram of a hydrogen atom, with the energy levels labeled. (b) Suppose light of 103 nm shines on the hydrogen atoms; what energy level will the hydrogen get excited to (hint: just guess and then compute the wavelength with the Rydberg formula to see if you’re correct)? (c) What three wavelengths will the hydrogen atoms emit as they relax back to the ground state? 14.16. When table salt is put in a candle flame, it burns bright yellow because of emission from the sodium atoms. This emission is at 590 nm. How much energy does the atom lose during this transition? 14.17. Consider a particle in a box. Its average velocity is clearly 0 because it’s staying in the box, but its instantaneous velocity varies over time, which we’ll estimate here. (a) Using the de Broglie wavelength equation, what is the approximate velocity of this particle as an equation? (It should be a function of m, n, L, and h.) (b) Compute the velocity of an electron in the n = 1 state of carotene (L = 2.4 nm). (c) Compute the velocity of an electron in the n = 11 state of carotene. 14.18. The energy levels for a particle in a 3-dimensional cubical box are E n x ,n y ,n z = h2 (n 2 8m L 2 x
+ n 2y + n 2z ), where m is the particle mass, L is the side length, and each of the n values is a non-negative integer (i.e. 1, 2, 3, ...). (a) What is the zero-point energy for a particle in this box? (b) What is the energy of the first excited state? (c) How many different quantum states (i.e. different sets of n x , n y , and n z values) correspond to this energy? (d) What is the next energy level? (e) How many different quantum states correspond to this energy? 14.19. Carbon monoxide (CO) vibrations are represented well as a quantum harmonic oscillator with a vibrational frequency of 6.4 · 1013 Hz. Suppose it is excited to its 3rd excited state (n = 3). (a) What wavelengths of light can it emit? (b) Vibrational transitions that only change the quantum number by 1 tend to couple more strongly to light than others, yielding stronger absorption and brighter emission; what is the wavelength for this transition? 14.20. Butadiene is a molecule with a conjugated π system, much like carotene. It has four electrons that are free to move and has a length of 0.553 nm. (a) Use the particle in a box model to compute the wavelength of its lowest energy absorption. (b) How close is this to the experimental result of 217 nm, given as a percent error? 14.21. Zoe’s pet rat got loose, but she knows that it’s somewhere in her bedroom, which is about 4 meters across. (a) Taking 2 m as the uncertainty in the rat’s location, what is the uncertainty in its velocity from the Heisenberg uncertainty
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principle (it weighs exactly 0.4 kg)? (b) Could its velocity be much higher than this value? (c) Zoe hears gentle snoring, so she believes that her rat is asleep, and is moving slower than this value; is this possible? (d) If Zoe had a machine that could measure the rat’s velocity extremely precisely, could this measurement result be lower than this value? Explain. 14.22. It was shown above that the quantum excitation frequency for a harmonic oscillator is the same as the classical resonance frequency. Let’s see if this is true for a particle in a box as well. (a) Find an equation for the classical frequency for a particle in a box, as a function of E, m, and L (hint: start by solving for the period, τ as a function of the velocity, v). (b) Show that a photon that excites a particle in a box from level n to level n + 1 has frequency h (2n + 1). (c) Approximate this photon frequency for large n, in which 8m L 2 2n + 1 ≈ 2n. Then, express the photon frequency as a function of E, m, and L (hint: solve for E n in terms of n and substitute). (d) Are the classical resonance and quantum transition frequencies essentially the same? Discuss. Puzzles 14.23. Researchers have created a “living laser” by expressing GFP in laboratory grown cells and then turning a population of the cells (uniformly dispersed in water) into a laser32 . Discuss what the researchers would have needed to assemble to create this laser, and things that they would have needed to consider. 14.24. In the Schrödinger’s cat experiment, suppose you open the box and find the cat is dead. Could you then measure its temperature (or do some other medical forensics) to find out when it died, and this would answer whether it had been in a superposition state or not?
32 Malte Gather and Seok Hyun Yun, “Single-cell biological lasers” Nature Photonics 5:406, 2011.
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Fig.15.1 Simulated appearance of two black holes orbiting each other, which produces gravitational waves. The swirl pattern arises from gravitational lensing of the background stars.
Opening question What does gravity do? Select all that are appropriate. (a) Keeps planets and moons in their orbits (b) Can either attract or repel objects (c) Bends light rays (d) Warps space and time (e) Propagates in waves
© Springer Nature Switzerland AG 2023 S. S. Andrews, Light and Waves, https://doi.org/10.1007/978-3-031-24097-3_15
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A new era of astronomy began at 9:50 am UTC on September 14, 2015. Throughout the entirety of human history before then, every direct observation of the universe beyond our solar system had been performed by observing electromagnetic radiation. People only observed visible light up until the late 1800s and then infrared, ultraviolet, and other bands more recently, but it was always some sort of electromagnetic radiation. This changed on that morning in September 2015 with the first direct detection of gravitational waves. Those waves arose over a billion light-years away, from two black holes that merged into one. Other gravitational waves have been observed since then and, undoubtedly, many more will be observed in the future. Gravitational waves are waves in the gravitational field, much as electromagnetic waves are waves in the electric and magnetic fields. They arise from accelerating masses, such as orbiting stars or black holes, or even just orbiting planets and moons. Gravitational waves are very difficult to detect due to their being naturally weak, and because strong ones tend to arise very far away. Einstein’s theory of general relativity showed that gravity bends space and distorts time, so gravitational waves are accurately described as ripples in the fabric of spacetime.
15.1
Gravity
Before discussing gravitational waves, it helps to back up a little and explore gravity first, which is the focus of this section.
15.1.1 Newtonian Gravity In the early 1600s, Galileo provided conclusive evidence for the heliocentric model of the solar system, in which the sun is at the center of the solar system and the planets go around it. This then raised the problem of what keeps the planets in their orbits. Half a century later, Isaac Newton famously solved the problem when he saw an apple fall off a nearby tree. The falling apple led Newton to realize that the Earth exerts a force on objects around it, such as apples on trees, and birds in the sky, and even the Moon and the sun. He realized that this force, which he called gravity1 , could attract the Moon to the Earth sufficiently strongly to cause the Moon to orbit the Earth. Likewise, the concept of gravity explained how other planets could orbit the sun, and even how other moons could orbit other planets. From these insights, Newton came up with his law of universal gravitation for describing the gravitational force between two objects. In modern notation, it is F=
Gm 1 m 2 r2
G = 6.674 · 10−11 m3 kg−1 s−2 .
(15.1)
1 Newton did not invent the word “gravity”, which comes from the Latin word gravitas for the quality of heaviness. However, he shifted its meaning and was the first to use it in the context of a force.
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Here, G is a number called the gravitational constant and has the value shown above, m 1 and m 2 are the masses of the objects, and r is the distance between them. This equation shows that gravitational force increases with larger object masses and decreases with larger separation between the objects. Note that this force is symmetric with respect to the two objects, meaning that each object pulls on the other one with the same amount of force. Also, masses are always positive, so the gravitational force is always positive, and gravity is always attractive. Anti-gravity, in which masses repel each other, is a nice science fiction concept but violates Newton’s law of gravitation and has never been observed. Example. The Earth’s mass is 5.97 · 1024 kg, the Moon’s mass is 7.35 · 1022 kg, and the distance between them is 3.84 · 108 m. What is the gravitational force acting on the Moon from the Earth?
Answer. Plugging these numbers into Newton’s law of gravitation shows that the force is 1.99 · 1020 N. This is the force of the Earth on the Moon, and of the Moon on the Earth.
15.1.2 Tides Newton realized that the distance dependence of his law of gravitation helped explain why the Earth has tides, including why many places (including Britain, where Newton lived) have two high tides per day. Leaving out the Earth for a moment, imagine three balls that are in a line, all reasonably close to each other and all falling toward the Moon. If the line of balls is perpendicular to the direction the balls are falling, as in the left panel of Figure 15.2, then this means that the balls are all about the same distance from the Moon. In this case, they experience the same gravitational force, so they fall at the same rate and stay the same distance from each other. On the other hand, if the line of balls is parallel to the direction they are falling, as in the middle panel of Figure 15.2, then they are at different distances from the Moon. In this case, the closest ball feels the
Fig. 15.2 Diagrams of tides. (Left, Middle) Balls falling toward the moon only spread out if they have different distance from the moon. (Right) The Earth and its oceans spreading out.
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most gravity and the farthest ball feels the least gravity, so they fall at different rates and spread out. To explain tides, the Earth is like the middle ball, while the oceans are like the outer balls, shown in the right panel of Figure 15.2. The part of the ocean that is closest to the Moon falls fastest, so it gets pulled away from the Earth. Likewise, the part of the ocean that is farthest from the Moon falls slowest, so the Earth pulls away from it. Together, these cause the ocean to be raised up on the sides of the Earth closest and farthest from the Moon in tidal bulges. The Earth makes one full rotation per day, which then explains why many places have two high and two low tides every day. Of course, the Earth doesn’t actually fall into the Moon, or the Moon into the Earth, which is due to to the Moon’s fast velocity around the Earth; both Earth and Moon are constantly falling toward each other, but the Moon’s velocity converts this falling into a nearly circular orbit. Real tides are more complicated than this due to the additional influence of the sun’s gravity, the inertia of the ocean water, the drag exerted on the ocean’s water by the Earth’s rotation, resonance effects, and other factors (see Section 6.5.6). However, this explanation still captures the essential aspects of the tide’s driving force.
15.1.3 Gravitational Fields Newton’s conception of gravity was an “action at a distance” model, meaning that one object exerts a gravitational force on another object over some intervening distance. Much as the action-at-a-distance concepts of electrical and magnetic forces got reinterpreted as electric and magnetic fields during the 19th century, the same happened with gravitation. Rather than thinking of a gravitational force acting between, say, the Earth and a falling apple, over the space between them, the idea was is to think of the Earth’s mass as creating a gravitational field around itself. Figure 15.3 illustrates this field. In this interpretation, the Earth’s gravitational field at the apple’s position exerts a force on the apple, and that force causes it to fall toward the Earth. Fig. 15.3 Gravitational field of the Earth. Longer arrows represent stronger gravity.
A benefit of this new interpretation is that gravity becomes a local interaction, in which objects are only influenced by the gravitational field at their position, rather than by masses that are far away. The gravitational field is a vector field and all objects
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within it experience a gravitational force in the field’s direction. In the example with the Earth, the field points toward the center of the Earth. The strength of the gravitational field comes directly from Newton’s law of gravitation, Eq. 15.1. Both sides of the equation are divided by an object’s mass, yielding the acceleration that an object experiences due to gravitational attraction, a=
GM . r2
(15.2)
Here, M is the mass of the object that’s producing the gravitational field. For example, the acceleration of an apple falling toward the Earth uses M as the Earth’s mass and r as the Earth’s radius (because we’re assuming the apple is close to the Earth’s surface), which is 6.37 · 106 m. Plugging these numbers into Eq. 15.2 gives the apple’s acceleration as 9.80 m/s2 . We’ve seen this value many times before. It is typically just called the Earth’s gravitational acceleration and denoted g. Note that the apple’s mass wasn’t necessary for this calculation; everything near the Earth’s surface falls with this same acceleration, regardless of its mass. If the gravitational field arises from multiple objects, then the separate fields add together to yield the total field. For example, the total gravitational field halfway between the Earth and Moon is the sum of the fields from the Earth, Moon, and sun, all measured at that point. To be thorough, one could also add in the fields other planets, although those influences would be extremely small. Example. What is the sun’s gravitational field at the Earth’s position? The sun’s mass is 1.99 · 1030 kg and it is 1.50 · 108 km from the Earth. Answer. Use the gravitational field equation with the sun’s mass and distance: a=
(6.674 · 10−11 m3 kg−1 s−2 )(1.99 · 1030 kg) GM = = 0.0059 m s−2 . r2 (1.50 · 1011 m)2
This means that the Earth is constantly accelerating toward the sun at a rate of about 0.6 cm/s2 . It doesn’t fall into the sun because its velocity around the sun keeps it in its orbit.
15.2
Gravitational Waves
15.2.1 First Direct Detection 1.3 billion years ago, when life on Earth consisted solely of single-celled organisms, two black holes in the southern sky were whirling around each other at high speed. The changing positions of the masses, each of which was about 30 times heavier than our sun, caused the gravitational field in the region to oscillate wildly. This produced
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waves in the gravitational field, calledgravitational waves2 , that propagated outward. The waves carried away energy, so the black holes fell closer together in what’s called an inspiral. This caused them to orbit even faster, making the waves stronger, and releasing more energy. This built up to create a powerful burst of gravitational waves until, suddenly, the black holes touched and merged together. At this point, the masses stopped orbiting each other, so the gravitational waves stopped, too. That burst of waves expanded outward at the speed of light and passed by the Earth 1.3 billion years later, on September 14, 2015. By this time, humans had evolved, they had developed gravitational wave observatories, and two observatories had just been upgraded for greater sensitivity. Those observatories detected the passing burst of gravitational waves, which were the first gravitational waves to be detected directly. Astronomers named this event GW150914, for gravitational waves and the date. Their results, shown in Figure 15.4, show the wave displacements on the y-axis as the amount of strain on the detectors (the distance they moved divided by their separation distance), which represents changes in the gravitational field at the Earth. Fig. 15.4 The first gravitational waves directly observed. The two colors show the waves as detected at two different observatories, of which the orange lines were shifted to account for the different observatory orientations and for the time taken for the wave to travel from one to the other.
The individual waves in this figure directly represent the positions of the two black holes as they orbited around each other. Their high frequency is remarkable. From the graph’s x-axis, the gravitational waves had a period of only about 0.02 s initially, which then decreased to about 0.004 s at the end, meaning that the frequency sped up from 50 Hz to 250 Hz, before the black holes finally touched and merged. These frequencies would be very low in the electromagnetic spectrum, and near the middle of the audible spectrum, but are extremely fast for such massive objects to orbit each other.
15.2.2 What are Gravitational Waves? As stated before, gravitational waves are waves in the gravitational field. Also, the gravitational field is the acceleration due to gravity. Gravitational waves can’t be observed with a single object but if you have two objects, say two balls, then a
2 These are called “gravitational waves” because the term “gravity waves” was already taken; those are normal water waves in which gravity provides the restoring force.
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passing gravitational wave would cause them to move back and forth relative to each other. Gravitational waves don’t affect the lengths of rigid objects, such as meter sticks, although they do apply slight compression and tension (pulling) forces on them. Gravitational waves are similar to electromagnetic waves in many ways. Both types of waves are waves in fields, both are able to propagate through empty space, and both propagate at the speed of light. Also, both are transverse waves and both have two types of linear polarization. However, gravitational and electromagnetic waves are waves in different kinds of fields, which has several important consequences. (1) They interact with different things. Electromagnetic waves are produced by moving electric charges and interact with other electric charges, whereas gravitational waves are produced by moving masses and interact with other masses. (2) Gravitational forces are much weaker than electromagnetic forces, so their waves are much weaker as well. Whereas we can see electromagnetic waves directly with our eyes, and indeed that’s all our eyes can see, only the biggest gravitational waves are detectable at all and even those require extraordinarily sensitive detectors. And (3) electric charges have both positive and negative types, such as for protons and electrons, whereas mass is always positive. This turns out to make the frequency and polarization of gravitational waves different from what one would expect based on electromagnetic waves.
15.2.3 Frequency The left side of Figure 15.5 shows a hydrogen chloride molecule that’s rotating about its center of mass. The hydrogen atom, in gray, has a positive electric charge and the chlorine atom, in green, has a negative electric charge, so the rotating molecule causes the electric field to rotate as well. It rotates once for each turn of the molecule, so the electromagnetic waves that this molecule emits, which are in the microwave region of the spectrum, have the same frequency as the molecule rotation rate (we’re ignoring quantum effects, which aren’t important to these arguments).
Fig. 15.5 Comparison of a HCl molecule (left) and a binary star (right). In the molecule, hydrogen is gray, chlorine is green, and the bond is shown as a thick line. Signs show electric charges in the molecule and mass values in the binary star.
The right side of the figure shows a binary star, which is really two stars that each orbit their common center of mass. Both of these stars have a “+” symbol to indicate that their mass is positive, as mass always is. However, the fact that both masses have
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the same sign means that this binary star looks about the same every half rotation, rather than every whole rotation as was the case for the HCl molecule. As a result, each gravitational wave corresponds to a half cycle of its source. In other words, gravitational wave frequencies are twice the star rotation frequencies. Looking back at Figure 15.4, the peak wave frequency of event GW150914 was about 250 Hz. This implies that the black holes orbited each other at about 125 Hz. Gravitational wave frequencies are also double the binary star frequency if two stars have different masses. This is because they orbit their center of mass, so each star still contributes roughly equally to the waves. Nevertheless, a mass asymmetry does add small amounts of higher harmonics to the waves, which are then useful for determining what the stars’ masses actually were. For example, the black holes described above didn’t actually have identical masses but were about 29 and 36 times heavier than the sun. A different pair of inspiraling black holes, detected on April 12, 2019, turned out to have one that was three times heavier than the other3 .
15.2.4 Polarization The equal signs of the masses also affects gravitational wave polarizations. Suppose four balls are arranged in a diamond pattern at the center of a binary star, as shown on the left side of Figure 15.6. Adding up the forces on each ball, from Newton’s law of gravitation (Eq. 15.2), shows that the gravitational forces are different on the different balls, which causes the diamond to distort. The two balls closer to the stars move outward toward the stars, while the other two balls move inward toward the center. The right side of the figure summarizes this distortion by showing what happens to a ring of balls, again showing that the motion is outward on one axis and inward on the other.
Fig. 15.6 (Left) Forces on balls between two stars. (Right) Gravitational distortion summarized as an ellipse.
As the binary star rotates, this gravitational distortion rotates as well. It also propagates away from the binary star, along the star’s axis, to create a spiral shaped gravitational wave, as shown in Figure 15.7. A stationary observer would observe the
3 See “GW190412: The first observation of an unequal-mass black hole merger” available at https://
www.ligo.org/science/Publication-GW190412/
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distortion going around in a circle, so this gravitational wave is circularly polarized4 . The rotation of the distortion over time is similar to that for circularly polarized light waves, but with the difference that this displacement is outward on one axis and inward on the other axis, rather than being a simple vector that points in a single direction5 . As an aside, note that the gravitational wave distortion is perpendicular to its propagation direction, meaning that gravitational waves are transverse waves.
Fig. 15.7 A rotating binary star on the left creates a gravitational field that distorts a circle into an ellipse. These distortions propagate away from the binary star as a circularly polarized gravitational wave. (This picture shows left-hand or counter-clockwise polarization.)
A binary star also emits gravitational waves in the plane of the stars’ orbit. If you’re a long way away and watching the two stars, it appears as though they go apart and together and apart and together, and so on, because you can’t see that they’re actually going around each other. Likewise, the gravitational waves alternate between having their distortion stretched and compressed along the plane of the stars’ orbit. These are called linearly polarized gravitational waves, of which there are two types. The “+” polarization alternates between being stretched vertically and horizontally, whereas the “x” polarization alternates between stretched along the two diagonal axes. Figure 15.8 summarizes these four types of polarization. As with electromagnetic radiation, the circular polarizations can be computed as sums and differences of the linear polarizations, and vice versa.
15.2.5 Energy Like all waves, gravitational waves transport energy. However, this fact was not obvious for many years, until the American physicist Richard Feynman came up
4 The direction of circular polarization is typically defined in astronomy as the rotation direction of the source, when looking in the direction of wave propagation. 5 In more technical terms, electromagnetic waves are typically dipole radiation. A dipole is a pair of objects with opposite sign that are separated by some distance, such as the HCl molecule in Figure 15.5. In contrast, gravitational waves are predominantly quadrupole radiation. A quadrupole has two-fold rotational symmetry, such as the binary star shown in Figure 15.5. Gravitational waves would be dipole radiation if mass could be negative, but it can’t. On the other hand, electromagnetic waves can be quadrupole radiation for symmetric molecules, such as rotating CO2 molecules.
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Fig. 15.8 Polarizations for gravitational waves. One full wave period is shown, which is half of the binary star period.
with a simple example that illustrated gravitational wave energy in 1957, called “the sticky bead argument.” In this argument, one considers two beads that are free to move along a rigid rod, although with some friction. As a gravitational wave passes by them, the gravitational field alternately pushes the beads together and apart, but the rod is rigid so it cannot deform. As the beads slide back and forth on the rod, the friction between the beads and the rod causes the system to heat up slightly. This thermal energy obviously had to come from somewhere, leading to the conclusion that it came from the gravitational waves. More recent work has shown that gravitational waves are extraordinarily weak for low mass and low frequency systems, such as the Moon orbiting the Earth, but can also carry an enormous amount of energy for large masses orbiting with high frequencies. In fact, the reason why binary black holes and neutron stars inspiral and then merge is because gravitational waves carry away large amounts of energy from their orbits. For example, the gravitational waves from the GW150914 event that was described above converted 3 entire solar masses worth of mass into gravitational wave energy (converting mass to energy according to Einstein’s famous equation E = mc2 ). During the brief fraction of a second before they finally merged, those two black holes emitted more than 50 times more gravitational wave energy than the entire rest of the visible universe emitted as light energy. An interesting consequence of stars needing fast orbits to produce large gravitational waves is that they have to be very close together, with separations of only a few thousand kilometers. This implies that the stars have to be very small. For example, two stars that are the size of our sun would collide long before they could get close enough to orbit each other at such high frequencies. Thus, binary stars that produce strong gravitational waves tend to be composed of white dwarfs, neutron stars, or black holes, all of which are stars that have used up their energy sources and then collapsed to varying degrees under their own gravity.
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Quantitatively, the emitted gravitational wave power for a binary system with circular orbits is given by P=
32G 4 m 21 m 22 (m 1 + m 2 ) , 5c5r 5
(15.3)
where G is the gravitational constant, m 1 and m 2 are the object masses, r is the orbit radius, and c is the speed of light. This equation shows the importance of the large masses and the small orbit radius (corresponding to a high orbit frequency). As r decreases toward zero, the emitted power increases toward infinity, stopping only when the two stars physically collide into each other. Although gravitational waves were first detected directly in 2015, the first experimental support for gravitational waves arose about 40 years earlier, based on the wave energy. Russell Hulse and Joseph Taylor, two American astronomers, observed a neutron star that was emitting signals with a periodic Doppler shift, from which they correctly deduced that it was orbiting another neutron star. Furthermore, they found that the orbital period was gradually speeding up at a rate that was consistent with them slowly inspiraling due to energy emission as gravitational waves. These two stars have continued inspiraling since this initial discovery and are predicted to merge in about 300 million years. Example. How much power does the Moon’s orbit around the Earth emit as gravitational waves? The Earth’s mass is 5.97 · 1024 kg, the Moon’s mass is 7.35 · 1022 kg, and the distance between them is 3.84 · 108 m. Answer. The Moon weighs enough less than the Earth that the Earth stays essentially stationary, while the Moon orbits at radius r = 3.84 · 108 m. This allows us to use Eq. 15.3 with these numbers: P=
32G 4 m 21 m 22 (m 1 + m 2 ) = 7 µW. 5c5r 5
This 7 µW of power, from the entire Earth-Moon system, is about the same power that a single fruit fly uses to fly. For comparison, the Earth transfers about 121 GW of power to the Moon through gravitational interactions with the Earth’s tides.
15.2.6 Momentum Gravitational waves also carry momentum. The primary consequence of this momentum does not arise in the waves themselves, but to the stars that created the waves in the first place. First of all, a rotating binary star has a large angular momentum, which is the momentum of the two stars going around in their orbits. This has to be removed for
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the stars to merge together. Circularly polarized gravitational waves fulfill this task by carrying angular momentum away. Also, gravitational waves transport normal linear momentum. As a binary star emits gravitational waves with momentum in one direction, this necessarily creates a “kick” that pushes the stars in the opposite direction. In some cases, this kick is strong enough to eject the stars out of their own galaxies6 .
15.3
Propagating Gravity
15.3.1 Speed of Gravity Suppose the sun suddenly stopped existing. It takes about 8 minutes for the light to get from the sun to the Earth, so the sun would not appear any different for 8 more minutes, and then it would suddenly go dark. Gravity also appears to travels at the speed of light, so we wouldn’t observe any change in the sun’s gravitational field for 8 more minutes either, and then it would suddenly turn off, too. The likelihood that gravity travels at the speed of light suggests a connection between the two types of waves. However, no direct connection has been identified so far, despite extensive efforts to find one. Instead, the reason they travel at the same speed is best explained by Einstein’s special theory of relativity, a theory that redefines our standard concepts of time and space for things that move at very high speeds. This theory identifies a “special speed” that has some interesting properties. First, when an object moves at this speed, then its speed relative to any observer is always this same special speed, regardless of the observer’s own speed (this makes no intuitive sense, but that’s what special relativity says). Second, all particles that don’t have mass (or, more properly, a rest mass), always travel at this special speed when in vacuum, and all particles that do have mass must travel at a slower speed. Light particles, meaning photons, don’t have a rest mass, so light travels at this special speed. For this reason, we call this special speed the speed of light. Gravity waves are believed to be quantized as gravitons, although there is no direct evidence for them. If they exist and also don’t have a rest mass, then they must also travel at the same special speed. In other words, gravity would travel at the speed of light. There were strong theoretical arguments for gravitons not having a rest mass, and hence gravity propagating at the speed of light, but this was not known for certain until very recently. It was spectacularly confirmed in 2017 when two neutron stars that were 130 million light-years away from Earth were observed to inspiral together and then merge. The event was observed as a series of gravitational waves and then, less than 2 seconds later, as a burst of gamma rays (Figure 15.9). The fact that the gravitational and electromagnetic waves were less than 2 seconds apart, after traveling for 130 million years, experimentally showed that their speeds are extraordinarily close to
6 Merritt,
D., Milosavljevi´c, M., Favata, M., Hughes, S. A., and Holz, D. E. “Consequences of gravitational radiation recoil” The Astrophysical Journal Letters, 607:L9, 2004.
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being identical. Most likely, their speeds actually are identical, and the 2 second difference arose from the time between the stars merging and their material exploding as a result of the merger. Fig. 15.9 Gamma ray signal (top) and gravitational wave signal (bottom) from two merging neutron stars that were 130 million light-years away.
15.3.2 Gravitational Near and Far Field* Recall from the first section of this chapter that gravity, like all forces, is symmetric, in which both objects pull equally hard on the other one. For example, the Earth exerts exactly the same gravitational attraction on the Moon as the Moon does on the Earth. The Moon moves more in response to this force because it weighs less, but the amount of force is exactly the same. However, the next section described the detection of gravitational waves from two black holes that were 1.3 billion lightyears away, measured by the forces that those waves exerted on objects on the Earth. Does this mean that the Earth exerted an equal force on those black holes, but with a delay of 1.3 billion more years? And what if the black holes, now single black hole because they merged, isn’t there anymore? Then, where does the force go? The answer to this problem lies in the difference between the gravitational near field and far field, which are analogous to the electromagnetic near and far fields that we met in Section 11.1.6. Gravitational effects that are less than a wavelength away are in the near field. This wavelength is typically quite large. For example, it takes the Earth one year to orbit the sun, and gravitational waves travel at the speed of light, so the wavelength that corresponds to Earth’s orbit is 1 light-year long. This is 1000 times longer than the distance to Pluto, so the entire solar system is well within this near field distance. On the other hand, gravitational effects that are more than a couple of wavelengths away are in the far field. For the Earth’s orbit, this includes all of the stars (except the sun). In general, gravitational waves are only fully developed in the far field. To be clear, Newton’s law of gravitation, Eq. 15.1, applies at all length scales. The difference between near and far field only applies to effects that arise from accelerating masses, including particularly gravitational waves from orbiting planets and stars. There are several differences between near and far field effects.
