Fundamentals of the Physical Theory of Diffraction [2ed.] 1118753666, 978-1-118-75366-8, 9781118753712, 1118753712

Table of contents :
Content: Fundamentals of the Physical Theory of Diffraction
Contents
Preface
Foreword to the First Edition
Preface to the First Edition
Acknowledgments
Introduction
1 Basic Notions in Acoustic and Electromagnetic Diffraction Problems
1.1 Formulation of the Diffraction Problem
1.2 Scattered Field in the Far Zone
1.3 Physical Optics
1.3.1 Definition of Physical Optics
1.3.2 Total Scattering Cross-Section
1.3.3 Optical Theorem
1.3.4 Introducing Shadow Radiation
1.3.5 Shadow Contour Theorem and the Total Scattering Cross-Section
1.3.6 Shadow Radiation and Reflected Field in the Far Zone. 1.3.7 Shadow Radiation and Reflection from Opaque Objects1.4 Electromagnetic Waves
1.4.1 Basic Field Equations and PO Backscattering
1.4.2 PO Field Components: Reflected Field and Shadow Radiation
1.4.3 Electromagnetic Reflection and Shadow Radiation from Opaque Objects
1.5 Physical Interpretations of Shadow Radiation
1.5.1 Shadow Field and Transverse Diffusion
1.5.2 Fresnel Diffraction and Forward Scattering
1.6 Summary of Properties of Physical Optics Approximation
1.7 Nonuniform Component of an Induced Surface Field
Problems
2 Wedge Diffraction: Exact Solution and Asymptotics. 2.1 Classical Solutions2.2 Transition to Plane Wave Excitation
2.3 Conversion of the Series Solution to The Sommerfeld Integrals
2.4 The Sommerfeld Ray Asymptotics
2.5 The Pauli Asymptotics
2.6 Uniform Asymptotics: Extension of the Pauli Technique
2.7 Fast Convergent Integrals and Uniform Asymptotics: The "Magic Zero" Procedure
Problems
3 Wedge Diffraction: The Physical Optics Field
3.1 Original PO Integrals
3.2 Conversion of PO Integrals to the Canonical Form
3.3 Fast Convergent Integrals and Asymptotics for the PO Diffracted Field
Problems. 4 Wedge Diffraction: Radiation by Fringe Components of Surface Sources4.1 Integrals and Asymptotics
4.2 Integral Forms of Functions and
4.3 Oblique Incidence of a Plane Wave at a Wedge
4.3.1 Acoustic Waves
4.3.2 Electromagnetic Waves
Problems
5 First-Order Diffraction at Strips and Polygonal Cylinders
5.1 Diffraction at a Strip
5.1.1 Physical Optics Part of a Scattered Field
5.1.2 Total Scattered Field
5.1.3 Numerical Analysis of a Scattered Field
5.1.4 First-Order PTD with Truncated Scattering Sources
5.2 Diffraction at a Triangular Cylinder. 5.2.1 Symmetric Scattering: PO Approximation5.2.2 Backscattering: PO Approximation
5.2.3 Symmetric Scattering: First-Order PTD Approximation
5.2.4 Backscattering: First-Order PTD Approximation
5.2.5 Numerical Analysis of a Scattered Field
Problems
6 Axially Symmetric Scattering of Acoustic Waves at Bodies of Revolution
6.1 Diffraction at a Canonical Conic Surface
6.1.1 Integrals for the Scattered Field
6.1.2 Ray Asymptotics
6.1.3 Focal Fields
6.1.4 Bessel Interpolations for the Field
6.2 Scattering at a Disk
6.2.1 Physical Optics Approximation.

Citation preview

Fundamentals of the Physical Theory of Diffraction

Fundamentals of the Physical Theory of Diffraction Second Edition

Pyotr Ya. Ufimtsev

Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Ufimtsev, Pyotr Yakovlevich. Fundamentals of the physical theory of diffraction / Pyotr Ya. Ufimtsev. – 2e. pages cm Includes bibliographical references and index. ISBN 978-1-118-75366-8 (cloth) 1. Electromagnetic waves–Diffraction. 2. Diffractive scattering. I Title. QC665.D5U35 2014 535′ .42–dc23 2013039736 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

Contents

Preface

xiii

Foreword to the First Edition

xv

Preface to the First Edition

xix

Acknowledgments

xxi

Introduction 1 Basic Notions in Acoustic and Electromagnetic Diffraction Problems 1.1 1.2 1.3

xxiii

1

Formulation of the Diffraction Problem / 1 Scattered Field in the Far Zone / 3 Physical Optics / 7 1.3.1 Definition of Physical Optics / 7 1.3.2 Total Scattering Cross-Section / 10 1.3.3 Optical Theorem / 11 1.3.4 Introducing Shadow Radiation / 12 1.3.5 Shadow Contour Theorem and the Total Scattering Cross-Section / 17 1.3.6 Shadow Radiation and Reflected Field in the Far Zone / 20 1.3.7 Shadow Radiation and Reflection from Opaque Objects / 22

v

vi

CONTENTS

1.4

Electromagnetic Waves / 23 1.4.1 Basic Field Equations and PO Backscattering / 23 1.4.2 PO Field Components: Reflected Field and Shadow Radiation / 26 1.4.3 Electromagnetic Reflection and Shadow Radiation from Opaque Objects / 28 1.5 Physical Interpretations of Shadow Radiation / 31 1.5.1 Shadow Field and Transverse Diffusion / 31 1.5.2 Fresnel Diffraction and Forward Scattering / 32 1.6 Summary of Properties of Physical Optics Approximation / 32 1.7 Nonuniform Component of an Induced Surface Field / 33 Problems / 36 2

Wedge Diffraction: Exact Solution and Asymptotics

49

2.1 2.2 2.3 2.4 2.5 2.6 2.7

Classical Solutions / 49 Transition to Plane Wave Excitation / 55 Conversion of the Series Solution to the Sommerfeld Integrals / 57 The Sommerfeld Ray Asymptotics / 61 The Pauli Asymptotics / 63 Uniform Asymptotics: Extension of the Pauli Technique / 68 Fast Convergent Integrals and Uniform Asymptotics: The “Magic Zero” Procedure / 72 Problems / 76

3

Wedge Diffraction: The Physical Optics Field

87

3.1 3.2 3.3

Original PO Integrals / 87 Conversion of PO Integrals to the Canonical Form / 90 Fast Convergent Integrals and Asymptotics for the PO Diffracted Field / 94 Problems / 100

4

Wedge Diffraction: Radiation by Fringe Components of Surface Sources 4.1 4.2 4.3

Integrals and Asymptotics / 104 Integral Forms of Functions f (1) and g(1) / 112 Oblique Incidence of a Plane Wave at a Wedge / 114 4.3.1 Acoustic Waves / 114 4.3.2 Electromagnetic Waves / 118 Problems / 120

103

CONTENTS

5 First-Order Diffraction at Strips and Polygonal Cylinders 5.1

vii

123

Diffraction at a Strip / 124 5.1.1 Physical Optics Part of the Scattered Field / 124 5.1.2 Total Scattered Field / 128 5.1.3 Numerical Analysis of the Scattered Field / 132

5.1.4 First-Order PTD with Truncated Scattering Sources j(1) / 135 h 5.2 Diffraction at a Triangular Cylinder / 140 5.2.1 Symmetric Scattering: PO Approximation / 141 5.2.2 Backscattering: PO Approximation / 143 5.2.3 Symmetric Scattering: First-Order PTD Approximation / 145 5.2.4 Backscattering: First-Order PTD Approximation / 148 5.2.5 Numerical Analysis of the Scattered Field / 150 Problems / 152 6 Axially Symmetric Scattering of Acoustic Waves at Bodies of Revolution 6.1

6.2

6.3

6.4

Diffraction at a Canonical Conic Surface / 158 6.1.1 Integrals for the Scattered Field / 159 6.1.2 Ray Asymptotics / 160 6.1.3 Focal Fields / 166 6.1.4 Bessel Interpolations for the Field u(1) / 167 s,h Scattering at a Disk / 169 6.2.1 Physical Optics Approximation / 169 6.2.2 Relationships Between Acoustic and Electromagnetic PO Fields / 171 6.2.3 Field Generated by Fringe Scattering Sources / 172 6.2.4 Total Scattered Field / 173 Scattering at Cones: Focal Field / 176 6.3.1 Asymptotic Approximations for the Field / 176 6.3.2 Numerical Analysis of Backscattering / 179 Bodies of Revolution with Nonzero Gaussian Curvature: Backscattered Focal Fields / 183 6.4.1 PO Approximation / 184 6.4.2 Total Backscattered Focal Field: First-Order PTD Asymptotics / 186 6.4.3 Backscattering from Paraboloids / 186 6.4.4 Backscattering from Spherical Segments / 192

157

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CONTENTS

6.5

Bodies of Revolution with Nonzero Gaussian Curvature: Axially Symmetric Bistatic Scattering / 196 6.5.1 Ray Asymptotics for the PO Field / 196 6.5.2 Bessel Interpolations for the PO Field in the Region 𝜋 − 𝜔 ≤ 𝜗 ≤ 𝜋 / 200 6.5.3 Bessel Interpolations for the PTD Field in the Region 𝜋 − 𝜔 ≤ 𝜗 ≤ 𝜋 / 200 6.5.4 Asymptotics for the PTD Field in the Region 2𝜔 < 𝜗 ≤ 𝜋 − 𝜔 Away from the GO Boundary 𝜗 = 2𝜔 / 201 6.5.5 Uniform Approximations for the PO Field in the Ray Region 2𝜔 ≤ 𝜗 ≤ 𝜋 − 𝜔, Including the GO Boundary 𝜗 = 2𝜔 / 202 6.5.6 Approximation of the PO Field in the Shadow Region for Reflected Rays / 205 Problems / 207

7

Elementary Acoustic and Electromagnetic Edge Waves 7.1

Elementary Strips on a Canonical Wedge / 212

on Elementary Strips / 213 Integrals for j(1) s,h Triple Integrals for Elementary Edge Waves / 217 Transformation of Triple Integrals into One-Dimensional Integrals / 220 7.5 General Asymptotics for Elementary Edge Waves / 225 7.6 Analytic Properties of Elementary Edge Waves / 230 7.7 Numerical Calculations of Acoustic Elementary Fringe Waves / 234 7.8 Electromagnetic Elementary Edge Waves / 237 7.8.1 Electromagnetic EEWs on the Diffraction Cone Outside the Wedge / 241 7.8.2 Electromagnetic EEWs on the Diffraction Cone Inside the Wedge / 243 7.8.3 Numerical Calculations of Electromagnetic Elementary Fringe Waves / 245 7.9 Improved Theory of Elementary Edge Waves: Removal of the Grazing Singularity / 245 7.9.1 Acoustic EEWs / 248 7.9.2 Electromagnetic EEWs Generated by the Modified Nonuniform Current / 253 7.10 Some References Related to Elementary Edge Waves / 256 Problems / 257

7.2 7.3 7.4

211

CONTENTS

8 Ray and Caustic Asymptotics for Edge Diffracted Waves

ix

261

8.1

Ray Asymptotics / 261 8.1.1 Acoustic Waves / 261 8.1.2 Electromagnetic Waves / 266 8.1.3 Comments on Ray Asymptotics / 267 8.2 Caustic Asymptotics / 269 8.2.1 Acoustic waves / 269 8.2.2 Electromagnetic Waves / 274 8.3 Relationships between PTD and GTD / 275 Problems / 276

9 Multiple Diffraction of Edge Waves: Grazing Incidence and Slope Diffraction

285

9.1 9.2

Statement of the Problem and Related References / 285 Grazing Diffraction / 286 9.2.1 Acoustic Waves / 286 9.2.2 Electromagnetic Waves / 290 9.3 Slope Diffraction in Configuration of Figure 9.1 / 292 9.3.1 Acoustic Waves / 292 9.3.2 Electromagnetic Waves / 295 9.4 Slope Diffraction: General Case / 296 9.4.1 Acoustic Waves / 296 9.4.2 Electromagnetic Waves / 299 Problems / 302

10

Diffraction Interaction of Neighboring Edges on a Ruled Surface

305

10.1 Diffraction at an Acoustically Hard Surface / 306 10.2 Diffraction at an Acoustically Soft Surface / 309 10.3 Diffraction of Electromagnetic Waves / 312 10.4 Test Problem: Secondary Diffraction at a Strip / 314 10.4.1 Diffraction at a Hard Strip / 314 10.4.2 Diffraction at a Soft Strip / 317 Problems / 318 11

Focusing of Multiple Acoustic Edge Waves Diffracted at a Convex Body of Revolution with a Flat Base 11.1 11.2

Statement of the Problem and its Characteristic Features / 325 Multiple Hard Diffraction / 327

325

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CONTENTS

11.3 Multiple Soft Diffraction / 328 Problems / 330 12

Focusing of Multiple Edge Waves Diffracted at a Disk

333

12.1 Multiple Hard Diffraction / 334 12.2 Multiple Soft Diffraction / 336 12.3 Multiple Diffraction of Electromagnetic Waves / 340 Problems / 341 13

Backscattering at a Finite-Length Cylinder

343

13.1

Acoustic Waves / 343 13.1.1 PO Approximation / 343 13.1.2 Backscattering Produced by the Nonuniform Component j(1) / 347 13.1.3 Total Backscattered Field / 352 13.2 Electromagnetic Waves / 354 13.2.1 E-polarization / 354 13.2.2 H-polarization / 360 Problems / 362 14

Bistatic Scattering at a Finite-Length Cylinder

365

14.1

Acoustic Waves / 365 14.1.1 PO Approximation / 366 14.1.2 Shadow Radiation as a Part of the Physical Optics Field / 368 14.1.3 PTD for Bistatic Scattering at a Hard Cylinder / 370 14.1.4 Beams and Rays of the Scattered Field / 376 14.1.5 PO Shooting-Through Rays and Their Cancellation by Fringe Rays / 381 14.1.6 Refined Asymptotics for the Specular Beam Reflected from the Lateral Surface / 382 14.2 Electromagnetic Waves / 386 14.2.1 E-Polarization / 386 14.2.2 H-Polarization / 388 14.2.3 Refined Asymptotics for the Specular Beam Reflected from the Lateral Surface / 390 Problems / 393

Conclusion

397

CONTENTS

xi

References

399

Appendix to Chapter 4: MATLAB Codes for Two-Dimensional Fringe Waves and Figures (F. Hacivelioglu and L. Sevgi)

411

Appendix to Chapter 6: MATLAB Codes for Axial Backscattering at Bodies of Revolution (F. Hacivelioglu and L. Sevgi)

431

Appendix to Section 7.7: MATLAB Codes for Diffraction Coefficients of Acoustic Elementary Fringe Waves (F. Hacivelioglu and L. Sevgi)

439

Appendix to Section 7.8.3: MATLAB Codes for Diffraction Coefficients of Electromagnetic Elementary Fringe Waves (F. Hacivelioglu and L. Sevgi)

443

(0) mod

⃗ Radiated by Modified Appendix to Section 7.9.2: Field dE (0) mod ⃗ Uniform Currents J Induced on Elementary Strips (P. Ya. Ufimtsev)

447

Index

451

Preface

The physical theory of diffraction (PTD) is a high-frequency asymptotic technique for the investigation of antennas and scattering problems. PTD was announced publically under this name for the first time in a report and a book by Ufimtsev, 1962a,b. An anniversary article (Ufimtsev, 2013a) contains comments on its origination and development. This monograph presents a complete and comprehensive description of modern PTD based on the concept of elementary edge waves. Its basic subject is the diffraction of acoustic and electromagnetic waves by perfectly reflecting objects. Here are new features of the revised version:

r New Sections 1.3.6 and 1.4.2 establish that the shadow radiation equals zero r r r r r

in the directions of the reflected rays, and the reflected field equals zero in the shadow direction. New Sections 1.3.7 and 1.4.3 extend the theory of shadow radiation and reflection to opaque objects. New Section 1.5 provides physical interpretations of the shadow radiation via Fresnel diffraction and forward scattering. New Section 2.7 develops the “magic zero” procedure to derive fast convergent integrals and uniform asymptotics, which are convenient for numerical and analytic analysis of the canonical wedge diffraction problem. Chapter 3 simplifies the physical optics (PO) approximation, introducing new functions 𝑣(0) (kr, 𝜓). s,h New Section 3.3 derives the fast convergent integrals and uniform asymptotics for the PO diffracted field.

xiii

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PREFACE

r New Section 4.3.2 extends the topic of polarization coupling, which is critical in PTD. It also clarifies the nature of this phenomenon.

r New Section 7.8.2 describes the diffracted field inside a perfectly reflecting wedge and explains its origination as a consequence of the equivalence theorem.

r New Section 7.8.3 presents numerical data for diffraction coefficients of electromagnetic elementary fringe waves.

r New Section 8.3 clarifies the relationships between PTD and GTD. r New Section 10.4 studies diffraction at a strip to test the asymptotic theory of the secondary diffraction derived in Chapter 10.

r New Section 14.1.5 studies the PO shooting-through rays and their cancellation r r r r

by fringe rays. It highlights the fundamental role of the nonuniform/fringe components introduced in PTD. An essential supplement to the main text is provided by the end-of-chapter problems followed by their solutions. They will be helpful in the study and application of PTD. MATLAB codes presented in the appendices allow for quick numerical calculations of fringe waves and axial backscattering at bodies of revolution. Compared to the first edition, more attention is given to the theory of electromagnetic waves. Additional smaller insertions and corrections are incorporated throughout the book.

The theory developed in the book may find various applications. Among them are problems associated with the design of microwave antennas, estimation of scattering cross-sections, identification of scattering objects, and propagation of waves in an urban environment. The most significant example of practical applications of PTD represents the design of the American F-117 stealth fighter and B-2 stealth bomber. In combination with numerical methods, PTD can be used for the development of efficient hybrid techniques for the investigation of complex diffraction problems. The book is intended for researchers working on antennas and scattering problems in industry and university laboratories. It can also be useful for teaching a variety of university courses, including topics on high-frequency asymptotic techniques in diffraction theory. University instructors and graduate students will benefit from this book as well. I am very grateful to Dr. Feray Hacivelioglu and Prof. Levent Sevgi, who produced MATLAB codes for numerical calculation of diffracted waves. Many thanks are also due to the reviewers for their valuable comments. Pyotr Ya. Ufimtsev Los Angeles, California August 2013

Foreword to the First Edition Ideas have consequences. Great ideas have far-reaching consequences.