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First, as we saw for electromagnetic waves, absorption of radiation feeds back to the transmitter in the near field but not in the far field. As a near field example, the Moon’s orbit around the Earth creates a rotating component to the local gravitational field, which points up when the Moon is overhead, west when the Moon is to the west, and so on. This rotating field creates the tides on the Earth, as we saw above. Because we are in the Moon’s near field, those tides are able to feed back to the Moon to affect its orbit. Indeed, they do so, constantly pulling the Moon forward in its orbit and speeding it up by 10−7 m/s each year. In contrast, the Earth is in the far field of the black holes that were 1.3 billion light-years away. Their gravitational waves also created a distortion to the local gravitational field, but in that case the black holes’ gravitational wave energy had been fully transferred to the gravitational field, so detecting them on Earth did not feed back to affect the black holes. A second difference is that the near field strength falls off much faster. The fact that people can use gravitational measurements to observe black hole orbits that are over a billion light-years away is impressive. However, this is almost routine by now. On the other hand, this detection would utterly inconceivable if the black holes didn’t emit gravitational waves, but we had to detect the gravitational field changes that arose solely from Newton’s law of gravitation. Finally, the near field has some curious extrapolation behaviors that don’t occur in the far field. This issue first arose in 1805 when the French mathematician PierreSimon Laplace realized that if gravity propagated at the speed of light, and light takes 8 minutes to get from the sun to the Earth, then the Earth would always be pulled toward where the sun appeared to be 8 minutes ago rather than where it is currently. In this case, the Earth wouldn’t be pulled toward the middle of its orbit, as is essential for the orbit to be stable, but would constantly be pulled forward and would spiral outward. This wasn’t observed, so Laplace decided that gravity must propagate at essentially infinite speed. In retrospect, Laplace was correct about the Earth being pulled toward the sun’s actual position rather than where it was 8 minutes ago (called its “retarded position”), but incorrect about the effectively infinite speed of gravity. This was corrected about a century later by several scientists (the Dutch mathematician Hendrik Lorentz, the French mathematician Henri Poincaré, and, of course, Albert Einstein), who realized that time and space are distorted for fast-moving objects. One consequence is that the gravitational field at any point in space can be computed from the instantaneous positions of the surrounding objects, without any delay, provided that they are moving at constant velocity7 . In effect, the gravitational field extrapolates the positions of objects into the future based on their current velocity. Figure 15.10 illustrates this extrapolation by showing the sun’s attraction toward the Earth, which is equivalent to the opposite attraction but easier to illustrate. It shows that the direction of the sun’s attraction toward the Earth is based on the sun’s most recent “sensing” of the Earth’s position and velocity, from 8 minutes earlier,
7 This
effect arises from the Lorentz invariance of static fields, which is a part of Einstein theory of special relativity. See Wikipedia “Speed of Gravity”.
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which is then extrapolated forward in time to make a prediction of the Earth’s current position. The sun is attracted toward this extrapolated position. It’s very close to the actual position, and would be the same except that the Earth actually follows a curved orbital trajectory. Of course, neither the sun nor Earth is “aware” of this extrapolation, but this is just how the mathematics turns out to behave for near field effects in special relativity. Fig. 15.10 Extrapolation effect in which the sun is attracted to the extrapolated position of the Earth, based on where the Earth was 8 minutes earlier. This figure is very exaggerated.
Sunlight also travels from the sun to the Earth at the speed of light, but it behaves differently. In its case, Earth is in the electromagnetic far field, so there’s no extrapolation effect. Instead, when we look at the sun in the sky, we see where it was 8 minutes ago, not where it is currently. This position shift is called solar aberration. The same effect also occurs for stars, which is called stellar aberration; it was one of the earliest methods used for determining the speed of light (Section 2.3.3).
15.4
Warped Space
The statement that gravitational waves represent ripples in spacetime takes a bit of explaining. It involves several steps in a logical argument, most of which are not obvious but are well grounded in Einstein’s general theory of relativity. This theory, which extends the special theory of relativity to include gravity, is sufficiently counter-intuitive that it has been investigated very thoroughly experimentally, and it has always been shown to be correct.
15.4.1 The Equivalence Principle The first step in the logical argument is Einstein’s equivalence principle, which states that gravity and acceleration are indistinguishable from each other. This comes from the observation that an object’s mass, say 1 kg, can actually mean two very different things. It can mean the object’s inertial mass, describing how hard it is to start the object moving, or to stop it if it’s already moving. It can also mean the object’s gravitational mass, describing how strongly it is gravitationally attracted toward the Earth or other massive objects. It turns out that inertial and gravitational masses are
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always exactly the same, regardless of the size, shape, or composition of the object. For this reason, we just call both of them “the mass”. An implication of the equal mass values is that it is impossible to distinguish between acceleration, which depends on inertial mass, and gravity, which depends on gravitational mass. Assuming you are sitting on a chair in a room right now, you probably think that you are being pulled down onto the chair, and the chair down to the floor, because gravity is pulling you and the chair downward. However, as shown in Figure 15.11, everything around you would seem exactly the same if there were no gravity, but the entire room were being accelerated upward, perhaps because it’s mounted on a spaceship. This would push the floor into the chair, and the chair into you, in exactly the same way.
Fig. 15.11 A person reading a book, who can’t tell if he is being pulled down by gravity, or pushed up by acceleration.
15.4.2 Gravitational Attraction of Light The second step in the argument is that light bends when it is in an accelerating reference frame. For example, suppose someone were to shine a laser pointer across the width of a car. If the car was stationary, or even moving at a constant speed, the laser light would simply shine across the car in a straight line. However, suppose the car was stopped when the light was turned on, but then accelerated very quickly so that it had moved forward by the time the light beam got to the far side. An outside observer would say that the light beam went straight but that it hit a point farther back in the car because the car drove forward. However, a person inside the car would say that the light beam appeared to curve backward. In the third step, we apply the equivalence principle to this light beam that is bending in an accelerating car. The argument is that if acceleration and gravity are indistinguishable from each other, and a light beam appears to bend in an accelerating car, then it must also bend due to gravity. In other words, gravity attracts light waves. This is a substantial leap, and one that many scientists did not agree with initially, but it has been experimentally verified. The first confirmation of this prediction was in 1919 when the English astronomer Arthur Eddington and his colleagues measured
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Fig.15.12 Light being bent by gravity. (Upper left) Starlight being bent by the sun’s gravity. (Lower left) Light from a distant blue galaxy being bent around a closer red galaxy. (Right) An Einstein ring created by light from a distant blue galaxy being bent around a closer red galaxy.
the positions of several stars during a solar eclipse. They found that the stars appeared to be farther away from the sun than they really were, which implied that the sun’s gravity had bent the star’s light. The upper left panel of Figure 15.12 shows this effect. It’s a little like a mirage, except that the light isn’t refracting, but goes at a constant speed the entire time. The bending of light by gravity is now widely observed in astronomy and is called gravitational lensing. It is most spectacular if a bright light source is directly behind a very massive object, such as a black hole or a galaxy, because then the distant light is redirected toward the Earth on all sides. This causes the distant object to appear as a circle around the closer object, called an Einstein ring. The bottom left panel of Figure 15.12 illustrates this and the right panel shows an example.
15.4.3 Curved Space The fourth step in the argument is to assert that light always takes the fastest possible path, which is Fermat’s principle. This principle was initially based on the common observation that light rays travel in straight lines, which are the fastest paths between two points; it also turns out that reflected and refracted light take the fastest possible paths as well (see Sections 8.7 and 9.6). Although it was an empirical principle initially, it was supported afterward by Huygens’s principle, which states that every point on a wavefront can be seen as a source for new waves. This claim that light always takes the fastest possible path seems to contradict the statement that gravity bends light. However, we have experimental proof that gravity does bend light and there are also good reasons for believing Fermat’s principle. Thus, the only possible way out of this paradox is to say that gravity bends space itself. The notion is that space is warped, so the fastest route through this warped space is sometimes in a curved path. This explains how gravity bends light. Visualizing warped space in three dimensions is practically impossible, but two dimensional models make reasonably good analogies. Here, space is envisioned as a stretchable membrane, like a trampoline, as shown in Figure 15.13. The membrane represents empty space when flat and curved space when a heavy ball (the yellow ball) forms a depression in it. A marble (the red ball) can then roll around this depression
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Fig. 15.13 Two dimensional visualization of gravity bending space.
in an elliptical trajectory, much like a planet orbits a star. Note that this marble does not go around the heavy ball because of any action-at-a-distance attraction between them, but only because of the local tilt of the fabric8 . If the marble were instead tossed in at high speed, it would follow a straighter trajectory. However, it would never be perfectly straight, but would still curve around the depression. This faster track represents the bending of light in a gravitational field, due solely to the curvature of space. If two balls circle around each other on the fabric, they create waves in the fabric that propagate outward. These are analogous to gravitational waves9 . Experimental support for curved space arises in Mercury’s orbit around the sun. Its orbit, like those of all planets, is not exactly circular but slightly elliptical. The perihelion of its orbit, which is the point on its orbit where it’s closest to the sun, is not quite in the same place at every orbit, but shifts about 0.16◦ every 100 Earth years. Of this, about 0.15◦ can be explained by gravitational tugs by other planets, but the last 0.01◦ did not have an adequate explanation until Einstein derived his theory of general relativity. His work showed that the sun’s curvature of space makes the orbital distance a tiny bit shorter than one would ordinary expect and agrees with the observed perihelion shift.
15.4.4 Ripples in Spacetime Throughout this discussion, we have kept space and time as separate concepts. However, working through these arguments more carefully shows that this cannot actually be true, but that our concepts of time and space get scrambled together in complicated ways. Mathematically, time becomes a fourth dimension, so people talk about space and time together as the single entity spacetime10 . As an example of these strange interactions, general relativity shows that time passes more slowly in high gravitational environments than in low gravitational environments. For example, the movie Interstellar correctly showed that one hour on a fictional water planet that is
8 Many
zoos and science museums have a similar contraption, in which the visitor rolls a coin into a curved funnel and watches the coin roll around many times before finally falling into the center. 9 See the excellent YouTube video by Steve Mould titled “Visualising gravitational waves”. 10 Although time is a fourth dimension, its mathematics are still different in some important ways from those of the three spatial dimensions, so time is not equivalent to the spatial dimensions.
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close to a black hole, and thus in a high gravity environment, is equivalent to 7 years on Earth, which is in a low gravity environment (on the other hand, this planet and everything on it would have been torn to shreds by tidal forces long before it got that close to the black hole). The conclusion that gravity bends space, or more accurately spacetime, implies that gravitational waves are ripples in spacetime (Figure 15.14). When a gravitational wave ripple encounters a pair of objects, such as two asteroids, it moves them closer and farther by a tiny bit. This motion could be considered to arise from changes in the local gravitational fields or, equivalently, from deflections in space itself. In this latter interpretation, the objects are stationary with respect to “the fabric of space,” but move because space deforms.
Fig. 15.14 Artistic conception of gravitational waves as ripples in spacetime.
15.5
Gravitational Wave Detection
15.5.1 Existing Observatories The first direct detection of gravitational waves, the GW150914 event in 2015, occurred almost exactly one hundred years after Einstein had predicted their existence. Their detection represents an impressive engineering feat because they only moved two mirrors relative to each other by about a thousandth of the width of a single proton. A single atom is enormous by comparison. There are about a half dozen functional gravitational wave observatories in the world and several others in progress. Of these, three have detected gravitational waves. Two of them are part of the United States’s LIGO project (Laser Interferometric Gravitational Wave Observatory), with one observatory in Livingston, Louisiana and the other in Hanford, Washington. The third is the Virgo Observatory near Pisa, Italy, which is a collaboration of laboratories from France, Italy, the Netherlands, Poland, Hungary, and Spain.
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Fig. 15.15 Michelson Interferometer. Light enters from the left, is split at the beam splitter, goes to the two mirrors, recombines at the beam splitter, and then goes left toward the source and forward to the detector. This is a repeat of Figure 3.17.
All three are essentially very large Michelson interferometers (Figure 15.15). As described in Section 3.3.3, a Michelson interferometer splits an incident beam of light into two beams, sends them down different “arms” of the interferometer that are typically perpendicular to each other, reflects those beams off mirrors at the ends of those arms, and then recombines the two beams back into one through superposition. The two beams combine with constructive interference if they travel exactly the same distances in the two arms, or with destructive interference if there is a half wavelength difference in the pathlengths. Michelson and Morley developed this type of interferometer to determine how the Earth’s motion would affect the apparent speed of light. To their surprise, they found that it had no effect at all, which later formed the key inspiration for Einstein’s equivalence principle. The LIGO and Virgo gravitational wave observatories use interferometers with infrared lasers (1064 nm) and have arms that are about 4 km long with mirrors at the ends that can move in response to the gravitational field (Figure 15.16). Gravitational waves move these end mirrors slightly. This changes how long it takes light to shine back and forth in the two arms slightly, which changes the level of constructive or destructive interference, and hence changes the brightness of laser light at a detector. Fig. 15.16 LIGO observatory in Hanford, Washington. The long pipes enclose the interferometer arms.
Essentially every conceivable source of experimental noise is a problem here, including vibrations from traffic on nearby roads, minor earthquakes, thermal motion of atoms in the mirrors, the statistics of counting laser photons, etc. The interferometers use many techniques to address these issues and thus boost sensitivity to gravitational waves. For example, the light doesn’t just make one pass down and back in each arm, but about 280 trips, thus increasing the effective arm length 280-fold. Also, the light paths are under extremely high vacuum, the laser power is boosted to very
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high intensity using sophisticated optics, and the mirrors have elaborate damping mechanisms to remove the effects of any external vibrations. In addition, the mirrors are polished to levels of precision that are measured in individual atoms and they reflect over 99.9999% of incident light11 .
15.5.2 Future Observatories The actual detection of gravitational waves has led to increased interest in developing new observatories. For example, one observatory is being built in Japan and another is in the planning process for construction in India. Also, several space-based interferometers are being designed, with the idea that they would be free from terrestrial vibrations, air, and most other noise sources on Earth. This would allow them to be sensitive to much lower gravitational wave frequencies. Figure 15.17 shows a gravitational wave spectrum. The colored bars show the frequencies and energy levels predicted for gravitational waves from different sources, while the curved black lines show the best possible sensitivities for various observatories.
Fig. 15.17 A gravitational wave spectrum showing likely sources of gravitational waves in colored bars and observatory sensitivities with black lines.
It shows that the LIGO observatories are only sensitive to relatively high energy and high frequency waves, which is a spectral region that tends to be produced by inspiraling stars and black holes that are fairly compact. That’s part of why those waves are the most studied, and are the primary focus of this chapter.
11 see
https://www.ligo.caltech.edu/page/ligo-technology.
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The LISA (Laser Interferometer Space Antenna) observatory will be space-based and is projected to be launched in the late 2030s. It should be able to detect much heavier binary star systems as well as binary galaxies, both of which orbit more slowly than compact binaries. Finally, the EPTA and IPTA observatories (European and International Pulsar Timing Array, respectively) extend these interferometer concepts out to a vastly larger size scale by timing the regular flashes of stars called pulsars that are thousands of light-years away from us, and inferring the presence of gravitational waves from irregularities in those pulse times. This method might be sensitive enough to observe the stochastic background of gravitational waves, which is analogous to the background noise of a crowd of people talking in a room. Information about this background would produce a better understanding of the statistics of stars and galaxies in the universe. It might also help elucidate the conditions during the very early universe, shortly after the Big Bang.
15.6
Summary
Newton provided the first description of gravity with his law of universal gravitation. It states that two objects attract each other with force F=
Gm 1 m 2 . r2
This accurately describes the orbits of the planets and moons, along with the driving force for the Earth’s tides. It is based on an action-at-a-distance concept, in which objects attract each other over the intervening distance between them. It was reinterpreted later to express gravity as a vector field in which all objects produce a gravitational field around themselves, and then objects accelerate in response to the gravitational field at their particular locations. Gravitational waves are waves in the gravitational field. They create distortions in the field that cause objects to oscillate closer together and farther apart. Much like electromagnetic waves, they can propagate through empty space, travel at the speed of light, are transverse waves, and have two polarizations. However, they are much weaker and interact with masses rather than electric charges. The fact that masses are always positive causes their frequency to be double the source (e.g. binary star) frequency. It also means that their displacements are outward on one axis and inward on the perpendicular axis. These displacements rotate in circularly polarized waves and oscillate back and forth between the two axis for linearly polarized waves; these have “+” and “x” polarizations. Gravitational waves transmit energy, where the total power emitted by a binary star with circular orbits is P=
32G 4 m 21 m 22 (m 1 + m 2 ) . 5c5r 5
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This is minuscule for systems with low masses and slow orbits, such as the Moon and Earth, but enormous for the final stages of inspiraling neutron stars or black holes. Gravitational waves also carry angular and linear momentum. While gravity propagates at the speed of light, this does not arise from direct connections between the two fields, but because the special theory of relativity shows that all massless particles must travel at this speed. A neutron star merger event that was observed with both gravitational and light waves had pulses that arrived within seconds of each other, showing that these waves traveled at very nearly the same speed, and likely at the same speed. Gravitational waves are only fully developed several wavelengths away from their source, which is called the far field. Here, absorption of radiation does not feed back to the source. Closer distances, often including sizes that are larger than the solar system, are the near field. Here, gravitation often acts as though it is instantaneous because the special theory of relativity shows that fields effectively extrapolate the positions of objects as though they were moving with a constant velocity. The equivalence principle states that gravity and acceleration are indistinguishable from each other. A beam of light appears to bend to an observer who is in an accelerating vehicle, which combines with the equivalence principle to show that gravity bends light. Combing this with Fermat’s principle, which states that light always takes the fastest possible path through space, implies that gravity must warp space. Finally, working through these arguments carefully shows that time gets distorted by gravity as well. Thus, gravitational waves warp space and time, making them ripples in spacetime. The LIGO observatories made the first direct detection of gravitational waves in 2015. They use extraordinarily precise Michelson interferometers that are able to detect the movements of mirrors by much less than the width of a single proton. This success has led to the planning and construction of several new gravitational observatories.
15.7
Exercises
Questions 15.1. Does gravity bend light? (a) Never (b) Yes, but only if the gravity is strong enough (c) Yes, but only certain light wavelengths (d) Yes for photons, but not for light waves (e) Always 15.2. Suppose you’re standing on a bathroom scale, and it shows your weight suddenly increase and then decrease. List the following potential causes in order from most likely to least likely:
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It was caused by a passing gravitational wave The Earth suddenly changed mass You suddenly changed mass You moved a little bit The scale is unreliable
15.3. You are looking at a binary star edge-on, with its axis of rotation pointed up and down. What gravitational wave polarization will you observe? (a) Up-down (b) Left-right (c) + (d) x (e) circular 15.4. What are gravitational waves? Select all appropriate answers. (a) A form of electromagnetic radiation (b) Waves in the gravitational field (c) Part of the strangeness of quantum mechanics (d) The dominant cause of the Earth’s tides (e) Waves that distort space itself 15.5. Which properties do gravitational waves possess? Select all appropriate answers. (a) Two linear polarizations (b) Longitudinal waves (c) Transport energy and momentum (d) Wavelengths and frequencies (e) Propagate at the speed of light
Problems 15.6. For the gravitational waves emitted by the Moon orbiting the Earth (T = 27 days), what are (a) the wave frequency in Hz and (b) the wavelength? (c) Express this wavelength in light-days. 15.7. Calculate how much gravitational wave power the Earth emits as it orbits the sun. 15.8. Using the gravitational field equation (Eq. 15.2), calculate the gravitational field exerted by the Moon, to 4 significant figures, at (a) the center of the Earth, (b) the surface of the Earth closest to the Moon, and (c) the surface of the Earth farthest from the Moon. (d) Calculate the difference between parts b and c. 15.9. Using the gravitational field equation (Eq. 15.2), calculate the gravitational field, to 5 significant figures, exerted by the Sun at (a) the center of the Earth,
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(b) the surface of the Earth closest to the Sun, and (c) the surface of the Earth farthest from the Sun. (d) Calculate the difference between parts b and c. (e) If you did the previous problem too, then compare the answers to explain why the Moon has a greater effect on Earth’s tides than the Sun. 15.10. Two stars, each of mass m, are orbiting each other in a binary star with orbital frequency f . The binary star has radius, r , which is half of the distance between the two stars. (a) What is the speed of each star, v, as a function of the other variables? (b) The fact that each star moves in a circle implies that it 2 is accelerating toward the circle’s center at rate a = vr (called the centripetal acceleration); equate this to the gravitational acceleration and solve for r in terms of G, m, and f . (c) Compute the distance between the black holes that were observed in the GW150914 event when their gravitational waves had a frequency of 250 Hz, assuming each black hole had 30 times the sun’s mass. (d) Compute the power emitted by these black holes at the same point of their inspiral. 15.11. Consider a binary star, of which each star has mass M and they are separated by distance d. An observer is distance r away from the center of this system, along an axis that is perpendicular to the star’s axis of rotation, and is measuring the gravitational field from the stars. (a) What field does this observer measure when the stars appear lined up, with one behind the other? (b) What field does this observer measure when the stars appear stacked up, with one above the other? (c) Compute the difference between the two prior answers, and approximate using the assumption that d r so that r only appears once in 1 1 2a 3a 2 the final equation (hint: (x+a) 2 ≈ x 2 (1 − x + x 2 ) when a x). (d) Does this field fall off faster or slower with distance than that of gravitational waves, which is proportional to r −1 ?
Puzzles 15.12. Suppose the sun blinked into and out of existence with a 1 second period (ignore the fact that this violates conservation of mass). (a) Would this produce gravitational waves, and if so, (b) would they be longitudinal or transverse, (c) what sort of polarization would they have, and (d) what would their frequency be? 15.13. Consider a rapidly rotating binary star with a 1 second period. Now, suppose one of the stars stopped existing but the other kept moving along its exact same orbit as if nothing had happened (ignore the fact that this violates conservation of momentum). (a) Would this single star produce gravitational waves, and if
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so, would they be (b) longitudinal or transverse, (c) what sort of polarization would they have, and (d) what would its frequency be? 15.14. List some challenges and possible solutions in building a gravitational wave laser.
A
Numbers
A.1
Scientific Notation
Numbers in physics are often very large or very small. For example, the speed of light is very nearly 3 hundred million meters per second. This can be written as 300,000,000 m/s. Writing the number in this way is good for impressing people with its large size but not particularly convenient for calculations. Instead, it’s usually preferable to use scientific notation, giving the speed of light as 3 · 108 m/s, which means that it’s the product of 3 and eight 10s. An equivalent interpretation is that 3 · 108 is the number 3 with the decimal place moved 8 places to the right. The terminology is that the 3 is called the digit term and the 108 is called the exponential term. As another example, Planck’s constant is 6.626 · 10−34 J s. This means that it equals 6.626 with the decimal moved 34 places to the left, giving 0.000000000000000 0000000000000000006626. Clearly, the scientific notation version is more convenient. In general, positive exponents show that the number should be made larger, so the decimal point is moved to the right, while negative exponents show that the number should be made smaller, so the decimal point is moved to the left. If the exponent is 0, such as in the number 6.2 · 100 , then the decimal point isn’t moved at all; this number would be the same as just 6.2. The digits term of the number is often made to be between 1 and 10 but doesn’t have to be. For example, the wavelength of green light is 550 nm (nm is nanometer). This can be converted to meters to give 5.5 · 10−7 m, but it might be more convenient to write it as 550 · 10−9 m, where we recognize the 10−9 m term as a nanometer.
© Springer Nature Switzerland AG 2023 S. S. Andrews, Light and Waves, https://doi.org/10.1007/978-3-031-24097-1
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Appendix A: Numbers
A.1.1
Scientific Notation on a Calculator
You can use scientific notation on a calculator in the same way that it is written. For example, you can enter 3 · 108 in a calculator by pressing the buttons: 3
x
1
xy
0
8 . However, a better approach is to use the “EE” button on a
calculator (sometimes labeled “Exp”), which is a shortcut for entering the power of 10. Using it, you would enter 3 · 108 as: 3 EE 8 , which requires fewer button pushes. More importantly, using the “EE” key leads to fewer mistakes. For example, suppose you want to know how long it takes light to go 1 meter. To do so, you would start with the definition of velocity, v = dt , and solve it for time to give t = dv and then plug in the numbers. Suppose you entered them as follows: 1
÷
3
x
1
0
xy
8 . The calculator would return a value of
3.33 · 107 , implying that it would take over ten million seconds for light to travel one meter, which is clearly wrong. Why did this fail? The answer is that the calculator interprets all multiplications and divisions sequentially, so it actually calculated 1 ( 13 ) · 108 , whereas you wanted to calculate 3·10 8 . One solution to this problem would be to use parentheses in the calculation. Alternatively, the better approach is to use the “EE” key, entering the equation as 1 ÷ 3 EE 8 . This gives the correct answer, which is 3.33 · 10−9 . In other words, light takes about 3 nanoseconds to go 1 meter.
A.2
More Calculator Advice
Sometimes you need to compute numbers using complicated equations. For example, the wavelengths of light that hydrogen atoms emit can be computed from the equation 1 1 1 . − =R λ n2 m2 Assuming you know the values for R, m, and n, you could enter the whole equation into a calculator at once while using lots of parentheses. This might work, but it also might not work. A safer approach is to calculate the equation in parts, writing down the intermediate results on paper as you go along.
A.3
Precision
Every number in science conveys three pieces of information: the value of the number, the units of the number, and the precision with which the number is known. The value is the best estimate for the number, the units are described in Appendix B, and the precision is discussed here. The precision can be described in several ways. It is often described with adjectives such as “exactly,” “about,” “roughly,” and “nearly.” For example, the speed of light
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in vacuum is exactly 299,792,458 m/s (because the length of a meter was defined using the speed of light), the speed of sound in air is about 340 m/s, and the speed of a water wave produced by a boat is roughly 1.5 m/s. The precision can also described quantitatively using uncertainties, which explicitly show the range for a number using a ± (plus-or-minus) symbol. For example, the mass of an adult cat is 4 ± 0.5 kg, meaning that most adult cats are between 3.5 and 4.5 kg. Some cats are outside of this range, of course, but most are within it. In this case, the uncertainty describes the amount of variation that one should expect. As another example, the mass of the Earth is (5.9722 ± 0.0006) · 1024 kg. Here, the uncertainty represents the extent of scientific knowledge. The Earth has some particular mass that is exact, but it is only known to a precision of ±0.0006 · 1024 kg. As yet another example, suppose you measure the length of a string with a ruler that is marked in centimeters and millimeters. It would be difficult to determine the length to better precision than about 1 mm with this ruler, so it would be reasonable to record the result as ±1 mm. Significant figures provide a third way to show the precision. A number has as many significant figures as there are digits in the number that are not just placeholders. For example, the number 1300 has two significant figures, which are the 1 and the 3; the two zeros at the end are just place-holders that show the correct power of 10. Also, the number 0.0000628 has three significant figures; the four leading zeros are place-holders that again show the correct power of 10. If the decimal point is shown, then any trailing zeros are not place-holders but are counted as significant figures. For example, 15.790 has five significant figures and 29,000.0 has 6 significant figures. The interpretation of significant figures is that the number’s value is reasonably precise up to the last figure that is shown. However, the value is not known (or not important) to greater precision. For example, it’s appropriate to say that the gravitational acceleration at the Earth’s surface is 9.8 m/s2 . In fact, it actually varies some, ranging from 9.764 m/s2 near the equator to 9.834 m/s2 near the poles. Expressing the gravitational acceleration as 9.8 m/s2 correctly implies that the actual value may differ from this by ±0.1 m/s2 .
A.3.1
Determining Precision in Calculations
A calculation is typically only as precise as its least precise number. For example, if some particular piece of paper is almost exactly 11 inches long and some other piece of paper is roughly 2 inches long, then their combined lengths must be roughly 13 inches long, where this solution uses the description from the number with the lower level of precision. The same principle applies when precision is given quantitatively. When adding or subtracting numbers, the rule is that the number of decimal places in the answer should be the same as the term that had the fewest decimal places. For example, let’s add 12.367 + 2.5,
434
Appendix A: Numbers 1 2. 3 6 7 + 2. 5 ? ? 1 4. 8 ? ?
The last two digits weren’t known for the second term, so I just filled them in with question marks, and then used the system that anything plus a question mark is a question mark. The result is that this sum is 14.8. You could also do this by adding 12.367 + 2.5 to get 14.867, and then dropping the last digits to get 14.8. Or, slightly better, you could round the result off rather than dropping digits, making the answer 14.9. However, the last digit is an estimate anyhow, so it doesn’t really matter. When multiplying or dividing, the rule is that the number of significant figures in the answer should be the same as the factor that had the fewest significant figures. For example, multiplying 11.32 · 0.02 produces 41.32 · 0.02 = 0.8264 → 0.8 Here, I did the math using all of the decimal places and then rounded the answer off to one significant figure, based on the one significant figure of the 0.02 factor, to give the final answer as 0.8. For calculations with multiple parts, it can be helpful to carry an extra significant figure or two along during the calculation and only the round off at the end, but this isn’t necessary. A nice thing about significant figures is that they let you use numbers that are only as precise as the least precise number in the entire calculation. For example, suppose you wanted to compute the speed of a wave in deep water that has a wavelength of about 2 m. The equation for this is v=
λg , 2π
where λ is the wavelength and g is the acceleration from gravity. You could worry about whether you should set g to 10 m/s2 , which is pretty close, or 9.8 m/s2 , which is more accurate, or even some more accurate value for where you live. You could also worry about whether you can enter π as 3.14, or if it should be 3.1415, or if it needs even more digits. However, the point is that it doesn’t matter. The 2 m wavelength only has one significant figure, so you only need one significant figure for the other numbers too. In practice, including one extra significant figure can improve the accuracy of the answer slightly, but more than that are almost never necessary1 .
1 There are rare cases where some numbers need to have extra significant figures. This arises, for example, when a small difference between two numbers is very important, but the numbers themselves are unimportant and/or poorly known.