The physical theory of diffraction (PTD) that Professor Ufimtsev introduced in the 1950s—a methodology for approximate evaluation at a high enough frequency of the scattering from a body, especially a body of complicated shape—has proven to be a truly great idea. The first form of PTD developed by Professor Ufimtsev, the vector form applicable to electromagnetic scattering from three-dimensional bodies, has played a key role in the development of modern low-radar-reflectivity weapons systems, such as the Lockheed F-117 stealth fighter and the Northrop B-2 stealth bomber, functioning both as a design tool and as a conceptual framework. These systems in turn have revolutionized the conduct of large-scale government-versus-government warfare and thus have helped to shape history. Ben Rich, who oversaw the F-117 project as head of Lockheed’s fabled Skunk Works, refers to Professor Ufimtsev’s work as “the Rosetta Stone breakthrough for stealth technology.” At Northrop, where I worked on the B-2 project, we were so enthusiastic about PTD that a co-worker and I sometimes broke into choruses of “Go, Ufimtsev” to the tune of “On, Wisconsin.” At both Lockheed and Northrop we referred to PTD as “industrial-strength” diffraction theory, to distinguish it from the approach to diffraction then being favored in the universities, which was not well enough developed to handle the problems of stealth design. Like many good theories, PTD is much easier to apply than to explain. But let us now nevertheless examine the inner workings of PTD and seek to understand why it is

xv

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FOREWORD TO THE FIRST EDITION

such a useful approach. First of all, PTD is based on two important principles, which it will be convenient to refer to here as the physical principle and the geometrical principle. The physical principle shows how the scattered field at a point outside a scattering body can be determined from an integral of appropriate field quantities over the surface of the body. In acoustics these quantities are the pressure at a hard surface, the normal velocity at a soft surface, both at an impedance boundary or the surface of a penetrable body. In electromagnetics they are the tangential magnetic field at the surface of a perfect conductor, the tangential magnetic and electric fields at an impedance boundary or the surface of a penetrable body. The geometrical principle states that at high enough frequency, when the wavelength is small enough compared to the critical dimensions of the scattering body, the surface integrals can be evaluated asymptotically to yield a description of the total field outside the body in terms of geometrical rays, including diffracted rays. The change in field amplitude along a ray can be calculated geometrically by tracing the divergence and convergence of ray bundles except in the regions surrounding (a) a geometrical shadow boundary, for which ray tracing predicts a field discontinuity across the boundary, and (b) a caustic, that is, a locus where adjacent geometrical rays meet or cross (such as, in the simplest case, a focal point), at which ray tracing predicts an infinite field. The correct value for the field in these regions, which shrink as frequency increases, can be found by using uniform asymptotic techniques to evaluate the surface integrals. One of the important features of PTD is this ability to calculate the field accurately in shadow boundary and caustic regions. It is especially important in low observables design because we are often interested in far-field scattering of a plane wave from a body with straight or slightly curved edges, a configuration for which parts of the far-field region lie in caustic regions. The other major advantages of PTD arise from the way the surface fields are handled. There is a uniform part that is defined everywhere on the surface and a nonuniform part that serves as a correction term. For electromagnetics the uniform part is usually, though not always, given by the physical optics (PO) approximation: namely, that the surface fields at a point are the same as if the point lay on an infinite plane surface tangent to the actual body at the point and with the same boundary conditions as at the point. For acoustics the uniform part is usually given by the analogous approximation. Because this acoustics approximation does not have a firmly established name and because other investigators have set the precedent, Professor Ufimtsev uses the terminology PO in both electromagnetics and acoustics throughout this book. Much of Chapter 1 is devoted to PO and its implications. The nonuniform fields for a nonpenetrable body—for example, a hard body in acoustics or a perfect conductor in electromagnetics—tend to be strongest near a diffracting feature such as an edge where two faces of a faceted surface meet, and these fields often diminish rapidly with distance from the feature. It should be emphasized here that this desirable behavior is a consequence of the judicious choice of the uniform part.

FOREWORD TO THE FIRST EDITION

xvii

The nonuniform surface fields are determined using the results of simpler scattering problems, often called canonical problems. Consider again, for example, an edge on a faceted surface. Let the body be a perfect conductor and the edge be straight with the wedge angle formed by the two faces constant along its length, let the illuminating field be a plane wave, and let us choose the PO fields as the uniform part. Then the canonical problem is diffraction of an appropriately oriented plane wave from an infinitely long wedge with perfectly conducting flat faces (even if the faces on the body of interest are not flat). This problem reduces to two scalar two-dimensional problems, one for an incident electric field normal to the edge, the other for an incident magnetic field normal to the edge, and exact solutions exist for these problems. The vector surface fields can be constructed from the two scalar solutions, and the nonuniform surface fields associated with the edge are then found by subtracting the physical optics fields of the canonical problem from the full solution. There now arises the problem of reconciling the uniform part and the nonuniform part, which is defined on a surface that may not exactly match the body surface. Professor Ufimtsev addresses this in Chapter 7, where he reduces the nonuniform part to a continuous array of elementary edge waves concentrated along the edge. These elementary edge waves are sources of diffracted rays and have a directivity pattern that is related to the canonical problem. In the parlance of engineering they would be called diffraction coefficients. The nonuniform contribution to the field diffracted from the edge is now given by an integral of the elementary edge waves over the length of the edge. But when we evaluate asymptotically the integral for the physical optics diffraction from a face, we see that it reduces to an integral along the illuminated part of the face perimeter plus possibly other localized terms (such as a specular reflection contribution). Thus, there are edge diffraction contributions from the uniform part of the surface field on both faces that meet at the edge (if both are illuminated) as well as from the nonuniform part, and these three terms give the total edge diffraction. Furthermore, it turns out that each element of the edge produces diffraction in essentially all directions. We can now, from this investigation of how the surface fields are modeled, extract these additional important features of PTD: 1. PTD can find accurately the reflection and diffraction from a body of complicated shape without having to match the entire body to canonical problems, just the regions that give rise to diffraction. 2. PTD minimizes the difficulty of reconciling the geometries of the body and of the canonical problem. 3. PTD yields diffracted rays in all directions from each element of a linear diffracting feature rather than just in directions on the well-known diffraction cone. The third point is extremely important in low-observables work, where the off-cone rays can sometimes yield the strongest fields in a region.

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FOREWORD TO THE FIRST EDITION

This book presents a thorough development of the fundamentals of PTD for both scalar and vector cases as applied to acoustics and electromagnetics, including important aspects of the theory only recently developed by Professor Ufimtsev. For acoustics it is, of course, the scalar theory that is of interest. For electromagnetics both scalar and vector theory should be of interest. Canonical problems are often two-dimensional, and two-dimensional problems can be reduced to scalar form. Emphasis in the book is on nonpenetrable bodies with “classical” boundary conditions at the surface: the Dirichlet and Neumann problems of applied mathematics, the corresponding soft and hard boundary problems of acoustics, and the perfect conductor problem of electromagnetics. PTD is, however, in principle readily extended to cases of a body with an impedance boundary condition at its surface and of a penetrable but opaque body, and has in fact been used extensively for such bodies, although much of the work is classified, proprietary, or otherwise restricted. The extension to translucent and transparent bodies is more challenging, not because of any shortcoming of PTD but because it can be necessary to deal with such complicated phenomena as diffracted waves that travel through the body and are then refracted out of the body. Much has been said and written about the relative merits of the two major modern approaches to diffraction theory: PTD on one hand, and on the other, Professor Joseph Keller’s geometrical theory of diffraction (GTD) and its modified versions, the uniform theory of diffraction (UTD) developed at the Ohio State University and the similar uniform asymptotic theory of diffraction (UAT). Both approaches are valid, each yields a ray description of the field (PTD as an end result, GTD as a starting point), each has its advantages, and the two have now been cross-fertilizing each other for half a century. The work of the next generation, I fervently hope, will be to mold these approaches and other contributions together into a single modern theory of diffraction from bodies. By his detailed exposition of the fundamentals of PTD in this volume, Professor Ufimtsev has not only produced a work of great contemporary value but also a compendium that can be extremely useful in this reconciliation process. Kenneth M. Mitzner November 2006

Preface to the First Edition The physical theory of diffraction (PTD) is a high-frequency asymptotic technique for the investigation of antennas and scattering problems. This monograph presents the first complete and comprehensive description of the modern PTD based on the concept of elementary edge waves (EEWs). Its subject is the diffraction of acoustic and electromagnetic waves by perfectly reflecting objects located in a homogeneous lossless medium. The basic idea of PTD is that the diffracted field is considered as the radiation generated by scattering sources (currents) induced on objects. Uniform and nonuniform scattering sources are introduced in PTD. Uniform sources are defined as sources induced on an infinite plane tangent to the object at a source point. Nonuniform sources are caused by any deviation of the scattering surface from the tangent plane. For large convex objects with sharp edges, the basic contributions to the scattered field are produced by uniform sources and by those nonuniform sources that concentrate near edges (often called fringe sources). The integration of uniform sources leads to the physical optics (PO) approximation for the scattered field. The PTD is the natural extension of the PO approximation by taking into account the additional field created by the nonuniform/fringe sources. The book provides high-frequency asymptotics for the scattering sources and for the scattered field in the far zone. Scattering characteristics are calculated for a variety of objects, such as strips, polygonal cylinders, cones, bodies of revolution with nonzero Gaussian curvature (including paraboloids and spherical segments), and finite circular cylinders with flat bases. The title of the book underlines the fact that a great deal of attention is to be given to scattering physics. The analytic expressions derived clearly explain the physical structure of a scattered field and describe, in detail, all of the reflected and diffracted xix

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PREFACE TO THE FIRST EDITION

rays and beams, as well as the fields in the vicinity of caustics and foci. Also, a new fundamental component of the field, shadow radiation, is introduced. It is shown that this component contains half of the total scattered power. The physical manifestations of shadow radiation are the well-known phenomena of Fresnel diffraction and forward scattering. Plotted numerical results supplement the theory and provide visualizations of the individual contributions of different parts of the scattering objects to the total diffracted field. Detailed comments explain all critical steps in the analytic and numerical calculations to facilitate their examination and utilization by readers. All chapters are followed by problems for independent investigation, which will be helpful in studying PTD, especially for students. This book is intended for researchers working on antennas and scattering problems in industrial and university laboratories. It can also be useful for teaching a variety of university courses that include topics on high-frequency asymptotic techniques in diffraction theory. University instructors and graduate students will benefit from this book as well. P. Ya. Ufimtsev Los Angeles, California June 2006

Acknowledgments

Work on this book was partially sponsored by the Center of Aerospace Research and Education at the University of California at Irvine. I highly appreciate support by the director of this center, Dr. Satya N. Atluri. Many thanks go to Dr. A.V. Kaptsov for his professional advice, which helped greatly in my work with FORTRAN and SIGMA-PLOT programs. During the preparation of this book I often appealed to my sons Ivan and Vladimir with requests to check and improve my English and to fix computer problems. I am thankful for their assistance. Thanks are also due to E.V. Jull, K.M. Mitzner, Y. Rahmat-Samii, and A.J. Terzuoli, Jr. for their review of the manuscript and valuable comments. This book includes, in revised form, materials from certain articles I wrote for the journals Zhurnal Tekhnicheskoi Fiziki (Russia), Journal of the Acoustical Society of America (United States), Annals of Telecommunications (France), and Electromagnetics (United States). I thank the editorial boards of the journals for their permission to use these materials. P. Ya. Ufimtsev

xxi

Introduction

The physical theory of diffraction (PTD) is an asymptotic high-frequency technique originated in earlier work by this author (Ufimtsev, 1957, 1958a,b,c, 1961). The results of initial journal publications on PTD were summarized in a monograph (Ufimtsev, 1962b), which became a bibliographical rarity a long time ago. To acquaint a new generation of readers with the original form of PTD, some sections of this monograph were updated and included in two books (Ufimtsev, 2003, 2009). Comments on origination and development of PTD were presented recently in an anniversary article (Ufimtsev, 2013a). The selected topics of the modern form of PTD were published in concise form in articles by Butorin and Ufimtsev (1986), Butorin et al. (1987), Ufimtsev (1989, 1991), Ufimtsev and Rahmat-Samii (1995), Ufimtsev (1998, 2006a,b, 2008a,b), and Hacivelioglu et al. (2011). This book presents the first complete and comprehensive description of modern PTD based on the concept of elementary edge waves (EEW). The theory is developed for acoustic and electromagnetic waves scattered by perfectly reflecting objects. For acoustic waves, soft (Dirichlet) or hard (Neumann) boundary conditions are imposed on scattering objects located in a homogeneous nonviscous medium. The absence of viscosity is justified for a fluid (such as air and water) in the linear approximation (Kinsler et al., 1982; Pierce, 1994). In diffraction problems for electromagnetic waves, the scattering objects are considered as perfectly conducting bodies located in vacuum. The assumption of infinite conductivity is acceptable for metallic objects detected by radar. The boundary condition related to electromagnetic waves states that on the surface of perfectly conducting bodies, the tangential component of the electric vector is equal to zero (Balanis, 1989, 2012). The diffraction theory of acoustic waves is scalar, and it is simpler than the vector theory of electromagnetic waves. Because of this, we investigate first an acoustic xxiii

xxiv

INTRODUCTION

diffraction problem in detail and then present its electromagnetic version, referring to similar elements in acoustic theory. This facilitates the study of electromagnetic problems. Note also that from a mathematical point of view, all two-dimensional diffraction problems have identical solutions for acoustic and electromagnetic waves. These problems are considered in the book for acoustic waves. The relationships between acoustic and electromagnetic diffracted waves are emphasized throughout the book. They are also formulated in the boxes located at the beginning of most chapters and sections. PTD has found various applications. Some related references are collected at the end of the book in the section “Additional References Related to the PTD Concept: Applications, Modifications, and Developments.” In particular, PTD was used successfully in the design of the American F-117 stealth fighter and B-2 stealth bomber (Browne,1991a,b; Rich, 1994; Rich and Janos, 1994; Grant, 2013; see also Mitzner’s foreword for three Ufimtsev books (2003, 2007, 2009). The present book contains only original results obtained by the author (some of them in collaboration with colleagues). The distinctive feature of PTD is that it belongs to the class of source-based theories. The scattered/diffracted field is considered as radiation by surface sources which are induced (due to diffraction) on the scattering objects by incident waves. In the case of electromagnetic waves and metallic scattering objects, these sources are surface electric charges and currents. In the case of acoustic waves, these sources are the surface distributions of the “acoustic pressure” on rigid objects, or the surface distributions of the “fluid velocity” on soft (pressure-release) objects. Compared to ray-based techniques, the advantage of this approach is that it allows calculation of the scattered field everywhere, including diffraction regions, such as foci and caustics, where the diffracted field does not have a ray structure. The central and original idea of PTD is the separation of surface sources into uniform and nonuniform components. This separation is a flexible procedure, based on an appropriate choice of canonical diffraction problems (Ufimtsev, 1998). In the present book (except Section 7.9), the uniform component is defined as the scattering sources induced on the infinite plane tangent to the object at a source point. In the case of incident waves with a ray structure, this component is determined according to the geometrical optics (GO) (geometrical acoustics, GA) for electromagnetic (acoustic) waves. The field found by integration of the uniform component is considered a highfrequency approximation for the scattered field. In acoustic diffraction problems, this approximation is interpreted as the Kirchhoff approximation (KA). In electromagnetic diffraction problems, it is known as the physical optics (PO) approach. In the present book we use the term physical optics for both electromagnetic and acoustic waves, just as in the work by Bowman et al. (1987, p. 29). The PTD is the natural extension of PO and takes into account the additional field generated by the nonuniform component, which has a diffraction nature and is caused by any deviation of the scattering surface from an infinite tangent plane. Another definition of the uniform and nonuniform scattering sources is introduced in Section 7.9. Here, the uniform component is defined as the field induced on the halfplane tangential to the illuminated face of the scattering edge (and to the edge itself). The nonuniform component is the difference between the exact field on the tangential