A.3 Precision
A.3.2
435
Propagating Uncertainties
Things get more complicated if you want to do calculations with numbers that are expressed with uncertainties, partly because they raise the question of what the uncertainties really mean. Do they represent the outer limits of the possible range, the standard deviations of multiple measurements, or something else? These definitions affect how the uncertainties should be used in calculations. Suppose the uncertainties represent the outer limits of the range. In this case, when you add or subtract numbers, you should add or subtract the regular number values as normal, and then always add the uncertainties. For example, 8±2 −3±1 5±3
To explain first term is at the bottom of its range, which is 8 − 2 = 6, and the second term is at the top of its range, which is 3 + 1 = 4. These combine to give a minimum answer of 6 − 4 = 2; this agrees with the lower limit given under the line, which is 5 − 3 = 2. Similarly, the largest possible answer arises from the top of the range for the first term and the bottom of the range for the second term, leading to (8 + 2) − (3 − 1) = 8. Again, this agrees with the upper limit under the line, which is 5 + 3 = 8. There are comparable rules for multiplication, division, and other math operations but they generally aren’t worth memorizing. Instead, the simple approach is to place each factor at the appropriate end of its range, as needed, to get the smallest and largest possible answers, and then use that range for the final uncertainty. Consider (5 ± 1) · (7 ± 1). The middle value is just 5 · 7 = 35. The smallest possible answer is 4 · 6 = 24 and the largest possible answer is 6 · 8 = 48. Thus, the answer could be anywhere in the range from 11 below 35 to 13 above 35. These are close enough that we can just split the difference and report the result as 35 ± 12. The final case that’s reasonably simple is for addition or subtraction when the uncertainties represent standard deviations rather than outer limits. Here, the rule is that you should add up the squares of the uncertainties, and then take the√square root 2 2 of √ that sum. Consider (8 ± 2) − (3 ± 1), as above. The uncertainty is 2 + 1 = 5 ≈ 2.2. Thus, the answer is 5 ± 2.2. Note that this uncertainty is larger than any of the uncertainties for the individual terms, but also smaller than the value computed above for the outer limits, which was 3.
436
Appendix A: Numbers
A.4
Exercises
Questions A.1. Which one of following values is not equal to 0.000042? (a) (b) (c) (d) (e)
4.2 · 10−5 42 · 10−6 0.042 · 10−3 4.2 · 105 0.000042
A.2. Without using a calculator, what is 9.1 ÷ 10000? (a) (b) (c) (d) (e)
9.1 · 10−4 9.1 · 104 0.000091 1100 9.1
A.3. How many significant figures are in the number 0.0003810? (a) (b) (c) (d) (e)
3 4 5 7 8
A.4. What is 89.3 ÷ 0.71? (a) (b) (c) (d) (e)
100 130 126 125.8 125.774647887
A.5. What is 6.2 · 10−8 + 110.3? (a) (b) (c) (d) (e)
116.5 1.165 · 10−8 110.300000062 110.3 6.2 · 10−8
A.4 Execises
437
Problems A.6. Write the following numbers using scientific notation. (a) 13400, (b) 0.0000 082, (c) 4.7. A.7. Arrange the following numbers in sequence, from smallest to largest: 5, 500 · 10−3 , 0.5 · 102 , 5 · 10−2 . A.8. Compute the following without using a calculator: (a) (3 · 108 )2 , (b) (c)
(6·10−34 )(3·108 ) 900·10−9
1 , 2·1014
.
A.9. Using a calculator, compute (a) (3.00 · 108 )(3.15 · 107 ), (b)
2.0·108 . 5.6·1014
A.10. Compute (a) 0.056 + 0.00734, (b) 1200.0 − 5, (c) 5.001 + 3.2 × 0.20000. A.11. Assuming uncertainties represent outer limits, compute (a) (11 ± 3)+(5 ± 2), (b) (11 ± 3) − (5 ± 2). A.12. Assuming uncertainties represent outer limits, compute (a) (11 ± 3) × (5 ± 2), (b) (11 ± 3) ÷ (5 ± 2). A.13. Assuming uncertainties represent standard deviations, compute (a) (11 ± 3) + (5 ± 2), (b) (11 ± 3) − (5 ± 2). Puzzles A.14. Suppose you have 100 bricks that are each 20 ± 1 cm long and you lay them down in a row with each brick touching the one next to it. (a) Taking the uncertainty as the outer limit, what is the length of the row of bricks, including its uncertainty? (b) Taking the uncertainty as the standard deviation, what is the length of the row of bricks, including its uncertainty? (c) Explain why the resulting uncertainty values are so different.
B
Units
B.1
Units Are Your Friends
How are you supposed to remember a long list of equations? The simple answer is that you aren’t. Memorizing a long list of equations is a quick route to frustration and is not what successful scientists or students do. Thinking about units is an important way to avoid memorizing equations. Suppose, for example, that you want to compute a velocity, v, from a wavelength, λ, and a frequency, f . You remember that there is an equation that relates these variables, but you can’t remember which term goes where. So, you try vλ = f
(incorrect).
To check if this is correct, compute the units on the left side: v is in m/s and λ is in m, and these combine to give m2 /s. Meanwhile, the units on the right side are s−1 . These are not the same, showing that your guess is incorrect. With more trial and error, you try one of v = λ f or λ =
v v or f = f λ
(correct).
In each of these cases, the units match on the two sides of the equation and, in fact, all of these rearrangements are correct. Thus, the units enabled you to find correct equations and to easily discard incorrect ones. Similarly, when you come up with your own equations, it’s a good idea to check that the units are valid. If they’re not, then the equation is certainly incorrect. This is a simple check that can catch a large fraction of mistakes.
© Springer Nature Switzerland AG 2023 S. S. Andrews, Light and Waves, https://doi.org/10.1007/978-3-031-24097-1
439
440
B.2
Appendix B: Units
The Metric System
Everyone used “customary” units of measure, which varied by region, up until near the end of the 18th century. Lengths were measured in British feet (30.48 cm), French feet (32.48 cm), Norwegian feet (31.37 cm), Japanese shaku (about 30 cm), or any of many other commonly used units, depending on where one lived. These units sometimes varied substantially by region within the country and also varied over time. This was inconvenient because there was no way to accurately describe how big something was without bringing the actual object along. It was also a problem for international trade and scientific communication. Furthermore, even within the systems of units of individual countries, it was difficult to convert values between different sizes of units because they weren’t separated by regular intervals. In the modern American system of volumes, for example, there are 3 teaspoons to a tablespoon, 16 tablespoons in a cup, 2 cups in a pint, 2 pints in a quart, 4 quarts in a gallon, and, roughly, 7.5 gallons in a cubic foot. These irregular intervals make conversion difficult. The metric system was developed to address these problems and is now used universally in science. It is also the primary system of measurement in most of the world, with the primary exception of the United States. It is often called the SI system, from the French Système International. The metric system is built upon a small collection of base units including the meter (m) for length, the kilogram (kg) for mass, and the second (s) for time2 . There are also derived units that are built from base units. For example, energy is often measured in joules (J), a derived unit, where 1 joule is equal to 1 kg m2 s−2 . Table B.1 lists the most important base and derived units. If you express all quantities with these base or derived units, then further unit conversion is generally not required. Table B.1 Metric base and derived units quantity
dimension
unit
symbol
in base units
length mass time frequency energy power force pressure
L M T T −1 L 2 M T −2 L 2 M T −3 L M T −2 L −1 M T −2
meter kilogram second hertz joule watt newton pascal
m kg s Hz J W N Pa
m kg s s−1 kg m2 s−2 kg m2 s−3 kg m s−2 kg m−1 s−2
2 This book uses the modern “mks” convention in which meters, kilograms, and seconds are the base units. This largely supplanted the older “cgs” convention, which is based on centimeters, grams, and seconds.
B.3 Unit Math
441
The “dimension” column in the table shows the dimension of the quantity, where L represents length, M represents mass, and T represents time. It can be useful, although the final column, “in base units” is equivalent and typically more useful. Larger and smaller versions of the base and derived units can be formed using the prefixes that are listed in Table B.2. These prefixes often make numbers more convenient, but it is generally best to stick with meters, kilograms, and seconds within calculations; if you start mixing in centimeters, milligrams, or other units, then you’ll need to keep track of the factors of 10. Thus, the best approach is typically to convert all numbers to base or derived units at the beginning of a calculation, do the calculation with those units, and then add the appropriate prefix to your answer at the end to make it more meaningful. Table B.2 Metric prefixes prefix tera giga mega kilo centi milli micro nano pico
abbreviation
value
expanded value
T G M k c m µ n p
1012
1,000,000,000,000 1,000,000,000 1,000,000 1000 0.01 0.001 0.000001 0.000000001 0.000000000001
109 106 103 10−2 10−3 10−6 10−9 10−12
In this table, the “expanded value” column might present numbers in a more familiar way than the “value” column but is actually less useful. For example, suppose you are given a number as 43 km and you want to enter it into your calculator in meters, as you should. You could enter it as 43 and then tack on three zeros, or you could enter it as 43 and then multiply by 1000. However, the best approach is to think of this number as 43 · 103 m, where the 103 part is the kilo part; enter this number in a calculator as 43, then the EE key, and then 3; see Appendix A.
B.3
Unit Math
As mentioned above, it is a good idea to check the units in equations to ensure that they agree on both sides. If they don’t agree, the equation is certainly incorrect. A simple approach is to substitute in the units for each variable, while dropping the unit prefixes. The prefixes aren’t necessary because they only change the value, but not the underlying unit. Here is an example using the relationship among velocity, wavelength, and frequency:
442
Appendix B: Units
v =λf
→
m = (m)(s−1 ) s
These units agree, confirming that the equation is reasonable (and in fact is correct). Here are the rules for unit math. • Ignore all numbers. This includes all values like 2, 4.7, and -58, all factors of π, and all metric prefixes. • Change derived units to base units if it will help. For example, change Hz to s−1 and W to J s−1 . On the other hand, it might not help. For example, changing J to kg m2 s−2 may or may not lead to a simpler result. • Countable objects are optional as units. For example, a wavelength can be expressed equivalently as 0.1 m, or as 0.1 m/wave. Likewise, a rotational frequency can be given equivalently as 2000 rotations/s or 2000 s−1 . • Addition and subtraction: units must be the same for both terms, and units stay the same at the end. For example, 2 apples + 3 apples = 5 apples; this is valid because the units are the same in both terms and then become the same in the end. However, 2 apples + 3 oranges is not valid because the terms don’t have the same units. 2 s + 3 s−1 is equally incorrect. • Multiplication and division: combine units. For each type of unit, such as “m” for meter, add up all of the exponent values in the numerator and subtract any of the exponent values in the denominator. For example, multiplying m by m gives m2 and dividing m by s gives m s−1 . Note that a unit with a negative exponent is equivalent to dividing by that unit, so m/s = m s−1 and m/s2 = m s−2 . Suppose you have the equation λ = hp , where h is measured in J s, and p is measured in kg m/s. You want to know the units for λ. The equation and variables have meanings, of course, but that is irrelevant here. λ=
h p
→
λ=
kg m2 s−2 s Js = =m kg m s−1 kg m s−1
Here, we first replaced the variables with their units, we then found that the joules in the numerator didn’t cancel with anything in the denominator so we expanded it in terms of the base units (J = kg m2 s−2 ), and we finally collected together all of the kg, m, and s terms to find the total number of each. As it turned out, all that was left was a meter unit, showing that λ must have units of meters. Once you are done simplifying, a fraction should only have a single numerator and a single denominator. For example, the unit m/s/s doesn’t make sense. Does this mean (m/s)/s, which simplifies to m/s2 , or does this mean m/(s/s), which simplifies to m? It’s unclear, so it’s best to avoid such constructions. This leads to the question of how to deal with fractions of fractions, which is a very common situation. Suppose, you want to divide m/s by m/wave. A simple approach is to replace each of the two fractions with a single row of units by using negative exponents, and to then count
B.4 Unit Conversion
443
up how many you have of each type of unit, m s m wave
B.4
=
m s−1 wave = wave s−1 = −1 m wave s
Unit Conversion
As mentioned above, a typical approach for solving a problem is to express all numbers with their values in meters, kilograms, and seconds (and joules, newtons, pascals, etc., as given in Table B.1), do the math with these base units, and then make the result more tidy by replacing any powers of 10 with the appropriate prefix. This unit conversion is simple if you work entirely with metric values, often allowing you to account for units on the fly while entering numbers into a calculator. For example, suppose you want to find the frequency of green light, which has a wavelength of 500 nm. To do this, rearrange the equation v = λ f to get f = λv . The velocity is 3 · 108 m/s and λ is 500 nm, so enter the following into a calculator f = 3e8 ÷ 500e-9 The result is 6 · 1014 . Because everything was entered in meters and seconds, the result is also in metric base units. In this case, the result is in s−1 which can be verified using unit math. This answer of 6 · 1014 s−1 is fully sufficient, but could also be rescaled by taking 12 factors of 10 off it and using the tera prefix to give the frequency as 600 THz. Unit conversion is more complicated if you aren’t working entirely in the metric system. Most people still instinctively start by reaching for their calculators, but this is generally a mistake. Instead, it’s better to start with paper and pencil, write down the unit conversion equation, and then use a calculator at the very end. The procedure for this unit conversion equation starts by rephrasing the question as a mathematical expression. For example, suppose the question is how many feet are in 1 meter? This is rephrased as 1m 1 Writing the right hand side as a fraction with a 1 in the denominator isn’t necessary, but can help avoid confusion later on. Having the left side there is nice because it reminds you of where you’re going. Note that this equation is indeed an equality, in that some number of feet is actually exactly the same thing as 1 meter. Next, multiply the right side of this equation by 1 as many times as needed until you get the correct units, where this “1” is a conversion factor. Units can be crossed off, as they cancel with each other. Continuing with this example, we convert meters to centimeters by cm multiplying by the fraction 100 1 m = 1 and then cross off meters: ? feet =
? feet =
1 m 100 cm · 1 1 m
444
Appendix B: Units
Continuing gives ? feet =
1 m 100 cm inch 1 1 foot · · · 12 1 1 m 2.54 cm inch
At the very end, enter all of these numbers into a calculator, which gives the result that 1 meter equals 3.28 feet. More complicated types of units make the task a little more complicated but not fundamentally different. For example, the density of water is 1 g/cm3 . What is it in pounds per cubic foot? Here, the unit conversion equation looks like: ?
12 12 2.54 2.54 12 g kg cm cm cm inch inch inch lbs 1 1 2.20 lbs 2.54 = · · · · · · · · 3 3 g 1 foot 1 foot 1 foot 1 inch 1 inch 1 inch 1 kg ft cm 1000
Plugging these numbers into a calculator gives the answer as 62.3 lbs/ft3 . As a final point, note that Google does an excellent job of unit conversion, making this entire task largely unnecessary. Nevertheless, unit conversion is a useful skill to have in case you need a number in certain units and you don’t have internet access at that moment. Table B.3 lists some common unit conversions. Table B.3 Some unit conversions 1 inch 1 foot 1 mile 1 acre 1 ft3 1 gallon
B.5
= = = = = =
2.54 cm 12 inches 5280 feet 43560 ft2 7.48 gallons 3.785 liters
1 hour 1 minute 1 day 1 kg 1 calorie 1 kilocalorie
= = = = = =
60 minutes 60 seconds 24 hours 2.205 lbs 4.184 J 4184 J
Exercises
Problems B.1. For each part, use the unit conversion method presented here and show your work. (a) How many km is 4000 miles? (b) How many nm is 17 µm (the width of a human hair)? (c) How many cycles per day is 97 MHz? B.2. For each part, use the unit conversion method presented here and show your work. (a) How many gallons are in 0.25 acre-feet (the annual water use of a typical family)? (b) How many km/liter are equal to 40 miles per gallon? (c) How many US$/gallon is equal to 1.28 euros/liter (the price of gasoline in France), assuming that 1 euro equals $1.14?
B.5 Exercises
445
B.3. Following are several equations. For each, simply calculate what the units are for the result, if possible; if it’s not possible, write “invalid equation.” Don’t worry about the numerical values. (a) E = h f (photon energy). h is in J s, f is in Hz. −1 (curved mirror equation). do and di are in m. (b) f = d1o + d1i (c) λ =
g 2π f 2
(deep water wavelength). g is in m/s2 , f is in Hz.
B.4. Following are several equations. For each, simply calculate what the units are for the result, if possible; if it’s not possible, write “invalid equation.” Don’t worry about the numerical values. (a) P = AσT 4 (Stefan-Boltzmann law). A is in m2 , σ is in W m−2 K−4 , T is in K. L (laser cavity decay time). L and R are in m, v is in m/s. (b) τ = v(1−R) (c) h r =
ha λ
(relative water depth). h a and λ are in m.
B.5. What is the mass of the Pacific Ocean? The density of seawater is 1029 kg/m3 , the volume of the Pacific Ocean is 7.1 · 1017 m3 , and the answer should be in kg. B.6. Suppose x is in J/photon and y is in J/s. How can x and y be combined to produce an answer in photons/s (e.g. x y, x/y, y/x, etc.)? B.7. Suppose x is in waves/m and you want to convert this to m/wave. How would you calculate the answer from x? B.8. Suppose σ is in W m−2 K−4 and T is in K. How can σ and T be combined to produce an answer in W m−2 ? B.9. Suppose x is in apples/pie, y is in slices/pie, and z is in slices/person. How can x, y, and z be combined to produce an answer in apples/person? B.10. Electricity is normally sold in units of kilowatt-hours (KWh). Simplify this to (a) a metric derived unit, (b) metric base units (ignore prefixes and other scaling, just focus on the type of unit).
C
Algebra
C.1
Solving Problems
Solving an algebra problem is like walking across a park that has many trails. You start at a parking lot, you choose a trail, and you follow it. When you get to a fork in the trail, you choose among the options and see where it leads. At any time, you can always retrace your steps. Provided that you stay on the trail, you can’t get truly lost and, with luck, you’ll end up at your desired destination. On the other hand, if you leave the trail, then you could very well get stuck in a swamp. In algebra, you start with an equation and you manipulate it in various ways to get to the desired solution. The way to stay on the trail during these manipulations is to follow the rules of algebra, of which all the important ones are listed below. Some trails are shorter and some are longer, but none are wrong. Beware of shortcuts! You may have learned various “helpful” shortcuts in algebra classes, such as the semimagical cross-multiply and divide method for simplifying fractions, or the FOIL technique for expanding binomials. These can work, but only if you remember them perfectly and understand their limitations. If you make a mistake with a shortcut, you will probably get stuck in one of the metaphorical swamps. As with walking in a park, it’s normal to take an inefficient route to the solution at first, but you will get faster with time as you start to recognize various waypoints. Provided that you follow the rules, you won’t get a wrong answer and, with perseverance, you will get to your desired solution. Suppose you are told to solve the following equation for y, x=
y−1 . y
Figure C.1 shows steps that you could take to solve this problem, going from the problem on the left to the solution on the right. It shows that there are many possible © Springer Nature Switzerland AG 2023 S. S. Andrews, Light and Waves, https://doi.org/10.1007/978-3-031-24097-1
447
448
Appendix C: Algebra x=
x=
y 1 − y y
y −1 y
x = 1−
xy = y − 1
1=
y −1 xy
−x = −1+
xy xy + x xy
y 1 − xy xy
1 y
1− x =
1 1 −1 = − + x xy
1=
1 1 − x xy
−1+
1 y
y (1− x ) = 1
y − xy = 1
−xy = −y + 1
−xy = − 1=
1 y
−x + 1 =
1 1 = x xy
y=
1 y
1 1− x
x = xy −x + 1 1 −1+
1 x
= xy
Figure C.1 Possible solution approaches for the following problem: solve x =
y−1 y
for y.
routes from problem to solution, all of which are equally correct. Some are obviously shorter than others, but it doesn’t actually matter which one is taken. Also note the arrow colors, which represent what was done in each step. Most steps use red arrows, in which the same thing was done to both sides of the equation, some have blue arrows to denote distributing or factoring, and a few have green arrows to denote multiplying by 1 or combining to make 1. That’s all. Nothing else was required.
C.2
Expressions and Equations
Algebra deals with two things, expressions and equations. An expression is just a number, or something that would evaluate to a number if you knew the values of all the variables. Examples of expressions are: 5, 10x + 3, and cos(x). None of these mean anything by themselves; they just are. An equation is a statement of equality. It says that the expression on the left is equal to the expression on the right. For example, the equation E = mc2 makes the statement that the energy of an object is equal to the value c2 times the mass of the object. This statement carries meaning.
C.2.1
Manipulating Expressions
The five important rules for manipulating expressions are listed below, all which can be performed on any mathematical expression. • Associative property. In addition and multiplication, you can combine terms in any order. a + b + c = (a + b) + c = a + (b + c) abc = (ab)c = a(bc) Example: 2 + 3 + 4 and 2 + (3 + 4) and (2 + 3) + 4 are the same; they all add to 9.
C.2 Expressions and Equations
449
Example: 2 · 3 · 4 and 2 · (3 · 4) and (2 · 3) · 4 are the same; they all evaluate to 24. • Commutative property. In addition and multiplication, you can reverse the order of the terms. a+b =b+a a − b = −b + a ab = ba a 1 1 =a = a b b b Example: you can convert 2 + 3 to 3 + 2 and it still adds to 5. Example: 2 · 3 is the same as 3 · 2, with both versions multiplying to 6. • Distributive property. Multiplication can be distributed over addition and subtraction, meaning that the a in the expression a(b + c) can be distributed over the sum to give ab + ac. The opposite of distributing is called factoring or, sometimes, combining like terms. a(b + c) = ab + ac b c b+c = + a a a Example: 2(3 + 4) distributes to 2 · 3 + 2 · 4. Either way, it evaluates to 14. Example: 2 · 4 + 2 · (−4) factors to 2(4 − 4). These evaluate to 0. Example: 5x 2 + 2x 2 factors to (5 + 2)x 2 , which is 7x 2 . Here, we are combining like terms. Example: consider (x + a)(x − a). Treat the (x − a) term as just a value and distribute it over the (x + a) term. This gives x(x − a) + a(x − a). Next, distribute over the (x − a) term to get x 2 − xa + ax − a 2 . Finally, notice that −xa + ax add to zero to get x 2 − a 2 . • Adding zero. You can add 0 to any expression, where this 0 is usually a number minus itself. The opposite of this is that if an expression is adding two things that combine to 0, then they can be removed. a =a+b−b Example: If y = 4(x + 1) + x, we can add 1 − 1 to the right hand side to get y = 4(x + 1) + x + 1 − 1. By associativity, this is y = 4(x + 1) + (x + 1) − 1. Combining like terms gives y = 5(x + 1) − 1. • Multiply by one. You can multiply any expression by 1, where this 1 is usually a number divided by itself. The opposite of this is that if two things multiply to 1, then they can be turned into 1 and possibly removed. (Caveat: if you remove x x , or something equivalent, then you should check that x isn’t 0). A typical use
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of this rule is that you can multiply the top and bottom of a fraction by the same thing. a=a·
b b
5 5x Example: If y = 1/x , you can multiply this by xx to get y = x/x . The x/x fraction equals 1, so remove it to get y = 5x. This answer will almost always be good enough, but it might be helpful to think about the possibility of x being 0. If this is 5 isn’t actually defined because dividing the case, then the starting expression, 1/x by zero isn’t allowed. Thus, to be thorough, you could state that x is not allowed to equal 0.
There is one manipulation that students often attempt to use, but is not valid. It goes by the name of “everything distributes” or “freshman’s dream.” It is the wishful thinking that exponents can distribute over addition and subtraction, but it’s just not true: (a + b)n = a n + bn √ √ √ a + b = a + b 1 1 1 = + a+b a b
C.2.2
Manipulating Equations
There is exactly one rule for manipulating equations: • You can do anything you want to one side of an equation, provided that you do the same thing to the other side too. For example, you can: add the same thing to both sides, multiply both sides by the same thing, take the reciprocal of both sides, square both sides, take the square root of both sides, etc. Note though, that you have to include the entirety of each side when doing this; for example, you can’t just multiply one term of a side by some number. Example: if 2x = x + 5, you can subtract x from each side to get x = 5. Example: if xy = 2x 2 , you can multiply both sides by x to get y = 2x 3 . 3 Example: if 1y = 5x 3 , you can take the reciprocal of both sides to get y = 5x . Unfortunately, there are a couple of caveats to this rule. First, you’re not allowed to divide both sides by zero. This is fairly obvious if the zero looks like a zero, but sometimes it doesn’t. In particular, if you divide both sides by x, then you have to consider the possibility that x might be zero, and then deal with that as a special case. The other caveat arises when you take the square root of both sides of an equation. The reason is that every number has both a positive and a negative square root (e.g.
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451
if x 2 = 25, then x could equal either 5 or −5). So, if you take the square root of both sides, then you need to think about whether you want the positive root, the negative root, or both roots. Typically, the best approach is to ignore these issues until the very end, and then go back to see if anything needs extra attention.
C.3
Exponents
Exponents represent repeated multiplication. For example x 2 is x · x. However, even with this understanding, they can sometimes get confusing. Thus, the following list gives the common manipulations with exponents. Each one can be reduced to the concept of repeated multiplication, but they are listed here for convenience. am m−n and 1 = a −n an = a an 108 3 2 5 Example: 2 · 2 = 2 and 103 = 105 and 1012 = 10−2 n n (ab)n = a n bn and ab = abn 2 2 Example: (2 · 3)2 = 22 · 32 = 36 and 10 = 10 =4 5 52 m n mn (a ) = a
• a m a n = a m+n and • •
Example: (108 )2 = 1016 √ 1/n n • a = a √ Example: 251/2 = 25 = 5 • a 0 = 1 and a 1 = a Examples: 50 = 1 and 51 = 5
C.4
Exercises
Questions C.1. Which of the following statements are incorrect? Choose all that are appropriate. (a) 5x 2 − 3x = x(5x − 3) + 9x (b) x 29x+5 = 9x 5 x2
x 5 (c) x 9x+5 = 9x + 9x 3 (d) 1/x x/3 = x (e) 5 + x y = yx − 3 + 8 2
2
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Problems C.2. Simplify to remove fractions of fractions: (a) C.3. C.4. C.5. C.6.
1 1/x ,
(b)
Solve for y: (a) x = y − 43, (b) x = 1/y, (c) 5x = Solve for r : V = 43 πr 3 Solve for λ: E = hc λ Solve for x: x 2 − 16 = 0
1/x x ,
2y x .
(c)
x/y 2 . y/x 2
D
Geometry
Geometry is much too large a field to summarize here. Instead, this appendix presents a few topics that are particularly useful for the problems given in this book.
D.1
Triangles
D.1.1
Similar Triangles
Triangles that have the same shape, but whose sizes may vary, are called similar triangles. These have the important properties that (1) all of their angles are the same and (2) there is a single scaling factor for all of their side lengths. Any pair of triangles that have the first property also have the second, and vice versa. Figure D.1 shows an example. The angles near the middle of the figure, both labeled a, are the same because they are created by the same lines. Also, the angles at the outsides of the figure, b and c, are the same because the outside edges are parallel to each other. The result that these angles are equal can be proven with geometry proofs, but it’s usually sufficient to verify equality for several simple cases (e.g. let c be 90◦ ) and to then assume that the result holds true more generally. The equal angles in each triangle implies that they are similar. The side lengths also show this same property, by showing that each edge of the triangle on the left is twice as long as the corresponding edge for the triangle on the right. Figure D.1 Two similar triangles.
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Appendix D: Geometry
Right Triangles and Trigonometry
Triangles that have a 90◦ angle, called a right angle and depicted with a little square in the corner, are called right triangles. The left side of Figure D.2 shows an example. Right triangles have a few special properties.
Figure D.2 Pythagorean theorem, showing the result on the left and a proof on the right.
First, the Pythagorean theorem3 gives the length of the hypotenuse, c, (the side opposite the right angle) from the lengths of the other two sides, a and b. It is c2 = a 2 + b2 . The left panel of the figure shows this result and the right panels of the figure shows a simple proof of this theorem. It shows that if four identical triangles are drawn at the corners of a box, then the remaining area is equal to c2 . The same four triangles can then be rearranged, making the remaining area equal to a 2 + b2 . The remaining areas must be equal in the two cases because the triangles occupy the same amount of area, thus proving the assertion that c2 = a 2 + b2 . Another property of right triangles is that the ratios of the side lengths are given by the trigonometry functions sine, cosine, and tangent. Consider one triangle point and call its angle θ, as shown on the left side of Figure D.3. The ratio of the length of the side that is opposite this angle to that of the hypotenuse is sin θ. Similarly, the ratio of the length of the side that is adjacent to this angle to that of the hypotenuse is cos θ. Finally, the ratio of the opposite to the adjacent lengths is tan θ. In summary, sin θ =
opposite hypotenuse
cos θ =
adjacent hypotenuse
tan θ =
opposite adjacent
These sine, cosine, and tangent trigonometry functions allow you to find triangle side lengths from the angles. For example, if you know that θ = 40◦ and that the adjacent side is 3 meters, then you can compute the triangle’s height by rearranging the tangent equation to give h = (3 m) tan 40◦ , which evaluates to 2.5 m. These equations are
3 The Pythagorean theorem is named for the ancient Greek philosopher Pythagoras (born c. 570 BCE), but it’s unknown whether he actually contributed to it. It was also known centuries earlier to the Babylonians and Indians and, at some point, to the ancient Chinese. See Wikipedia “Pythagorean theorem.”
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especially convenient if the hypotenuse has length 1, as shown on the right side of the figure. In that case, the opposite side has length sin θ and the adjacent side has length cos θ.
Figure D.3 Right triangle and trigonometry.
Sometimes you want to reverse this process to find the angles from the side lengths. For example, suppose you have the equation y = sin θ, you know that y = 0.5, and you want to know what θ is equal to. You clearly need to solve for θ by doing the same thing to both sides of the equation, but the question is what thing needs to be done. In this case, it is the arcsine function, which is simply the inverse of the sine function. That is, arcsin(sin θ) = θ. In this problem, you would take the arcsine of both sides of the equation to give arcsin(y) = arcsin(sin θ) which then simplifies to give arcsin(y) = θ. Plugging in y = 0.5 gives the result that θ = 30◦ . Similarly, the arccosine and arctangent functions are the inverses of the cosine and tangent functions. The arcsine function is sometimes written as sin−1 and, similarly, arccosine as cos−1 and arctangent as tan−1 . I discourage this notation because it causes confusion between whether the −1 is an exponent or denotes the inverse function but, unfortunately, it’s quite common. In particular, the sin−1 style notation is almost universally used on calculators. On calculators, you typically get the inverse functions by pressing the “second function” key and then the sine, cosine, or tangent keys.
D.2
Perimeters, Areas, and Volumes
For a two-dimensional shape, such as a rectangle, triangle, or circle, the perimeter is the total length of its outside edge and the area is the size of the enclosed space. Figure D.4 shows some examples, using P for perimeter and A for area. For circles, the perimeter is also called the circumference. Circle sizes often use the number π, which is an irrational number that is approximately π = 3.14159265359.... For three-dimensional shapes, the surface area is the area of the shape’s surface and the volume is the amount of enclosed space. The perimeter is no longer meaningful here. Figure D.5 shows a few examples.
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Figure D.4 Some perimeters and areas for a rectangle, triangle, and circle. Figure D.5 Areas and volumes for a box and sphere.
D.3
Exercises
Questions D.1. Consider a right triangle with base x, height y, and interior angle θ as shown in the following figure. Which of the following are incorrect? Choose all that are appropriate.
(a) sin θ = xy (b) tan θ = xy (c) cos θ = √ 2x 2 √ x +y x 2 +y 2 1 (d) cos θ = y (e)
sin θ cos θ
=
y x
Problems D.2. Consider a brick that has side lengths a, b, and c. What is the length of a diagonal line that goes from one corner to the corner that’s farthest away from it. D.3. The right triangle shown below has sides of length 3 m and 4 m. (a) What is the triangle’s perimeter? (b) What is the angle shown as θ in the figure?
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Puzzles D.4. Is the Pythagorean theorem true for a triangle that’s not flat, but drawn on the surface of a sphere? (Hint: try to come up some simple triangle on a sphere where it’s clearly wrong.)
E
Additional Resources
Following are resources that I have found particularly useful for learning about the topics covered in this book. This list is far from exhaustive, but should be a good start for further reading.
Online Resources • Google Scholar — An excellent search engine for academic research papers. • Hyperphysics — Online physics textbook that covers an impressive array of topics. • Physics LibreTexts — Another excellent online physics textbook. • Khan Academy — Short online lectures about a wide range of topics. • Wikipedia — General purpose encyclopedia for nearly everything. • Wolfram MathWorld — Excellent mathematics resource, often at an advanced level.
History • Opticks: A Treatise of the Reflections, Refractions, Inflections & Colors of Light (Isaac Newton, 1704. Available in many modern editions, e.g. 1952 Dover Edition) — A surprisingly readable account of Newton’s research on optics. • Theories of Vision from Al-Kindi to Keplar (David C. Lindberg, 1976, University of Chicago Press: Chicago) — A detailed history of the early theories of vision. • The Conceptual Development of Quantum Mechanics (Max Jammer, 1966, McGraw-Hill Book Company: New York) — A detailed academic history of quantum mechanics by an eminent physicist. Substantial quantum mechanics knowledge is assumed.
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Appendix E: Additional Resources
Introduction to Light and Waves • Introduction to Light: The Physics of Light, Vision, and Color (Waldman, 1983, Prentice-Hall, Inc. Republished 2002 by Dover.) — Light and waves for undergraduate non-science majors, with a substantial emphasis on optics. • Light Science (Thomas D. Rossing and Christopher J. Chiaverina, 1999, SpringerVerlag, New York) — Another book on light and waves for undergraduate nonscience majors, with an emphasis on optics and art. • Seeing the Light (D.S. Falk, D.R. Brill, and D. G. Snork, 1986, John Wiley & Sons: New York) — Yet another book on light and waves for undergraduate non-science majors, again with an emphasis on optics. The book is somewhat dated. • Waves, Sound, and Light (McDougal Littell Science, 1996, McDougal Littell: Evanston, IL.) — A middle school book about waves, sound, and light. While aimed at a young audience, this book covers the key concepts in a clear manner.
More Advanced Light and Waves • College Physics: Putting it all together (Ron Hellings, Jeff Adams, and Greg Francis, 2018, University Science Books: Mill Valley, CA) – A reasonably comprehensive undergraduate physics textbook. This one presents optics by focusing on the convergence and divergence of spherical waves, which is an intriguing contrast to the conventional ray optics approach that is covered here and in most other books. • Introduction to Electrodynamics, 4th edition (David J. Griffiths, 2017, Cambridge University Press: Cambridge) — An excellent undergraduate textbook on electricity and magnetism, which goes beyond the level of introductory physics texts. This gives the mathematics for light waves, among other relevant topics. • Physics for Engineers and Scientists, 3rd edition (Hans C. Ohanian and John T. Markert, 2007, W. W. Norton & Company: New York) — Another reasonably comprehensive undergraduate physics textbook that has excellent chapters on light, waves, quantum mechanics, and other topics. • Physics: Principles with Applications, 7th Edition (Douglas C. Giancoli, 2014, Prentice Hall, Upper Saddle River, NJ) — Yet another reasonably comprehensive undergraduate physics textbook, with both calculus and non-calculus versions. It has excellent chapters on light, waves, quantum mechanics, and other topics.
Water Waves • Tides: The Science and Spirit of the Ocean (Jonathon White, 2017, Trinity University Press, San Antonio, TX) — An accessible book about the science and culture of tides and related phenomena, such as tidal waves.
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Optics and Color • Color: A course in mastering the art of mixing colors (Betty Edwards, 2004, Tarcher/Penguin publishing: New York) — A practical guide to color mixing for artists. • Color science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (G. Wyszecki and W.S. Stiles, 1982, Wiley: New York) — A technical reference book about color. • Colorimetry: Understanding the CIE system (Janos Schanda, 2007, WileyInterscience: Hoboken, NJ) — A technical description of the CIE color system. • Insight into Optics (O.S. Heavens and R.W. Ditchburn, 1991, John Wiley and Sons: New York.) — An optics book that has short sections that are more selfcontained than usual. This makes it hard to read but a good reference book. • Introduction to Modern Optics, 2nd edition (Grant R. Fowles, 1989, Dover Publications, Inc.: New York.) —- An advanced but somewhat old fashioned book on optics for upper level undergraduates in physics. This focuses primarily on wave aspects of light but also goes into photons and some quantum mechanics. • Optics, 4th edition (E. Hecht, 2002, Pearson: San Francisco) — A thorough textbook about optics for upper level undergraduates or graduate students. • Optics Made Clear: The Nature of Light And How We Use It (William L. Wolfe, 2007, The International Society for Optical Engineering: Bellingham, WA.) — Another optics book that has short sections that are more self-contained than usual, again making it hard to read but a good reference book. This book is full of odd facts that can be fun to browse. • Rainbows, Halos, and Glories (Robert Greenler, 1980, Cambridge University Press: Cambridge.) — A delightful book about optical atmospheric phenomena, such as rainbows, halos, and glories, describing the physical cause of each effect, substantially from the author’s own research. This book is well illustrated with diagrams and photographs.
Modern Physics • Modern Physics, 3rd edition (Raymond Serway, Clement Moses, Curt Moyer, 2004, Cengage Learning: Boston.) — A relatively non-technical introduction to a wide variety of modern physics topics. • Introduction to Quantum Mechanics, 3rd edition (David J. Griffiths and Darrell F. Schroeter, 2018, Cambridge University Press: Cambridge.) — A thorough introduction to quantum mechanics that is aimed for upper level undergraduate physics students. • QED: The Strange Theory of Light and Matter (Richard P. Feynman, 1985, Princeton University Press: Princeton, NJ) — A popular science book about quantum mechanics and interactions between light and matter. • Quantum Chemistry, 2nd edition (Donald McQuarrie, 2007, University Science Books: Mill Valley, CA.) — This is one of the more accessible quantum mechanics textbooks, which is aimed for upper level undergraduate physical chemistry
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students. This book focuses almost exclusively on spatial wave functions rather matrix mechanics or bra-ket notation. • Astrophysics for People in a Hurry (Neil de Grasse Tyson, 2017, W.W. Norton & Company, Inc.: New York) — A light reading but also also very informative book about stars, light, relativity, and the cosmos. • Gravitational waves (Tony Rothman, 2018, American Scientist 106:96) — A nice article that describes gravitational waves. • Warping spacetime (Kip Thorne, Chapter 5 in The Future of Theoretical Physics and Cosmology: Celebrating Stephen Hawking’s 60th Birthday, edited by Gibbons, Rankin and Shellard, Cambridge University Press, Cambridge, UK, 2003) — An accessible introduction to black holes primarily, but also gravitational waves, wormholes, and time travel.
F
Answers to Odd-Numbered Problems
Chapter 1 1.1 a. 1.3 e. 1.5 d. 1.7 (a) extramission; (b) infinite speed; (c) particles. 1.9 There is no evidence of rays emitted by eyes; Light is known to travel at a finite speed and stars are very far away, so the extramission theory cannot explain how we can see stars; Light is known to enter people’s eyes, of which one piece of evidence is that our eyes feel pain when looking at a very bright light such as the sun; Our eyes are not depleted of any substance when we look far out into space. 1.11 Atomic emission shows distinct colors; photoelectric effect (ultraviolet photons eject electrons from metal surfaces); some chemical reactions can only be started by ultraviolet light; sunburns only happen in ultraviolet light; blackbody radiation can only be explained when assuming particle nature of light.
Chapter 2 2.1 b. 2.3 e. 2.5 a,c,e. 2.7 b. 2.9 See Figure 2.3. 2.11 (a) String, drum head, all electromagnetic (light, radio, UV, IR, etc.), gravitational, seismic S-waves; (b) Slinky (when pushing on end), sound, seismic P-wave. 2.13 (a) Any value between 500 and 570 nm, of which 540 is close to the middle of the range; (b) smaller; (c) much larger; (d) much larger; (e) much larger; (f) much smaller; (g) much smaller; (h) similar. 2.15 (a) 406 miles per hour. Use v = d/t, where d = 6500 miles and t = 16 hours. This is similar to the speed of a commercial airliner. (b) 53%. Compute with (460 mph)/(760 mph) · 100%. 2.17 (a) 565 times faster. Divide 192 · 103 m/s by 340 m/s. This is very fast. (b) No, it’s not relevant because there’s no sound in space. (c) 0.064%. Divide 192 · 103 m/s by 3 · 108 m/s and multiply by 100%. Despite being very fast, it’s still vastly slower than the speed of light. 2.19 (a) 513 miles/hour. Use v = d/t with d = 5900 miles and t = 11.5 hours. (b) 76%. Divide 513 mph by 670 mph and multiply by 100%. Commercial aircraft speeds are substantially © Springer Nature Switzerland AG 2023 S. S. Andrews, Light and Waves, https://doi.org/10.1007/978-3-031-24097-1
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limited by the speed of sound. (c) 7.7 · 10−5 %. Divide 513 mph by 6.7 · 108 mph and multiply by 100%. Airplanes are vastly slower than light. 2.21 (a) 2.26 · 108 m/s. Use v = c/n, where c = 3 · 108 m/s and n = 1.33. (b) 2 · 108 m/s. Use v = c/n, where c = 3 · 108 m/s and n = 1.33. (c) 1.24 · 108 m/s. Use v = c/n, where c = 3 · 108 m/s and n = 2.42. 2.23 (a) 0.012 s. Rearrange v = d/t to t = d/v. The sound travels to the moth and back, which is d = 4 m, and v = 340 m/s. (b) 0.0068 m, which is 0.68 cm. Rearrange v = λ f to λ = v/ f with v = 340 m/s and f = 50 · 103 Hz. (c) Similar to the size of a typical moth body. 2.25 For all parts, rearrange v = λ f to λ = v/ f and use v = 3 · 108 m/s and the given frequency. (a) Radio, 3.16 m, from λ = (3 · 108 m/s)/(94.9 · 106 Hz). (b) Microwave, 0.12 m or 12 cm, from λ = (3 · 108 m/s)/(2450 · 106 Hz). (c) Microwave, 0.12 m or 12 cm, from λ = (3 · 108 m/s)/(2.45 · 109 Hz). It’s the same as a microwave oven. (d) Infrared, 9.40 · 10−7 m or 940 nm, from λ = (3 · 108 m/s)/(3.19 · 1014 Hz). 2.27 (a) 88 m. Rearrange v = λ f to λ = v/ f and use v = 1500 m/s and f = 17 Hz. (b) 1.0 hours. Rearrange v = d/t to t = d/v and use d = 5500 · 103 m and v = 1500 m/s to get t = 3667 s. Divide by 60 to get minutes and another 60 to get hours. 2.29 3.0 · 109 Hz. Rearrange v = λ f to f = v/λ and use v = 3 · 108 m/s and λ = 10 · 10−2 m. 2.31 0.24 s. Rearrange v = d/t to t = d/v and use v = 3 · 108 m/s and d = 72000 · 103 m; this distance is doubled because the signal has to go to the satellite and back. (b) 5.9 s. Use the same t = d/v equation but now v is the speed of sound, 340 m/s, and d = 2 · 103 m. 2.33 This is identical to the car analogy that was introduced in the chapter but now with actual cars, so the same equation applies. Rearrange v = λ f to λ = v/ f and use v = 55 mph 2.35 (a) Transverse, horizontal or in-plane. (b) Longitudinal, no polarization. (c) Transverse, vertical or out-of-plane. (d) None, water waves combine transverse (vertically polarized) and longitudinal motions, meaning a combination of pictures II and III. (e) S-wave for I, P-wave for II, and S-wave for III.
Chapter 3 3.1 c. 3.3 b, d. 3.5 e. 3.7 c, e. 3.9 (a) See Figure 3.3. (b) and (c) See Figure 3.6. 3.11 (a) See Figure 3.14. (b) Thicker. 3.13 (a) At any of the nodes, which are the bridge ends and the center of the bridge. (b) At either antinode, which are 1/4 and 3/4 of the way along the bridge. 3.15 (a) See Figure 3.19. (b) Not go in the room. These low notes have waves that are much larger than the doorway, so they don’t go through the doorway appreciably. (c) Go in the room and fill it. These medium notes have wavelengths that are similar to the door width, so they diffract strongly. (d) Go in the room and travel straight across it. The high notes have much shorter wavelengths than the door width, so they go through the door but do not diffract strongly. 3.17 (a) 130 cm, which is 1.30 m. This is twice the length of the guitar string. (b) 143 m/s. Use v = λ f , where λ = 1.30 m and f = 110 Hz. (c) 220 Hz. Use f = v/λ, where v is unchanged and λ is half of its prior value. (d) 58 cm. Use λ = v/ f , where v is unchanged and f = 123 Hz. Then divide by 2 because the string length is half of the wavelength from the equation λ = 2L/n. 3.19 (a) 3.0 m. The fundamental mode
Appendix F: Answers to Odd-Numbered Problems
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wavelength is twice as long as the cavity, from λ = 2L/n with n = 1. (b) 113 Hz. Use f = v/λ. Plug in v = 340 m/s and λ = 3.0 m.
+ =
3.21 3.23 (1) It would shine light rays in a different direction than the incoming light waves, so their destructive interference would only be at a very few points at best. Those points could conceivably be at some object’s surface but that doesn’t help. It needs to be at both of your eyes (and everyone else’s eyes) all at once for an object to appear dark. (2) It has no way to detect the incoming light waves. (3) Even if it could detect the incoming light waves, it couldn’t measure the incoming light wavelength, phase, and polarization accurately enough.
Chapter 4 4.1 e. 4.3 (a) Sound waves in air drive oscillations in wine glass vibrations. (b) Strong coupling (increased primarily by better positioning of speaker and glass). (c) Weak damping (sound waves cannot break cheap wine glasses because they have too much damping). 4.5 It does not change. String waves are 1-dimensional so they don’t spread out, so the energy density stays the same. 4.7 Damping. The walls absorb sound over a broad frequency range, which only occurs with damping.
4.11 (a) Energy. (b) 4.3 · 1010 J, using 1 kW = 1000 J, 1 hr = 4.9 3600 s, and 1 W s = 1 J. (c) 1400 W, by dividing the annual energy by 3.15 · 107 s per year. 4.13 100 times weaker. The sun’s power spreads out as it moves away from the sun. The area that this power goes through increases as the square of the distance from the sun (e.g. the surface area of a sphere is 4πr 2 ), so increasing the distance by a factor of 10 increases the area by a factor of 102 . 4.15 (a) 0.30 W/m2 . Divide 15 W = 0.30 W/m2 . (b) 159 W/m2 . the power by the sphere surface area, ρ3D = 4π(2 m)2 −3
2·10 W 2 Divide the power by the illuminated area, ρ = π(2·10 −3 m)2 = 159 W/m . (c) The laser pointer. 4.17 (a) Green or blue-green. The spectrum peak is at slightly less than 500 nm. (b) Radiation is similar at all visible wavelengths, so sunlight looks white. (c) About: 0% at 300 nm, 70% at 500 nm, 0% at 1400 nm, and 100% at 1600 nm. The transmission coefficient is the ratio of the light at the Earth’s surface to the light above the atmosphere. 4.19 (a) Blue. The green, blue, and violet colors are reflected strongly, while the red, orange, and yellows are not, producing an overall blue color. (b) Red, orange, and yellow. This absorption turns out to arise from methane in Uranus’s atmosphere. (c) 50%. Draw vertical lines at 380 and 740 nm to delineate the visible region, then a horizontal line at the height where there is as much area (corresponding to reflected visible light energy) in the visible portion above the line as below it; this height is at about 50%.
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Chapter 5 5.1 d. 5.3 b. 5.3 (a) 40,706 Hz. Use Eq. 5.1 or 5.2 using vo = 6.0 m/s and c = 340 m/s. (b) 41,412 Hz. The Doppler effect gets doubled due to the echo. 5.7 (a) Red. (b) Emits at f s = 4.572 · 1014 Hz and observed at f o = 4.540 · 1014 Hz. Convert from wavelength to frequency with f = λc , where c = 3 · 108 m/s. (c) 2.1 · 106 m/s. Use Eq. 5.2 and solve for v to get v = c fsf . Here, f = 4.572 · 1014 Hz − 4.540 · 1014 Hz = 0.032 · 1014 Hz. Note the significant figures. Both frequencies were computed to four significant figures but the numbers were close enough to each other that only the last two of them were left after the subtraction. As a result, the answer is most correctly given with just two significant figures. This problem can also be solved with Eq. 5.3. 5.9 (a) No sound. The plane is supersonic and she is ahead of the shock wave. (b) 54 Hz. Use Eq. 5.3 with f s = 120 Hz, c = 340 m/s and vs = 415 m/s. 5.11 275 Hz. The 28,000 km/s receding speed is 2.8 · 107 m/s, which is 9.33% the speed of light. This is large enough to use the relativistic Doppler shift equation, Eq. 5.7. Here, β = −0.0933 and f s = 250 Hz. 5.13 (a) 415 Hz. Rearrange Eq. 5.3 to ffos = 1 ∓ vcs . Defining f a = 520 Hz and fr = 345 Hz as approaching and receding speeds, their sum is
fs fa
+
fs fr
= 2, from which f s = 415 Hz. (b) 68.6 m/s.
Rearrange Eq. 5.3 some more to get vs = c(1 − ffas ) = 68.6 m/s. 5.15 (a) 220 Hz. They are not moving relative to each other. (b) 220 Hz. Again, they are not moving relative to each other. (c) 2.01 m. This is a moving source problem (Figure 5.4 and Eq. 5.3), where Alex is moving in the medium in the direction that is away from Sal at vs = 103 m/s and c = 340 m/s. An observer who is stationary in the medium would hear Alex at 169 Hz, which is λ = 2.01 m; Sal is moving but measures the same wavelength. (d) 1.08 m. This is the same as part c but with the opposite sign. (e) Alex would not hear Sal at all because the wind speed is faster than the speed of sound, so no waves would propagate to Alex. 5.17 (a) 1.25 Hz. Use f = T1 . (b) 0.25 Hz. Subtract the camera frequency from the rope frequency. (c) 2.25 Hz. Add the camera and rope frequencies. (d) f o = f s − f c . (e) The camera is rotating faster than the string, making the video of the string appear to rotate backward.
Chapter 6 6.1 c. 6.3 a,d. 6.5 c,d. 6.7 (a) No. The harmonics are integer multiples of the fundamental frequency, not integer octaves. For example, the 3rd harmonic is between 1 and 2 octaves above the fundamental. (b) The 4th harmonic. 6.9 (a) Inside. These are capillary waves, in which long waves are slow and short waves are fast. (b) Outside. These are gravity waves, in which short waves are slow and long waves are fast. 6.11 (a) All the odd harmonics: 1, 3, 5, 7, etc. This is because all even harmonics have a node at the center, whereas all odd harmonics have an antinode at the center. (b) 3:1, 5:1, 7:1, etc. The frequencies increase linearly for standing waves on a string, so the n’th harmonic has frequency n times faster than the fundamental, which is a ratio of n:1. In this case, only the odd harmonics are amplified, so only those ratios are listed.
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(c) Reasonably consonant. The numbers in the ratios are reasonably small. 6.13 (a)
gλ 2.79 m/s. Use the deep water wave speed, v = 2π with g = 9.8 m/s2 and λ = 5 m. (b) 12.49 m/s, using the same equation. Big ships can go very fast. (c) 0.39 m/s, again with same equation. Length strongly affects duck swimming. Longer water birds (e.g. mergansers, cormorants, and geese) generally swim faster than short ducks (e.g. coots). When short ducks need to swim fast, they usually skim over the water, rather than swimming through the water √ (watch ducklings sometime). (d) 0.99 m/s. Use the shallow water wave speed, v = gd. Boats can’t go fast in shallow water. 6.15 (a) 12 s. From the definition of velocity, use t = dv , where d = 70 km and v = 6 km/s. (b) 6 km. This is from λ = vf , where v = 6 km/s and f = 1 Hz. (c) 32 m/s (64 mph). The tsunami wavelength is much √longer than the water depth, so use the shallow water wave speed equation, v = gd, where g = 9.8 m/s2 and d = 100 m. (d) 2200 s, which is 36 minutes. Use t = dv , where d = 70 km and v = 32 m/s. 6.17 (a) 0.386 m. The pipe has a node at each end, so λ = 2L for the fundamen. tal mode. Convert to frequency with vsound = λ f and rearrange to get L = vsound 2f γk B T Here, f = 440 Hz. (b) 0.317 m. The speed of sound equation is vsound = m
where γ = 1.4, k B = 1.38 · 10−23 m2 kg s−2 K−1 , m = 4.8 · 10−26 kg, and T = 193 K. From these, the speed of sound is 279 m/s. Plug this into the equation for the Hz frequency as before. pipe length from part (a), L = vsound 2 f , using the same 440 5 BT 6.19 (a) 8360 m/s. Use the speed of sound equation, v = γk m air , plugging in γ = 3 , k B = 1.38 · 10−23 m2 kg s−2 K−1 , T = 5778 K, and m air = 1.90 · 10−27 kg. Ignore the pressure. (b) About 25 times faster. Divide 8360 m/s by 340 m/s. 6.21 (a) Short waves are attenuated more than long waves. Also, long waves tend to diffract more around a person’s head. (b) 0.06 waves. Compute wavelength with λ = vf = 3.4 m/wave, using v = 340 m/s and f = 100 Hz. Dividing 0.2 m by 3.1 m/wave gives 0.06 waves. (c) 2.9 waves. Use the same equation but with f = 5000 Hz. (d) When the phase difference is only part of a wave, it’s easy to tell which ear is receiving it first. However, when the difference is multiple waves, it’s hard to match up waves heard with one ear with those heard by the other ear. Also, faster waves take faster signal processing. 6.23 15,800 times more energy. The√ magnitude difference here is 2.8 and a magnitude difference of√1 corresponds to 1000 times more energy, so a magnitude difference of 2.8 is ( 1000)2.8 = 15, 800 times more energy. 6.25 nv , exactly as for as open-ended pipe. The boundary conditions are different f = 2L from an open ended pipe, but the fundamental mode still represents a half wave, and likewise for higher harmonics. Thus, the natural frequencies are the same.
Chapter 7 7.1 b,c,e. 7.3 No. The sun is not a point light source, so your shadow spreads out and turns fuzzier at increasing distances. The Moon is so far away that your shadow is completely indistinguishable by that point. 7.5 Always in syzygy. They have to be lined up for an eclipse to happen. 7.7 The astronaut would view a new earth, meaning
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that the Earth is completely dark. Also, the Earth would block the astronaut’s view of the Sun, essentially creating a solar eclipse. 7.9 (a) 49.6◦ . Rearrange the equation h h m ◦ to θ = arctan soblique = arctan 1.00 tan θ = soblique 0.85 m = 49.6 . (b) 8.7 m. Rearrange
h the equation tan θ = soblique to h = soblique tan θ = (7.4 m) tan 49.6◦ = 8.7 m. 7.11 (a) A circle. (b) An ellipse. (c) The oblique projection. It has the same width as the orthographic projection, but is longer. 7.13 (a) Make the light source smaller than the ball. (b) Make the light source larger than the ball. (c) The penumbra radius is always larger than the ball’s radius. (d) Impossible; the penumbra cannot be smaller than the ft)(5 ft) ball. 7.15 (a) 720 ft. Rearrange the equation hdii = hdoo to do = hho di i = (3 0.25/12 ft = 720 ft) = 48 ft. (b) 48 feet. Rearrange the equation hdii = hdoo to h o = dodhi i = (720 ft)(4/12 5 ft ft. 7.17 It will make the image slightly blurrier than normal because the light has more ways to get to any individual point on the film. Also, each point source of light, such as a star, will appear as a tiny square.
Chapter 8 8.1 e. 8.3 b. 8.5 (a) Wall, paper, rock, table top, etc. (b) Mirror, water surface, glass surface, etc. 8.7 (a) Concave. (b) Convex; (c) Flat. (d) Concave. (e) Flat, concave, and convex. (f) Concave. (g) Concave and convex. 8.9 (a) 270 m. (b) On the ground, 270 m south of the base of the building. (c) Real image. (d) Closer to the building. To see this, draw a picture and notice that as the sun moves higher in the sky, the angle of incidence for each ray that hits the mirror increases; this means that the angle of reflection increases as well, causing the rays to hit the ground closer to the building. 8.11 (a) 80 cm. Her eyes are 160 cm above the floor; drawing a picture shows that the light that goes from her feet to her eyes must reflect off the mirror at a position that is half the height between her feet and eyes, which is 80 cm above the floor (formally, this result arises from the law of reflection, which asserts equal incident and reflected angles, and then use of similar triangles shows that the distances are equal below and above the reflection point). (b) 165 cm. This is halfway between her eyes the top of her head. (c) 85 cm. Note that this is exactly half of the woman’s height. (d) No. We didn’t use the distance to the mirror to find the answer, which implies that F
C
(b) 15 cm. From the mirror equation, it doesn’t affect the answer. 8.13 (a) 1 1 1 1 = − . (c) − . Use the magnification equation, M = − ddoi . (d) Yes. 8.15 (a) di f do 2 √ 10.79 ns. The light travels distance 1 m to the wall and then 5 m to you, using the Pythagorean theorem, which is√ 3.23 m. Divide by the speed of light to get the time. (b) 9.43 ns. The light travels 2 2 m which is 2.83 m; then divide by the speed of light. (c) 10.79 ns. This is the same as part a. (d) No. See Figure 8.23. (e) Point B. By symmetry, reflection at point B obeys the law of reflection, whereas reflection elsewhere does not. 8.17 The following diagram shows two parallel rays that start at distance x from a plane and reflect off a parabolic mirror and go to the focus. From the geometric definition of a parabola, length a = a and b = b , so each ray
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has length x from its source to the focus. This is true for all parallel rays, so they a
a'
x–a
b' b
x–b
. 8.19 For the upper mirror, f = 10 cm and all get redirected to the focus. do = 10 cm − 1 cm = 9 cm. From Eq. 8.6, di = −90 cm, so the image is 90 cm above the top of the upper mirror. From Eq. 8.7, its magnification is M = 10. For the lower mirror, f = 10 cm, the object distance is do = 10 cm + 90 cm = 100 cm, where the 10 cm arose from the change of mirror location and the sign on 90 cm changed due to the different mirror orientation. From the equations, di = 11.1 cm and M = 0.111. The image is real and its position is 11.1 cm above the lower mirror, making it float in space just above the back of the upper mirror. Its total magnification is M = 10 × 0.111 = 1.1, so it’s just slightly larger than the original object. 8.21 (a) It is fuzzy because the sun isn’t a point source but covers some area in the sky, so rays from different parts of the sun have shadow edges at different locations (see Chapter 7). (b) Smaller than the actual sun. Convex mirrors provide a wide field of view, so everything in the view has to appear smaller. This can also be shown using a diagram which shows that the sun’s (virtual) image is at the mirror focus; it appears the same size as the actual sun when seen from the mirror surface, but appears smaller when seen from farther away. (c) Sharper edge than a direct shadow from the sun because the image of the sun is smaller. 8.23 (a) 2 images; one from the left mirror (L) and one from the right mirror (R). (b) 2 images; each hits both mirrors, but in the opposite order, so they are LR and RL. (c) 1 image; the light reflects three times as RLR or LRL, but both options put the image in the same location. (d) Infinite. Every image can reflect off another mirror to create yet more images. 8.25 If the water were perfectly smooth, the sun’s reflection would appear as a dot, at the location that obeys the law of reflection between the sun’s position and the angle to our eyes. Here, the water has small ripples which make it slightly not flat. A slight tilt of the water surface redirects light by twice that amount, by the law of reflection, causing the sun’s reflection to spread out. This spread has an equal angle in all directions, because experience shows that it’s independent of the wave direction, so one might expect it to appear as a circle. However, there is a lot more water surface in a given field of view when considering the water in the direction to the sun, rather than perpendicular to this direction, so the water on this axis has more chances to get the necessary tilt angles. As a result, the water in the direction to the sun appears brightest.