INTRODUCTION

xxv

wedge and this new uniform component. This type of separation of the surface field allows formulation of the advanced version of PTD, which is free of the grazing singularity (Section 7.9). The localization principle related to the behavior of a high-frequency diffracted fields is used to determine the asymptotic approximations for the nonuniform component. In particular, according to this principle, the nonuniform sources induced in the vicinity of sharp curved edges are asymptotically identical to the nonuniform sources induced on a tangential wedge near the tangency point. Because these sources concentrate in the vicinity of the scattering edge, they are often called fringe sources, and the diffracted fields generated by these sources are termed fringe waves. The fundamental role of the fringe waves is emphasized and demonstrated throughout the book Thus, the wedge diffraction is the basic canonical problem for the investigation of edge waves, and it is studied in detail in this book. Exact and asymptotic expressions for two-dimensional edge waves are derived in Chapters 2, 3, and 4. These results are then used in Chapter 5 to construct simple asymptotic expressions for the field diffracted at strips and polygonal cylinders. Notice that two-dimensional diffraction problems for acoustically soft (hard) scat⃗ tering objects are equivalent to electromagnetic problems where the electric vector E ⃗ (magnetic vector H) is parallel to the generatrix of perfectly conducting objects. Due to this equivalence, some results obtained by Ufimtsev (1962b) for two-dimensional electromagnetic problems are transferable for acoustic problems, with proper redefinitions of physical quantities. For the same reason, the asymptotics derived in Chapter 5 for acoustic waves are also valid for electromagnetic waves diffracted at perfectly conducting strips and trilateral cylinders. A new physical interpretation of classical physical optics is introduced in Chapter 1. The scattered PO field is separated into the reflected field and shadow radiation. The first part contains ordinary reflected rays and beams and dominates in the geometrical optics region. The shadow radiation is equivalent to the field scattered at a blackbody (of the same shape and size as the actual scattering object) and dominates in the vicinity of the shadow region (Figs. 1.4 and 14.6). Manifestations of the shadow radiation are the well-known phenomena Fresnel diffraction and forward scattering. The shadow contour theorem established in Section 1.3.5 states that different objects with identical shadow boundaries on their surfaces generate identical shadow radiation. This theorem significantly facilitates the approximate estimation of scattering at complex objects (Alekseev et al., 2007). It is also shown here that the shadow radiation contains half of the total power scattered by perfectly reflecting objects. Thus, the new formulation of the PO field elucidates the scattering physics and explains the nature of the fundamental diffraction law according to which the total scattering cross-section of large (compared to the wavelength) perfectly reflecting objects equals double the transverse area of geometrical optics shadow zone behind the object. A significant part of this book is devoted to the theory of elementary edge waves and to its applications. An elementary edge wave is a wave radiated by surface sources induced in the vicinity of an infinitesimal element of the edge. The high-frequency asymptotic expressions found for elementary edge waves allows one to investigate

xxvi

INTRODUCTION

diffraction at arbitrary curved edges with large radii of curvature (compared to the wavelength). Elementary edge waves can also be interpreted as elementary edge-diffracted rays. The PO field as well can also be understood as the linear superposition of the other type of elementary rays. Because of this, PTD can be considered as a ray theory on the level of elementary rays. Even in diffraction regions such as geometrical optics boundaries, foci, and caustics, the wave field can be represented in terms of elementary rays. Ordinary reflected and diffracted rays are found in PTD by the asymptotic evaluation of field integrals and can be interpreted as the beams of elementary rays generated in the vicinity of stationary points. Such a possible interpretation of PTD goes back to the intuitive Huygens’ principle, which was rigorously formulated by Helmholtz in terms of elementary spherical waves/rays (Baker and Copson, 1939). The general theory of elementary waves is utilized in the book to solve a variety of diffraction problems. Backscattering and bistatic scattering at bodies of revolution are considered in Chapter 6. Ray and caustic asymptotics are derived in Chapter 8. Slope and multiple diffraction at large objects are investigated in Chapters 9 and 10. The results of these chapters are utilized in Chapters 11 and 12 to analyze the focusing of multiple edge waves on the symmetry axis of bodies of revolution. An example of the disk diffraction problem (whose exact asymptotic solution is known) establishes that PTD provides correct expressions for the first term in the total asymptotic expansion for each multiple edge-diffracted wave. This result is a matter of principle because it provides validation of PTD. Also notice other examples of theoretical and experimental validation of PTD in diffraction problems for electromagnetic waves (Nefedov and Fialkovsii, 1972; Ufimtsev, 1962b, 2003, 2009). Chapters 13 and 14 derive the PTD asymptotics for the field scattered at a finite cylinder under oblique incidence of a plane wave. Together with the numerical results illustrated in the figures, they explain the physical structure of the scattered field. New features of the theory are emphasized here. They concern the necessity to calculate high-order terms in the PO field as well as radiation by nonuniform component of the scattering sources caused by smooth bending of a cylindrical surface. The theory developed in the book can find various applications. Among them are the problems associated with the design of microwave antennas, the estimation of scattering cross-sections, the identification of scattering objects, and the propagation of waves in an urban environment. In combination with numerical methods, it can be used for the development of efficient hybrid techniques for the investigation of complex diffraction problems. The book can also be useful for teaching a variety of university courses, including topics on high-frequency asymptotic techniques in diffraction theory. The problems (together with their solutions) following each chapter will be helpful in studying PTD, especially for students. MATLAB codes presented in the appendices allow for quick numerical calculations of fringe waves and axial backscattering at bodies of revolution. The International System of Units (SI) and the time dependence of exp (−i𝜔t) for wave fields and sources are used in this book. Readers who prefer the dependence exp(j𝜔t) can easily transform the book equations to this time format by simple √ √ replacement of the positive imaginary unit i = −1 by the negative unit −j = − −1.

1 Basic Notions in Acoustic and Electromagnetic Diffraction Problems 1.1

FORMULATION OF THE DIFFRACTION PROBLEM

In this book the physical theory of diffraction (PTD) is developed for both acoustic and electromagnetic waves diffracted at perfectly reflecting objects. In two-dimensional problems, this theory is valid for both electromagnetic and acoustic waves. First we present the theoretical fundamentals for acoustic waves and then for electromagnetic waves. In the linear approximation, the velocity potential u of harmonic acoustic waves satisfies the Helmholtz wave equation (Kinsler et al., 1982; Pierce, 1994): ∇2 u + k2 u = I.

(1.1)

Here k = 2𝜋∕𝜆 = 𝜔∕c is the wave number, 𝜆 the wavelength, 𝜔 the angular frequency, c the speed of sound, and I the source strength characteristic. The time dependence is assumed to be in the form exp(−i𝜔t) and is suppressed below. The acoustic pressure p and the velocity 𝑣 of fluid particles, caused by sound waves, are determined through the velocity potential (Kinsler et al., 1982; Pierce, 1994), p = −𝜌

𝜕u = i𝜔𝜌 u, 𝜕t

𝑣⃗ = ∇u,

(1.2)

Fundamentals of the Physical Theory of Diffraction, Second Edition. Pyotr Ya. Ufimtsev. © 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

1

2

BASIC NOTIONS IN ACOUSTIC AND ELECTROMAGNETIC DIFFRACTION PROBLEMS

where 𝜌 is the mass density of a fluid. The power flux density of sound waves, which is the analog of the Poynting vector for electromagnetic waves, equals ⃗ = p𝑣⃗ = p ∇u. P

(1.3)

Its value averaged over the period of oscillations T = 2𝜋∕𝜔 equals ⃗ av = 1 Re(p∗ 𝑣). P ⃗ 2

(1.4)

Two types of boundary conditions are imposed on the surface of perfectly reflecting objects: the Dirichlet condition, u = 0 or

p=0

(soft),

(1.5)

for objects with a soft (pressure-release) surface, and the Neumann condition, 𝜕u = n̂ ⋅ ∇′ u = 0 (hard), 𝜕n

(1.6)

for objects with a hard (rigid) surface. Here u is the total field that is the sum of incident and scattered waves. The symbol n̂ stands for a unit outward vector, which is normal to the scattering surface S (Fig. 1.1). The gradient operator ∇′ is applied to coordinates of the integration/source point Q. To complete formulation of the diffraction problem and to ensure the uniqueness of its solution, the wave equation and boundary conditions above are supplemented by the Sommerfeld radiation condition for the scattered field: ( lim r

) 𝜕u − iku = 0 𝜕r

with r → ∞,

(1.7)

where r is the distance from the scattering object to the observation point. P r nˆ

ˆ m r ˆ m Q

Figure 1.1 Scattering surface S. Here r is the distance between the observation point P (which can be ̂ is directed in the far zone) and the integration point Q (on the surface of the scatterer). The unit vector m from point Q to point P.

3

SCATTERED FIELD IN THE FAR ZONE

In the International System (SI) of units, the quantities introduced above have the following dimensions: [r] = m,

[t] = s,

[p] = kg∕(m ⋅ s2 ),

[𝑣] ⃗ = [c] = m∕s, [u] = m2 ∕s,

[𝜔] = 1∕s,

[𝜌] = kg∕m3 ,

⃗ ] = kg∕s3 . [P

(1.8)

Standard notation is used here: m for meter, kg for kilogram, and s for second. The pressure unit, the pascal (Pa), equals 1 newton/m2 . The SI unit of the power flux density is 1 watt/m2 = 1 joule/(1 s ⋅1 m2 ). In scattering problems which admit to an electromagnetic interpretation, the quantity u above plays the role of electric field intensity ([E] = volt∕m) or magnetic field intensity ([H] = ampere∕m), depending on the polarization of electromagnetic waves. Their power flux density, called the Poynting vector, is defined as ∗

⃗ = E⃗ × H⃗ ⋅ P ⃗ av = 1 Re[E⃗ × H⃗ ]. P 2

1.2

(1.9)

SCATTERED FIELD IN THE FAR ZONE

The scattered field is determined by the Helmholtz integral expressions (Baker and Copson, 1939):

us = −

1 𝜕u eikr ds, ∫ 4𝜋 S 𝜕n r

uh =

1 𝜕 eikr u ds, ∫ 4𝜋 S 𝜕n r

(1.10)

where the integrals are taken over the scattering surface S. The function us describes the field scattered by an acoustically soft object, while the function uh relates to the field scattered by an acoustically hard object. The field quantities u and 𝜕u∕𝜕n in the integrands belong to the total field on the object surface (i.e., to the sum of the incident and scattered fields). These quantities represent the surface sources of the scattered field induced by the incident wave. We denote them by the symbols js =

𝜕u , 𝜕n

jh = u,

(1.11)

similar to those used for induced currents in the electromagnetic version of PTD (Ufimtsev, 2003, 2009). The quantity eikr ∕r in Equation (1.10) represents the Green’s function of a homogeneous medium (i.e., the fundamental solution of the wave equation) and n̂ is a unit outward vector normal to the surface S. In the far field, where r ≫ kd2 (d is a characteristic linear dimension of the object), the field expressions (1.10) can be simplified. We choose the origin of the coordinate

4

BASIC NOTIONS IN ACOUSTIC AND ELECTROMAGNETIC DIFFRACTION PROBLEMS

Q(r', ϑ ', φ' ) ˆ m

P(R, ϑ , φ )

r

r'

R

Ω z y

nˆ

S

x Figure 1.2 S is the surface of the scattering object, Q is the integration point (with the spherical coordinates r′ , 𝜗′ , 𝜑′ ) on the surface S, P is the observation point (with the coordinates R, 𝜗, 𝜑) in the far zone, and Ω is the angle between the directions from the origin to the integration and observation points.

system somewhere inside the object, as shown in Figure 1.2. Under the conditions R ≫ r′ , R ≫ kd2 , we have r ≈ R − r′ cos Ω,

eikr eikR −ikr′ cos Ω ≈ e r R

(1.12)

with cos Ω = cos 𝜗 cos 𝜗′ + sin 𝜗 sin 𝜗′ cos(𝜑 − 𝜑′ ).

(1.13)

In addition, ( ) 𝜕 eikr eikr 1 eikr = ∇′ ⋅ n̂ = − ik − ∇r ⋅ n̂ 𝜕n r r r r

̂ with ∇r = m,

(1.14)

or in view of Equation (1.12), eikR −ikr′ cos Ω 𝜕 eikr ̂ ⋅ n̂ ). (m ≈ −ik e 𝜕n r R

(1.15)

Finally, we obtain the following approximations for the field in the far-away point P: us = −

′ 1 eikR js e−ikr cos Ω ds, 4𝜋 R ∫S

(1.16)

uh = −

′ ik eikR ̂ ⋅ n̂ ) ds, j e−ikr cos Ω (m 4𝜋 R ∫S h

(1.17)

̂ and n̂ are unit vectors. In this book we develop asymptotic approximations where m first for the surface sources js,h and then for the scattered field (1.16), (1.17). Expressions (1.16) and (1.17) can be written in the generic form us,h = u0 Φs,h

eikR , R

(1.18)

SCATTERED FIELD IN THE FAR ZONE

5

where the functions Φs = −

′ 1 j e−ikr cos Ω ds, 4𝜋u0 ∫S s

Φh = −

′ ik ̂ ⋅ n̂ ) ds j e−ikr cos Ω (m 4𝜋u0 ∫S h

(1.19)

represent the directivity patterns of the scattered field and u0 is the complex amplitude of the incident wave at the origin of the coordinates (R = 0). Notice that in the vicinity of the scattering object located in a far zone from an external source, the incident wave can be approximated by the equivalent plane wave with amplitude u0 . According to Equations (1.2) to (1.4), the power flux density of the scattered field is determined by ⃗ sc = i𝜔𝜌u∇u. P

(1.20)

In the far field, ∇u ≈ iku ⋅ R̂

with R̂ = ∇R.

(1.21)

Therefore, the power flux density averaged over the oscillation period T = 2𝜋∕𝜔 equals | | ̂ ⃗ sc = 1 Re[p∗ 𝑣] P ⃗ = 12 Re[(i𝜔𝜌 u)∗ (iku] ⋅ R̂ = 12 k2 Z |u2 | ⋅ R, av 2 | |

(1.22)

Z = 𝜌c

(1.23)

where

is the characteristic impedance of the medium. Usually, the far field is characterized by the bistatic cross-section 𝜎 introduced through the relation Psc av =

𝜎 ⋅ Pinc av 4𝜋R2

,

(1.24)

where 1 2 | inc |2 Pinc av = 2 k Z |u |

(1.25)

is the power flux density of the incident wave. This definition suggests the following interpretation. The bistatic cross-section is the area 𝜎 of a hypothetical plate perpendicular to the direction of the incident wave. This plate intercepts the incident power Pinc av ⋅ 𝜎 and distributes it uniformly into the entire surrounding space with a power flux density equal to the actual power flux density scattered by the object in the direction of observation. Because the scattered power depends on the direction of scattering, the

6

BASIC NOTIONS IN ACOUSTIC AND ELECTROMAGNETIC DIFFRACTION PROBLEMS

scattering cross-section 𝜎 is a function of this direction. The term bistatic means that the direction of scattering can be arbitrary. When the scattering direction coincides with the direction to the source of the incident wave, the quantity 𝜎 is called the backscattering or monostatic cross-section. Thus, according to Equations (1.22) and (1.24), 𝜎 = 4𝜋R2

Psc av Pinc av

= 4𝜋R2

|usc |2 |uinc |2

.

(1.26)

Where the field scattered from a smooth convex surface has a ray structure, the bistatic cross-section is predicted by the geometrical optics (geometrical acoustics) and equals 𝜎 = 𝜋𝜌1 𝜌2 .

(1.27)

Here 𝜌1 and 𝜌2 are principal radii of curvature of the scattering surface at the reflection point. It is also assumed that this surface is perfectly reflecting (soft or hard). Two interesting features of this quantity should be emphasized. First, the expression (1.27) is universal. It is applicable for both acoustic and electromagnetic waves because the ray structure does not depend on the nature of the waves and is determined totally by the geometry of the scattering surface. If the geometry is the same, the divergence of reflected rays will be the same for both acoustic and electromagnetic rays. Also, the modulus of reflection coefficient for any perfectly reflecting surfaces (soft or hard for acoustic waves, or perfectly conducting for electromagnetic waves) equals unity. However, just these two factors, the ray divergence and the reflection coefficient, totally determine the amplitude of reflected rays, and eventually the bistatic cross-section. Formula (1.27) can be generalized for imperfect reflecting surfaces: 𝜎 = |ℜ|2 𝜋𝜌1 𝜌2 ,

(1.28)

where ℜ is the reflection coefficient, which can be different for acoustic and electromagnetic waves. The second interesting and not obvious feature of Equation (1.27) is the following. This expression does not depend on the angle between the incident and reflected rays at the same reflection point (Fig. 1.3). In other words, it is constant for any bistatic angles, including the zero angle related to backscattering. This property of scattering from perfectly reflecting objects follows from the theory of Fock (1965), as shown by Ufimtsev (1999). It was also noted by Crispin and Siegel (1968) and by Ruck et al. (1970). The theory of Fock (1965) is more general. It is also valid for imperfect scattering surfaces characterized by the reflection coefficient ℜ. In this case, Fock’s theory leads straight to Equation (1.28), where ℜ depends on the bistatic angle as well as on the boundary conditions.

PHYSICAL OPTICS

7

Figure 1.3 Scattering from the same reflection point (at the same reflecting object) for various bistatic angles. The bistatic cross-section 𝜎 of this perfectly reflecting object is constant for all of these angles and equals the monostatic cross-section.

1.3

PHYSICAL OPTICS

This high-frequency approach is widely used in acoustic and electromagnetic diffraction problems.

1.3.1

Definition of Physical Optics

In electromagnetics, the term physical optics (PO) is usually applied to the highfrequency asymptotic technique introduced by Kirchhoff (1883) and Macdonald (1912). In this book it is also applied in acoustical problems, although the term Kirchhoff approximation is more common there (Brill and Gaunaurd, 1993; Moser et al., 1993; Menounou et al., 2000). Basic features of this approximation are well known and exposed in many references (Ruck et al., 1970; Knott et al., 1993; Ufimtsev, 1999, 2008a,b; Boag, 2004; Vico-Bondia et al., 2010; see also Section 1.6). Physical optics is a constituent part of the physical theory of diffraction (PTD) developed in this book to study diffraction at objects placed in a free homogeneous space. Extension of PO for opaque objects is presented in Sections 1.3.7 and 1.4.3. PO and PTD for metallic bodies located in inhomogeneous ionized media (plasma) have been developed in (Yarygin, 1972; Pereversev and Ufimtsev, 1976). According to PO, the field induced on the surface of a scattering object is determined by geometrical optics (geometrical acoustics). The physics behind this is as follows. Geometrical optics describes a wave field in the limiting case when a wavelength tends to zero (Born and Wolf, 1980). With respect to such a small wavelength, the scattering surface at the reflection point can be considered approximately as a tangential plane. Therefore, the surface field induced at the tangential infinite plane is a good high-frequency approximation for true scattering sources induced on a large scattering object. Two such planes, P1 and P2 , tangent at points Q1 and Q2 , are shown in Figure 1.4. These points are located on the “illuminated” side of the object. Notice that according to geometrical optics, the field is zero in the shadow region, including points on the object surface.

8

BASIC NOTIONS IN ACOUSTIC AND ELECTROMAGNETIC DIFFRACTION PROBLEMS

P1 shadow cross-section Q1 shadow region

Sil

z

Q2 P2

Figure 1.4 Surface fields induced on scattering objects at points Q1 and Q2 are asymptotically identical to the fields induced at tangential planes P1 and P2 , respectively. Sil is the illuminated part of the object surface. A dark plate behind the object displays the cross-section of the geometrical shadow region.

Thus, reflection from a tangential plane is an appropriate canonical (“fundamental”) problem. Its exact solution can be found easily using geometrical optics as well as by image theory. The total field generated by an external source above an infinite reflecting plane (Fig. 1.5) is the sum of the incident and reflected fields. The latter can be interpreted as the field created by the image source. On an acoustically soft plane, the total field is zero as a result of the boundary condition (1.5), but its normal derivative equals 𝜕us 𝜕uinc =2 , 𝜕n 𝜕n

(1.29)

due to the exact solution of this problem. On an acoustically hard plane, the normal derivative of the total field is zero as a result of boundary condition (1.6), and the field itself equals uh = 2uinc ,

(1.30)

as follows from the solution of this reflection problem. In the general PTD, these quantities are interpreted as the uniform components of induced sources on a smooth convex scattering surface: j(0) s =2

𝜕uinc , 𝜕n

= 2uinc . j(0) h

(1.31)

source *

nˆ

image * Figure 1.5 Reflection from an infinite plane.