Chapter 9 9.1 c. 9.3 (a) Yes. (b) At the critical angle, which is about 48.8◦ . (c) No. (d) Yes. (e) Yes, but only the jellyfish has a strong reflection; it’s angle is shallow enough for total internal reflection, whereas the fish’s reflection on the surface is minimal. 9.5 Hot air has a lower refractive index than cold air. Patches of air with these different refractive indices causes light rays to shift around by small amounts, causing objects to appear to
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. 9.9 (a) 22.2◦ . Rearrange move, or to shimmer. 9.7 (a) n 1 sin θ1 = n 2 sin θ2 . (b) Snell’s law to solve for the angle of refraction, θ2 = arcsin( nn 21 sin θ1 ), and plug in n 1 = 1.00, n 2 = 1.52, and θ1 = 35◦ . (b) 25.6◦ . Use the same equation, but now n 1 = 1.52, n 2 = 1.33, and θ1 = 22.2◦ . (c) 25.5◦ . Again, use the same equation, now with n 1 = 1.00, n 2 = 1.33, and θ1 = 35◦ . Note that the answers for the last two parts are the same. 9.11 (a) Yes. If you look in with a grazing ray going upward, your view will refract to the critical angle, which is 48.8◦ from the normal for the front face. This ray has an angle of 41.2◦ from the normal for the top of the water. That’s less than the critical angle, so the view escapes through the top. (b) 61.2◦ . A grazing ray from the top enters the water with the critical angle of 48.8◦ , which is an incident angle 41.2◦ relative to the front face. Use Snell’s law with n 1 = 1.33, more than the n 2 = 1.00, and θ1 = 41.2◦ . 9.13 (a) 1.25 m/s. The depth is much
wavelength, so use the deep water wave speed equation, vdeep ≈ (1.56 m/s2 )λ with λ = 1 m. (b) 1.25 Hz. Use f = λv = (1.25 m/s)/(1 m). (c) 0.49 m/s. The water is much shallower than the wavelength, so use the shallow water equation, vshallow = gd = (9.8 m/s)(0.05 m). (d) 2.55. Divide 1.25 m/s by 0.49 m/s. (e) 0.39 m. Use λ = v ◦ f = (0.49 m/s)/(1.25 Hz). (f) 19.9 . Rearrange Snell’s law, n 1 sin θ1 = n 2 sin θ2 1 . (b) xi = to θ2 = arcsin( nn 21 sin θ1 ) = arcsin( 2.55 sin 60◦ ). 9.15 (a) −2.5 cm and h i = 2 cm. Use the thin lens equation, x1i = 1f + x1o , with f = −5 cm and xo = −5 cm to get the location. Use the lens magnification equation, M = xxoi to get the magnification. The result is 0.5, which is multiplied by the 4 cm object height to get the image height. (c) Yes. 9.17 (a) 52 cm behind the lens. This can be solved with the thin lens equation, but it’s easier to recognize that the object is at the −2F point, so the image is at the 2F point. (b) Real. (c) 3.486 ns. Use t = dv . Path A has two segments in air of 51.5 cm, on each side of the lens, and one segment in glass of 1 cm. Using c = 2.998 · 108 m/s, the time spent in each air segment is 1.718 ns. The time in glass uses v = (2.998 · 108 m/s)/1.5 = 1.999 · 108 m/s and d = 1 cm to give t = 0.0500 ns. Add the segments to get 3.486
ns. (d) 3.485 ns. Path B is essentially fully in the air, and each segment has length (52 cm)2 + (5 cm)2 = 52.240 cm. The total length is 104.48 cm, and divide by the speed to light to get 3.485 ns. (e) They are the same, to within round-off error. Also, they should be the same due to Fermat’s principle of least time. 9.19 16.0 cm. The apparent depth equation, dd21 = nn 21 applies equally well to looking from water to air, but now the refractive indices need to be reversed. Thus, dd21 = 1.33 1 = 1.33, so the mayfly is 1.33 times higher than it appears. It appears at 12 cm, so it’s really at 16.0 cm.
Chapter 10 10.1 a. 10.3 d. 10.5 At night, people only see with their rod cells because these cells are generally more sensitive than cones. However, rod cells are insensitive to red light, so red things appear black at night. On the other hand, rods are sensitive to
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yellow light, so yellow fire engines show up well at night. 10.7 Any traffic signals with red in them, or, worse, red and green combined. Examples are traffic lights, stop signs, and red brake lights on a car. A single flashing light is particularly problematic because color-blind people can’t tell if it’s a flashing red or flashing yellow light. 10.9 (a) Red. (b) Red. (c) Black. (d) Red. 10.11 (a) Blue. (b) Black. (c) Green or black. 10.13 When the sun is overhead, there is relatively little air between us and it, so little light is blocked and the sun appears white. When the sun is close to the horizon, there is a lot of air between us and it, so much more cyan light is removed, leaving a red color. 10.15 Yes. It would work if the movie projector showed one image with red light the other with actual yellow light (e.g. 570 nm), as opposed to a mixture of red and green. Also, the 3D movie glasses would need relatively narrow band filters that transmitted only red for one eye and only yellow for the other eye. At this point, different images would be presented to the two eyes, so they would see the two images separately, without interference. The fact that both eyes use the same types of cones to see those images is unimportant.
Chapter 11 11.1 c. 11.3 e. 11.5 a. 11.7 a, b, c, d, e. 11.9 This is identical to Figure 11.7: 11.11 No. Looking directly at the sun is dangerous because it concentrates a great deal of energy on a small part of the eye’s retina, which then gets burned. Blocking UV light reduces the energy some, but a lot of visible light energy remains, still burning the retina. 11.13 (a) Temperature, mass, height, number of objects, energy, etc. (b) Wind strength and direction, uphill steepness and direction, vehicle speed and direction, electric field at some location, etc. (c) temperature map, disease prevalence map, height of a string as a function of string position, height of water surface as function of position, etc. (d) water flow map, wind speed and direction map, hill steepness and direction map, electric field, magnetic field, etc. 11.15 Red. This is because even more of the bluer components would be scattered away than with our current atmosphere, leaving just the redder components. 11.17 (a) Shortwave, FM radio, microwave, visible, gamma ray. (b) Gamma ray, visible, microwave, FM radio, shortwave. (c) Shortwave, FM radio, microwave, visible, gamma ray. 11.19 (a) Vertical. They need to block the horizontally polarized light to reduce glare. (b) No further effect. After the first pair of sunglasses, all light is vertically polarized, and all of this is transmitted by the second pair of sunglasses. 11.21 (a) 50%. The first polarizer blocks half of the light, and all of the remaining light goes through the second one. (b) 0%. The first polarizer blocks half of the light, and the second blocks the remaining light. (c) 12.5%. The first polarizer blocks half of the light. For the second, use Malus’s law with θ = 45◦ to give 50% transmission. The light emerges at 45◦ and then hits the third polarizer, which is rotated another 45◦ , so Malus’s law again gives 50% transmission. Multiply all three 50% values to give a total transmission of 12.5%. 11.23 (a) The following images show the electric fields in red and mag-
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netic fields in blue for an electromagnetic wave in a cavity, for 8 time points in a cycle. (b) Out of phase. (c) In the magnetic field. (d) In the electric field. 11.25 No, this doesn’t make sense (and also isn’t true). If you look at a tiny spot of the sky during the daytime, the blue light rays that you see in that spot were scattered by air molecules that are in the column of air that extends from your eyes and through that spot. Standing in the bottom of a well doesn’t change this, and doesn’t block these blue light rays. Thus, the sky appears just the same when standing in the bottom of a well as when standing outside.
Chapter 12 12.1 d, e, a, c, b. 12.3 b,c. 12.5 (a) Black. (b) Black. 12.7 Highly reflective materials couple less strongly to radiation, so they absorb less sunlight during the day and emit less radiation at night. This reduces temperature fluctuations, so less heating is needed at night and less cooling during the day. 12.9 The person with the space blanket will stay warmer because it reflects her radiation back to her, whereas the person in the garbage bag will get colder because the bag will radiate his heat away from him. 12.11 A person’s heat has to get out somehow if the person isn’t going to overheat. This heat typically results in infrared emission. There are a few ways to mask this. One is to use a reflective sheet only on the side facing the enemy, so heat can escape in the other direction. Alternatively, a thermally conductive sheet will disperse the heat over a larger area, so it’s not as hot and emits less radiation. Yet another approach is to scatter different warm objects around so that the enemy can’t tell which one is the person. 12.13 (a) 1473 K. Add 273 to convert from ◦ C to K. (b) 1.967 µm. Use Wien’s displacement law. (c) Infrared. (d) Shorter, from T dependence of Wien’s displacement law. 12.15 33.2 mm2 . Solve the StefanBoltzmann law for A to get A = σTP 4 and substitute in P = 100 W and T = 2700 K, which gives 3.32 · 10−5 m2 . Convert units using 106 mm2 = 1 m2 . 12.17 (a) 5 W. Multiply 100 W of incident radiation by vis. = 0.05. (b) 299 W. Use the Stefan-Boltzmann law, with T = 273 K, and multiply the result by I R = 0.95. (c) Cools off. 12.19 2.77 W. Use the Stefan-Boltzmann law with A = 0.017 m2 and T = 273 K to get the emission of 5.35 W. Repeat but with T = 303 K to get the absorption of 8.12 W. Take the difference. 12.21 (a) 6.03 · 10−8 K. Use the equation 23 30 kg. (b) 4.8 · 104 m, which is 48 given, T = 1.2·10 M , with M = M = 1.99 · 10 km. Use Wien’s displacement law. (c) 2945 m. Use the radius equation given, R = 1.48 · 10−27 M and the solar mass. (d) 2.05 · 10−29 W. Use the Stefan-Boltzmann law with the temperature from above and surface area of π R 2 = 2.73 · 107 m2 . 12.23 (a) Bluish-white from the color temperature scale in Figure 12.4. (b) 8.57 · 106 m. Rearrange the Stefan-Boltzmann law to A = σTP 4 and insert numbers to get A =
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9.23 · 1014 m2 . The surface area is A = 4π R 2 and solve for R. (c) 4.54 · 108 kg/m3 . 4 3 The white dwarf density is M V , where V is the volume and is equal to 3 π R = 21 3 3 −6 3 2.64 · 10 m . (d) 454 kg. Multiply the star density by 1 cm = 10 m . 12.25 (a) = 1 . (d) 342 W/m2 . 12.27 (a) 254 Avisible = π R 2E . (b) Atotal = 4π R 2E . (c) AAvisible total √ 4 K. Rearrange the Stefan-Boltzmann law to T = 4 P/Aσ and substitute in P = 239 W and A = 1 m2 . (b) -18◦ C. Subtract 273 to convert from K to ◦ C. (c) 11.4
. (b) µm. Use Wien’s displacement law with T = 254 K. 12.29 (a) The numbers calculated here are close to Figure 12.7 values for “Incoming solar radiation”, “Reflected solar radiation”, “Outgoing longwave radiation”, and “Surface radiation”. Other numbers differ. (c) Figure 12.7 includes several heat transfers to the atmosphere: incoming solar radiation absorption by the atmosphere, thermals, and evapotranspiration, which we ignored. Because of this larger heat transfer to the atmosphere, they also calculated a larger “Downwelling radiation”. 12.31 6V σT 4 /c. The inside surface of the teapot has area 4πr 2 . Substituting this into the StefanBoltzmann law, shows that the surface emits 4πr 2 σT 4 power. The light that is emitted travels distance 2r at the speed of light, c, so it spends 2rc time within the pot. Multiplying this time by the power gives the radiation energy within the pot, which is 8πr 3 σT 4 /c. To express this in terms of the volume, use V = 43 πr 3 , leading to the result 6V σT 4 /c. 12.33 (a) P2 = 2P1 . The top layer emits P1 upward, so it also emits P1 downward. To stay at steady state, it must absorb 2P1 from below, meaning that the 2nd layer emits 2P1 upward. (b) P3 = 4P1 . (c) Pn = 2(n−1) P1 . (d) 16548 W. Use the Stefan-Boltzmann law with T = 273 + 462 = 735 K. (e) 5.4 layers. Rearrange the Pn equation to give n − 1 = log2 PPn1 and substitute in numbers. The result is n − 1 = 4.4, so n = 5.4.
Chapter 13 13.1 d. 13.3 c. 13.5 b. 13.7 (a) 5.45 · 1014 Hz. Use the frequency-wavelength relation, c = λ f . (b) 3.61 · 10−19 J. Use the Planck-Einstein relation, E = h f . (c) Lower frequency. (d) Lower energy. 13.9 (a) The photoelectric effect is the emission of electrons from a material by light. (b) Einstein explained that electrons were ejected by individual photons, which was only possible if the photons had high enough energies, which supported the particle explanation for light. (c) Essentially all electronic light sensors are based on the photoelectric effect, including photomultiplier tubes, night-vision goggles, and digital cameras. 13.11 3.00 · 1010 W. The spacecraft speed needs to be 0.2 × 3 · 108 m/s = 6 · 107 m/s. At this speed, its momentum will be 120,000 kg m/s. Each photon has momentum h/λ = 6.25 · 10−29 kg m/s, but each one imparts twice this amount on the spacecraft because it bounces off and returns back toward Earth, so each photon imparts 1.25 · 10−28 kg m/s momentum. Dividing the spacecraft momentum by the photon momentum gives 9.60 · 1032 photons. Each photon has energy equal to E = hc/λ = 1.87 · 10−20 J, and multiplying by
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the number of photons gives total light energy of 1.80 · 1013 J. Divide this by 10 minutes, which is 600 seconds, to get 3.00 · 1010 W, which is the necessary laser power. 13.13 (a) 3.98 · 10−19 J. Use E = hc/λ. (b) 3.44 · 1021 photons. 1367 W is the same thing as 1367 J/s, meaning that 1367 J of energy is incident in 1 second. Divide total joules by joules per photon from before to get the number of photons, (1367 J)/(3.98 · 10−19 J/photon) = 3.44 · 1021 photons. (c) 1.33 · 10−27 kg m/s (these units are the same as J s/m). Use p = h/λ. (d) 4.56 · 10−6 kg m/s. Multiply the number of photons by the momentum per photon. (e) 7.2 · 10−6 kg m/s. Use p = mv with m as the ant’s mass and v as the ant’s velocity. (f) The ant. (g) The sunlight. Silvering the sidewalk would cause the sunlight to exert twice as much momentum, from both the photon hitting and the photon rebounding. This would double the light momentum to 9.12 · 10−6 kg m/s, which is more than the ant’s momentum. 13.15 (a) 2.47 · 1015 Hz. Rearrange the Planck-Einstein relation to f = E/h. 1 , where σt is the life(b) 5.0 · 107 Hz. Use the spectral width equation σ f = 4πσ t h = 3.3 · 10−26 time. You can also use the energy-time uncertainty to give σ E = 4πσ t J. Then, convert to frequency with the Planck-Einstein relation f = E/h. (c) 2 · 10−6 %. Divide the spectral width by the frequency and then multiply by 100%. (d) 13.17 (a) Larger coherence length. The wavelength is in the microwave region, so λ ∼ 10 cm, and the wave peaks are at least 8,611 km wide. (b) Yes, it is reasonable to say that each photon is 8,611 km across because the coherence length is a valid measure for photon size. 13.19 (a) 6L 2 σT 4 . Use the Stefan-Boltzmann equation, P = AσT 4 , d d where A = 6L 2 . (b) 2L 3c . Rearrange v = t and replace v by c to get t = c . Substi2L 3 4 tute in d = 3 . (c) 4L σT . Multiply the emission power with the photon lifetime. b (d) hcT b . Find the wavelength from Wien’s displacement law, λ = T and convert 4σT 3 b hc . Divide the 3 σT 4 b 3 σT 3 b energy in the box by the energy per photon to get photons = 4LhcT = 4L hc . 22 3 Then divide by the box volume to get the density. (f) 8.93 · 10 photons/m . Plug in
to energy with the Planck-Einstein relation, E =
hc λ
=
hcT b
. (e)
numbers. (g) 274 times more air molecules. Divide the air molecule density by the photon density. (h) 22.4 nm. Invert the photon density to get 1.12 · 10−23 m3 /photon and take the cube root to get 2.24 · 10−8 m. 13.21 It will have the energy of a blue photon. The Planck-Einstein equation, E = h f refers to the frequency that is detected, not the frequency that the photon was emitted at. When someone bumps into normal objects, the bump is bigger when going faster because of the higher relative speed. For photons, the relative speed is always the speed of light, but the bump is still bigger when going faster, now due to the Doppler shift.
Chapter 14 14.1 d. 14.3 a. 14.5 No, it’s not a valid concern. Fluorescent dyes only absorb ultraviolet light, and don’t emit it. Thus, their clothes don’t glow in the UV. 14.7 This is a
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475
duplicate of Figure 14.27: . (b) The metastable state. (c) Longer wavelength. 14.9 (a) 0.463 nm. The de Broglie wavelength is λ = hp and p = mv,
h . Plug in h = 6.626 · 10−34 J s, m = 9.11 · 10−31 kg, and v = 1570 · 103 so λ = mv m/s. (b) The wavelength is about twice as large as the atom separation. (c) The wavelength is longer than the atom separation, so each electron is spread out over many (b) (c) . (d) The hydrogen atom is atoms. 14.11 (a) bouncing back and forth, as in a classical oscillator. 14.13 (a) 656.3 nm, by plugging numbers into the Rydberg formula. (b) Red. (c) 121.5 nm. (d) 91.16 nm. (e) Ultraviolet. (f) No. This is the smallest possible n 1 and the largest possible n 2 , so the 1 − n12 difference is as large as possible. (g) 2.179 · 10−18 J, or 13.6 eV. Use the n2 1
2
photon energy E = hc . λ . 14.15 (a) This is a duplicate of Figure 14.17: The energy levels and n values are important for this problem but the transitions and energy values are not. (b) n = 3. From the Rydberg formula, if n 1 = 1 and n 2 = 3, then λ = 103 nm. (c) 656 nm, 121 nm, and 103 nm. Transitions from n = 3 are 3 → 2 and then 2 → 1, and 3 → 1, which give these wavelengths, from the Rydhn . The wavelengths for a particle in a box are λ = 2L berg formula. 14.17 (a) 2Lm n and the momentum for a particle is p = λh . Also, momentum is p = mv. Combine h hn these to get v = mλ = 2Lm . (b) 1.52 · 105 m/s. Plug numbers into the previous equa−34 tion with h = 6.626 · 10 J s, n = 1, L = 2.4 · 10−9 m, and m e = 9.109 · 10−31 6 kg. (c) 1.67 · 10 m/s. Plug in the same numbers but now n = 11. 14.19 (a) 4.69 µm, 2.34 µm, and 1.56 µm. Harmonic oscillator energies are E n = h f (n + 21 ), so E = h f (n 1 + 21 ) − h f (n 2 + 21 ) = h f n. The possible n values from n = 3 are 1, 2, and 3, for which the transition energies are 1, 2, and 3 times h f , and the transition frequencies are 1, 2, and 3 times f . Convert to wavelengths with λ = cf . (b) 4.69 µm, from part (a). 14.21 (a) 6.6 · 10−35 m/s. Rearrange the uncertainty princih h , to σ p ≥ 4πσ and plug in numbers to get σ p ≥ 2.63 · 10−35 kg m/s. ple, σx σ p ≥ 4π x Divide by the mass to get the velocity. (b) Yes. The uncertainty principle is an inequality, so uncertainties can be arbitrarily larger than the Heisenberg limit. In practice, rats often move at several meters per second. (c) No. The uncertainty principle states that the rat’s velocity is unknown to within 6.6 · 10−35 m/s, so Zoe can’t know that it’s moving slower than that. (d) Yes. The uncertainty principle describes a lack of prior knowledge. However, any particular measurement is allowed to result in a precise value, including a value that’s less than the velocity uncertainty (however, a velocity measurement that’s this precise would cause the position uncertainty to be larger than the room size, thereby potentially catapulting the rat out of the room). 14.23 They needed to assemble a laser cavity by adding a reflecting mirror at one end and a partially reflecting mirror at the other. They also needed a strong pump light source that was tuned to the GFP absorption wavelength. Other considerations include: is GFP efficient at converting absorbed light into fluorescence or is too much absorbed
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light lost to non-radiative energy loss (this efficiency is called quantum yield); is the metastable state sufficiently long-lived to achieve population inversion; are the cells sufficiently transparent in the emitted light wavelengths that laser light can be amplified without excessive losses; and is GFP a sufficiently stable protein that it won’t degrade excessively with repeated optical cycling.
Chapter 15 15.1 e. 15.3 c. 15.5 a, c, d, e. 15.7 196.4 W. Use the gravitational wave power equation, 32G 4 m 2 m 2 (m +m )
1 2 1 2 P= , plugging in G = 6.674 · 10−11 m3 kg−1 s−2 , c = 2.998 · 108 5c5 r 5 m/s, m Ear th = 5.792 · 1024 kg, m Sun = 1.9891 · 1031 kg, and r = 1.496 · 1011 m. 15.9 (a) 5.9316 · 10−3 m s−2 . This is the same as the last problem, but use the sun’s mass, 1.3275 · 1020 kg, and the Earth-sun distance, r = 1.496 · 1011 m. (b) 5.9321 · 10−3 m s−2 . Same approach but subtract the Earth’s radius. (c) 5.9311 · 10−3 m s−2 . Same approach but add the Earth’s radius. (d) 5 · 10−7 m s−2 . (e) The difference between the gravitational accelerations on the two sides of the Earth is about 4.4 times GM GM higher from the moon than from the sun. 15.11 (a) a = (r +d/2) 2 + (r −d/2)2 . Add up the gravitational fields from the two stars, accounting for their different distances. (b) a = 2 Gr M 2 . Again, add up the two fields, this time using distance r to both stars. (c)
a = d r
+
3G Md 2 . r4
3d 2 ), 2r 2
Approximate the first answer to a =
which simplifies to a = 3d 2 r2
2 GM (2 + 3d ). r2 r2
GM (1 − dr r2
+
3d 2 ) + Gr M 2 (1 + 2r 2
Subtract the second answer to
get a = Gr M and simplify. (d) This near field falls off as r −4 , which is much 2 · −1 faster than the r dependence of gravitational waves. 15.13 (a) Yes, the gravitational field changes periodically, so it would create gravitational waves. (b) A mixture of longitudinal and transverse. It’s fully transverse for an observer on the axis of rotation, who sees a circularly polarized wave. On the other hand, an observer in the plane of rotation, would see a longitudinal wave from the star moving closer and farther and a transverse wave from the star moving up and down. (c) See the previous answer. (d) 1 Hz. It’s the same as the star rotation period because the gravitational field changes at this frequency. The frequency needed to be doubled for two stars because they are symmetric about their center of mass, but this single star isn’t symmetric.
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Appendix A A.1 d. A.3 b. A.5 d. A.7 5 · 10−2 , 500 · 10−3 , 5, 0.5 · 102 . A.9 (a) 9.45 · 1015 . (b) 3.6 · 10−7 . A.11 (a) 16 ± 5. (b) √ 6 ± 5. A.13 (a) 16 ± 3.6. Add 11 and 5 to get 16. Compute the uncertainty as 32 + 22 . (b) 6 ± 3.6. Subtract 11 − 5 to get 6 and compute the uncertainty the same way as before.
Appendix B B.1 (a) 6437.4 km. Work: ? km = 1 km 1000 m
4000 miles 1
·
5280 ft 1 mile
= 6437.4 km. (b) 17,000 nm. Work: ? nm =
2.54 12 inch c m 1m cm · · · 100 1 ft 1 in 9 17 µm 10 nm · 1m · = 6 1 10 µm 1m
·
−1 106 MHz s 60 s · 1 17, 000 nm. (c) 8.4 · 1012 cycles/day. Work: ? cycles/day = 97 · 1 · 1 MHz min 24 60 min hr −1 −1 12 · 1 day = 8.4 · 10 day . B.3 (a) J. Work: E = h f = J s Hz = J s s = J. 1 hr (b) m. Work: f = ( d1o + d1i )−1 = (m−1 + m−1 )−1 = (m−1 )−1 = m. (c) m. Work:
λ= ? kg
g s−2 = ms−2 = m. B.5 7.3 · 1020 kg. Use 2π f 2 7.1·1017 3 1029 kg = 7.3 · 1020 ocean m · 1 = 1 · 1 1 ocean 3 m
the same unit conversion method: kg. B.7 1/x. B.9 x z/y.
Appendix C C.1 b, d. C.3 (a) y = x + 43. (b) y = x1 . (c) y =
5x 2 2 .
C.5 λ =
hc E.
Appendix D D.1 is √ a, d. D.3 (a) 12 m. Use the Pythagorean theorem to find that the hypotenuse 32 + 42 = 5, meaning 5 m, and then add up the sides to get 12 m. (b) 36.9◦ . Use any inverse trig function with the appropriate pair of sides. Using the two legs, for ◦ example, θ = arctan 43 m m = 36.9 .
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Figure Credits
Chapter 1 1.1 Used with permission from Gerry Cordon, 2009. 1.2 (L) Wikimedia Commons “Empedocles in Thomas Stanley History of Philosophy.jpg” from Thomas Stanley, 1655. Public domain. (R) SSA, 2018. 1.3 © “Superboy” cover, issue #8 ©DC Comics, May 1950. Used with permission. 1.4 (L) Cropped from Wikimedia Commons “Democritus2.jpg”. Public domain. (M) Alchetron “Kanada (philosopher)” CC BY-SA 3.0. (R) SSA, 2018. 1.5 (L) Wikimedia Commons “Ibn al-Haytham.png” CC BY-SA 4.0, created by user Sopianwar. Many internet sites assign this image to other Arab philosophers, including Ibn Sahl. However, this image was used on the 2003 series Iraqi 10,000 dinar note as Ibn Al-Haytham, suggesting that it really does depict him. (R) SSA, 2018; palm tree is from pixabay.com, CC0. 1.6 (L) Refraction by a block of acrylic, SSA, 2019. (M) Wikimedia Commons “Frans Hals - Portret van René Descartes.jpg”, Portrait of René Descartes after Frans Hals, c. 1649-1700, Public domain. (R) Wikimedia Commons “Christiaan Huygens-painting.jpeg”, Portrait of Christiaan Huygens by Caspar Netscher, 1671, Public domain. 1.7 Wikimedia Commons “GodfreyKneller-IsaacNewton-1689.jpg”, Portrait of Isaac Newton by Barrington Bramley, 1992, after a painting by Sir Godfrey Kneller, 1689, Public domain. 1.8 (L) Wikimedia Commons “LifeOfThomasYoung1855PeacockG.jpg”, frontispiece from The Life of Thomas Young, M.D., F.R.S., &tc. (1855) by George Peacock. Public domain. (R) Wikimedia Commons “Double slit interference.png”, showing double-slit interference of sunlight passing through two slits ∼ 1 cm long and ∼ 0.5 mm apart by Aleksandr Berdnikov, 2016. CC BY-SA 4.0. 1.9 (L) Wikimedia Commons “James-Clerk-Maxwell-1831-1879.jpg”. Public domain. (R) SSA, 2018. 1.10 (L) Public domain (from Kiel University webpage http://www.uni-kiel. de/grosse-forscher/index.php?nid=planck&lang=e). (R) SSA, 2020. 1.11 Wikimedia Commons “Einstein patentoffice.jpg” showing Einstein in 1904 or 1905, about when he was working on the photoelectric effect. Image was cropped from a photograph by Lucien Chavan. Public domain. 1.12 SSA, 2018. © Springer Nature Switzerland AG 2023 S. S. Andrews, Light and Waves, https://doi.org/10.1007/978-3-031-24097-1
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Chapter 2 2.1 Originally from Wikimedia Commons “1989 Loma Prieta earthquake seismogram.jpg” created by the United States Geological Survey, Public domain; modified by SSA, 2019 to improve image quality. 2.2 SSA, 2018. 2.3 SSA, 2018. 2.4 SSA, 2019. 2.5 SSA, 2019. 2.6 SSA, 2019. 2.7 SSA, 2019. 2.8 SSA, 2019. 2.9 SSA, 2018. 2.10 SSA, 2019. 2.11 SSA, 2019.