PHYSICAL OPTICS

9

These expressions define the induced sources only on the illuminated part of the scattering object. On the shadowed part, these components are set to zero. By substituting Equation (1.31) into Equation (1.10), one obtains expressions for the scattered field at any distance from the scatterer (Fig. 1.4):

(0) uPO s ≡ us = −

uPO ≡ u(0) = h h

1 eikr j(0) ds, s ∫ 4𝜋 Sil r

1 𝜕 eikr j(0) ds. h 4𝜋 ∫Sil 𝜕n r

(1.32)

These expressions represent the scalar physical optics approximation, also known in acoustics as the Kirchhoff approximation. In the present book we use the term physical optics for both acoustic and electromagnetic waves. The symbols u(0) s and u(0) are introduced in Equation (1.32) to emphasize that these fields are generated by h the uniform component of the surface sources induced. Thus, physical optics, which deals with these uniform components, is a constituent part of the general PTD. The physical optics of Equation (1.32) possesses a special property related to the field scattered in the direction toward the source of the incident wave. According to Equations (1.16) and (1.17), the PO far field is determined as u(0) s =−

1 eikR 𝜕uinc −ikr′ cos Ω ds, e 2𝜋 R ∫Sil 𝜕n

(1.33)

=− u(0) h

′ ik eikR ̂ ⋅ n̂ ) ds. uinc e−ikr cos Ω (m 2𝜋 R ∫Sil

(1.34)

The field incident on the scattering object (being a substantial distance from the source) can be represented in the form i

uinc = const eik𝜙 .

(1.35)

The unit vector ∇𝜙i = k̂ i indicates the direction of the incident wave, and the unit ̂ = ∇r = k̂ s shows the direction of scattering. In the case of backscattering, vector m the equality k̂ s = −k̂ i is valid. Note also that 𝜕uinc = ∇uinc ⋅ n̂ = ikuinc (∇𝜙i ⋅ n̂ ) = ikuinc (k̂ i ⋅ n̂ ). 𝜕n

(1.36)

̂ ⋅ n̂ = −(k̂ i ⋅ n̂ ) into Equations The substitution of Equation (1.36) and the quantity m (1.33) and (1.34) leads to the equation (0) u(0) s = −uh = −

′ ik eikR uinc e−ikr cos Ω (k̂ i ⋅ n̂ ) ds. 2𝜋 R ∫Sil

(1.37)

10

BASIC NOTIONS IN ACOUSTIC AND ELECTROMAGNETIC DIFFRACTION PROBLEMS

Hence, in the frame of the physical optics approximation, the backscattered fields created by soft and hard objects (of the same shape and size) have equal magnitudes and differ only in sign. 1.3.2

Total Scattering Cross-Section

The power flux density of the scattered waves is defined by Equation (1.22). By the integration of this quantity over the object surface, one can find the total power scattered in all directions. In the PO approximation, the total power scattered from an acoustically soft object equals Ptot = 12 Re

∫Sil

( sc )∗ ( tot ) ps 𝑣⃗s ⋅ n̂ ds,

(1.38)

where ̂ =2 𝑣⃗tot s ⋅n

𝜕uinc , 𝜕n

(1.39)

inc = −i𝜔𝜌uinc . The and in accordance with the boundary condition (1.5), psc s = −p incident wave in the vicinity of the scattering object can be approximated by the plane wave (Fig. 1.4):

uinc = u0 eikz .

(1.40)

Then ̂ =2 𝑣⃗tot s ⋅n

𝜕uinc = 2iku0 eikz (̂z ⋅ n̂ ), 𝜕n

(

psc s

)∗

= i𝜔𝜌u∗0 e−ikz

(1.41)

and

Ptot = −k𝜔𝜌|u0 |2

2

∫Sil

(̂z × n̂ ) ds = k2 ZA|u0 | ,

(1.42)

where A is the area of the object’s projection on a plane perpendicular to the direction of propagation or, in other words, the area of the shadow cross-section (Fig. 1.4). In view of Equation (1.25), the power flux density of the incident wave equals 1 2 2 Pinc av = 2 k Z|u0 | .

(1.43)

11

PHYSICAL OPTICS

The total cross-section is defined by the ratio 𝜎 tot =

Ptot Pinc av

(1.44)

and equals

𝜎 tot = 2A.

(1.45)

This result is also valid for hard objects and for perfectly conducting objects which scatter electromagnetic waves. It can easily be verified for hard objects. Indeed, in this case, Ptot = 12 Re

( ∫Sil

ptot h

)∗ (

) ̂ 𝑣⃗sc h ⋅ n ds,

inc ptot = 2i 𝜔𝜌 u0 eikz , h = 2p

(1.46)

̂ 𝑣⃗sc ⃗inc ⋅ n̂ ) = −iku0 eikz (̂z ⋅ n̂ ). h ⋅ n = −(𝑣

(1.47)

Substitution of Equation (1.47) into (1.46) leads to Equations (1.42) and (1.45). 1.3.3

Optical Theorem

There is a specific connection between the total scattering cross-section and the far scattered field in the shadow/forward direction. In the PO approximation, the far-field expressions (1.18) and (1.19) take the form PO uPO s = u0 Φs

eikR , R

= u0 ΦPO uPO h h

eikR , R

(1.48)

where ΦPO s =−

1 𝜕uinc −ikr′ cos Ω ds, e 2𝜋 u0 ∫Sil 𝜕n

(1.49)

ΦPO =− h

′ ik ̂ ⋅ n̂ ) ds, uinc e−ikr cos Ω (m 2𝜋 u0 ∫Sil

(1.50)

with cos Ω as defined in Equation (1.13). The incident wave (1.40) propagates in the z-direction. For the observation point in the forward direction, we have 𝜗 = 0, ̂ = ẑ , r′ cos 𝜗′ = z′ , and cos Ω = cos 𝜗′ , m ′ 𝜕uinc = ∇uinc ⋅ n̂ = iku0 eikz (̂z ⋅ n̂ ) 𝜕n

(1.51)

12

BASIC NOTIONS IN ACOUSTIC AND ELECTROMAGNETIC DIFFRACTION PROBLEMS

and also PO ΦPO s (𝜗 = 0) = Φh (𝜗 = 0) = −

ik ik (̂z ⋅ n̂ ) ds = A, 2𝜋 ∫Sil 2𝜋

(1.52)

where A is the area of the shadow cross-section (Fig. 1.4). Comparison of Equation (1.52) with Equation (1.45) shows that

𝜎 tot =

4𝜋 ImΦ(𝜗 = 0). k

(1.53)

This equation is well known as the optical theorem (Born and Wolf, 1980). 1.3.4

Introducing Shadow Radiation

Shadow radiation was introduced for electromagnetic waves by Ufimtsev (1968). It was later investigated again by Ufimtsev (1990) and discussed by him (1996). A significant part of the results related to electromagnetic shadow radiation was included in Theory of Edge Diffraction in Electromagnetics (Ufimtsev, 2003, 2009). In the present section, the notion of shadow radiation is introduced in conjunction with scalar waves. Consider again the reflection from an infinite perfectly reflecting plane (Fig. 1.6) located in a homogeneous medium. The scattered field in the region z > 0 is determined by the Helmholtz integral expression (Baker and Copson, 1939): ) ( ikr 1 𝜕utot eikr tot 𝜕 e − ds, u u = 4𝜋 ∫S 𝜕n r 𝜕n r sc

(1.54)

where utot = uinc + usc is the total field, ds = dx dy is a differential area of the infinite plane S (z = 0), and r is the distance between the integration and observation points. source *

P(x,y,z)

z r S

nˆ x

nˆsh

image * Figure 1.6 Reflection of waves from an infinite plane S in a homogeneous medium. The source is in the region z > 0. The total field in the shadow region (z < 0) equals zero.

PHYSICAL OPTICS

13

Let the reflecting plane be acoustically soft. Then, on its surface, inc usc s = −u ,

𝜕utot s

=2

𝜕n

𝜕uinc 𝜕uinc 𝜕uinc = + . 𝜕n 𝜕n 𝜕n

(1.55)

Therefore, Equation (1.54) can be rewritten as sc = usc utot,sc s s,1 + us,2 ,

(1.56)

where ) ( 1 𝜕 eikr 𝜕uinc eikr − ds, uinc 4𝜋 ∫S 𝜕n r 𝜕n r ) ( 1 𝜕 eikr 𝜕uinc eikr = − ds. −uinc 4𝜋 ∫S 𝜕n r 𝜕n r

usc s,1 =

(1.57)

usc s,2

(1.58)

To evaluate the integral in Equation (1.57), we utilize the Helmholtz equivalency theorem (Baker and Copson, 1939). According to this theorem, the field u created by the acoustic source at point P in a homogeneous medium (Fig. 1.7) can be represented as the radiation generated by the equivalent sources uinc and 𝜕uinc ∕𝜕N distributed over a closed imaginary surface Σ of volume V: { ) ( ikr 0 𝜕 uinc eikr 1 inc 𝜕 e − ds = u u(P) = 4𝜋 ∫Σ 𝜕N r 𝜕N r uinc (P)

when

P inside V,

when

P outside V. (1.59)

Here it is supposed that the source of the incident wave is located inside the volume V. One should emphasize the following wonderful property of this theorem. The field at point P inside or outside V does not depend on the shape of its surface Σ. One can deform this surface in any way, but the result of the integration in Equation (1.59) will Nˆ

Σ

* source

nˆ

V

Nˆ Figure 1.7 The equivalency principle. Σ is an arbitrary imaginary surface covering volume V of a free homogeneous medium, n̂ and N̂ are the inward and outward unit vectors normal to Σ, respectively, and a source of the incident wave is inside V.

14

BASIC NOTIONS IN ACOUSTIC AND ELECTROMAGNETIC DIFFRACTION PROBLEMS

Nˆ nˆ * source

r R

HR V

nˆ Nˆ

r P

SR P

Figure 1.8 Surface of integration Σ = SR + HR in Equation (1.60). A source of the incident wave is inside the volume V.

be the same: u(P) = 0 if P ∈ V and u(P) = uinc (P) if P ∉ V. Of course, it is assumed that during such a deformation of the surface Σ, it does not intersect point P. To evaluate integral (1.57), we first apply the equivalency theorem (1.59) to the closed surface Σ = SR + HR (Fig. 1.8). Here SR is a circular plate with a radius R, which is a part of the infinite plane S shown in Figure 1.6, and HR is a hemisphere with the same radius R. It is supposed that the source of the incident wave is inside the volume V. According to the equivalency theorem (1.59), 1 4𝜋 ∫SR +HR

{ ) ( ikr 0 𝜕uinc eikr inc 𝜕 e − ds = u 𝜕N r 𝜕N r uinc (P)

when

P inside V,

when

P outside V (1.60)

or after replacement of N̂ by −̂n, 1 4𝜋 ∫SR +HR

{ ) ( ikr inc eikr 0 𝜕 e 𝜕u − ds = uinc 𝜕n r 𝜕n r −uinc (P)

when

P inside V,

when

P outside V. (1.61)

One can show that the field at observation point P generated by the equivalent sources distributed over HR vanishes when the radius R of HR tends to infinity. Note also that with R → ∞, the surface SR is transformed into the infinite plane S. Taking these observations into account, we finally obtain the following values for the function (1.57): { usc s,1 =

0

in the region z > 0,

−uinc

in the region z < 0.

(1.62)

PHYSICAL OPTICS

15

The physical meaning of the field usc is clear. It cancels the incident wave in the s,1 region z < 0, creating the complete shadow there. That is why we call this field the shadow radiation and denote it by ush : ) ( ikr 1 𝜕uinc eikr inc 𝜕 e u = − ds. u 4𝜋 ∫S 𝜕n r 𝜕n r sh

(1.63)

With respect to this equation, the surface S can be interpreted as perfectly absorbing (i.e., black), as it does not reflect the incident wave (Ufimtsev, 1968, 1990, 1996, 2008a, 2009). , we introduce new denotations To clarify the physical meaning of the function usc s,2 in Equation (1.58), 𝜕urefl s

inc urefl s = −u ,

𝜕n

=

𝜕uinc , 𝜕n

(1.64)

and apply the equivalency theorem to the surface Σ = SR + HR∗ shown in Figure 1.9. It is supposed that the image source (i.e., the source of the reflected field) is inside the volume V ∗ . According to the equivalency theorem, 1 u(P) = 4𝜋 ∫SR +H ∗ R { 0 = urefl (P)

( refl

u

𝜕 eikr 𝜕urefl eikr − 𝜕n r 𝜕n r

when

P inside V ∗ ,

when

P outside V ∗ .

) ds

(1.65)

P nˆ

z SR

HR*

P

V*

* image source

R

nˆ

Figure 1.9 The equivalency theorem applied to the reflected field. Here, as in Figure 1.8, SR is a circular plate on the plane z = 0 with radius R, and HR∗ is a hemisphere with the same radius R.

16

BASIC NOTIONS IN ACOUSTIC AND ELECTROMAGNETIC DIFFRACTION PROBLEMS

When R → ∞, the integral over HR∗ tends to zero, SR is transformed into the infinite plane S, and therefore ( usc s,2

=

urefl s {

=

1 = 4𝜋 ∫S

𝜕 urefl s 𝜕n

refl eikr 𝜕us eikr − r 𝜕n r

urefl s

in the region z > 0,

0

in the region z < 0.

) ds

(1.66)

Thus, the function (1.58) represents the reflected field in the region z > 0. The field (1.54) scattered by the hard infinite plane S can also be represented as the sum of the reflected field and the shadow radiation: refl sh usc h = uh + u ,

where

( urefl h

𝜕 urefl h 𝜕n

1 = 4𝜋 ∫S

(1.67)

refl eikr 𝜕uh eikr − r 𝜕n r

) ds

(1.68)

and on the plane S, inc urefl h =u ,

𝜕urefl h 𝜕n

=−

𝜕uinc . 𝜕n

(1.69)

The shadow radiation does not depend on the boundary conditions and is the same for both soft and hard planes. It is defined by Equation (1.63). The above definitions of reflected and shadow radiations are applicable to the PO field scattered by arbitrary soft and hard objects. In this case, however, the integration surface in (1.63), (1.66), and (1.68) must be specified as the illuminated side (Sil ) of the object (Fig. 1.4). Thus, in general, sh uPO = urefl s,h + u , s,h

(1.70)

where ) refl eikr 𝜕us,h eikr − ds, r 𝜕n r ) ( ikr 1 𝜕uinc eikr inc 𝜕 e = − ds u 4𝜋 ∫Sil 𝜕n r 𝜕n r (

urefl s,h ush

1 = 4𝜋 ∫Sil

𝜕 urefl s,h 𝜕n

(1.71)

(1.72)

PHYSICAL OPTICS

17

and urefl and 𝜕urefl ∕𝜕n are as defined by Equations (1.64) and (1.69). As shown below s,h s,h in Sections 1.3.7 and 1.4.3, the PO representation in the form (1.70) is valid for any large opaque objects characterized by arbitrary reflection coefficients (Ufimtsev, 2008a). Also, note other important relationships among the PO field, reflected field, and shadow radiation. Substituting the boundary values (1.64) and (1.69) into Equation (1.71), one finds that for any scattering direction,

refl urefl s = −uh .

(1.73)

Then, according to Equations (1.70) and (1.73), one can represent the shadow radiation in the form

ush =

1 ( PO u 2 s

) . + uPO h

(1.74)

In addition, it follows from Equations (1.70), (1.73), and (1.74) that

urefl s =

1 ( PO u 2 s

) , − uPO h

urefl h =

1 ( PO u 2 h

) . − uPO s

(1.75)

The significance of Equations (1.74) and (1.75) is that they allow one to find the shadow radiation and reflected fields if the PO soft and hard fields are known. Shadow radiation (1.72) can be interpreted as the field scattered by a blackbody (Kirchhoff, 1883). For electromagnetic waves, a notion similar to that of a blackbody was introduced by Macdonald (1912), and the electromagnetic shadow radiation was investigated by Ufimtsev (1968, 1990, 2008a, 2009). 1.3.5

Shadow Contour Theorem and the Total Scattering Cross-Section

Among the properties of the shadow radiation, the most significant are the shadow contour theorem and the total power of shadow radiation. They have already been established for electromagnetic waves (Ufimtsev, 1968, 1990, 1996, 2009) and will now be verified for acoustic waves. Let us compare the shadow radiation generated by two scattering objects with different shapes but with the same shadow contour (Fig. 1.10). Their illuminated sides are S1 and S2 .

18

BASIC NOTIONS IN ACOUSTIC AND ELECTROMAGNETIC DIFFRACTION PROBLEMS

n1

n2

S1

S2

Figure 1.10 Two different objects with the same shadow contour (dotted lines).

According to Equation (1.72), ush 1 = ush 2

1 4𝜋 ∫S1

1 = 4𝜋 ∫S2

( uinc (

𝜕 eikr 𝜕uinc eikr − 𝜕n1 r 𝜕n1 r

𝜕 eikr 𝜕uinc eikr − uinc 𝜕n2 r 𝜕n2 r

) ds, (1.76)

) ds.

Taking these quantities, their difference can be written as ush 1

− ush 2

1 = 4𝜋 ∫S1 +S2

( inc

u

𝜕 eikr 𝜕uinc eikr − 𝜕n r 𝜕n r

) ds,

(1.77)

where n̂ = n̂ 1 , n̂ = −̂n2 is the external normal to the surface S1 + S2 (Fig. 1.11). As a result of the Helmholtz equivalency principle (1.59), the quantity (1.77) equals zero for observation points P outside the volume closed by the surface S1 + S2 . Therefore, sh ush 1 = u2 .

(1.78)

n

n

S1 S2

Figure 1.11 Surface S1 + S2 in a homogeneous medium. All sources and observation points are outside the volume closed by this surface.

19

PHYSICAL OPTICS

This equation represents the shadow contour theorem:

The shadow radiation does not depend on the entire shape of a scattering object and is determined completely only by the size and geometry of the shadow contour.

This theorem emphasizes the role that the shadow contour plays in diffraction theory and explains the well-known results (Maggi, 1888; Rubinowicz, 1917; Kottler, 1923a,b; Ufimtsev, 1990), where the surface integral (1.72) is reduced to a linear integral over the shadow contour plus (or minus) the incident wave. Now let us evaluate the total power of the reflected field and shadow radiation. The total power of the reflected field can be written as 1 Prefl s,h = 2 Re

∫Sil

( refl )∗ ( refl ) ps,h 𝑣⃗s,h ⋅ n̂ ds,

(1.79)

where according to Equations (1.2), (1.41), (1.64), and (1.69), ̂ = 𝑣⃗inc ⋅ n̂ = iku0 eikz (̂z ⋅ n̂ ), 𝑣⃗refl s ⋅n

inc = −i𝜔𝜌u0 eikz , prefl s = −p

(1.80) prefl h

inc

=p

ikz

= i𝜔𝜌u0 e ,

𝑣⃗refl h

⋅ n̂ = −𝑣⃗

inc

⋅ n̂ = −iku0 e (̂z ⋅ n̂ ). ikz

Substitution of these quantities into Equation (1.79) results in 1 2 refl 2 inc ⋅A Prefl s = Ph = 2 k ZA|u0 | = P

(1.81)

and

𝜎 refl,tot = A,

(1.82)

where A is the area of the shadow region cross-section (Fig. 1.4). These equations show that the total power of the reflected field exactly equals the power of the intercepted incident rays. The scattering object only distributes them in the surrounding space. By changing the shape of the illuminated surface Sil , one can significantly decrease the backscattering by deflection of the reflected rays from the direction back to the source of the incident wave. This is the first basic idea used in stealth technology (Ufimtsev, 1996; Lynch, 2004; Alekseev et al., 2007). The total power of the shadow radiation is determined by Psh,tot = 12 Re

∫Sil

(pinc )∗ [𝑣⃗inc ⋅ (−̂n)] ds.

(1.83)

20

BASIC NOTIONS IN ACOUSTIC AND ELECTROMAGNETIC DIFFRACTION PROBLEMS

The minus sign in front of n⃗ is chosen because on the surface Sil of blackbodies, no radiation or reflection exists in the positive normal direction. According to Equations (1.2) and (1.40), pinc = i𝜔𝜌u0 eikz

and

(𝑣⃗inc ⋅ n̂ ) = iku0 eikz (̂z ⋅ n̂ ).

(1.84)

Substitution of Equation (1.84) into Equation (1.83) leads to Psh,tot = 12 k2 ZA|u0 |2 = Pinc A

(1.85)

and

𝜎 sh,tot = A.

(1.86)

Thus, the shadow radiation power equals the reflected power, and their sum exactly equals the total scattered power (1.42). This result elucidates the physics behind the fundamental diffraction law (1.45). It shows that objects with soft and hard boundary conditions reveal a dual nature. They can be interpreted as if they are simultaneously perfectly reflecting (with reflection coefficients ℜs = −1 and ℜh = 1) and perfectly absorbing (with ℜ = 0), that is, black. This law can now be written in the form

𝜎 tot = 𝜎 refl,tot + 𝜎 sh,tot = 2A,

(1.87)

where A is the area of the shadow region cross-section. From the equation 𝜎 refl,tot = A, it is also clear that the total power of reflected waves does not depend on the object shaping if the area of shadow cross-section remains constant. However, this power can be decreased by absorbing coatings; this is the second basic idea of stealth technology. In contrast, the shadow radiation cannot be decreased by any absorbing coatings, and it can be used for bistatic detection of large opaque objects with a small backscattering cross-section (Ufimtsev, 1996). 1.3.6

Shadow Radiation and Reflected Field in the Far Zone

We refer to Figures 1.2 and 1.4 and choose an incident wave as the plane wave (1.40). Also, we apply the far-field approximations (1.12) to (1.15) and the quantity ′ ′ ′ 𝜕 𝜕uinc = ∇′ u0 eikz ⋅ n̂ = u0 ′ eikz (̂z ⋅ n̂ ) = iku0 eikz (̂z ⋅ n̂ ). 𝜕n 𝜕z

(1.88)

PHYSICAL OPTICS

21

Then the shadow radiation (1.72) and reflected field (1.71) can be approximated by ush = −iku0

′ ′ 1 eikR ̂ ⋅ n̂ ) + (̂z ⋅ n̂ )]eik(z −r cos Ω) ds, [(m 4𝜋 R ∫Sil

refl urefl h = −us = iku0

′ ′ 1 eikR ̂ ⋅ n̂ ) + (̂z ⋅ n̂ )]eik(z −r cos Ω) ds. [−(m ∫ 4𝜋 R Sil

(1.89) (1.90)

In the backscattering case, when 𝜗 = 𝜋,

cos Ω = − cos 𝜗′ ,

̂ = −̂z, m

r′ cos Ω = −z′ ,

(1.91)

it follows from Equations (1.89) and (1.90) that ush = 0,

refl urefl h = −us = iku0

′ 1 eikR (̂z ⋅ n̂ )e2ikz ds. 2𝜋 R ∫Sil

(1.92)

However, in the forward direction, when 𝜗 = 0,

cos Ω = cos 𝜗′ ,

̂ = ẑ , m

r′ cos Ω = z′ ,

(1.93)

Equations (1.89) and (1.90) state that ush = −iku0

1 eikR A eikR (̂z ⋅ n̂ ) ds = iku0 , 2𝜋 R ∫Sil 2𝜋 R

refl urefl h = us = 0.

(1.94) (1.95)

These equations reveal the fundamental features of the shadow radiation and reflected field in the far zone. In a high-frequency approximation, the shadow radiation equals zero in the backscattering direction (𝜗 = 𝜋); however, it is maximal in the forward/shadow direction (𝜗 = 0). To the contrary, the reflected field is zero in the forward direction. However, in the backscattering direction, it is maximal and equals refl urefl h = −us = −iku0

1 eikR A 2𝜋 R

(1.96)

if the illuminated side of the object is a plate perpendicular to the incident wave direction. In this particular case, the magnitude of the reflected field is the same as that of the shadow radiation in the forward direction. Notice that application of the optical theorem (1.53) to the field (1.94) leads directly to Equation (1.87) for the total cross-section of soft and hard scattering objects. Now consider shadow radiation in the directions of reflected rays. They are generated on the object surface in the vicinity of stationary points of the phase factor exp[ik(z′ − r′ cos Ω)] in Equation (1.90). Discrete stationary points produce ordinary reflected rays, while continuous sets of stationary points (stationary lines and spots)

22

BASIC NOTIONS IN ACOUSTIC AND ELECTROMAGNETIC DIFFRACTION PROBLEMS

ˆ m nˆ

α

kˆ inc z Figure 1.12 Geometry of reflection.

generate reflected beams (Ufimtsev, 1996, 2009). The integrand in Equation (1.89) for shadow radiation has the same stationary points. However, their contributions to the scattered field vanish asymptotically because the integrand terms cancel each other. Indeed, according to Figure 1.12, ẑ ⋅ n̂ = cos 𝛼,

̂ ⋅ n̂ = cos 𝛽, m

𝛽 = 𝜋 − 𝛼.

(1.97)

̂ ⋅ n̂ = −(̂z ⋅ n̂ ) and according to Equation (1.89), one immediately obtains Therefore, m ush = 0. That is, in a high-frequency approximation, the shadow radiation equals zero in the direction of reflected rays. 1.3.7

Shadow Radiation and Reflection from Opaque Objects

According to GO, the field on the illuminated surface of a scattered object is determined by the local reflection coefficient, ℜ: 𝜕u 𝜕uinc = (1 − ℜ) . 𝜕n 𝜕n

u = (1 + ℜ)uinc ,

(1.98)

Hence, the PO field is given by u=

] [ 1 𝜕 eikr 𝜕uinc eikr − (1 − ℜ) ds. (1 + ℜ)uinc 4𝜋 ∫Sil 𝜕n r 𝜕n r

(1.99)

In this case, separation of the field is straightforward: u = urefl + ush ,

(1.100)

where the reflected component refl

u

1 = 4𝜋 ∫Sil

( ℜu

inc

𝜕 eikr 𝜕uinc eikr +ℜ 𝜕n r 𝜕n r

) ds

(1.101)

and the shadow radiation is determined by Equation (1.72). Transition to the particular cases of acoustically soft and hard objects is obvious: ℜ = −1 for soft objects and

ELECTROMAGNETIC WAVES

23

ℜ = 1 for hard objects. In view of Equations (1.12) to (1.15) and (1.88), the reflected field in the far zone can be represented as refl urefl h = −us = iku0

′ ′ 1 eikR ̂ ⋅ n̂ ) + (̂z ⋅ n̂ )]ℜeik(z −r cos Ω) ds. [−(m 4𝜋 R ∫Sil

(1.102)

̂ = ẑ , and hence urefl = 0 in the forward direction From Equation (1.93) we have m (𝜗 = 0) for any opaque objects.

1.4 1.4.1

ELECTROMAGNETIC WAVES Basic Field Equations and PO Backscattering

In this section we present briefly basic notions used in the book to describe electromagnetic waves. We study the diffraction of electromagnetic waves primarily at perfectly conducting bodies that are large compared to the wavelength. It is assumed ⃗ and that the waves and scattering objects are in free space (vacuum). The electric (E) ⃗ magnetic vectors (H) of the wave field are determined as ⃗ e ) + k2 A ⃗ m, ⃗ e] − ∇ × A ⃗ = i Z0 ⋅ [∇(∇ ⋅ A E k i ⃗ m ) + k2 A ⃗ e. ⃗ m] + ∇ × A [∇(∇ ⋅ A H⃗ = kZ0

(1.103) (1.104)

√ Here √ Z0 = 1∕Y0 = 𝜇0 ∕𝜀0 = 120𝜋 ohms is the impedance of free space, k = 𝜔 𝜀0 𝜇0 = 2𝜋∕𝜆 is the wave number, and ikr ⃗je e d𝑣 ⃗e = 1 A 4𝜋 ∫ r

and

ikr ⃗m = 1 ⃗jm e d𝑣 A 4𝜋 ∫ r

(1.105)

are the electric and magnetic vector-potentials. They are the solutions of the equation ⃗ e,m = −⃗je,m , ⃗ e,m + k2 A ΔA

(1.106)

where ⃗je (⃗jm ) is the electric (magnetic) current density of the field sources. Note that Equations (1.103) to (1.106) are more convenient for calculation than those usually accepted in books on engineering electromagnetics. Indeed, Equations (1.105) and (1.106) have exactly the same form for both electric and magnetic potentials. The second terms (outside the brackets) in Equations (1.103) and (1.104) do not contain the factors 1∕𝜇0 and 1∕𝜀0 , which eventually disappear in the integral field expressions.

24

BASIC NOTIONS IN ACOUSTIC AND ELECTROMAGNETIC DIFFRACTION PROBLEMS

In the far zone from a scattering object, one can use the following approximation [similar to Equations (1.16) and (1.17)] associated with Figure 1.2: ikR ⃗je,m e−ikr′ cos Ω d𝑣, ⃗ e,m = 1 e A 4𝜋 R ∫

(1.107)

where cos Ω = cos 𝜗 cos 𝜗′ + sin 𝜗 sin 𝜗′ cos(𝜑 − 𝜑′ ). This approximation leads to the field components: ) ( E𝜗 = Z0 H𝜑 = ik Z0 Ae𝜗 + Am 𝜑 , ) ( E𝜑 = −Z0 H𝜗 = ik Z0 Ae𝜑 − Am 𝜗 .

(1.108) (1.109)

The radial components ER and HR are on the order of 1∕R2 and are neglected here. The coordinate system is shown in Figure 1.2. The scattering cross-section is determined by Equations (1.9) and (1.26) as |E |2 | ⃗ sc | 𝜎 = 4𝜋R2 | | . |E | | ⃗ inc |

(1.110)

Equation (1.45) for the total scattering cross-section is also valid for electromagnetic waves. In the case of an incident wave with linear polarization, it can be represented in the form of Equation (1.53) as 𝜎 tot =

) ( sc 4𝜋 E −ikR , ⋅ Re Im k Einc

(1.111)

where Esc is the field scattered in the forward direction. The PO approximation for the far field scattered by perfectly conducting objects is defined by ikR ⃗j(0) e−ikr′ cos Ω ds, ⃗e = 1 e A 4𝜋 R ∫Sil

⃗ m = 0, A

(1.112)

where Sil is the illuminated side of the scattering surface and ⃗j(0) = 2[̂n × H⃗

inc

]

(1.113)

is the uniform component of the surface electric current induced by the incident wave on the illuminated side of the scattering object. A paper by Ufimtsev (1999) describes in detail the properties of the PO approximation for electromagnetic waves (see also Ruck et al., 1970).

ELECTROMAGNETIC WAVES

25

y Sil

γ

z

ϑˆ P(R, ϑ , φ ) Backscattering at point P with coordinates 𝜗 = 𝜋 − 𝛾 and 𝜑 = −𝜋∕2 .

Figure 1.13

Consider the PO approximation for backscattering illustrated in Figure 1.13. The incident plane wave is given in two forms: as E-polarized, Exinc = E0x eik(y sin 𝛾+z cos 𝛾) ,

(1.114)

Hxinc = H0x eik(y sin 𝛾+z cos 𝛾) .

(1.115)

and as H-polarized,

Observation point P is located in the plane x = 0 in the direction 𝜗 = 𝜋 − 𝛾, 𝜑 = −𝜋∕2. Following Ufimtsev [2003, 2009; Equations (4.1.12) and (4.1.13)], the PO backscattered field in the far zone is determined by Ex(0) = − Hx(0) =

′ ik eikR Einc e−ikr cos Ω (k̂ i ⋅ n̂ ) ds, 2𝜋 R ∫Sil x

′ ik eikR H inc e−ikr cos Ω (k̂ i ⋅ n̂ ) ds. 2𝜋 R ∫Sil x

(1.116) (1.117)

Recall that the electromagnetic field components of the plane wave and the PO spherical wave in the far zone are connected by the relationships Hxinc

=−

E𝜗inc Z0

,

Hx(0)

=

E𝜗(0) Z0

.

(1.118)

Hence, Equation (1.117) can be written as E𝜗(0) = −

′ ik eikR E𝜗inc e−ikr cos Ω (k̂ i ⋅ n̂ ) ds. ∫ 2𝜋 R Sil

(1.119)

It follows from Equations (1.116) and (1.119) that in the PO approximation, the field backscattered by convex perfectly conducting objects does not depend on the polarization of the incident wave.

26

BASIC NOTIONS IN ACOUSTIC AND ELECTROMAGNETIC DIFFRACTION PROBLEMS

Comparison of Equations (1.116) and (1.117) with Equation (1.37) reveals the fundamental relationships that exist between the PO approximations for backscattered acoustic and electromagnetic waves:

1.4.2

Ex(0) = u(0) s

if

Exinc = uinc ,

(1.120)

Hx(0) = u(0) h

if

Hxinc = uinc .

(1.121)

PO Field Components: Reflected Field and Shadow Radiation

Utilizing the vector equivalency theorems (Ufimtsev, 2003, 2009) and the idea of Section 1.3.4, one can represent the PO electromagnetic field in a form similar to (1.70). Suppose that the incident plane wave (Fig. 1.4) ⃗ inc = Z0 [H⃗ E

inc

⃗ 0 eikz × ẑ ] = E

(1.122)

propagates in free space. According to PO, this wave excites on the object the electric current ⃗je = 2[̂n × H⃗

inc

].

(1.123)

⃗je = ⃗je,refl + ⃗je,sh ,

(1.124)

We separate this current as

where ⃗je,refl = ⃗je,sh = n̂ × H⃗

inc

.

(1.125)

Then we formally set on the object the magnetic current ⃗jm = ⃗jm,refl + ⃗jm,sh = 0

(1.126)

with the components ⃗ inc , ⃗jm,refl = n̂ × E

⃗jm,sh = −̂n × E ⃗ inc .

(1.127)

According to the theory of blackbodies (Ufimtsev, 1968, 2003, 2009), the currents ⃗je,sh and ⃗jm,sh generate its shadow radiation. Therefore, the currents ⃗je,refl and ⃗jm,refl

ELECTROMAGNETIC WAVES

27

generate the reflected component of the PO field. Thus, the PO field can be represented in a form similar to Equation (1.70): ⃗ (0) = E ⃗ refl + E ⃗ sh , ⃗ PO ≡ E E

H⃗

PO

≡ H⃗

(0)

= H⃗

refl

sh

+ H⃗ .

(1.128)

In the far zone [where R ≫ A∕𝜆 and A is the area of the shadow region cross-section (Fig. 1.4)] the PO components are determined according to Equations (1.108) and (1.109) as E𝜗refl =

inc ′ ik eikR ⃗ inc ] ⋅ 𝜑}e {Z [n × H⃗ ] ⋅ 𝜗̂ + [̂n × E ̂ −ikr cos Ω ds, 4𝜋 R ∫Sil 0

(1.129)

E𝜑refl =

inc ik eikR ̂ −ikr′ cos Ω ds, ⃗ inc ] ⋅ 𝜗}e {Z [n × H⃗ ] ⋅ 𝜑̂ − [̂n × E 4𝜋 R ∫Sil 0

(1.130)

E𝜗sh =

inc ′ ik eikR ⃗ inc ] ⋅ 𝜑}e {Z [n × H⃗ ] ⋅ 𝜗̂ − [̂n × E ̂ −ikr cos Ω ds, 4𝜋 R ∫Sil 0

(1.131)

E𝜑sh =

inc ik eikR ̂ −ikr′ cos Ω ds, ⃗ inc ] ⋅ 𝜗}e {Z [n × H⃗ ] ⋅ 𝜑̂ + [̂n × E 4𝜋 R ∫Sil 0

(1.132)

⃗ refl ] [∇R × E , Z0

(1.133)

H⃗

refl

=

H⃗

sh

=

⃗ sh ] [∇R × E . Z0

Here r′ is the spherical coordinate of an integration point, cos Ω = ⃗r′ ⋅ ∇R, and 𝜗̂ and 𝜑̂ are the unit vectors corresponding to the spherical coordinates 𝜗 and 𝜑 of the field point (R, 𝜗, 𝜑). The coordinate system is shown in Figure 1.2. Basic properties of the shadow radiation for electromagnetic waves have been studied by Ufimtsev (2003, 2009). It was shown that these waves satisfied the shadow contour theorem and complementary principle for black screens. Their directivity patterns were calculated for black half-planes, black strips, and blackbodies of revolution. The total scattering cross-section of blackbodies was found to be the same for both electromagnetic and acoustic waves, as given by Equation (1.86). Also, one can show that the electromagnetic PO components have the same general properties as those of the acoustic components:

r The electromagnetic shadow radiation is equal to zero in the backscattering direction. Proof of this statement is given in Problem 1.10.

r The electromagnetic reflected field is equal to zero in the forward direction. Proof is given in Problem 1.10.

r The electromagnetic shadow radiation in the forward direction is maximal and equal to Exsh = Z0 Hysh = E0x

ik eikR A , 2𝜋 R

(1.134)

28

BASIC NOTIONS IN ACOUSTIC AND ELECTROMAGNETIC DIFFRACTION PROBLEMS

where A is the area of the shadow region cross-section (Fig. 1.4). Proof is given in Problem 1.10. r The electromagnetic shadow radiation equals zero in the directions of reflected rays. Proof is given in Problem 1.11. We emphasize that these statements are correct only with respect to the high-frequency approximation of the scattered field in the far zone. 1.4.3 Electromagnetic Reflection and Shadow Radiation from Opaque Objects Macdonald (1912) formulated PO for imperfectly conducting objects. We consider here a more general problem: scattering of electromagnetic waves from arbitrary opaque objects with local reflection coefficients. In particular, this formulation is applicable for metallic objects covered by absorbing layers. The geometry of the problem is shown in Figures 1.2 and 1.4. In addition to the outward normal n̂ on the object surface, we introduce two orthogonal tangent vectors ̂t and 𝜏̂ in such a way that ̂t × 𝜏̂ = n̂ . We also place vector 𝜏̂ into the incidence plane containing the normal n̂ and the local incident and reflected rays with unit vectors k̂ inc and k̂ refl (Fig. 1.14). The incident wave (1.122) can be decomposed into two parts at each reflecting point: ⃗ inc + E ⃗ inc , ⃗ inc = E E 1 2

H⃗

inc

inc

inc

= H⃗ 1 + H⃗ 2 ,

(1.135)

[ inc ] ⃗ inc = Z0 H⃗ × ẑ . E 2 2

(1.136)

where [ inc H⃗ 1

=

⃗ inc ẑ × E 1 Z0

] ,

inc

⃗ inc and H⃗ are parallel to vector ̂t and perpendicular to the incidence Here vectors E 2 1 plane, ) ( ⃗ inc = 0, n̂ ⋅ E 1

(

inc )

n̂ ⋅ H⃗ 2

= 0.

nˆ kˆ refl

kˆ inc

τˆ ˆt Figure 1.14 Local unit vectors and the incident and reflected rays.