Chapter 3 3.1 Pixabay by user MabelAmber, 2016, Pixabay license (free for commercial use). 3.2 SSA, 2019. 3.3 SSA, 2018. 3.4 Wikimedia Commons “Ile de ré.JPG” by Michael Griffon taken at Phares des Baleines on Île de Ré, 2011, CC BY 3.0. 3.5 (L) Pixabay, uploaded 2015. Shows a Southeast Asian long-tail boat. Pixabay license (free for commercial use). (R) SSA, 2019. 3.6 SSA, 2019. 3.7 Pixabay, by user hpgruesen, 2013. CC0. 3.8 Wikimedia Commons “Tsunami by hokusai 19th century.jpg” by Katsushika Hokusai, c. 1830-1832, Public domain. 3.9 SSA, 2019. 3.10 SSA, 2018. 3.11 SSA, 2019. 3.12 SSA, 2018. Photographed at Gasworks Park, Seattle, WA. 3.13 SSA, 2019. 3.14 (L) SSA, 2019. (R) SSA, 2019. 3.15 (L) Max Pixel, “Ball Soapy Water Colorful Soap Bubble” CC0. (M) Pexels, “Purple, black, and brown cicada on green leaf” by Egor Kamelev, 2018. Pexels license (free for commercial use). (R) Cropped from image from Pixabay by user sbl0323, 2012, CC0. 3.16 (L) SSA, 2018, (R) Wikimedia Commons “Fabry-perot-natrium-d” by user StefanPohl, 2013. Public domain. 3.17 SSA, 2018. 3.18 From Google Earth showing March 25, 2018 image of São Martinho do Porto, Portugal, copyright 2019 DigitalGlobe; use is permitted by GoogleEarth terms of service. 3.19 SSA, 2019. 3.20 SSA, 2019. 3.21 SSA, 2018. 3.22 SSA, 2018. 3.23 SSA, 2019. 3.24 SSA, 2018. 3.25 ©Chris Jones, with permission; photograph was taken at some point before 1973. Image shows HeNe laser light diffraction around a steel ball bearing hanging from a magnetized needle. Picture was taken on 35 mm black and white film, by projecting the image directly on the film using a camera body without a lens. 3.26 (L) “Hubble’s New Shot of Proxima Centauri, our Nearest Neighbor” provided by ESA/Hubble and NASA without copyright. (R) Photograph of 1/15 scale model of Hubble Space Telescope, 1983, NASA image 8330773, provided by NASA without copyright. 3.27 (L) SSA, 2019. (R) Used with permission from Sergio Emilio Montúfar Codoñer, photographed at Plantario Ciudad de La Plata, Buenos Aires, Argentina, 2015. 3.28 (L) Pxhere, 2017, CC0. (M) Wikimedia commons “Anna’s hummingbird.jpg” showing male Anna’s hummingbird, by Robert McMorran, 2008, public domain. (R) Pixabay, 2015, CC0. 3.29 (L) PublicDomainPictures.net “Blue Morpho Butterfly” by Vera Kratochvil, CC0. (R) Wikimedia commons “Morpho sulkowskyi wings.jpg” from Radislav Potrailo et al. Nature Comm. 6:7959, 2015. CC BY 4.0. 3.30 (L) Wikimedia Commons “Eastern Bluebird-27527-2.jpg” by Ken Thomas, 2007, public domain. (M) Wikimedia Commons “Anodorhynchus hyacinthinus -Disney -Florida-8”, showing a Hyacinth Macaw, also known as Hyacinthine Macaw, at Disney’s Animal Kingdom Park, by
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Hank Gillette, 2007. CC BY-SA 3.0. (R) Wikimedia Commons “Peacock feathers closeup.jpg” by Alex Duarte, 2008, public domain.
Chapter 4 4.1 Pexels; photographed Jan 25, 2018 by George Desipriss in Crete, Greece, Public domain. 4.3 Shutterstock image 1006734022 by user UfaBizPhoto. 4.2 SSA, 2020. 4.4 SSA, 2020. 4.5 Data from Wikimedia “Fluorescent lighting spectrum peaks labelled”, CC BY-SA 3.0, redrawn by SSA, 2020. 4.6 SSA, 2020, using iPhone app “Spectrum”. 4.9 (L) From Behrooz, Wang, Gordaninejad, Smart Materials and Structures 23:045014, 2014. (R) SSA, 2020, assuming a 2550 K blackbody. 4.7 SSA, 2020; the spectrum represents the E2 note on a standard tuned guitar, plucked near the center of the string. 4.8 SSA, 2020. 4.10 Patent application “Protective beer bottle with high luminous transmittance”, 2009; not subject to copyright restrictions. 4.11 (L) From Shea L. Valley, Handbook of Geophysics and Space Environments, 1965, McGraw-Hill, New York, NY. (R) Wikimedia Commons “Atmosfaerisk spredning.png” by user Maksim, 2006, and improved by user GifTagger, 2014. Public domain. 4.12 SSA, 2020, with data from S. A. Prahl, Optical absorption spectrum of Beta-carotene in hexane (2012), http://omlc.ogi.edu/spectra/PhotochemCAD/data/ 041-abs.txt. 4.13 SSA, 2020. 4.14 SSA, 2020. 4.15 Wikimedia “Beta-carotin.svg” by user NEUROtiker, 2007. Public domain. 4.16 SSA, 2018. 4.17 SSA, 2018. 4.18 Wikimedia Commons “Image-Tacoma Narrows Bridge1”, a picture of the first Tacoma Narrows Bridge. CC BY-2.0. 4.19 pxhere.com, 2017, CC0. 4.20 SSA, 2020. 4.21 Pixabay, by user PIRO4D, 2016, Pixabay license, free for commercial use. Ex. 4.16 NOAA data from National Buoy Data Center station 51001 on April 27, 2020 at 1550Z, provided without copyright. Ex. 4.17 Wikimedia “Solar Spectrum.png” prepared by Robert A. Rohde, CC BY-SA 3.0. Ex. 4.19 Measured by Christophe Pellier, 2018, and downloaded from his blog “Planetary Astronomy”. The Uranus intensity, arising from sunlight reflected off Uranus, was divided by the intensity of star HD9986, which is very similar to the Sun, to create a reflectance spectrum. The y-axis was rescaled by SSA, 2020, so that the visible albedo would be about 0.5, in agreement with experiment.
Chapter 5 5.1 NASA and ESA “Hubble Ultra Deep Field 2014” provided without copyright. 5.2 Wikimedia Commons “Christian Doppler.jpg”, Public domain. 5.3 SSA, 2020. 5.4 SSA, 2018. 5.5 SSA, 2020. 5.6 Wikimedia, “Doppler ultrasound of systolic velocity (Vs), diastolic velocity (Vd), acceleration time (AoAT), systolic acceleration (Ao Accel) and resistive index (RI) of normal kidney” by Kristoffer Hansen, Michael Nielsen and Caroline Ewertsen, 2015, CC BY 4.0. 5.7 SSA, 2020. 5.8 SSA, 2020. 5.9 (L) SSA, 2018. (R) Cropped and rotated from Wikimedia Commons “Supersonic-bullet-shadowgram-Settles.tif” by user Settles1, 2014, CC BY-SA 3.0.
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5.10 (L) Cropped from pxhere image 811630, CC-0. (R) Cropped from pxhere image 1403931, CC-0. 5.11 Wikimedia Commons “Advanced Test Reactor.jpg” showing bright blue glowing fuel plates from the Idaho National Laboratory’s Advanced Test Reactor (ATR) core, submerged in water for cooling, 2009, Argonne National Laboratory. CC BY-SA 2.0.
Chapter 6 6.1 Pixabay by user brankee, 2016, Pixabay license (free for commercial use). 6.2 SSA, 2018. 6.3 SSA, 2018. 6.4 SSA, 2020. 6.5 SSA, 2020; sources: Wikimedia Commons “Animal hearing frequency range.svg”, Wikipedia “Perception of infrasound”. Pigeons: Heffner, Henry E., et al. Behavior research methods 45:383, 2013. Spiders (jumping spider): Shamble, Paul S., et al. Current Biology 26:2913, 2016. Moth (greater wax moth): Yong, Ed “Moth smashes ultrasound hearing records”, Nature News, May 8, 2013. 6.6 Wikimedia Commons “CRL Crown rump lengh 12 weeks ecografia Dr. Wolfgang Moroder.jpg”, showing an ultrasound image of the foetus at 12 weeks of pregnancy in a sagittal scan by user West ga obgyn, 2006. CC BY-SA 3.0. 6.7 SSA, 2020. Sound samples were downloaded from Philharmonia (philharmonia.co.uk) and then processed with Mathematica to yield waveforms and spectra. 6.8 SSA, 2020. 6.9 SSA, 2020. This table was computed assuming 12 tone equal temperament. 6.10 SSA, 2020. 6.11 SSA, 2020, using Mathematica demonstration “Normal Modes of a Circular Drum Head” by Adam Smith, 2011, and frequency data from “Acoustics and Vibration Animations” webpage by Daniel Russell, 1998. 6.12 Pixabay, 2016, CC0. 6.13 publicdomainpictures.net “Ripples On A Pond 3739” by Vince Mig, CC0. 6.14 SSA, 2020. 6.15 PublicDomainPictures.net “Breaking Waves 4” by Kevin Phillips, CC0. 6.16 PxHere, 2017, CC0. 6.19 SSA, 2018. 6.20 SSA, 2022. 6.17 (L) Wikimedia Commons,“Deep water wave after three periods.png” by user Kraaiennest, 2008. CC BY-SA 4.0. (R) Wikimedia Commons “Shallow water wave after three wave periods.gif” by user Kraaiennest, 2008. CC BY-SA 4.0. 6.18 SSA, 2020. 6.21 Wikimedia Commons “M2 tidal constituent.jpg” by R. Ray, NASA. Public domain. 6.22 NOAA data, made available at https://tidesandcurrents.noaa.gov. Not copyrighted. 6.23 Republished from Alfè, D., M. J. Gillan, and G. D. Price. “Temperature and composition of the Earth’s core” Contemporary Physics 48:63-80, 2007, with permission from Dario Alfè.
Chapter 7 7.1 Pixabay by user Nanananini, 2015. CC0. 7.2 PxHere by Magdalena Roeseler, 2017. CC BY 2.0. 7.3 SSA, 2022. 7.4 SSA, 2022. 7.5 SSA, 2022. 7.6 (L) SSA, 2022. (R) NASA “The Eclipse 2017 Umbra Viewed from Space”, image iss052e056122 (Aug. 21, 2017). Provided by NASA without copyrighted. 7.7 (L) NASA “2017 total solar eclipse”, showing a total solar eclipse on Monday, August 21, 2017 above Madras, Oregon. Provided by NASA without copyright. (M) ESA/NASA “Solar
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eclipse from the International Space Station” by Expedition 43 Flight Engineer Samantha Cristoforetti, March 20, 2015. Provided by NASA without copyright. (R) Wikimedia Commons “Annular solar eclipse 2012.jpg” by user Nakae, showing an annular solar eclipse, Tokyo, Japan, 2012. CC BY 2.0. 7.8 SSA, 2022. 7.9 NASA, “Total lunar eclipse” by Fred Espenak, 2000. provided by NASA without copyright. 7.10 SSA, 2022. 7.11 Originally from Athansius Kircher, Ars Magna Lucis Et Umbrae, 1645. Copyright expired. See http://www.essentialvermeer.com/camera_ obscura/co_one.html. 7.12 SSA, 2022. 7.13 SSA, 2017. Photographed just before total solar eclipse on August 21, 2017 in Corvallis, OR. 7.14 Top row: (1) Wikimedia “Euglena sp.jpg” showing Euglena cells collected in Wakefield, Quebec, by user Deuterostome, 2011. CC BY-SA 3.0. (2) Wikimedia “Dugesia subtentaculata 1.jpg” showing an asexual specimen of Dugesia subtentaculata from Santa Fe, Montseny, Catalonia by Eduard Solà, 2008. CC BY-SA 3.0. (3) Wikimedia “Nautilus pompilius (detail).jpg” showing a chambered nautilus at Pairi Daiza, Brugelette, Belgium, by Hans Hillewaert, 2008. CC BY-SA 4.0. (4) Wikimedia “Arca zebra (zebra ark) (San Salvador Island, Bahamas) 1 (16188350021).jpg” showing a zebra ark clam by James St. John, 2011. CC BY 2.0. Bottom row: SSA, 2022.
Chapter 8 8.1 From flickr.com by Dominic Alves “C-curve - Anish Kapoor”, 2009, CC BY 2.0. 8.2 SSA, 2018. 8.3 (L) Wikimedia Commons “Waves breaking on the sea wall at Teignmouth (0161).jpg” by user Nilfanion, 2013. CC BY-SA 4.0. (R) Cropped from “Stop Sign in a Shallow Lake” by Brent Hanson (USGS), 2009, published by the Dakota Water Science Center, showing flooding of the James River at the North Dakota-South Dakota state line. Public domain. 8.4 SSA, 2018. 8.5 SSA, 2019. 8.6 (A) Wikimedia Commons “Buoy seal.jpg” showing a sea lion on buoy number 14 in San Diego Bay by user RadicalBender. Public domain. (B) SSA, 2019. (C) Wikimedia Commons “Reflector on tribranch.jpg” by user Jensens, 2008. Public domain. (D) Retroreflector placed on moon by Apollo 14 astronauts. Provided by NASA without copyright. 8.7 SSA, 2019. 8.8 (L) SSA, 2018, (R) SSA, 2018. 8.9 SSA, 2018. 8.10 (L) SSA, 2018. (R) SSA, 2019. 8.11 (1) Wikimedia Commons “Satellite dish in Austria.JPG” by user High Contrast, 2011. CC BY 3.0 DE. (2) Wikimedia Commons “ParabolicMicrophone.jpg” by J. Glover, Atlanta, Georgia, 2007, showing a parabolic microphone being used during an American football game. CC BY-SA 3.0. (3) Wikimedia Commons “"Sungril" solar cooker photo1.jpg” by user Fluffyarse, 2011, showing a Sungril brand solar cooker. CC0 1.0. (4) pxhere.com, uploaded 2017. CC0. 8.12 SSA, 2018. 8.13 (L) SSA, 2019, (M) SSA, 2018. (R) SSA, 2018. 8.14 SSA, 2018. 8.15 SSA, 2018. 8.16 Wikimedia Commons “Ritchey-Chrétien.png” by user Tamasflex, 2011. CC BY-SA 3.0. 8.17 (L) SSA, 2019, showing a mirror in Griffith Park, Los Angeles. (R) SSA, 2018. 8.18 SSA, 2019. 8.19 MaxPixel, CC0. 8.20 SSA, 2018. 8.21 Modified from Wikimedia Commons “Cysteine ball-andstick.png” by user MarinaVladivostok, 2013. CC0 1.0. 8.22 SSA, 2019. 8.23 (L) SSA, 2019. (R) SSA, 2019. Ex. 8.8 Cropped from Wikimedia Commons “Denge acoustic
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mirrors -March2005.jpg” showing acoustic mirrors near Greatstone-on-Sea, Kent, Great Britain, by Paul Russon, 2005. CC BY-SA 2.0. Ex. 8.9 Wikimedia Commons “Walkie-Talkie - Sept 2015.jpg” by user Colin, 2015. CC BY-SA 4.0. Ex. 8.20 Icons from icon-icon.com. “Best free icons for personal and commercial use.” Ex. 8.24 Wikimedia Commons “Ggb in soap bubble 1.jpg” by user Brocken Inaglory, 2007. CC BY-SA 3.0. Ex. 8.25 Wikimedia Commons “Sun reflection at Lake Bowen.jpg” by user NqcRz, 2014. CC BY-SA 4.0. Ex. 8.26 Wikimedia Commons “Light pillars over Laramie Wyoming in winter night.jpg” by user Wyogold, 2013. CC BY-SA 3.0.
Chapter 9 9.1 Pexels, photograph by James Wheeler near Vancouver, Canada, 2013 using a 14.0 mm lens (a wide angle lens). CC0. 9.2 (L) SSA, 2019. (M) SSA, 2022 (R) SSA, 2022. 9.3 (L) Jeffrey Beale, from US highway 50 in Prowers County, Colorado, 2013. Downloaded from flickr. CC BY-SA 2.0. (R) SSA, 2022. 9.4 SSA, 2022. 9.5 SSA, 2022. 9.6 (L) SSA, 2022. (M) SSA, 2022. (R) Wikimedia Commons “US Navy 110607-N-XD935-191 Navy Diver 2nd Class Ryan Arnold, assigned to Mobile Diving and Salvage Unit 2, snorkels on the surface to monitor multi.jpg” by US Navy Mass Communication Specialist 1st Class Jayme Pastoric, 2011. Public Domain. 9.7 SSA, 2022. 9.8 Top, left to right: (1) SSA, 2022. (2) pxhere.com “Close-up of fiber optic, end glow cable” by user chonnes, 2020. CC0. (3) SSA, 2022. (4) Pixabay by user gr8effect, 2016. Pixabay license (free for commercial use). Bottom: SSA, 2022. 9.9 (L) Wikimedia Commons “Traffic sign (speed) during night.jpg” by user Konstantin, 2005. CC BY-SA 3.0. (M) SSA, 2022. (R) flickr by James St. John “Eyeshine in Felis catus (domestic cat) 1” 2020. CC BY 2.0. 9.10 (L) SSA, 2022. (R) Wikimedia Commons “Dissecting-the-Nanoscale-Distributions-and-Functions-ofMicrotubule-End-Binding-Proteins-EB1-and-ch-pone.0051442.s011.ogv”, which is a video. This figure shows a frame at 17.143 s of the “merge” portion of the video. From Movie S4 of Nakamura et al. “Dissecting the Nanoscale Distributions and Functions of Microtubule-End-Binding Proteins EB1 and ch-TOG in Interphase HeLa Cells” PLOS ONE 7:e51442, 2012. CC BY 3.0. 9.11 SSA, 2022. 9.12 SSA, 2022. 9.14 SSA, 2022. 9.15 SSA, 2022. 9.16 SSA, 2022. 9.17 SSA, 2022. 9.18 Cropped from stocksnap.io “Flower Bloom Free Stock Image” by user “Nature’s Beauty”. CC0. 9.19 Cropped from flickr “Person hält Apple iPhone 12 Pro Max in Pazifikblau mit drei Kameras und matter Rückseite vor weißem Hintergrund” by Marco Verch, 2020; annotations by SSA, 2022. 9.20 SSA, 2022. 9.21 SSA, 2022. 9.22 SSA, 2022. 9.23 Insect image is from MaxPixel. CC0. Rest is SSA, 2022. 9.24 SSA, 2022. 9.25 (L) Imaged at Dartmouth College electron microscope facility, directed by Max Guinel, on a FEI XL-30 FEG ESEM. Public domain. (R) Wikimedia file “Coronaviruses 004 lores.jpg” from the Center for Disease Control and Prevention Public Health Image Library #4814. Public domain. 9.26 (L) Wikimedia Commons “Prism rainbow schema.png” by user Joanjoc commonswiki, 2005. CC BY-SA 3.0. (R) Wikimedia Commons “Színszóródás prizmán2.jpg” by Zátonyi Sándor, 2003. CC BY-SA 3.0. 9.27 SSA, 2022. 9.28 SSA, 2022. 9.29 Wikimedia Commons “Rain-
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bow in Budapest.jpg” by user Takkk, 2006, showing a rainbow in Budapest taken with a 55 mm focal length lens. CC BY-SA 3.0. 9.30 SSA, 2022. 9.31 SSA, 2022. Ex. 9.3 All icons from icon-icon.com. “Best free icons for personal and commercial use.” Ex. 9.6 Cropped from Wikimedia Commons “Fata Morgana Example.jpg” taken near Brisbane, Australia by user Timpaananen, 2012. CC BY-SA 3.0. Ex. 9.13 SSA, 2022.
Chapter 10 10.1 Last Painter on Earth painted by James Doolin, 1983. Permission for use in this book granted by Lauren Doolin McMillen, 2018. 10.2 SSA, 2018. 10.3 SSA, 2018, using cone data from Stockman, A., D.I.A. MacLeod, and N.E. Johnson “Spectral sensitivities of the human cones” JOSA A 10 (1993): 2491 and rod data from Wyszecki, G., and Stiles, W. S. (1982). Color Science: concepts and methods, quantitative data and formulae (2nd ed.) Wiley:New York, Table I(4.3.2). 10.4 (L) SSA, 2018; (R) Wikimedia Commons “MICHAEL JACKSON, Afterimage.jpg” by Dimitri Parant, 2010, CC BY-SA 2.0. 10.5 Wikimedia Commons “Eight Ishihara charts for testing colour blindness, Europe Wellcome L0059161.jpg” from the Wellcome Image Library, file L0059161. CC BY 4.0. 10.6 Turner, P.L., and M.A. Mainster “Circadian photoreception: aging and the eye’s important role in systemic health” British Journal of Ophthalmology (2008). 10.7 (L) SSA, 2018, using CIE standard observer for human spectrum; used CIE standard observer Y and Z spectra for dog, but shifted Z to peak at 430 nm in agreement with Miller P.E. and C.J. Murphy (1995) “Vision in dogs” JAVMA 207:1623 and also expanded Z to give sensitivity down to about 315 nm. (M) Max Pixel “Golden retriever puppy in the pool” CC0. (R) SSA, 2018, using RGB data for same Max Pixel image; both R and G values were set to G/2, thus removing red and color balancing by shifting green to yellow. The more accurate approach of using the dog spectrum shown in the left panel yielded essentially the same result. 10.8 (L) Wikimedia Commons “Potentilla reptans sl1.jpg” by user Stefan.lefnaer at Donaupark, Vienna-Donaustadt, Austria, 2016, CC BY-SA 4.0. (R) Wikimedia Commons “Flower in UV light Potentilla reptans.jpg” by user Wiedehopf20, 2012, CC BY-SA 4.0. The visible light was excluded with the Baader U-Filter 2" (T=300-390nm), the so called Venus-Filter for astronomy. 10.9 (L) Newton, I. Opticks, 1704. From Book One, part II, proposition VI, p. 155. (R) SSA, 2018. 10.10 SSA, 2018. 10.11 SSA, 2018. 10.12 SSA, 2018. 10.13 SSA, 2018. 10.14 (L) Cropped from Wikimedia Commons “Gumballs (8739287019).jpg” by Ben CollinsSussman, 2013, CC BY-2.0. (M) SSA, 2018. (R) From Ocean Optics, Inc. website https://oceanoptics.com/application/food-beverage-quality-control/ “Reflected color of sugar-rich gumballs”; permission granted, 2018. 10.15 (L) From http://www. nwplants.com/business/catalog/pop_tre.html by Wallace Hansen, 2012, CC BY-SA 3.0. (M) Freerange Stock “Japanese Maple Bloodgood Wet Leaves” by Eric Yuen, Equalicense. (R) From http://www.nwplants.com/business/catalog/cor_sers.html by Wallace Hansen, 2012, CC BY-SA 3.0. 10.16 Wikimedia Commons “HSV color solid cylinder saturation gray.png” by user SharkD, 2010, CC BY-SA 3.0. 10.17 SSA,
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2018 using same data as in Fig. 10.3. 10.18 SSA, 2018 using data from Mathematica ChromaticityPlot. 10.19 SSA, 2018 using data from mathematica ChromaticityPlot.
Chapter 11 11.1 From pxhere.com, 2017. CC0. 11.2 (L) Wikimedia “Annual Average Temperature Map.png”, by Robert A. Rohde for Berkeley Earth, 2014. CC BY 4.0. (R) Wikimedia “Ocean surface currents.jpg” by NOAA, 2008. Public domain. 11.3 SSA, 2018. 11.4 SSA, 2018. 11.5 From Good Free Photos (https://www.goodfreephotos. com) “Green Northern Lights across the sky”, showing Slaven’s Roadhouse, Alaska. CC0. 11.6 Wikimedia Commons “Maxwell’s Laws.jpg” by Ad Meskens, 2010, showing a plaque on George Street, Edinburgh. CC BY-SA 3.0. 11.7 SSA, 2018. 11.8 SSA, 2020. 11.9 SSA, 2020. 11.10 SSA, 2020. 11.11 From freeimages.co.uk, “Communications tower”, free image. 11.12 Wikimedia Commons “UV Vis IR Portrait.jpg” by user Spigget, 2007. CC BY-SA 3.0. 11.13 Pixabay (https://pixabay. com), 2016. CC0. 11.14 SSA, 2020. 11.15 SSA, 2020. 11.16 (L) From Unsplash by Romain Dupraz showing Noville, Switzerland. Free image. (R) Used with permission from Malvern Panalytical “Laser Diffraction Particle Size Measurement of Food and Dairy Emulsions”, 2005. 11.17 Wikimedia Commons, “Peyto Lake-Banff NPCanada.jpg”, showing Peyto Lake, Banff National Park, Alberta, Canada, by Tobias Alt, 2005. GNU Free Documentation License v. 1.2. 11.18 (L,M) SSA, 2018. (R) Pixabay; sunset landscape in Phoenix, Arizona by user Samuriah, 2015. CC0. 11.19 SSA, 2020. 11.20 SSA, 2020. 11.21 SSA, 2020. 11.22 SSA, 2020. 11.23 SSA, 2020. 11.24 Used with permission from Bryan Palmintier, 2020. Photos taken at Fern Lake, Rocky Mountain National Park, CO, showing a greenback cutthroat trout. 11.25 SSA, 2020. 11.26 (L) SSA, 2020. (R) Shutterstock. “Cross Polarization” by Dave Turner, photo 72219862. ex. 11.22 Used with permsission from Bryan Palmintier, 2020. Photo taken on Yampa River, CO. ex. 11.24 Photographed by Mars river Spirit, May 19, 2005, from Gusev Crater. Image by NASA/JPL/Texas A&M/Cornell, Public domain.
Chapter 12 12.1 A false-color image of the Sun photographed at 30.4 nm by the Atmospheric Imaging Assembly of NASA’s Solar Dynamics Observatory, on August 19, 2010. Public domain. 12.2 Pexels, by Taafi Saft, 2017. Free for all uses. 12.3 SSA, 2020. 12.4 SSA, 2020. Colors were computed from integrating the Plank blackbody formula over the xyz color space and then converting to rgb values, following instructions by Simon Woods, 2014, posted at https://mathematica.stackexchange.com/ questions/57389/convert-spectral-distribution-to-rgb-color. Color temperature values for the lighting examples given here vary widely between authors, so attempts were made to choose representative values. See Wikipedia “Color temperature”. 12.5 SSA, 2021. 12.6 SSA, 2021. 12.7 From Trenberth, Kevin E. and John T. Fasullo,
Appendix G: Figure Credits
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“Tracking Earth’s energy: from El Niño to global warming” Surveys in geophysics 33:413-426, 2012. Used with permission. 12.8 SSA, 2021. CO2 data are from Lüthi, Dieter, Martine Le Floch, Bernhard Bereiter, Thomas Blunier, Jean-Marc Barnola, Urs Siegenthaler, Dominique Raynaud et al. “High-resolution carbon dioxide concentration record 650,000-800,000 years before present.” Nature 453:379-382, 2008, downloaded from ftp://ftp.ncdc.noaa.gov/pub/data/paleo/icecore/antarctica/epica_ domec/edc-co2-2008.txt. Temperature data are from Jouzel, Jean, Valérie MassonDelmotte, Olivier Cattani, Gabrielle Dreyfus, Sonia Falourd, Georg Hoffmann, Benedicte Minster et al. “Orbital and millennial Antarctic climate variability over the past 800,000 years” Science 317:793-796, 2007, downloaded from ftp://ftp.ncdc.noaa. gov/pub/data/paleo/icecore/antarctica/epica_domec/edc3deuttemp2007.txt. Ice age shading is based on temperatures above or below −5◦ C relative to present. HumanNeanderthal split is based on DNA evidence described in Wikipedia “Neanderthals” and elsewhere, which generally suggests dates between 500 and 300 thousand years ago, although there are also many studies that place this date back to 800 thousand years ago or even earlier. The “Dawn of civilization” is taken to be about 6000 years ago.