(1.137)

ELECTROMAGNETIC WAVES

29

At the reflection point, the reflected waves are determined by the local reflection coefficients, ℜ1 and ℜ2 , as ⃗ inc , ⃗ refl = ℜ1 E E 1 1

refl

inc

H⃗ 2 = ℜ2 H⃗ 2 .

(1.138)

One can verify that in accordance with the geometrical optics, [

refl ]

n̂ × H⃗ 1

[ inc ] = −ℜ1 n̂ × H⃗ 1 ,

[

] [ ] ⃗ refl = −ℜ2 n̂ × E ⃗ inc . n̂ × E 2 2

(1.139)

The equivalent electric and magnetic currents on the object surface are defined in general as ⃗ ⃗je = [̂n × H],

⃗jm = −[̂n × E ⃗ ].

(1.140)

Their components generating the shadow radiation are determined by [ ( inc inc )] inc ⃗je,sh = n̂ × H⃗ + H⃗ = [̂n × H⃗ ], 1 2 )] [ ( inc ⃗ inc ]. ⃗ 2 = −[̂n × E ⃗jm,sh = − n̂ × E ⃗ +E 1

(1.141)

The current components generating the reflected field are defined as [ ( refl )] ⃗je,refl = n̂ × H refl + H⃗ , 2 1

)] [ ( refl ⃗jm,refl = − n̂ × E ⃗ refl . ⃗ +E 1 2

(1.142)

In view of Equations (1.138) and (1.139), they can be presented in the form [ ] [ inc ] ⃗je,refl = −ℜ1 n̂ × H inc + ℜ2 n̂ × H⃗ , 2 1 [ ] [ ] ⃗ inc + ℜ2 n̂ × E ⃗ inc . ⃗jm,refl = −ℜ1 n̂ × E 1 2

(1.143)

It is obvious that the currents (1.141) create the same shadow radiation as in the case of perfectly conducting objects, and it is given by Equations (1.131) and (1.132). The currents (1.143) generate the reflected field: E𝜗refl =

E𝜑refl =

{ [ ( inc inc )] ik eikR Z n̂ × ℜ2 H⃗ 2 − ℜ1 H⃗ 1 ⋅ 𝜗̂ 4𝜋 R ∫Sil 0 ( [ )] } ⃗ inc − ℜ1 E ⃗ inc ⋅ 𝜑̂ e−ikr′ cos Ω ds, + n̂ × ℜ2 E 2 1

(1.144)

{ [ ( inc inc )] ik eikR Z0 n̂ × ℜ2 H⃗ 2 − ℜ1 H⃗ 1 ⋅ 𝜑̂ ∫ 4𝜋 R Sil ( [ )] } ⃗ inc − ℜ1 E ⃗ inc ⋅ 𝜗̂ e−ikr′ cos Ω ds. − n̂ × ℜ2 E 2 1

(1.145)

30

BASIC NOTIONS IN ACOUSTIC AND ELECTROMAGNETIC DIFFRACTION PROBLEMS

Thus, in the general case for opaque objects, the PO scattered field can also be represented in the form

⃗ =E ⃗ refl + Esh , E

⃗] [∇R × E H⃗ = . Z0

(1.146)

In the case of perfectly conducting objects (ℜ1 = −1, ℜ2 = 1), these expressions transform exactly into Equations (1.129) and (1.130). One can show that according to Equations (1.144) and (1.145), the reflected field in the forward direction (𝜗 = 0) equals zero in the general case, for arbitrary reflection coefficients ℜ1 and ℜ2 . Notice that the reflected field can be reduced significantly in some directions by absorbing coatings with small reflection coefficients. However, this is not the case with shadow radiation. Its power equals the total power incident on a scattering object, and it does not depend on the reflection coefficients. This quantity can be considered as the diffraction limit of the reduction of scattering by application of absorbing coatings on a scattering object. However, in principle it is possible to reduce the shadow radiation by filling in the shadow zone with the field restoring the incident wave (Ufimtsev, 1996). Theoretically, this can be done, for example, for objects with appropriate surface tensor impedance (Erokhin and Kocherzhevsky, 1974, 1979; Kildal et al., 1996) and for “cloaks” with specific tensor permittivity/permeability (Kwon and Werner, 2010), which allow an incident wave to bend an object smoothly and fill in the shadow zone. An object placed inside the volume shielded by such a cloak would be invisible for external observers or detectors. This idea may be of interest in stealth technology when designing objects to be invisible to radar. It is pertinent here to mention basic principles of existing stealth technology: appropriate shaping, use of radar-absorbing materials, and passive and active cancellation of the scattered field. Detailed descriptions of these principles are available (Ruck et al., 1970; Skolnik, 1970; Bhattacharyya et al., 1991; Knott et al., 1993; Lynch, 2004; Alekseev et al., 2007). Comments on these principles from a physical point of view are presented in a survey by Ufimtsev (1996). The shaping idea is to turn the rays reflected by objects away from the direction to radar. Absorbing coatings reduce the power of reflected rays. These techniques have been developed to reduce backscattering, that is, to design objects of low observability to ordinary monostatic radar. A next step in the development of stealth technology is the design of objects invisible to bistatic radars, which can register the scattered field in all directions around the object. The cloaking idea may be of interest for this purpose. It is based on the form invariance of Maxwell’s equations under coordinate transformations (Pendry et al., 2006; Kwon and Werner, 2010; Ozgun and Kuzuoglu, 2010). Realization of this idea requires the construction of appropriate metamaterials (Engheta and Ziolkovski, 2006). In practice, it may be extremely difficult to create such materials in the microwave spectrum for actual objects (e.g., airplanes, vehicles). These materials must be passive (non absorbing) and adaptable to changes in an object’s orientation, radar frequency, and polarization.

PHYSICAL INTERPRETATIONS OF SHADOW RADIATION

1.5 1.5.1

31

PHYSICAL INTERPRETATIONS OF SHADOW RADIATION Shadow Field and Transverse Diffusion

One can suggest the following physical interpretation of the shadow radiation. In the deep-shadow region, at a finite distance from the scattering object, this radiation can be considered as a wave that asymptotically cancels the incident wave: ush ∼ −uinc with k → ∞ (Macdonald, 1912, p. 332). Due to the transverse diffusion, this radiation penetrates the surrounding space (Ufimtsev, 2009, Sec. 5.5; Hacivelioglu et al., 2011, Sec. 5, Fig. 20). For large objects with edges located at the shadow contour, it can be appreciable even far from the forward direction (Ufimtsev, 1990, 2009). However, in this region the shadow radiation cannot be considered to be a satisfactory approximation to a field scattered by real objects. Indeed, away from the forward direction, shadow radiation exists in the form of edge-diffracted waves diverging from the shadow contour and not depending on an object’s material. Obviously, these waves cannot serve as a reasonable approximation to the edge-diffracted waves arising at actual objects with edges. Moreover, in the case of smooth objects, no edge waves exist. An actual region where the shadow radiation can be a reasonable approximation to a scattered field is limited by a main shadow lobe and its near-side lobes in the directivity pattern of the far scattered field. For large convex and smooth objects, the angular size of this region approximately equals Δ𝜗 = 1∕ka, where a is the radius of curvature of the object surface (Ufimtsev, 1990). The following comments are pertinent to clarify the notion of shadow radiation. Two different approaches exist in diffraction theory. One of them, used above in the formulation of PO, is based on the equivalency principle. According to this principle, the total field everywhere (around the scattering object and even inside the object) is considered as the sum of the incident wave and the scattered field. A real scattering object is actually replaced by equivalent sources or currents induced on its surface, or, formally, as if the scattering object no longer exists. Instead of the object, there are equivalent sources distributed in free space and radiating without restraint in all directions. If these sources are defined correctly, the equivalency principle provides the correct solution to the scattering problem. In the PO approximation, only the first term of the high-frequency asymptotic expansion for the shadow field is correct: in a deep shadow (Macdonald, 1912, p. 332) and in close proximity to the shadow boundary (Ufimtsev, 2003, 2009). The total field arising due to diffraction is considered in another approach. It is more representative for the field in the shadow region close to a scattering object. This field appears there due to the diffusion process in the vicinity of the shadow boundary and subsequent interaction of the diffusion field with the object surface. In the case of smooth convex objects, the shadow field consists of creeping waves and surface-diffracted rays (Franz and Depperman, 1952; Depperman and Franz, 1954; Keller, 1962; Fock and Wainshtein, 1963; Fock, 1965). For objects with sharp edges and flat faces, the shadow field consists of edge waves (Sommerfeld, 1896, 1935; Macdonald, 1902; Ufimtsev, 1965, 2003, 2009). All possible forms of a shadow field can be considered as the result of transverse wave diffusion and interaction of the diffusion field with an object’s surface (Ufimtsev, 1996).

32

1.5.2

BASIC NOTIONS IN ACOUSTIC AND ELECTROMAGNETIC DIFFRACTION PROBLEMS

Fresnel Diffraction and Forward Scattering

Well-known manifestations of shadow radiation are the phenomena of Fresnel diffraction and forward scattering (Ufimtsev, 1996, 2008a). Indeed, in the optical spectrum of electromagnetic waves, the dark bands bordering the geometrical optics shadow of opaque objects (on a white screen located behind the object) are simply the result of interference of the shadow radiation with the incident wave from a point source. This phenomenon was discovered and termed diffraction by Grimaldi (1665). Later, this phenomenon was associated with Fresnel, who suggested the first quantitative theory of diffraction. Note also that if a source region exceeds one-half of the first Fresnel zone, the diffraction images generated by different point sources overlap and blur a clear diffraction picture. Perhaps for this reason, some of Grimaldi’s contemporaries could not reproduce his diffraction experiments. One should mention the important role of Newton (1704), who published his own observations of diffraction and authoritatively confirmed Grimaldi’s discovery (Pelosi et al., 1998). The interpretation of Fresnel diffraction proposed is in agreement with Fock’s theory (1965). He established that the diffracted field in the vicinity of the shadow boundary behind an object consists of two basic components. The principal component does not depend on the electromagnetic properties of the scattering object and is determined only by its geometry. Its interference with the incident wave results in Fresnel diffraction. The second component is small, depends on the material properties, and looks like a background where the of Fresnel diffraction bands are located. As shown in the preceding sections, shadow radiation does not depend on the material properties of objects and is strong in the vicinity of the shadow boundary. Hence, it can be identified as the principal field component discovered by Fock. It is interesting that the history of diffraction as a science began in the seventeenth century with an investigation of just this phenomenon: interference of the shadow radiation with an incident wave (work of Grimaldi, Newton, Young, and Fresnel). Forward scattering is enhancement of a scattered field in directions approaching the shadow boundary behind an object. This phenomenon is the result of co-phase interference of elementary waves [integrands in Equation (1.72)] in the forward direction that actually represents the focal line for these waves. Forward scattering has been investigated extensively both experimentally and theoretically (Glaser, 1985; Willis, 1991). The numerical data for the scattered field presented by Bowman et al. (1987) and Ufimtsev (2003, 2009), as well as Figure 14.6 in this book, clearly illustrate the existence of this phenomenon. The analysis above reveals the nature of this phenomenon as shadow radiation, which is inherent in scattering related to all large opaque objects. 1.6 SUMMARY OF PROPERTIES OF PHYSICAL OPTICS APPROXIMATION The PO approximation describes properly both all rays reflected away from geometrical optics boundaries and the diffracted field near these boundaries, as well as near foci, and caustics. Reflected rays are revealed by the asymptotic evaluation of

NONUNIFORM COMPONENT OF AN INDUCED SURFACE FIELD

33

PO integrals. These integrals correctly predict the magnitude and position of main and near-side lobes in the directivity pattern of a far scattered field. However, these surface integrals consume significant computer time. Their transformation into line integrals reduces computer time and is a subject of continuing research (Maggi, 1888; Rubinowicz, 1917; Asvestas, 1985a,b, 1986, 1995; Gordon, 1994, 2003; Gordon and Bilow, 2002; Meincke et al., 2003; Pippi et al., 2004; Albani, 2011, Hacivelioglu et al., 2013). The backscattered field in PO approximation possesses special properties. According to Equation (1.37), acoustic fields scattered by soft and hard objects (of the same shape and size) differ only in sign. According to Equations (1.116) and (1.119), backscattering from perfectly conducting objects does not depend on linear polarization of the incident electromagnetic wave. Fundamental equivalency relationships (1.120) and (1.121) exist between the PO approximations for backscattered acoustic and electromagnetic waves. Separation of a PO field into a reflected field and shadow radiation elucidates the scattering physics. The well-known phenomena of Fresnel diffraction and forward scattering can be considered to be manifestations of shadow radiation. This separation also explains the fundamental law of diffraction theory, according to which the total scattering cross-section of large perfectly reflecting objects is double the area of their shadow cross-section. Also, in far-field approximation, it is found that the shadow radiation equals zero in the backscattering direction as well as in the directions of reflected rays, while the reflected field equals zero in the forward shadow direction. Drawbacks to PO include the following. It is not self-consistent. When the observation point approaches the scattering surface, the PO integrals do not reproduce the initial values for the surface field. Also, the PO field does not satisfy rigorously the boundary conditions and the reciprocity principle. As shown in Sections 7.6 and 7.8.2, the PO approximation predicts the nonzero fields inside perfectly reflecting objects and the spurious rays shooting through such objects (Section 14.1.5). The reason for these shortcomings is the geometrical optics approximation for the surface field, which does not include its diffraction components. The PO shortcomings are overcome in the PTD, which improves PO by taking into account the diffracted surface field.

1.7

NONUNIFORM COMPONENT OF AN INDUCED SURFACE FIELD

Surface acoustic fields u or 𝜕u∕𝜕n, as well as electric and magnetic currents, induced by an incident wave on scattering objects can be considered as the sources of a scattered field. As noted in the Introduction, the central and original idea of PTD is the separation of these sources into uniform and nonuniform components: j = j(0) + j(1) .

(1.147)

The uniform acoustic component j(0) is defined by Equation (1.31), and the uniform s,h electromagnetic component is determined by Equation (1.113). From a physical point

34

BASIC NOTIONS IN ACOUSTIC AND ELECTROMAGNETIC DIFFRACTION PROBLEMS

Q3 Q1 Q2

Figure 1.15 Uniform components of the field at points Q1 , Q2 , and Q3 on the scattering objects are identical to those on the infinite tangential planes, shown by dotted lines.

of view, this component represents the surface field induced on the infinite plane tangent to an object (Figs. 1.4 and 1.15). In the case of an incident plane wave, this field is distributed uniformly over the tangent plane. Its amplitude is constant and its phase is a linear function of the plane coordinates. That is why this component is called uniform. According to definitions (1.31) and (1.113), it can also be called the geometrical optics or ray component. In contrast, the nonuniform component j(1) is the diffraction part of the surface field. It is caused by diffraction due to any deviation of the scattering surface from that of the tangential infinite plane. These deviations can be a smooth or sharp bending, sharp edges, discontinuity of curvature, discontinuity of material properties, apertures, small bumps and deeps, and so on (Fig. 1.16). If the scattering object is convex and smooth and its dimensions and radii of curvature are large compared to the wavelength, the nonuniform component induced concentrates near the boundary between the illuminated and shadowed surfaces (Fig. 1.4). This component is described asymptotically by the Fock functions (Fock, 1965). From a physical point of view it represents creeping waves that radiate surface-diffracted rays. If the object possesses sharp edges, the nonuniform component concentrates in their vicinity (Fig. 1.17) and is described asymptotically by the Sommerfeld functions (Sommerfeld, 1935) presented in Chapter 2. This form of nonuniform component radiates edge waves, often called fringe waves. Similarly, the nonuniform sources near vertices radiate vertex waves. Taking into account the diffraction and nonuniform components of surface fields, PTD overcomes the PO shortcomings and provides more accurate asymptotic results for high-frequency scattered fields. The PTD separation of surface fields into uniform and nonuniform components has proved to be very productive and is often used in diffraction theory. This concept is quite flexible, as it can be extended for objects with other boundary conditions. It is also used successfully in hybrid techniques in combination with direct numerical methods. A proper choice of the uniform component depends on specific properties of the problem under investigation and can essentially facilitate its solution. See, for example, an article by Ufimtsev (1998) and related references (Ufimtsev, 1996, 2003, 2009), as well as the section “Additional References Related to the PTD Concept: Applications, Modifications, and Developments” toward the end of this book.

NONUNIFORM COMPONENT OF AN INDUCED SURFACE FIELD

35

smooth bending

sharp convex bending

sharp concave bending

sharp edges

discontinuity of curvature

ρ=∞

ρ 0. Then ∞

I(x, y) = i

∫−∞

∞

e−it𝜉 d𝜉

∫−∞

f (𝑤)ei𝑤𝜉 d𝑤

√ √ 2 2 with t = k cos 𝜑0 and f (𝑤) = (1∕ k2 𝑤2 )ei(|y| k −𝑤 −𝑤x) . According to the Fourier integral, ∞

∞

1 e−it𝜉 d𝜉 f (𝑤)ei𝑤𝜁 d𝜁 = f (t), ∫−∞ 2𝜋 ∫−∞ ∞

I(x, y) =

∫−∞

e−ik𝜉 cos 𝜑0 d𝜉

∞

∫−∞

eikr i2𝜋 ik(|y| sin 𝜑0 −x cos 𝜑0 ) e . d𝜁 = r k sin 𝜑0

PROBLEMS

37

As a result, usc s = u0

ik sin 𝜑0 I(x, y) = −u0 2𝜋

{

eik(y sin 𝜑0 −x cos 𝜑0 ) e−ik(y sin 𝜑0 +x cos 𝜑0 )

for for

y > 0, y < 0.

Thus, the surface sources induced by the incident wave on the soft plane inc radiate the scattering wave usc s = −u , which completely cancels the incident wave below the plane (y < 0). In other words, these sources block the region y < 0 from the incident wave. 1.2 Solve a scattering problem similar to Problem 1.1 but for a hard reflecting = 2uinc . plane where jh = utot h Solution

According to Equation (1.10), ∞

usc h =

∞

1 𝜕 eikr uinc d𝜉 d𝜁 . ∫−∞ 𝜕n r 2𝜋 ∫−∞

In view of Equation (1.14), 𝜕 eikr 𝜕 eikr =− 𝜕n r 𝜕y r ∞

usc h = −u0

∞

1 𝜕 eikr 1 𝜕 e−ik𝜉 cos 𝜑0 d𝜉 = −u0 I(x, y). ∫−∞ r 2𝜋 𝜕y ∫−∞ 2𝜋 𝜕y

This double integral I(x, y) was calculated in Problem 1.1. Hence, i 𝜕 ik(|y| sin 𝜑0 −x cos 𝜑0 ) e k sin 𝜑0 𝜕y { ik(y sin 𝜑 −x cos 𝜑 ) 0 0 e for y > 0, = u0 −e−ik(y sin 𝜑0 +x cos 𝜑0 ) for y < 0.

usc h = −u0

Thus, usc cancels the incident wave completely below the plane (y < 0). h 1.3 The incident wave Ezinc = E0z exp[−ik(x cos 𝜑0 + y sin 𝜑0 )] excites the surface inc electric current ⃗je = 2[̂y × H⃗ ] on the illuminated side (y = +0) of a perfectly conducting infinite plane (Fig. P1.1). Begin with Equations (1.103) to (1.105) and calculate the scattered field Ezsc generated by these currents above and below the plane. Consider the total field Eztot = Ezinc + Ezsc in the region y < 0 and realize the blocking role of the surface currents jz .

38

BASIC NOTIONS IN ACOUSTIC AND ELECTROMAGNETIC DIFFRACTION PROBLEMS

inc

⃗ = ikZ0 H, ⃗ Solution First find H⃗ . According to the Maxwell equation ∇ × E √ Hxinc = −Y0 sin 𝜑0 Ezinc with Y0 = 1∕Z0 , Z0 = 𝜇0 ∕𝜀0 = 120𝜋. Thus, jez = 2Y0 E0z sin 𝜑0 e−ik𝜉 cos 𝜑0 , sin 𝜑0 1 eikr eikr d𝜉 jz e−ik𝜉 cos 𝜑0 d𝜉 d𝜁 = E0z Y0 d𝜁 ∫−∞ r ∫−∞ r 4𝜋 ∫−∞ 2𝜋 ∫−∞ ∞

Aez =

= E0z Y0

∞

∞

∞

sin 𝜑0 I(x, y). 2𝜋

The double integral I(x, y) was calculated in Problem 1.1. The scattered field Ezsc is found with Equation (1.103) as Ezsc = ikZ0 Aez . As a result, Ezsc = −E0z eik(|y| sin 𝜑0 −x cos 𝜑0 ) . Compare this electromagnetic field with the analogous acoustic field in Problem 1.1. Realize the equivalence between two-dimensional acoustic and electromagnetic scattering problems. Thus, in the region below the plane (y < 0), the scattered wave Ezsc = −E0z e−ik(y sin 𝜑0 +x cos 𝜑0 ) = −Ezinc cancels the incident wave completely. 1.4

Solve a problem analogous to Problem 1.3 but with the incident wave Hzinc = H0z exp[−ik(x cos 𝜑0 + y sin 𝜑0 )]. Start with Equation (1.104). Solution On a perfectly conducting plane, this wave induces the electric current jex = 2H0z e−ik𝜉 cos 𝜑0 that radiates the field ∞

Hzsc = −

∞

𝜕 e 1 𝜕 eikr e−ik𝜉 cos 𝜑0 d𝜉 Ax = −H0z d𝜁 ∫−∞ r 𝜕y 2𝜋 𝜕y ∫−∞

= −H0z

1 𝜕 I(x, y). 2𝜋 𝜕y

The derivative 𝜕I(x, y)∕𝜕y was calculated in Problem 1.2. As a result, we have Hzsc = ±H0z eik(|y| sin 𝜑0 −x cos 𝜑0 )

with y > 0, y < 0.

Thus, below the plane (y < 0) the scattered field cancels the incident wave completely. Compare this electromagnetic field with the analogous acoustic field found in Problem 1.2. Realize the equivalence between two-dimensional acoustic and electromagnetic scattering problems. 1.5

Suppose that the incident wave uinc = u0 exp[ik(x cos 𝜙0 + y sin 𝜙0 )] hits the soft strip shown in Figure 5.1. Use Equation (1.71) to calculate the reflected part of the PO field scattered by this strip in the far zone.

PROBLEMS

39

(a) Express the far field in closed form. (b) Estimate the field in the directions 𝜙 = 𝜙0 , 𝜙 = 𝜋 − 𝜙0 , 𝜙 = 𝜋 + 𝜙0 , and 𝜙 = −𝜙0 . (c) Compute and plot the directivity pattern of the reflected field, setting a = 2𝜆 and 𝜙0 = 45◦ . Confirm the field values established in part (b). Solution (a) For the illuminated side of the soft strip, n̂ = −̂x, 𝜕urefl s 𝜕n

inc urefl = −u0 eik𝜂 sin 𝜙0 , s = −u

𝜕uinc 𝜕uinc =− = −u0 ik cos 𝜙0 eik𝜂 sin 𝜙0 , 𝜕n 𝜕x

=

eikr eikr eikr 𝜕 eikr 𝜕 eikr = ∇′ ⋅ n̂ = −∇ ⋅ n̂ = ∇ ⋅ x̂ = , 𝜕n r r r r 𝜕x r √ r = x2 + (y − 𝜂)2 + 𝜁 2 . Equation (1.71) takes the form (

a

∞

eikr 𝜕 eik𝜂 sin 𝜙0 d𝜂 d𝜁 + ik cos 𝜙0 ∫−∞ r 𝜕x ∫−a ) a ∞ ikr e × eik𝜂 sin 𝜙0 d𝜂 d𝜁 . ∫−a ∫−∞ r

urefl s = u0

1 4𝜋

−

According to Equiation (3.7), ∞

∫−∞

[ √ ] eikr d𝜁 = i𝜋H0(1) k x2 + (y − 𝜂)2 . r

√ We calculate the field in the far zone where 𝜌 = x2 + y2 ≫ ka2 . In this region one can use the asymptotic form (2.29) for the Hankel function, H0(1)

[ √ ] √ 2 2 k x + (y − 𝜂) ∼

√

2

𝜋k x2 + (y − 𝜂)2

eik

√ x2 +(y−𝜂)2 −i(𝜋∕4)

e

In the far zone, √ y x2 + (y − 𝜂)2 ≈ 𝜌 − 𝜂 = 𝜌 − 𝜂 sin 𝜙, 𝜌 [ √ ] √ 2 ik(𝜌−𝜂 sin 𝜙) −i(𝜋∕4) (1) 2 2 H0 k x + (y − 𝜂) ≈ e , e 𝜋k𝜌

.

40

BASIC NOTIONS IN ACOUSTIC AND ELECTROMAGNETIC DIFFRACTION PROBLEMS

and the field approximation is given by a

ei(k𝜌+𝜋∕4) (cos 𝜙0 − cos 𝜙)ik eik(sin 𝜙0 −sin 𝜙)𝜂 d𝜂, urefl s = u0 √ ∫ −a 2 2𝜋k𝜌 that is, urefl s = u0 i(cos 𝜙0 − cos 𝜙)

sin[ka(sin 𝜙0 − sin 𝜙)] ei(k𝜌+𝜋∕4) . √ sin 𝜙0 − sin 𝜙 2𝜋k𝜌

(b) It follows from this expression that

urefl s

⎧ 0 in the forward direction 𝜙 = 𝜙 , 0 ⎪ i(k𝜌+𝜋∕4) e ⎪ in the specular direction 𝜙 = 𝜋 − 𝜙0 , ⎪ u0 2ika cos 𝜙0 √ 2𝜋k𝜌 ⎪ ei(k𝜌+𝜋∕4) ⎪ in the backscattering = ⎨ u0 i cot 𝜙0 sin(2ka sin 𝜙0 ) √ 2𝜋k𝜌 ⎪ ⎪ direction 𝜙 = 𝜋 + 𝜙0 , ⎪ ⎪ 0 in the direction 𝜙 = −𝜙0 that is opposite to the specular ⎪ reflection 𝜙 = 𝜋 − 𝜙0 . ⎩

Do you expect the connection of this result, urefl s = 0, in the direction 𝜙 = −𝜙0 , with the general property of the shadow radiation that ush = 0 in the backscattering direction? Imagine that the strip is hit by the wave incident from the direction 𝜙 = −𝜙0 . Then the urefl s above can be considsh ered as the shadow radiation (u ) that according to Section 1.3.6 equals zero in the backscattering direction 𝜙 = −𝜙0 . (c) Represent urefl s in the form (5.34) and calculate the normalized scattering cross-section 10 log(𝜎s ∕kl2 ) defined by Equation (5.45) with l = 2a. Plot the results in polar coordinates with 0 ≤ 𝜙 ≤ 360◦ . 1.6

Solve a problem similar to Problem 1.5 but for a hard strip. Find the reflected part of the scattered field in the far zone. Solution (a) The geometry of the problem is shown in Fig. 5.1. The incident wave is given as uinc = u0 exp[ik(x cos 𝜙0 + y sin 𝜙0 )]. The reflected field is described by Equation (1.71). On the illuminated side of the strip, 𝜕urefl h

inc = eik𝜂 sin 𝜙0 , urefl h =u ik𝜂 sin 𝜙0

= u0 ik cos 𝜙0 e

𝜕n .

=−

𝜕uinc 𝜕uinc = 𝜕n 𝜕x

PROBLEMS

41

We then apply the related manipulations shown in Problem 1.5 and obtain 𝜕 eikr 𝜕 eikr = , 𝜕n r 𝜕x r

√ 𝜕uinc eikr 2𝜋 i(k𝜌+𝜋∕4) ik𝜂(sin 𝜙0 −sin 𝜙) e , d𝜁 ≈ u0 ik cos 𝜙0 e ∫−∞ 𝜕n r k𝜌 √ 𝜌 = x2 + y2 , √ ∞ 𝜕 eikr 2𝜋 i(k𝜌+𝜋∕4) ik𝜂(sin 𝜙0 −sin 𝜙) urefl e . d𝜁 ≈ u0 ik cos 𝜙 e ∫−∞ 𝜕n r k𝜌 ∞

Substituting these approximations into (1.71) and integrating over 𝜂, one obtains urefl h = −u0 i(cos 𝜙0 − cos 𝜙)

sin[ka(sin 𝜙0 − sin 𝜙)] ei(k𝜌+𝜋∕4) . √ sin 𝜙0 − sin 𝜙 2𝜋k𝜌

with the urefl found in Problem 1.5. It is seen that Compare this urefl s h refl refl uh = −us , in agreement with the general Equation (1.73). (b) urefl = 0 in the directions 𝜙 = 𝜙0 and 𝜙 = −𝜙0 , h ei(k𝜌+𝜋∕4) in the specular direction 𝜙 = 𝜋 − 𝜙0 , urefl h = −u0 i2ka cos 𝜙0 √ 2𝜋k𝜌 ) ei(k𝜌+𝜋∕4) ( urefl in the backscattering = −u i cot 𝜙 sin 2ka sin 𝜙 √ 0 0 0 h 2𝜋k𝜌 direction 𝜙 = 𝜋 + 𝜙0 . in the form (5.34) and calculate the normalized scattering (c) Represent urefl h cross-section 10 log(𝜎h ∕kl2 ) defined in (5.45) with i − 2a. Set a = 2𝜆 and 𝜙0 = 45◦ . Plot the results in polar coordinates with 0 ≤ 𝜙 ≤ 360◦ . Confirm the field values established in part (b). 1.7 Suppose that the incident wave uinc = u0 exp[ik(x cos 𝜙0 + y sin 𝜙0 )] hits the soft strip shown in Fig. 5.1. Use Equation (1.72) and calculate the shadow radiation part of the PO field scattered in the far zone. (a) Express the far field in closed form. (b) Estimate the field in the directions 𝜙 = 𝜙0 , 𝜙 = 𝜋 − 𝜙0 , and 𝜙 = 𝜋 + 𝜙0 . (c) Apply the optical theorem (5.16) to the total PO field uPO = ush + urefl s in the direction 𝜙 = 𝜙0 and give the geometrical interpretation of the total scattering cross-section. Take urefl s from Problem 1.5. (d) Compute and plot the directivity pattern of the shadow radiation, setting a = 2𝜆 and 𝜙0 = 45◦ . Confirm the field values established in part (b).

42

BASIC NOTIONS IN ACOUSTIC AND ELECTROMAGNETIC DIFFRACTION PROBLEMS

Solution (a) The geometry of the problem is shown in Fig. 5.1. The shadow radiation is determined by Equation (1.72). It is calculated in the far zone where √ 𝜌 = x2 + y2 ≫ ka2 . All appropriate approximations for the integrals in Equation (1.72) are similar to those shown in Problems 1.5 and 1.6. Applying them, one finds that ush = u0 i(cos 𝜙 + cos 𝜙0 )

sin[ka(sin 𝜙0 − sin 𝜙)] ei(k𝜌+𝜋∕4) . √ sin 𝜙0 − sin 𝜙 2𝜋k𝜌

(b) According to this result, ei(k𝜌+𝜋∕4) ush = u0 i2ka cos 𝜙0 √ in the forward direction 𝜙 = 𝜙0 , 2𝜋k𝜌 ush = 0 in the direction of the specular reflection 𝜙 = 𝜋 − 𝜙0 and in the backscattering direction 𝜙 = 𝜋 + 𝜙0 . These field zeros are in complete agreement with the general properties of the shadow radiation presented in Section 1.3.6. (c) The reflected part of the PO field is found in Problem 1.5. Thus, ei(k𝜌+𝜋∕4) sh refl PO , uPO s = u + us = u0 Φs (𝜙, 𝜙0 ) √ 2𝜋k𝜌 ΦPO s (𝜙, 𝜙0 ) = i2 cos 𝜙0

sin[ka(sin 𝜙0 − sin 𝜙)] . sin 𝜙0 − sin 𝜙

According to Equation (5.16), 𝜎sPO =

2 ImΦPO s (𝜙0 , 𝜙0 ) = 4a cos 𝜙0 = 2A, k

in agreement with the general theorem (1.45), (1.87). Here the quantity A = 2a cos 𝜙0 is the cross-section of the geometrical shadow region. (d) Represent ush in the form of Equation (5.34) with Φsh (𝜙, 𝜙0 ) = i(cos 𝜙 + cos 𝜙0 )

sin[ka(sin 𝜙0 − sin 𝜙)] . sin 𝜙0 − sin 𝜙

Calculate the normalized scattering cross-section 10 log(𝜎 sh ∕kl2 ) defined in (5.45) with l − 2a. Set a = 2𝜆 and 𝜙0 = 45◦ . Plot the results in polar coordinates with 0 ≤ 𝜙 ≤ 360◦ . Confirm the field values established in part (b). 1.8

The incident wave Ezinc = E0z exp[ik(x cos 𝜙0 + y sin 𝜙0 )] hits the perfectly conducting strip shown in Figure 5.1. Calculate the reflected part of the PO scattered field.

PROBLEMS

43

(a) Begin with Equations (1.103) to (1.105). Recall that the field reflected from a perfectly conducting surface can be interpreted as radiation generated by both electric and magnetic surface currents. On the illuminated side of the strip, these currents are defined as ⃗je,refl = n̂ × H⃗

inc

⃗jm,refl = n̂ × E ⃗ inc .

,

Construct the integral expressions for the electric and magnetic potentials of the reflected field. (b) Express the far field in closed form. Compare the result of this electromagnetic problem with the one found for its acoustic analog in Problem 1.5. Confirm the equivalency relationship stated at the beginning of this chapter. (c) Estimate the field in the directions 𝜙 = 𝜙0 , 𝜙 = 𝜋 − 𝜙0 , 𝜙 = 𝜋 + 𝜙0 , and 𝜙 = −𝜙0 . Solution (a) According to their definition, the magnetic currents on the strip contain = jm,refl = 0 and jm,refl = E0z eik𝜂 sin 𝜙0 . To determine the components jm,refl x z y the electric surface currents, we need to know the incident magnetic inc ⃗ inc = ikZ0 H⃗ field. The latter is found from the Maxwell equation ∇ × E as Hxinc = Y0 E0z sin 𝜙0 eik𝜂 sin 𝜙0 , Hyinc = −Y0 E0z cos 𝜙0 eik𝜂 sin 𝜙0 , Hzinc = 0, = 0, je,refl = 0, je,refl = Y0 E0z cos 𝜙0 eik𝜂 sin 𝜂𝜙0 . Y0 = 1∕Z0 . Hence, je,refl x y z According to Equation (1.105), a

∞

Aex,y = 0,

Aez =

1 eikr eik𝜂 sin 𝜙0 d𝜂 Y0 E0z cos 𝜙0 d𝜁 , ∫−∞ r ∫−a 4𝜋

Am x,z = 0,

Am y =

1 eikr eik𝜂 sin 𝜙0 d𝜂 E0z d𝜁 , ∫−∞ r ∫−a 4𝜋

a

r=

∞

√ x2 + (y − 𝜂)2 + 𝜁 2 .