Chapter 13 13.1 J. Adam Fenster, University of Rochester, 2013. Used with permission from J.A. Fenster. 13.2 SSA, 2021. 13.3 SSA, 2021. 13.4 SSA, 2020. 13.5 (L) Wikimedia Commons “Photomultiplier-2 hg.jpg” by Hannes Grobe, 2020, CC BY-SA 4.0. (R) Wikimedia Commons “Photomultiplier schema en.png” by German user Benutzer:Jkrieger, 2007 and translation by user Dietzel65, 2014. Public Domain. 13.6 SSA, 2021. 13.7 (L) Wikimedia Commons “Crookes radiometer.jpg” by user Timeline, 2005. CC BY-SA 3.0. (M) SSA, 2021. (R) SSA, 2021. 13.8 Wikimedia Commons “IKAROS solar sail.jpg” by Andrzej Mirecki, 2011. CC BY-SA 3.0. 13.9 SSA, 2021. 13.10 SSA, 2021. 13.11 SSA, 2021. 13.12 Photograph cropped from Wikimedia Commons “Drone view of ocean waves (Unsplash).jpg” by Caleb Jones, 2016. CC0 1.0 Universal Public Domain Dedication. Additional fraphics by SSA, 2021. 13.13 SSA, 2021.
Chapter 14 14.1 Wikimedia “Neon Dragon at Museum of Neon Art.jpg” showing artwork on the exterior of the Museum of Neon Art in Glendale, CA, photographed by user AndrewKeenanRichardson, 2016. CC0. 14.2 SSA, 2021. Represents an electron with v = 116 Å/fs and wave packet width σ = 1 Å. 14.4 SSA, 2021. 14.3 SSA, 2021. Represents a particle with mass 0.122 dyg (deciyoctogram, which is 10−28 kg) moving at velocity 8.66 Å/fs and starting with a wave packet width of σ = 0.5 Å. The de Broglie wavelength is 0.628 Å. The spacing between the different time points is to scale. 14.5 SSA, 2021. Computed with Mathematica. Particles are electrons, with
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mass 0.009109 dyg, energies 12, 24, and 36 aJ (the 24 aJ electron has a deBroglie wavelength of 1 Å), and initial spread of σ = 2 Å. The barrier is 26 aJ high and 2 Å thick. The figure x-axis extends from −10 to 40, with the electrons starting at x = 0 and the barrier centered at x = 10. Initial wave functions are at time 0, intermediate ones at t = 0.15 fs, and final ones at t = 0.3 fs. 14.6 (L) SSA, 2021. Figure is not to scale. (M) SSA, 2021. Molecule drawn using https://molview.org. Figure is roughly to scale; see Nguyen, Gulacyk, Kreglewski, and Kleiner, Coordination Chemistry Reviews 436:213797, 2021. (R) Wikimedia “Silicium-atomes.png” by Guilllamue Baffou, 2005. CC BY-SA 3.0. 14.7 Wikimedia “Austentite ZADP.jpg” by user “Cm the p”, 2006. Public domain. Figure shows zone axis diffraction pattern of twinned austenite in steel (a face-centered cubic structure), taken with JEOL 200CX transmission electron microscope. 14.8 Wikimedia “Buckminsterfullerene-3D-balls.png” by user Jynto, 2011. CC0 1.0 Public domain. 14.9 Modified from Wikimedia “Betacarotin.svg” by user NEUROtiker, 2007. Public domain. 14.10 SSA, 2021. 14.11 SSA, 2021. 14.12 SSA, 2021. 14.13 SSA, 2021. 14.14 SSA, 2019. 14.15 SSA, 2021. Energy levels are drawn to scale. 14.16 SSA, 2021. All panels use the same scale. 14.17 (L) SSA, 2021. (R) Wikimedia “JF Lab Physics.png” by user Bradyd9, 2015. CC BY-SA 4.0. 14.18 Periodic table from Wikimedia “Simple Periodic Table Chartblocks.svg” by user Double sharp, 2021. CC BY-SA 4.0. Modified by SSA, 2021. 14.19 SSA, 2021. 14.20 SSA, 2021. 14.21 SSA, 2021. 14.22 SSA, 2021. 14.23 (L) SSA, 2021. (R) SSA, 2021. Absorption data are from Wikipedia “Fraunhofer Lines,” although line intensity is largely ignored here. 14.24 (L) SSA, 2021. (R) SSA, 2021. Emission data are from Hyperphysics at http://hyperphysics.phy-astr.gsu.edu/hbase/ quantum/atspect.html. Only medium and strong lines are shown, but brightness is not depicted otherwise. 14.25 SSA, 2021. 14.26 Left to right: (A) Wikimedia “Compact fluorescent lamp.jpg” by Levente Fulop, 2010. CC BY 2.0. (B) Pixabay, by Victoria Borodinova / 8971, 2019. Pixabay License (free for commercial use). (C) Wikimedia “GFP 1ema ribbon fluor.png” by user Dcrjsr, 2013. CC BY 3.0. (D) Wikimedia “C. elegans.jpg” by Dan Dickinson, Goldstein lab, UNC Chapel Hill, 2013. CC BY-SA 3.0. (E) Wikimedia “Cosmic Blue Tetra Fish” by Robert Kamalov, 2018. CC BYSA 4.0. 14.27 SSA, 2021. 14.28 SSA, 2021. 14.29 SSA, 2021. 14.30 SSA, 2021. 14.31 SSA, 2021. 14.32 SSA, 2021. 14.33 (L) Wikimedia “Liquid helium Rollin film.jpg” by Alfred Leitner, 1963. Public domain. (R) Wikimedia “Meissner effect p1390048.jpg” by Mai-Linh Doan, 2007. CC BY-SA 2.5.
Chapter 15 15.1 LIGO “Two black holes merge into one”, developed by Simulating eXtreme Spacetimes (SXS), 2016. Image available for any purpose without prior permission. 15.2 SSA, 2018. 15.3 SSA, 2018; Earth illustration from clipart-library.com, public domain. 15.4 Caltech/MIT/LIGO Lab, Cropped from “Gravitational Waves, As Einstein Predicted”, showing the signals of gravitational waves detected by the LIGO observatories at Livingston, Louisiana, and Hanford, Washington, 2016. Image available for any purpose without prior permission. 15.5 SSA, 2022. 15.6 SSA, 2022.
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15.7 SSA, 2018, partly drawn with Mathematica. 15.8 SSA, 2022. 15.9 Cropped from image provided by HEASARC/NASA, 2017, credited to LIGO, Virgo, Fermi, INTEGRAL, NASA/DOE, NSF, EGO, and ESA; provided by NASA without copyright. 15.10 SSA, 2022. 15.11 SSA, 2018. 15.12 (L) SSA, 2018. (R) Cropped from image provided by ESA, Hubble, and NASA “A horseshoe Einstein ring from Hubble” showing gravitational lensing by galaxy LRG 3-757. Provided by NASA without copyright. 15.13 SSA, 2018. 15.14 Designed by SSA, 2018, background is “Hubble’s Cosmic Fireflies” by NASA, showing a cluster of galaxies called Abell 2163, provided by NASA without copyright. 15.15 SSA, 2018. 15.16 LIGO “LIGO Hanford”, 2008. Image available for any purpose without prior permission. 15.17 Wikimedia “Gravitational-wave detector sensitivities and astrophysical gravitational-wave sources.png” uploaded 2013. Created from software described in Moore, Cole, and Berry, “Gravitational-wave sensitivity curves” Classical and Quantum Gravity 32: 015014 (2015). CC BY-SA 1.0.
Appendix D D.1 to D.5 SSA, 2022.
Appendix F All figures SSA, 2022.
Appendix H H.1, H.2, H.3, H.4, H.5, H.6, H.7, H.8, H.9, H.10, H.11, H.12, H.13, H.14 and H.15 SSA, 2022. H.16 Wikimedia “Simple Periodic Table Chart-blocks.svg” by user Double sharp, 2021. CC BY-SA 4.0.
List of abbreviations (L) (M) (R) SSA CC0 CC BY 4.0 CC BY-SA 4.0
left panel middle panel right panel copyright by Steven S. Andrews Creative Commons No Rights Reserved Creative Commons Attribution 4.0 Creative Commons Attribution ShareAlike 4.0 International
H
Useful Facts and Figures
Table H.1 Fundamental constants Constant
Symbol
Approximate value
Speed of light in vacuum Elementary charge Planck’s constant Gravitational constant Electric permittivity of space Electron mass Proton mass Rydberg constant Boltmann’s constant Stefan-Boltzmann constant Wien’s displacement constant Avagadro’s number
c e h G
0 me mp R∞ kB σ b NA
2.998 · 108 m/s 1.602 · 10−19 C 6.626 · 10−34 J s 6.674 · 10−11 m3 kg−1 s−2 8.854 · 10−12 C2 N−1 m2 9.109 · 10−31 kg 1.673 · 10−27 kg 1.097 · 107 m−1 1.381 · 10−23 J/K 5.670 · 10−8 W m−2 K−4 2.898 · 10−3 m K 6.022 · 1023 mol−1
© Springer Nature Switzerland AG 2023 S. S. Andrews, Light and Waves, https://doi.org/10.1007/978-3-031-24097-1
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Table H.2 More useful numbers Constant Speed of sound in air Speed of sound in fresh water Speed of sound in seawater Density of water Gravitational acceleration Sun: Mass Radius Surface temperature Earth: Mass Radius
Symbol
Value
Notes
340 m/s 1481 m/s
Typical range is 330-350 m/s At 20◦ C
1500 m/s
Typical range is 1480-1540 m/s
g
1000 kg m−3 9.80 m s−2
At 4◦ C Average for Earth’s surface
M R
1.989 · 1030 kg 6.96 · 108 m
T M⊕ R⊕
5778 K 5.972 · 1024 kg 6.37 · 106 m
Moon: Mass Radius
7.348 · 1022 kg 1.74 · 106 m
Earth-Sun distance Earth-Moon distance
1.496 · 1011 m 3.844 · 108 m
Radius is larger at equator than poles The interior is much hotter Radius is larger at equator than poles Radius is larger at equator than poles Between object centers Between object centers
Table H.3 Refractive indices (from Table 2.2) medium
n
Vacuum Air Ice Water PMMA (Acrylic) Glass (crown & window glass) Sapphire Cubic zirconia (fake diamond) Diamond
1 (by definition) 1.0003 1.31 1.33 1.49 1.52 1.77 2.16 2.42
Appendix H: Useful Facts and Figures
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Table H.4 Greek alphabet Name
U.C.
L.C.
Name
U.C.
L.C.
Alpha Beta Gamma Delta Epsilon Zeta Eta Theta Iota Kappa Lambda Mu
A B E Z H I K
M
α β γ δ ε ζ η ϑ ι κ λ μ
Nu Xi Omicron Pi Rho Sigma Tau Upsilon Phi Chi Psi Omega
N O P T ϒ X
ν ξ o π ρ σ τ υ ϕ χ ψ ω
Table H.5 Metric base and derived units (from Table B.1) Quantity
Dimension
Unit
Symbol
In base units
Length Mass Time Frequency Energy Power Force Pressure
L M T T −1 L 2 M T −2 L 2 M T −3 L M T −2 L −1 M T −2
meter kilogram second hertz joule watt newton pascal
m kg s Hz J W N Pa
m kg s s−1 kg m2 s−2 kg m2 s−3 kg m s−2 kg m−1 s−2
Table H.6 Metric prefixes (from Table B.2) Prefix
Abbreviation
Value
tera giga mega kilo centi milli micro nano pico
T G M k c m µ n p
1012 109 106 103 10−2 10−3 10−6 10−9 10−12
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Appendix H: Useful Facts and Figures
Figure H.1 Optics timeline (from Figure 1.12).
Figure H.2 Wave terminology (from Figure 2.3). wavelength, λ nodes
antinodes
Figure H.3 Standing waves (from Figure 3.11).
Figure H.4 Electromagnetic waves (from Figure 11.7).
Figure H.5 Visible spectrum (from Figure 2.4).
Appendix H: Useful Facts and Figures
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Figure H.6 Electromagnetic spectrum (from Figure 11.10). Figure H.7 Scientific pitch notation (from Figure 6.8).
Table H.7 Musical note frequencies (from Figure 6.9) Note
Hz
Note
Hz
Note
Hz
C1
32.7
C2
Hz
65.4 C3
Note
130.8 C4
Hz
Note
261.6 C5
Hz
Note
523.3 C6
Hz
Note
1046.5
C7
2093.0
C 1
34.6
C 2
69.3 C 3
138.6 C 4
277.2 C 5
554.4 C 6
1108.7
C 7
2217.5
D1
36.7
D2
73.4 D3
146.8 D4
293.7 D5
587.3 D6
1174.7
D7
2349.3
D 1
38.9
D 2
77.8 D 3
155.6 D 4
311.1 D 5
622.3 D 6
1244.5
D 7
2489.0
E1
41.2
E2
82.4 E3
164.8 E4
329.6 E5
659.3 E6
1318.5
E7
2637.0 2793.8
F1
43.7
F2
87.3 F3
174.6 F4
349.2 F5
698.5 F6
1396.9
F7
F 1
46.2
F 2
92.5 F 3
185.0 F 4
370.0 F 5
740.0 F 6
1480.0
F 7
2960.0
G1
49.0
G2
98.0 G3
196.0 G4
392.0 G5
784.0 G6
1568.0
G7
3136.0
G 1
51.9
G 2
103.8 G 3
207.7 G 4
415.3 G 5
830.6 G 6
1661.2
G 7
3322.4
A1
55.0
A2
110.0 A3
220.0 A4
440.0 A5
880.0 A6
1760.0
A7
3520.0
A 1
58.3
A 2
116.5 A 3
233.1 A 4
466.2 A 5
932.3 A 6
1864.7
A 7
3729.3
B1
61.7
B2
123.5 B3
246.9 B4
493.9 B5
987.8 B6
1975.5
B7
3951.1
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Appendix H: Useful Facts and Figures
wave speed, v (m/s)
frequency, f (Hz) gravity waves capillary waves
v=
2πγ ρλ
v=
gλ 2π
minimum speed λ = 1.7 cm v = 0.23 m/s
wavelength, λ (m) Figure H.8 Deep water wave speed (from Figure 6.14).
Figure H.9 Concave mirrors (from Figure 8.15). Figure H.10 Convex mirrors (from Figure 8.17). F
Figure H.11 Lens principal rays (from Figure 9.14).
C
Appendix H: Useful Facts and Figures
497
Figure H.12 Color models (from Figures 10.9, 10.10, and 10.12).
blue scattering
scattered color scattering type
Rayleigh
red Tyndall
Mie
white scattering geometric
scattering amount
1 10-2 10-4 10-6 10-8 10-10
light wavelengths blue: 450 nm red: 650 nm
1 nm
10 nm
100 nm particle diameter
Figure H.13 Scattering (from Figure 11.14).
Figure H.14 Color temperature (from Figure 12.4).
1 μm
10 μm
498
Figure H.15 Blackbody spectra (from Figure 12.3).
Figure H.16 Periodic table.
Appendix H: Useful Facts and Figures
Index
A440 (music), 135 Aberration chromatic, 237 solar, 419 stellar, 28, 419 Absolute square, 360 Absolute temperature, 130 Absorption, 50, 384 Absorption coefficient, 86 Absorption spectrum, 86 Accommodation (eye), 231 Achromat, 237 Achromatic lens, 237 Acoustic Doppler velocimetry, 105 Active sonar, 132 Adding zero, 449 Additive color mixing, 258 Aerophones, 138 Aether, 20, 57 Afterimage, 253 Air density, 129 molecule mass, 130 pressure, 129 Airy disk, 66 Airy, Sir George Biddell, 66 al-Haytham, Ibn, 5 Albedo, 316 Algebra, 447 © Springer Nature Switzerland AG 2023 S. S. Andrews, Light and Waves, https://doi.org/10.1007/978-3-031-24097-1
Algerian scale, 136 Aluminum foil, 302 AM radio, 283 Amino acid, 201 Ammonia, 367 Amplitude, 18 Anechoic chamber, 98 Angle of incidence, 214 Angle of refraction, 214 Angular Doppler effect, 120 Angular magnification, 235 Angular momentum, 415 Animal hearing ranges, 131 Ant, 354 Antarctica, 321 Antenna, 89 Anterior chamber, 230 Antibonding orbital, 382 Anti-gravity, 407 Antinode, 51 Antireflective coatings, 56 Antisolar point, 239 Antumbra, 171 Aperture, 227 Aphrodite, 3 Apochromat, 238 Apparent depth, 216, 245 Arago, Dominique-François-Jean, 65 Arago-Poisson spot, 64, 70 499
500
Arca zebra clam, 177 Area, 455 Aristotle, 3, 257 Aspen, 263 Associative property, 448 Atmosphere spectrum, 85 Atmosphere (unit), 129 Atmospheric phenomena corona, 66 light pillar, 210 rainbow, 238 Atomic orbital, 377 Atomism, 4 Attenuation, 95 Aurora, 277 Average speed, 24 Babinet, Jacques, 65 Babinet’s principle, 65, 71 Back radiation, 317 Balmer series, 385 Balmer, Johann, 379 Banjo, 137 Base units, 440 Bat, 40, 119, 120, 131 Bathroom shower, 73 Battle ropes, 78 Beam splitter, 57 Beating pattern, 49, 112 Beatles (music group), 136 Bees, 296 Beetle, 67 Bell, 139 Bell, John, 394 Betelgeuse, 310, 355 Big Bang, 108, 325, 426 Binary star, 102, 107, 411, 429 Binoculars, 219, 235 Bioluminescent, 388 Bird vision, 255 Birefringence, 298 Blackbody, 309 Blackbody radiation, 309 Blackbody spectrum, 309, 333
Index
Black hole inspiral, 409, 414 temperature, 325 Black light, 285, 387 Bloch, Felix, 361 Blood cells, 236 Blood color, 385 Blood moon, 171 Bluebird, eastern, 68 Blue jay, 68 Blueshift, 106 gravitational, 108 Bluetooth, 283 Bohr, Niels, 345, 361 Boltzmann, Ludwig, 312, 332 Boltzmann’s constant, 130 Bonding orbital, 382, 401 Born, Max, 345, 368 Bose-Einstein condensate, 396 Bose, Satyendra, 396 Boson, 396 Boundary, 50 Boundary condition, 50, 52, 138, 371 Bradley, James, 28 Brass instruments, 139 Breakthrough starshot, 353 Brewster’s angle, 220, 297 Brewster, Sir David, 220, 297 Brix, 244 Brushed metal, 302 Buckminsterfullerene, 370 Buckyball, 370 Buffalo, NY, 152 Bulbous bow, 48 Bunson, Robert, 93 Butadiene, 402 Butane, 399 Buttercup, 67 Butterfly, 68, 72, 256 Calcite, 298 Calcium carbonate, 298 Calculator advice, 432 Calorie, 77 Cambrian explosion, 177
Index
Camera, 227 depth of field, 228 digital sensor, 227, 338 f-number, 229 film, 227 focal plane, 228 lens, 228 obscura, 173 pinhole, 173 smartphone, 229 Camera obscura, 173 Camera-type eyes, 230 Canary Islands, 394 Capillary waves, 140 Car headlights, 206 Car thermal emission, 314 Carbon dioxide, 319, 413 Carbonic acid, 321 Carbon monoxide, 402 Carotene, 86, 89, 250, 346 matter waves, 371 spectrum, 86 Cartesian form, 225 Cartesian space, 265 Casein micelle, 289 Cat, 255 Cataracts, 254 Cat’s eye retroreflector, 219 Cavity, 52, 371 polarization, 304 Cell blood, 236 cone, 250 euglena, 176 lymphocyte, 236 photoreceptor, 250 rod, 250 Cell phone, 283 Celsius (unit), 130 Center of mass, 411 Central maximum, 61 Central Processing Unit (CPU), 40 Chambered nautilus, 176 Chameleon, 255 Charge-Coupled Device (CCD), 338
501
Chemical bond, 381 Cherenkov, Pavel, 116 Cherenkov radiation, 116 Cherry tree, 263 Chile, 161 Chiral, 201, 299 Chlorophyll, 75, 97 Chord, 137 Chordophones, 137 Chrétien, Henri, 196 Christmas laser lights, 63 Chromatic aberration, 237 Chromaticity diagram, 266 Chromatic scale, 134 Chu, Steven, 343 CIE standard observer, 266 CIE xy color space, 266 Ciliary muscles, 231 Circular birefringence, 299 Circular dichroism, 89 Circular motion, 148 Circular polarization, 298 Circumference, 455 Clam arca zebra, 177 Clapotis, 52 Clarinet, 138 Cloud droplet, 287 CMYK color model, 260 Coherence length, 347, 389 Collapse (quantum), 345, 391 Collective excitation, 350 Color, 249 complementary, 259 primary, 258 secondary, 258 tertiary, 258 Color blind, 253 Color blind glasses, 254 Color model, 258 CMYK, 260 HSV or HSB, 264 LMS, 265 RGB, 258 RYB, 262
502
XYZ, 266 Color space, 265 Color temperature, 311, 322 Color wheel, 257 Coma, 196 Combining like terms, 449 Commutative property, 449 Compact Disc (CD), 63 Complementary colors, 259 Complementary Metal Oxide Semiconductor (CMOS), 338 Complex conjugate, 360 Complex numbers, 360 Compound eyes, 256 Compton, Arthur, 338 Compton scattering, 338 Cone cell, 250 Conjugated π system, 373, 384 Consonance, 135 Constellation Canis Major, 324 Canis Minor, 325 Orion, 118, 310, 312 Constructive interference, 47 Continental drift, 153 Continuous spectra, 82 Converging lens, 221 Cool roof initiative, 323 Cooper pair, 396 Coot, 142 Copenhagen interpretation, 345, 392 Cornea, 230 Corn syrup, 201 Corona, 66 Coronavirus, 236 Cosine, 454 Cosmic microwave background, 325 Cosmological redshift, 108 Coupling, 88 matter waves, 383 radiative, 315 Creeping cinquefoil, 256 Crest, 18 Crete, Greece, 75 Critical angle, 217
Index
Crooke’s radiometer, 341 Crossed polarizers, 292 Crown glass, 238 Crust, 152 Cubic zirconia, 244 Current ripples, 149 Customary units, 440 Cymbal, 139 da Vinci, Leonardo, 3 Damping, 95 Dan bau, 137 Dark adaptation, 252 Darwin, Charles, 314 Davisson, Clinton, 368 de Broglie frequency, 358, 401 de Broglie relations, 358 relativistic, 359 de Broglie wavelength, 358, 400 de Broglie, Louis, 10, 358 Decay constant, 349 Decibels, 82 Decimal places, 433 Deep water waves, 141 Deer vision, 255, 270 Democritus, 4 Department of Energy, 323 Depth of field, 228 Derived units, 440 Descartes, René, 6, 214 Destructive interference, 47 Detroit, MI, 152 Deuterium, 367, 369 Diamond, 40, 219 Didgeridoo, 138 Diffraction, 7, 58, 283 and interference, 60 Babinet’s principle, 65 electron, 368 grating, 62, 68 holes and obstacles, 58 photon, 346 pinhole camera, 174 spikes, 65 X-ray, 63
Index
Diffraction grating, 62, 68, 368 Diffraction-limited resolution, 234, 235 Diffraction spikes, 65 Diffuse reflection, 184 Digit term, 431 Diopter, 232 Dipole, 413 Dipole radiation, 413 Disappearing filament pyrometer, 313 Dispersion, 53, 236 matter waves, 362 normal vs. anomalous, 237 Dispersive, 141 Dissonance, 136 Distributive property, 449 Diverging lens, 221 Division by zero, 450 Djembe, 139 DNA, 338, 343 Dog vision, 255, 271 Dogwood, 263 Dolphin vision, 255 Doolin, James, 249 Doppler cooling, 343 Doppler effect, 101, 102 acoustic velocimetry, 105 angular, 120 binary star, 415 blueshift, 106 broadening, 119 cooling, 343 echolocation, 120 general equation, 111 moving observer, 103, 108 moving source, 103, 110 photon, 355 power, 113 radar, 105 redshift, 106 reflection, 111 relativistic, 114 shift, 109 supersonic, 114 traffic analogy, 104, 110 ultrasound, 105
503
Doppler, Christian, 102, 107 Doppler shift, 109 Double-pane window, 220 Double rainbow, 239 Double-slit experiment, 60, 344 Draupner wave, 48 Drum, 139 Earth albedo, 316, 326 average temperature, 317 climate, 316 energy budget, 316, 326 global warming, 319, 327 greenhouse effect, 318, 326 orbit, 418 phases, 172 structure, 152 Earthquake, 153 epicenter, 23 focus, 23 hypocenter, 23 p-waves, 23 s-waves, 23 Echo Doppler effect, 111 Echolocation, 40, 131, 158 Doppler effect, 120 Eclipse, 170 lunar, 171 solar, 170 Eddington, Arthur, 420 Eddy, 148 Edinburgh, Scotland, 278 Egypt, 5 Einstein, Albert, 9, 21, 26, 58, 76, 116, 333, 345, 359, 360, 393, 397, 414, 416, 418, 419 Einstein ring, 421 El Chichón volcano, 178 Electric field, 275 Electric guitar, 137, 159 Electromagnet, 277 Electromagnetic waves, 20 discovery, 8
504
gamma rays, 286 infrared, 284 long radio, 282 microwave, 283 polarization, 291 radio, 282 skipping, 283 ultraviolet, 284 visible light, 284 X-rays, 285 Electron, 275 Electron avalanche, 337 Electron cloud, 377 Electron microscope, 69, 235, 402 Electron-volt, 368 Elephant, 131 Elliot, Barney, 92 Ellipse, 208 Emission, 385 Emissivity, 315 Emmetropization, 231 Empedocles, 3 Energy, 76, 359 attenuation, 95 definition, 76 density, 79 harmonic oscillator, 375 hydrogen orbitals, 378 kinetic, 77, 359 levels, 372 mass-energy, 76 particle in a box, 372 photon, 335 potential, 77, 359 thermal, 77 units, 77 wave, 76, 78 zero-point, 372 Energy-time uncertainty, 349, 389 Energy transfer abrupt, 94 between energy types, 76 damping, 95 irreversible, 77 non-resonant, 94
Index
resonance, 87 reversible, 93 English Channel, 206 Enologist, 243 Entanglement, 393 Epicenter, 23, 154 EPR paradox, 393 EPTA, 426 Equal temperament tuning, 161 Equation, 448 Equilibrium position, 125 Equivalence principle, 419 Euclid, 3, 185 Euglena, 176 European Pulsar Timing Array, 426 Evanescent wave, 220, 366, 375 Everything distributes, 450 Evil eye, 3 Exoplanet, 106 Exponential term, 431 Exponents, 451 Expression (math), 448 Extramission theory, 2 Eye accommodation, 231 anterior chamber, 230, 250 arca zebra clam, 177 blue color, 290 chambered nautilus, 176 ciliary muscles, 231 cornea, 230, 250 evolution, 177 fovea, 251, 255 iris, 230, 252 lens, 230, 250 planaria, 176 pupil, 230, 252 retina, 230, 250 signaling, 250 Eyepiece lens, 233, 234 Eye spot, 176 Factoring, 449 Fahrenheit (unit), 130 Far field
Index
electromagnetic, 280 gravitational, 417 Fata Morgana, 243 Federal Communications Commission, 283 Feedback near field, 281 negative, 343 positive, 93, 149, 318, 320 Fermat, Pierre de, 202 Fermat’s principle, 202, 239, 421 Fermion, 396 Fermi velocity, 401 Feynman, Richard, 203, 413 Fiber optics, 31, 284 Field, 274 electric, 275 far, 280, 417 gravitational, 408 magnetic, 276 near, 279, 417 radiation, 280 scalar, 274 vector, 274, 408 Firefly, 336, 388 First harmonic, 53 First overtone, 53 Fission, 366 Fizeau, Hippolyte, 28 Flash Boys (book), 30 Flashdark, 74 Flint glass, 238 Flow separation, 149 Fluorescence, 386, 400 lifetime, 387 Fluorine molecule, 384 Flute, 54, 138 FM radio, 283 f-number, 229 Focal length, 191 pinhole camera, 175 Focal plane, 228 Focus, 189 Folly, 180 Food calories, 77
505
Foosball, 200 Force, 125, 127 gravitational, 406 measurement, 343 photon, 341 reaction, 182 restoring, 125 Fourier, Jean-Baptiste Joseph, 46 Fourier transform, 46, 348 Fovea, 251 Frankland, Edward, 386 Fraunhofer lines, 94, 107 Fraunhofer, Joseph von, 93 Free particle, 361 Frequency, 32, 33 Frequency downshift, 149 Frequency-wavelength relationship, 33 Freshman’s dream, 450 Fresnel, Augustin-Jean, 64, 219 Fresnel equations, 220 Fringes, 60 Front running, 31 Frosted glass, 286 Fruit fly, 256 Frustrated total internal reflection, 220, 366 Fuller, Buckminster, 370 Fundamental frequency, 53 Fundamental mode, 53 Fusion, 366 FWHM, 349 Galaxy GN-z11, 27, 106 NGC 1357, 119 Galilean telescope, 235 Galileo Galilei, 28, 235, 406 Gamma rays, 286 Gamut, 267 Gasoline, 399 Gaussian form, 225 Geodesic dome, 370 Geometric scattering, 287 Geometry, 453 Geostationary orbit, 27
506
Index
Germanium, 40 Greenhouse gas, 317 Germer, Lester, 368 Greenland, 321 Ginko, 263 Ground state, 373 Glacial lakes, 290 Group velocity, 145, 362 Global warming, 319, 327 Guitar, 73, 74, 137, 351 Glow-in-the-dark, 388 spectrum, 83 Glucose, 201, 299 Gnat, 340 Haidinger’s brush, 296 Golden Gate Bridge, 209 Hall of Mirrors, 182 Grackle, 68 Hard boundary, 50 Gravitational Hard X-ray, 285 acceleration, 142, 409 Harmonic oscillator, 374, 401, 402 blueshift, 108 Harmonica, 138 constant, 407 Harmonics, 53 field, 408 Harp, 137 force, 406 Harriot, Thomas, 214 lensing, 405, 421 Haze, 288 mass, 419 Heat capacity ratio, 129 particles, 350 Heat radiation, 284 redshift, 108 Heisenberg uncertainty principle, 390, waves, 21, 41, 350, 405, 410 400, 403 Gravitational waves, 405, 410 Heisenberg, Werner, 345, 390 background, 426 Heliocentric model, 406 behaviors, 411 Helium, 369 detection, 423 liquid, 396 energy, 413 spectrum, 385 frequency, 411 Helium atom, 383 momentum, 415 Heptatonic, 136 near and far field, 417 Hero of Alexandria, 3, 202 polarization, 412, 429 Hertz (unit), 34 power, 415 Hertz, Heinrich, 8, 34, 278 properties, 411 Hokusai, Katsushika, 49 spectrum, 425 Holocene extinction event, 321 speed, 416 Honeybee, 256 Hooke, Robert, 6 Graviton, 350, 416 House fly, 256 Gravity, 406, 420 HSB model, 264 Gravity waves, 141 HSV model, 264 Grazing incidence, 216 Hubble Space Telescope, 66, 101, 106, Grazing reflection, 185 Great Wave off Kanagawa, The, 49 196 Greece, 134 eXtreme Deep Field, 101 Greeks, ancient, 2 Hubble, Edwin, 106 Green Fluorescent Protein (GFP), 387, Hückel, Erich, 361 403 Hue, 264 Greenhouse effect, 318, 326 Hull speed, 159
Index
Hulse, Russell, 415 Human hair, 65, 73 Hummingbird, 67 Huygens, Christiaan, 6, 59 Huygens’s principle, 59, 240, 421 Hydrocarbons, 399 Hydrogen atom, 377 discharge tube, 379 molecule, 383 spectrum, 385 Hydrogen chloride, 374 Hypersonic weapon, 325 Hypocenter, 23, 153 Hypotenuse, 454 Icarus, 342 Ice age, 321 Ideal gas law, 130 Idiophone, 139 IKAROS, 342 Image, 186 convex mirror, 197 distance, 188 inverted, 192 magnification, 188, 195 magnified, 187 pinhole camera, 174 real, 192, 223 reduced, 187 upright, 188 virtual, 187, 216, 224 Image stabilization electronic, 230 optical, 230 Incidence angle of, 214 grazing, 216 normal, 216 Incident angle, 184 Incident ray, 184 Incident waves, 50, 182 India, 4, 137 Inertia, 125 Inertial mass, 419
507
Inferior mirage, 213 Inflation phase (universe), 108 In focus, 228 Infrared radiation, 284 Infrared thermometer, 313 Infrasound, 131 Inner core, 152 In phase, 47 Inspiral, 410 Intensity spectrum, 81 Interference, 7, 54 constructive, 47 destructive, 47 double-slit, 60 electron, 368 Fermat’s principle, 203, 240 filters, 56 molecule, 370 photon, 345, 346 single-slit, 63 structural coloration, 67 thin-film, 54, 67 Interference filters, 56 Interferometer, 57, 346, 424 Interglacial period, 321 Intergovernmental Panel on Climate Change, 320 International Pulsar Timing Array, 426 International Space Station, 170 International Telecommunication Union, 282 Interstellar movie, 422 Intromission theory, 3 Inverse trigonometry functions, 455 Invisible light, 20 Io, 28 Ion, 285, 379 Ionization energy, 379 Ionized state, 379 Ionizing radiation, 285 Ionosphere, 283 IPTA, 426 Iran, 136 Iridescent feathers, 68 Iris, 230
508
Iron oxide, 385 Irreversible process, 77 Ishihara chart, 254 Ishihara, Shinobu, 254 Islamic Golden Age, 5 Isotope, 319
Index
Law of universal gravitation, 406 LCD, 294, 342 Leaf color, 263 Lens, 221 achromatic, 237 apochromat, 238 aspheric, 222 converging, 221 James Webb Space Telescope, 66, 196 coordinates, 222 Janssen, Jules, 386 cylindrical, 221 Japan, 137 diverging, 221 Japanese maple, 263 electron, 236 Japan tsunami, 150, 159 erecting, 235 Jeans, James, 333 eyepiece, 233, 234 Joule (unit), 77 focal length, 222 Joule, James Prescott, 77 focus, 222 Jump rope, 52 multiple, 232 Jupiter, 28, 179 objective, 233, 234 Io, 28 principal rays, 223 spherical, 221 Kaleidoscope, 209 thin, 221 Kanada, 4 thin lens equation, 225 Kapoor, Anish, 181 Lewis, Michael, 30 Kazoo, 139 Light Keck Observatory, 196 invisible, 20 Kelvin (unit), 130 perception, 36 Kelvin wake pattern, 145 rays, 3 Kinetic energy, 77 refractive index, 29 Kirchoff, Gustav, 93 speed, 26 speed in a medium, 29 Lake Erie, 152, 160 visible, 20 Land, Edwin, 292 Light adaptation, 252 Laplace, Pierre-Simon, 418 Lighthouse, 224 Laser, 388, 400 Light Detection and Ranging (LIDAR), gravitational wave, 430 230 living, 403 Lightning, 285 medium, 388 Light pillar, 210 pumping, 388 Light source Laser Interferometer Space Antenna cool white color, 312 (LISA), 426 fluorescent, 81, 311 Laser Interferometric Gravitational Wave halogen, 311 Observatory (LIGO), 423 incandescent, 84, 311, 324 Laser tweezers, 342 LED, 311 Last Chance Canyon, CA, 249 warm white color, 312 Laundry brightener, 387 Light-year, 40 Law of reflection, 184
Index
Line of purples, 267 Line-of-sight propagation, 283 Line spectra, 82 Liquid crystal display, 292, 294 LMS color space, 265 Lockyer, Normal, 386 Lodestone, 274 Loma Prieta earthquake, 17 London, 206 Longitudinal waves, 22 Long radio waves, 282 Lorentz, Hendrik, 418 Lorentz invariance, 418 Los Angeles, 161 Loudness, 133 Louis XIV, King, 182 Love, Augustus Edward Hough, 155 Love waves, 154 Low angle reflection, 296 Luciferin, 336, 388 Luminiferous aether, 20, 57 Luminosity, 266 Lunar eclipse, 171 partial, 171 penumbral, 171 total, 171 Lycopene, 98 Lyman, Theodore IV, 379 Lymphocyte, 236 Mach cone, 115 Mach, Ernst, 115 Magic mirror illusion, 208 Magnetic field, 276 Magnetic resonance imaging, 397 Magnifying glass, 224 Major scale, 136 Malus, Étienne-Louis, 293 Malus’s Law, 293 Mantis shrimp, 256, 296 Mantle, 152 Many worlds interpretation, 392 Marimba, 139 Mars, 179 atmosphere, 288, 304, 318
509
greenhouse effect, 318 terraform, 319 Marvin the Martian, 118 Mass, 334, 420 Mass extinction, 321 Mass flow, 148 Matter waves, 20, 357 dispersion, 362 Fermat’s principle, 203 interference, 368 traveling, 361 velocity, 362 Maxwell’s equations, 278, 333 Maxwell, James Clerk, 8, 277 Mbira, 139 Measurement problem, 391 Mechanical waves, 19, 124 Medical ultrasound, 132 Medium (EM waves), 29 Medium (mechanical waves), 19 Membranophones, 139 Mercury, 422 Merganser, hooded, 146 Merian, J.R., 160 Merian’s formula, 160 Metal atom, 385 Metastable state, 388 Meter, 29 Methane, 319, 399 Metric prefixes, 441 Metric system, 440 Mexico, 178 Michelson, Albert, 58 Microscope, 233 electron, 69, 235, 402 TIRF, 221 Microwave link, 31 Microwave oven, 59, 71, 90, 283 Microwaves, 283 Middle C (music), 135 Mie, Gustave, 288 Mie scattering, 288 Milk, 288, 289 Millennium Bridge, 92 Millimeter wave scanner, 284
510
Minor scale, 136 Mirage, 213 Mirror acoustic, 206 center of curvature, 191 coma, 196 concave, 189 concave spherical, 191 convex, 197 elliptical, 208 equation, 194 focal length, 191, 195, 197 focus, 189 force, 353 full-length, 207 hyperbolic, 196 minimum size, 189 multiple, 198 optical axis, 190 parabolic, 190, 208 plane, 186 principal rays, 193, 197 requirements, 183 sculpture, 181 soap bubble, 209 spherical aberration, 196 symmetry, 199 three-mirror anastigmat, 196 Mode number, 52 Molar (unit), 87 Molar extinction coefficient, 87 Molecular bond, 381 Molecular orbital, 381 Molecular rotations, 90 Molecular vibrations, 91, 284, 374 Momentum, 125, 339, 358 angular, 415 gravitational waves, 415 photon, 339 Moon, 157 albedo, 327 apparent size, 170 dust, 338 eclipse, 170 Europa (of Jupiter), 323
Index
gravitational wave power, 415 orbit, 148, 171, 180, 406 phases, 172 recession, 186 retroreflector, 186 temperature, 323, 327 Titan (of Saturn), 119 Morely, Edward, 58 Mormon Tabernacle, 208 Mother-of-pearl, 68 Mould, Steve, 422 Mount Washington (NH), 120 Mo Zi, 3 MRI instrument, 397 Multiply by one, 449 Music, 133 consonance, 135 dissonance, 136 instruments, 137 intervals, 135 scales, 136 scientific pitch notation, 134 Musical road, 42 Myopia, 231 Nanodiamond, 331 Narcissus, 182 Natural frequency, 53, 374 Natural line width, 349 Natural notes, 134 Nautilus chambered, 176 Near field electromagnetic, 279 gravitational, 417 Near-sightedness, 231 Neon spectrum, 386 Neon tetra, 68 Neutrino, 367 Neutron, 369 Newton (unit), 127 Newton, Isaac, 6, 57, 81, 406 Newton’s rings, 56 Nichols radiometer, 341 Night vision goggles, 284, 337
Index
Nitrogen molecule, 384 NMR instrument, 397 Node, 51 Noether, Emmy, 76 Noise-canceling headphones, 48, 72 Non-linear effects, 45, 149, 339 Non-radiative transition, 386 Nondispersive, 127 Normal incidence, 216 Normal lens, 228 Normal modes, 53, 90, 137, 370 Northern lights, 277 Nuclear decay, 366 fission, 366 force, 369, 370 fusion, 366 reaction, 116, 286, 366 Numbers, 431 Object, 174, 186 distance, 188 Objective lens, 233, 234 Oblique projection, 167 Octave, 134 Oculus dexter, 232 Oculus sinister, 232 Oil sheen, 54 One-dimensional waves, 23 Opsin, 251 Optical activity, 299 axis, 222 cavity, 389 density, 86 fiber, 218 rotation, 299 trapping, 342 Optical axis, 190 Orbital atomic, 377 molecular, 381 Organ, 138, 159 Orthographic projection, 167 Ostriches, 14
511
Outer core, 152 Out of focus, 228 Out of phase, 47 Overtone, 53 Owl, 255 Oxygen molecule, 384 Oyster, 68 Ozone, 285 Ozone layer, 285 Pacific Ocean, 445 Palmieri, Luigi, 386 Parabola, 208 Parabolic mirror, 190 Parallax, 188 Paris, 28 Parker Solar Probe, 40 Parrot, 67, 255, 270 Particle in a box, 371, 402, 403 Particle theory of light, 4 Particle-wave debate, 6 Particle-wave duality, 9, 334, 344, 350, 370 Pascal (unit), 129 Paschen, Friedrich, 379 Passive sonar, 132 Pauli exclusion principle, 373, 396 Pauli, Wolfgang, 373 Peacock, 68 Peak, 18 Pedal tone, 139 Peephole, 235 Pendulum, 124 Pentatonic scale, 136 Penumbra, 169 Perfect fifth, 135 Perihelion, 422 Perimeter, 455 Period, 32 Periodic table, 380 Peripheral vision, 251 Periscope, 208 Persian scale, 136 Phase shift, 55, 73 Phase velocity, 145, 362
512
Phoenix, Arizona, 290 Phone display, 259 Phonon, 350 Phosphor, 387 Photochemistry, 338 Photoelectric effect, 9, 336 Photomultiplier tube, 337 Photon, 332, 334 density, 355 energy, 335 length, 354 momentum, 339 pressure, 355 properties, 334 size, 346 Photonic crystal, 68 Photoreceptor cell, 250 Photosynthesis, 97, 338 Phototaxis, 176 Piano, 137 Pickup (guitar), 137 Pigeon, 68, 131 Pigment, 67, 260 Pinhole camera, 173, 229 Pitch, 133 Pixel, 259, 295 Planaria, 176 Planck-Einstein relation, 335 Planck, Max, 9, 332 Planck’s constant, 335, 358 Planck’s law, 333 Plane mirror, 186 Planet exoplanet, 106 Mars, 288, 318 Mercury, 422 Saturn, 119 Venus, 318 Planing (boat), 115 Plankton, 388 Plato, 3 Plum tree, 263 Podolsky, Boris, 393 Poincaré, Henri, 418 Poisson, Siméon, 64
Index
Polarity, 44, 50 Polarization, 22, 279, 291 circular, 298 circular dichroism, 89 diagonal, 297 Fresnel equations, 220 horizontal, 291 low angle reflection, 296 multiple polarizers, 292 sources, 295 vertical, 291 Polarized sunglasses, 292, 297, 303 Polarizer, 291 Polaroid, 292 Poplar, 263 Population inversion, 388 Porcelain, 287 Porpoise, 131 Positive feedback, 93 Positron, 367 Potential energy, 77 Potential energy surface, 363 Potential energy well, 371 Pound (unit), 127 Power, 78 density, 79 wave, 79 Precision, 432 Presbyopia, 232 Pressure, 129 Primary colors, 258 Primary waves Huygens, 59 seismic, 23, 154 Principal quantum number, 377 Principal rays, 193, 197, 223 Prism, 236 Projection, 166 oblique, 167 orthographic, 167 Propagating uncertainties, 435 Propane, 399 Ptolemy, 3 Pulsar, 426 Pulse (waves), 44
Index
Pupil, 230 P-waves, 23, 154 Pyrometer, 313 Pythagoras, 133, 454 Pythagorean theorem, 454
boundaries, 182 diffuse, 184 Doppler effect, 111 frustrated total internal, 220 law of, 184 partial, 182 Quadrupole, 413 specular, 184 Quadrupole radiation, 413 total internal, 217, 218 Quantum water wave, 183 decoherence, 395 Refraction, 3, 155, 212 mechanics, 359 angle of, 214 teleportation, 394 density gradients, 213 tunneling, 366 direction, 213 wierdness, 389 water waves, 244 Quasiparticle, 350 Refractive index, 29, 34, 212, 281 Relative velocity, 103 Rabbit ear antenna, 302 Relativity, 58, 359 Radiation field, 280 de Broglie relations, 359 Radio waves, 282 Doppler effect, 114 Radiometer, 340 general, 108 Radium, 286 general theory, 419 Rainbow, 238 momentum, 359 Ramsay, Sir William, 386 special theory, 416 Ray diagram Resonance, 87 concave mirror, 192 electromagnetic waves, 89 convex mirror, 197 engineering, 91 plane mirror, 188 in resonance, 87 principal rays, 193, 197 matter waves, 382 shadow, 167 out of resonance, 87 Rayleigh-Jeans law, 333 reversible, 93 Rayleigh, Lord (J. W. Strutt), 155, 289, Resonate, 87 333 Rest mass, 334 Rayleigh scattering, 171, 289, 296 Restoring force, 125 Rayleigh waves, 155 Retarded position, 418 Ray-tracing, 4 Retina, 230 Real image, 192 Retinal, 250, 252 Red-eye artifact, 219 Retroreflector, 186 Redshift, 106 cat’s eye, 219 cosmic microwave background, 325 corner-cube, 186 cosmological, 108 Moon, 186 gravitational, 108 RGB color model, 258 Reflected angle, 184 Richter magnitude scale, 154 Reflected ray, 184 Right angle, 454 Right triangles, 454 Reflection, 3, 50, 182 Rip current, 148 at interface, 50
513
514
Ripples, 140 Ritchey-Chrétien telescope, 196 Ritchey, George, 196 Ritsu scale, 137 Rock flour, 289 Rod cell, 250 Rogue waves, 48 Roller coaster, 363 Rømer, Ole, 28, 42 Rosen, Nathan, 393 Rotational modes, 90 ROY G BIV, 257 Runaway greenhouse effect, 318 Rust, 385 RYB color model, 262 Rydberg constant, 380 Rydberg formula, 380, 401 Rydberg, Johannes, 379 Ryo scale, 137 Sahl, Ibn, 214 Sardine, 68 Satellite, 338 Saturation, 264 Saturn, 99 Saxophone, 139 Scalar, 274 Scanning electron microscopes, 235 Scanning tunneling microscope, 367 Scattering, 286 efficiency, 287 geometric, 287 large objects, 286 medium objects, 288 rainbows, 239 Rayleigh, 171 small objects, 289 Schlieren, 243 Schrödinger’s cat, 391, 403 Schrödinger equation, 360 Schrödinger, Erwin, 360 Scientific notation, 431 Scientific pitch notation, 135 Second harmonic, 53 Secondary colors, 258
Index
Secondary waves Huygens, 59 seismic, 23, 154 Seiche, 151, 160 Seismic waves, 19, 154 Seismogram, 155 Seismograph, 155 Selective absorption, 295 Semitone (music), 134 Shadow, 165 Shadow-based vision, 176 Shallow water waves, 142 Shock wave, 115 Shortwave radio, 283 SI system, 440 Significant figures, 433 Similar triangles, 174, 226, 453 Sine, 454 Sine wave, 18 Sinusoidal wave, 18 Ski moguls, 149 Skin cancer, 285 Sky color, 290, 311 Sky polarization, 296 Smartphone, 281, 283 Snellius, Willebrord, 214 Snell’s law, 214, 240 Snell’s window, 217 Snow blindness, 303 Snow drifts, 149 Snow emissivity, 315, 324 Soap bubble, 56, 72 Sodium, 94 Sodium D line, 94 Sodium emission, 119 Sodium spectrum, 386 Soft boundary, 50 Soft X-ray, 285 Solar aberration, 419 Solar eclipse, 170 annular, 170 partial, 170, 175 total, 170 Solar sail, 341, 354 Solar wind, 277
Index
Soliton, 143 Sonar, 132 Sonic boom, 115 Sound, 128 clicks, 348 phonon, 351 refraction, 214 Sound localization, 160 Sound spectrum, 131 Sound waves, 19 Southern lights, 277 South pole, 160 Space blanket, 323 Spacecraft Breakthrough starshot, 353 Cassini, 119 Helios-2, 119 Huygens, 119 IKAROS, 342 Voyager 1, 355 Spacetime, 422 Special relativity, 58 Spectral pyrometer, 313 Spectroscopy, 83 Spectrum absorption, 86 atmosphere, 85 atmospheric gases, 85 blackbody, 309, 333 carotene, 86 colored glass, 85 continuous, 82 earthquake, 84 eye sensitivity, 252 fluorescent light, 82 gravitational waves, 425 guitar, 83 gumball color, 261 helium, 385 hydrogen, 83, 385 incandescent light, 84 intensity, 81 line, 82 line widths, 349 musical instruments, 133
515
neon, 386 sodium, 94, 386 sound, 81, 131 sun, 94 transmission, 84 wave packet, 348 Specular reflection, 184 Speed, 24 Speed of gravity, 416 Speed of light, 26, 416 Speed of sound, 129 Spherical aberration, 196 Spherical harmonics, 378 Spin (quantum), 334, 373 Spirograph toy, 146 Spitzer Space Telescope, 196 Spyglass, 235 Square root, 450 SR-71 Blackbird airplane, 40, 323 Standing wave, 51 antinode, 51 coffee, 88 driven string, 88 harmonics, 53 idiophone, 139 matter waves, 370 membrane instrument, 139 natural frequencies, 53 node, 51 normal mode, 53 stringed instrument, 137 wavelength, 51 wind instrument, 138 Star Alnilam, 312 Betelgeuse, 118, 310, 355 binary, 411 neutron, 414 Procyon B, 325 Proxima Centauri, 40, 66 pulsar, 426 rotation, 107 S5-HSV-1, 106 Sirius, 118, 324 US-708, 120
516
white dwarf, 414 Stark effect, 251 Stationary waves, 160 Steel pan (instrument), 139 Stefan, Josef, 312, 332 Stefan-Boltzmann constant, 312 Stefan-Boltzmann law, 312, 315, 316 Stellar aberration, 28, 419 Steller’s jay, 68 Stimulated emission, 388 Stock trading, 30 Stokes drift, 148 Stokes, George, 148 Strain, 410 Strawberry, 250 String wave, 19, 126, 159 Strong nuclear force, 369 Structural coloration, 67, 72 Strutt, John William (Ld. Rayleigh), 155, 289, 333 Submarine communication, 282 Subtractive color mixing, 260 Sugar maple, 263 Sun apparent size, 170 eclipse, 170 photograph, 307 solar constant, 326 spectrum, 94, 100, 384, 386 speed of sound, 160 temperature, 311 Sunburn, 285, 338 Sun glitter, 210 Sunscreen, 285 Superconductivity, 397 Supercontinent, 153 Superfluid, 396 Superior mirage, 213, 243 Superposition, 44 beating pattern, 49, 112 constructive interference, 47 decomposition, 46 destructive interference, 47 different frequencies, 46 diffraction, 58
Index
Huygens’s principle, 59 interference, 54 matter waves, 376 oscillations in time, 47 polarization, 297, 413 principle, 44 reflection at boundaries, 50 standing waves, 50 structural coloration, 67 thin-film interference, 54 wave packet, 347 Supersonic motion, 114 Surface area, 455 Surface normal, 184, 214 Surface tension, 140 Surface waves, 23 Suspension bridge, 73 S-waves, 23, 154 Swiftlet, 158 Symmetry breaking, 148 gravitational, 407 rotational, 412 time-translation, 76 translational, 148 Système International, 440 Syzygy, 179 Tacoma Narrows Bridge, 91 Tangent, 454 Tapetum lucidum, 219 Taylor, Joseph, 415 Tectonic plates, 153 Telephoto lens, 228 Telescope, 234 diffraction spikes, 65 Very Long Baseline Array, 354 Television, 283 Temperature, 130 color, 311 Tension, 127 Terahertz gap, 284 Terraform, 319 Tertiary colors, 258 Theories of light, 1
Index
extramission, 2 intromission, 3 modern understanding, 10 particle-wave duality, 9 particles, 4 timeline, 11 waves, 6 Thermal energy, 77 Thermal radiation, 9, 308 color temperature, 311 coupling, 315 peak emission, 309 power, 312 two-way, 315 Thin film interference, 366 Thin lens, 221 Thin lens equation, 225 Thin-film interference, 54, 67 Three-dimensional waves, 24 Tidal bulges, 408 Tides, 150, 407 Timbre, 133 Time-translation symmetry, 76 Timpani, 139 TIRF microscopy, 221 T?hoku tsunami, 150 Tonic, 136 Total internal reflection, 218 Traffic analogy Doppler effect, 104 frequency-wavelength, 32, 35 Translational symmetry, 148 Translucent, 287 Transmission electron microscopes, 236 Transmission spectra, 84 Transverse waves, 22, 279 Triangles, 453 right, 454 similar, 453 Trigonometry, 454 Trochoid, 146 Trough, 18 Trumpet, 138 Tsunami, 39, 150 Tunneling, 366
517
Turbulence, 148 TV remote control, 284 Two-cycle engine smoke, 290 Two-dimensional waves, 23 Two-way thermal radiation, 315 Tyndall, John, 289 Tyndall scattering, 289 Ultrafast laser, 349 Ultrasound, 131 Ultrasound imaging, 132 Ultraviolet catastrophe, 333 Ultraviolet light, 285 Umbra, 169 Umbrella inversion, 367 Uncertainties, 433 Uncertainty principle, 390 Uncertainty propagation, 435 Undertow, 148 Unit, 439 acre-foot, 444 atmosphere, 129 base, 440 brix, 244 calorie, 77 celsius, 130 checking equations, 26 conversion, 443 customary, 440 decibels, 82 derived, 440 diopter, 232 electron-volt, 368 Fahrenheit, 130 food calorie, 77 hertz, 34 joule, 77 kelvin, 130 kilowatt-hour, 98, 445 math, 441 metric, 440 molar, 87 newton, 127 none, 29 optical density, 86
518
pascal, 129 pound, 127 pounds per square inch, 129 rpm, 34 SI system, 440 watt, 79 Unitless, 29 Universe age, 108 inflation phase, 108 Unpolarized light, 291 Uranus spectrum, 100 Vector, 274 Velocity, 24 relative, 103 Venus, 179 albedo, 318 atmosphere, 327 greenhouse effect, 318 sailing past, 342 transit, 179 Versailles, France, 182 Very Long Baseline Array, 354 VHF radio, 283 Vibrational modes, 91 Vietnam, 137 Violin, 54, 73, 89, 128, 137 Virgo Observatory, 423 Virtual image, 187, 216 Visible light, 20, 284 Vision extramission, 2 intromission, 3 shadow-based, 176 Vitamin A, 250 Volume, 455 von Jolly, Philip, 332 Voyager 1 spacecraft, 40, 355 Wake, 46, 115, 145 Warped space, 421 Wasp, 56, 296 Water
Index
density, 141 rotations, 90 surface tension, 140 vibrations, 91, 376 Water waves, 20, 140 base, 146 breaking, 143, 149 capillary, 140 deep vs. shallow, 141 evolution, 148 forces, 147 mass flow, 148 motion, 146 rip current, 148 rogue, 48 seiches, 151 sets and lulls, 50 Stokes drift, 148 tides, 150 tsunamis, 150 undertow, 148 wind generated, 148 Watt (unit), 79 Watt, James, 79 Wave base, 146 Wave components, 46 Wave function, 345, 360 Wave packet, 334, 347, 361 Wave speed, 24 capillary, 141 deep water, 141 group velocity, 145 light, 26 mechanical, 128 membrane, 158 phase velocity, 145 shallow water, 142 sound, 129 string, 127 tsunami, 150 water, 142 Wave theory of light, 6 Wavelength, 19 Waves amplitude, 18
Index
crest, 18 definition, 18 dispersive, 141 electromagnetic, 20, 278 energy, 18, 76, 78 evanescent, 220, 366, 375 frequency, 32, 33 gravitational, 21, 120, 350, 405, 410 gravity, 141 light, 278 longitudinal, 22 matter, 20, 357 mechanical, 19, 124 one-dimensional, 23 peak, 18 period, 32 polarization, 22, 291 power, 79 properties, 17 pulses, 44 seismic, 19, 154 shock, 115 sinusoidal, 18 sound, 19, 128 speed, 24 standing, 50 string, 19, 126 superposition, 44 surface, 23 three-dimensional, 24 transverse, 22 trough, 18
519
two-dimensional, 23 velocity, 24 water, 20, 140 wavelength, 19 Wedgwood, Josiah, 314 Wedgwood, Thomas, 314 Welding arc, 285 Western musical scale, 133 Whale, 41, 131 Whirly tube toy, 160 Wi-Fi, 283 Wide angle lens, 228 Wien’s displacement law, 310 Wien, Wilhelm, 310, 332 William Tell Overture, 42 Wimpole estate, 180 Wolf interval, 136 X-ray crystallography, 285 X-ray diffraction, 63 X-rays, 286, 339 X-ray vision, 301 Xylophone, 139 XYZ color space, 266 Yampa River, Colorado, 304 Young, Thomas, 7, 57, 60, 64, 344 Zero-point energy, 372, 391 Zoom lens, 228