(b) According to Equation (3.7), the integral over 𝜁 is expressed through the Hankel function. In the far zone this function is approximated as (see Problem 1.5) [ √ ] √ √ 2 ik(𝜌−𝜂 sin 𝜙) −i(𝜋∕4) (1) 2 2 e , 𝜌 = x2 + y2 . e H0 k x + (y − 𝜂) ≈ 𝜋k𝜌 After substituting this approximation into the potentials Aez and Am y , the latter will contain the simple integral a

∫−a

eik𝜂(sin 𝜙0 −sin 𝜙) d𝜂 =

2 sin[ka(sin 𝜙0 − sin 𝜙)] . k sin 𝜙0 − sin 𝜙

44

BASIC NOTIONS IN ACOUSTIC AND ELECTROMAGNETIC DIFFRACTION PROBLEMS

As a result, Aez =

sin[ka(sin 𝜙0 − sin 𝜙)] ei(k𝜌+𝜋∕4) 1 , Y0 E0z cos 𝜙0 √ k sin 𝜙0 − sin 𝜙 2𝜋k𝜌

Am y =

sin[ka(sin 𝜙0 − sin 𝜙)] ei(k𝜌+𝜋∕4) 1 . E0z √ k sin 𝜙0 − sin 𝜙 2𝜋k𝜌

⃗ refl , follow Equation (1.103). First, evaluate the quantities To calculate E e ⃗ m . Because 𝜕Ae ∕𝜕z = 0 and 𝜕Am ∕𝜕z = 0 , we ⃗ e and ∇ × A ⃗ ∇(∇ ⋅ A ) + k2 A z y have ⃗ e = k2 Ae ẑ , ⃗ e ) + k2 A ∇(∇ ⋅ A z ⃗m = ∇×A

𝜕Am y 𝜕x

ẑ ≈ ikAm y

𝜕𝜌 x m ẑ = ikAm y 𝜌 ẑ = ik cos 𝜙Ay ẑ . 𝜕x

Thus, Ezrefl = ik(Z0 Aez − cos 𝜙Am y ), Ezrefl = Eoz i(cos 𝜙0 − cos 𝜙)

sin[ka(sin 𝜙0 − sin 𝜙)] ei(k𝜌+𝜋∕4) , √ sin 𝜙0 − sin 𝜙 2𝜋k𝜌

E𝜙refl = E𝜌refl = 0. Compare this electromagnetic field with the acoustic field found in Problem 1.5. Realize the equivalence between two-dimensional acoustic and electromagnetic scattering problems. (c) According to these expressions:

Ezrefl

⎧ 0 in the forward direction 𝜙 = 𝜙 , 0 ⎪ i(k𝜌+𝜋∕4) ⎪ E i2ka cos 𝜙 e in the direction of specular reflection 0 √ ⎪ 0z 2𝜋k𝜌 ⎪ ⎪ 𝜙 = 𝜋 − 𝜙0 , ⎪ =⎨ ei(k𝜌+𝜋∕4) E i2 cot 𝜙0 sin(2ka sin 𝜙0 ) √ in the backscattering ⎪ 0z 2𝜋k𝜌 ⎪ ⎪ direction 𝜙 = 𝜋 + 𝜙0 , ⎪ ⎪ 0 in the direction 𝜙 = −𝜙0 that is opposite to the specular ⎪ reflection 𝜙 = 𝜋 − 𝜙0 . ⎩

Compare these results with those in Problem 1.5.

PROBLEMS

45

1.9 The incident wave Ezinc = E0z exp[ik(x cos 𝜑0 + y sin 𝜑0 )] hits the perfectly conducting strip shown in Figure 5.1. Calculate the shadow radiation part of the PO scattered field. (a) Start with Equations (1.103) and (1.105). The surface currents are defined as ⃗je,sh = n̂ × H inc ,

⃗jm,sh = −̂n × E ⃗ inc .

Apply the appropriate results of Problem 1.8 and find the far-field approximations for the electric and magnetic potentials of the reflected field. (b) Utilize the appropriate results of Problem 1.8 and find the far-field approximation for the shadow radiation. (c) Estimate the field in the directions 𝜙 = 𝜙0 , 𝜙 = 𝜋 − 𝜙0 , and 𝜙 = 𝜋 + 𝜙0 . (d) Apply the optical theorem (5.16) to the total field Eztot = Ezrefl + Ezsh in the direction 𝜙 = 𝜙0 and provide the geometrical interpretation of the total ⃗ refl in Problem 1.8. scattering cross-section. Take the data for E z Solution (a) Notice that ⃗je,sh = ⃗je,refl = n̂ × H inc ,

⃗jm,sh = −⃗jm,refl = −̂n × E ⃗ inc .

Therefore, the electric potentials of the shadow radiation will be the same as for the reflected field found in Problem 1.8. Also, the magnetic potentials of the shadow radiation will differ only in sign from those found in Problem 1.8 for the reflected field. Thus, we have Aez =

sin[ka(sin 𝜙0 − sin 𝜙)] ei(k𝜌+𝜋∕4) 1 , Y0 E0z cos 𝜙0 √ k sin 𝜙0 − sin 𝜙 2𝜋k𝜌

sin[ka(sin 𝜙0 − sin 𝜙)] ei(k𝜌+𝜋∕4) 1 Am . = − E √ 0z y k sin 𝜙0 − sin 𝜙 2𝜋k𝜌 (b) The field expression in terms of the potentials is the same as in Problem 1.8, Ezsh = ik(Z0 Aez − cos 𝜙Am y ). Hence, Ezsh = E0z i(cos 𝜙0 + cos 𝜙)

sin[ka(sin 𝜙0 − sin 𝜙)] ei(k𝜌+𝜋∕4) . √ sin 𝜙0 − sin 𝜙 2𝜋k𝜌

46

BASIC NOTIONS IN ACOUSTIC AND ELECTROMAGNETIC DIFFRACTION PROBLEMS

Compare this electromagnetic field with the acoustic field found in Problem 1.7. Realize the equivalence between two-dimensional acoustic and electromagnetic scattering problems. (c) According to this equation, ei(k𝜌+𝜋∕4) in the forward direction 𝜙 = 𝜙0 . Ezsh = E0z i2ka cos 𝜙0 √ 2𝜋k𝜌 Also, Ezsh = 0 in the specular direction (𝜙 = 𝜋 − 𝜙0 ) and in the backscattering direction (𝜙 = 𝜋 + 𝜙0 ), that is, in total agreement with the properties of the shadow radiation. (d) The total PO field equals EzPO = Ezsh + Ezrefl . Taking Ezrefl in Problem 1.8, we have ei(k𝜌+𝜋∕4) EzPO = E0z ΦPO (𝜙, 𝜙0 ) √ 2𝜋k𝜌 with ΦPO (𝜙, 𝜙0 ) = i2 cos 𝜙0

sin[ka(sin 𝜙0 − sin 𝜙)] . sin 𝜙0 − sin 𝜙

Then we employ Equation (5.16) for the total scattering cross-section, which is valid for both acoustic and electromagnetic waves. As a result, 𝜎 tot = 4ka cos 𝜙0 = 2A,

A = 2ka cos 𝜙0 ,

where A is the geometrical cross-section of the shadow region behind the strip. 1.10

Evaluate the electromagnetic shadow radiation and reflected field in the forward and backscattering directions. The basic equations are given in Section 1.4.2, and the geometry of the problem is shown in Figure 1.2 and 1.4. Solution form

Utilize Equations (1.129) to (1.132). Take the incident wave in the

Exinc = Z0 Hyinc = E0x eikz ,

Eyinc = 0,

Hxinc = 0,

Hyinc = Y0 E0x eikz .

Start with an investigation of the field in the forward direction, 𝜗 = 0. To approach this direction, first put the observation point into the half-plane

PROBLEMS

47

𝜑 = 0, where 𝜑̂ = ŷ . Then move the observation point along this half-plane to the positive z-axis, where 𝜗̂ = x̂ . In view of these comments, e−ikr H⃗

′ cos Ω

′

= e−ikz ,

inc −ikr′ cos Ω

e

⃗ inc e−ikr′ cos Ω = E0x eikz′ x̂ e−ikz′ = E0x x̂ , E ′

′

= H0y eikz ŷ e−ikz = H0y ŷ ,

Z0 [̂n × H⃗

inc

̂ −ikr′ cos Ω = E0x [̂n × ŷ ] ⋅ x̂ = −E0x [̂x × ŷ ] ⋅ n̂ = −E0x (̂z ⋅ n̂ ), ] ⋅ 𝜗e

Z0 [̂n × H⃗

inc

] ⋅ 𝜑̂ = E0x [̂n × ŷ ] ⋅ ŷ = −E0x [̂y × ŷ ] ⋅ n̂ = 0,

⃗ inc ] ⋅ 𝜑̂ = E0x [̂n × x̂ ] ⋅ ŷ = E0x n̂ ⋅ [̂x × ŷ ] = E0x (̂z ⋅ n̂ ), [̂n × E ⃗ inc ] ⋅ 𝜗̂ = E0x [̂n × x̂ ] ⋅ x̂ = E0x n̂ ⋅ [̂x × x̂ ] = 0. [̂n × E Now Equations (1.129) to (1.132) transform into ⎫ ik eikR [−(̂z ⋅ n̂ ) + (̂z ⋅ n̂ )] ds = 0, ⎪ ∫ 4𝜋 R Sil ⎪ ⎪ ikR ikR ik e ik e = Exsh = −E0x (̂z ⋅ n̂ ) ds = E0x A ,⎬ 2𝜋 R ∫Sil 2𝜋 R ⎪ ⎪ ⎪ = Eyrefl = 0, E𝜑sh = Eysh = 0. ⎭

E𝜗refl = Exrefl = E0x E𝜗sh E𝜑refl

for

𝜗=0

To evaluate the backscattering, move the observation point along the 𝜑 = 0 half-plane to the negative direction of the z-axis. In this case, 𝜑̂ = ŷ and 𝜗̂ = −̂x and Equations (1.129) to (1.132) transform into ⎫ ⎪ ⎪ ⎪ ikR ′ ik e = −Exsh = E0x [(̂z ⋅ n̂ ) − (̂z ⋅ n̂ )]ei2kz ds = 0,⎬ ⎪ 4𝜋 R ∫Sil ⎪ ⎪ = Eyrefl = 0, E𝜑sh = Eysh = 0. ⎭

E𝜗refl = −Exrefl = E0x E𝜗sh E𝜑refl 1.11

′ ik eikR (̂z ⋅ n̂ )ei2kz ds, 2𝜋 R ∫Sil

for

𝜗=𝜋

Evaluate the electromagnetic shadow radiation in the directions of reflected rays. The basic equations are given in Section 1.4.2, and the geometry of the problem is shown in Figure 1.2, 1.4, and P1.2. Solution the form

Utilize Equations (1.131) and (1.132). Take the incident wave in

Exinc = Z0 Hyinc = E0x eikz ,

Eyinc = 0,

Hxinc = 0,

Hyinc = Y0 E0x eikz .

48

BASIC NOTIONS IN ACOUSTIC AND ELECTROMAGNETIC DIFFRACTION PROBLEMS

ϑˆ

ρ

kˆ refl

nˆ

γ

y

ρ

kˆ inc

φ Z

Sil

X

Figure P1.2 Unit vectors at the reflection point. They are in the incident plane (𝜌, z).

Without loss of generality, one can consider the observation point in the ̂ region 0 < 𝜑 < 𝜋∕2 and 𝜋∕2 < 𝜗 < 𝜋. Orientations of the unit vectors n̂ , 𝜗, ̂kinc = ẑ , and k̂ refl at the reflection point on the object surface are shown in √ Figure P1.2. Here the vector k̂ refl represents the reflected ray, 𝜌 = x2 + y2 , 𝜌̂ = x̂ cos 𝜑 + ŷ sin 𝜑. The angle of incidence equals 𝛾. ̂ and 𝜑̂ are determined as The components of the unit vectors n̂ , 𝜗, nx = sin 𝛾 cos 𝜑,

ny = sin 𝛾 sin 𝜑,

𝜗x = − cos 𝜑 cos 2𝛾, 𝜑x = − sin 𝜑,

nz = − cos 𝛾,

𝜗y = − sin 𝜑 cos 2𝛾,

𝜑y = cos 𝜑,

𝜗z = − sin 2𝛾,

𝜑z = 0.

Equations for the shadow radiation take the form E𝜗sh = E0x

′ ′ ik eikR {[̂n × ŷ ] ⋅ 𝜗̂ − [̂n × x̂ ] ⋅ 𝜑}e ̂ ik(z −r cos Ω) ds, 4𝜋 R ∫Sil

E𝜑sh = E0x

ik eikR ̂ ik(z′ −r′ cos Ω) ds. {[̂n × ŷ ] ⋅ 𝜑̂ + [̂n × x̂ ] ⋅ 𝜗}e 4𝜋 R ∫Sil

In view of the expressions above for the unit vectors, [̂n × ŷ ] ⋅ 𝜗̂ = − cos 𝛾 cos 𝜑,

[̂n × x̂ ] ⋅ 𝜑̂ = − cos 𝛾 cos 𝜑,

[̂n × ŷ ] ⋅ 𝜑̂ = − sin 𝜑 cos 𝛾,

[̂n × x̂ ] ⋅ 𝜗̂ = sin 𝜑 cos 𝛾.

sh = 0 in the directions of the reflected rays. Now it is seen that E𝜗,𝜑

2 Wedge Diffraction: Exact Solution and Asymptotics

The relationships us = Ez and uh = Hz exist between the acoustic and electromagnetic fields in the two-dimensional wedge diffraction problem. Here Ez ⃗ and H) ⃗ that are parallel to the edge of and Hz are the components (of vectors E the wedge.

2.1

CLASSICAL SOLUTIONS

Diffraction at a wedge with a straight edge and infinite planar faces is an appropriate canonical problem to derive asymptotic expressions for edge waves scattered from arbitrary curved edges. In the particular case of the wedge, which is a semi-infinite half-plane, the exact solution of this canonical problem was found by Sommerfeld (1896), who constructed branched wave functions. Analysis of this work by Ufimtsev (1998) shows that Sommerfeld also developed almost everything that was necessary to obtain the solution for a wedge with an arbitrary angle between its faces. However, he missed the last step, which led directly to this solution. This more general solution was found by Macdonald (1902) using the classical method of separation of variables in the wave equation. Later, Sommerfeld also solved the wedge diffraction problem by his method of branched wave functions and derived simple asymptotic expressions for the edge-diffracted waves (Sommerfeld, 1935). Fundamentals of the Physical Theory of Diffraction, Second Edition. Pyotr Ya. Ufimtsev. © 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

49

50

WEDGE DIFFRACTION: EXACT SOLUTION AND ASYMPTOTICS

r

r0

φ φ0

φ =0 2π – α

φ =α Figure 2.1 A wedge is excited by the filamentary source located at the line r = r0 , 𝜑 = 𝜑0 .

Because the wedge diffraction problem is the basis for the construction of PTD, its solution is considered here in detail. First we derive this solution in the form of infinite series and then convert it to Sommerfeld integrals convenient for asymptotic analysis. The material of this chapter, with the exception of Sections 2.6 and 2.7, is a scalar version of the theory developed by the author for electromagnetic waves (Ufimtsev, 1962b, 2003, 2009). The geometry of the problem is shown in Figure 2.1. A wedge with infinite planar faces 𝜑 = 0 and 𝜑 = 𝛼 is located in a homogeneous medium. It is excited by a cylindrical wave. The source of this wave is a radiating filament with coordinates r = r0 and 𝜑 = 𝜑0 . This is a two-dimensional problem where 𝜕∕𝜕z ≡ 0. The field outside the wedge (0 ≤ 𝜑 ≤ 𝛼) satisfies the Helmholtz wave equation Δu + k2 u = I0 𝛿(r − r0 , 𝜑 − 𝜑0 )

(2.1)

and the boundary conditions us = 0

(2.2)

𝜕u∕𝜕n = 0

(2.3)

or

on the faces 𝜑 = 0 and 𝜑 = 𝛼. In the case of electromagnetic waves, these boundary conditions are appropriate for a perfectly conducting wedge, and the function us ⃗ , while function uh is the represents the z-component of electric field intensity E ⃗ In the case of acoustic waves, condition z-component of magnetic field intensity H. (2.2) relates to an acoustically soft wedge, and (2.3) relates to an acoustically hard wedge. Outside the immediate vicinity of the source, the field u satisfies the homogeneous wave equation Δu + k2 u = 0.

(2.4)

51

CLASSICAL SOLUTIONS

For the two-dimensional problem, the Laplacian operator is defined by 1 𝜕 𝜕2 1 𝜕2 + + 2 2. 2 r 𝜕r r 𝜕𝜑 𝜕r

Δ=

(2.5)

Using the classical method of separation of variables, we set in Equation (2.4) u = R(r)Φ(𝜑)

(2.6)

and substitute this u into Equation (2.4). After simple manipulations, the latter can be separated into two equations: d2 R 1 dR + + x dx dx2

( 1−

𝑣2l x2

) R=0

with x = kr,

d2 Φ + 𝑣2l Φ = 0. d𝜑2

(2.7)

(2.8)

The function Φ and the separation constants 𝑣l are determined from the boundary conditions. In the case of soft boundary conditions, according to Equations (2.2) and (2.8), Φ = {sin 𝑣l 𝜑},

𝜋 𝑣l = l , 𝛼

l = 1, 2, 3, … ,

(2.9)

and in the case of hard boundary conditions, in accordance with Equations (2.3) and (2.8), Φ = {cos 𝑣l 𝜑},

𝜋 𝑣l = l , 𝛼

l = 0, 1, 2, 3, … .

(2.10)

The Bessel and Hankel functions ⎧ J (kr) ⎫ ⎪ ⎪ 𝑣l R=⎨ ⎬ (1) ⎪ H𝑣l (kr) ⎪ ⎭ ⎩

(2.11)

represent the solution of the radial Equation (2.7). The reason to select the positive indexes 𝑣l ≥ 0 in Equations (2.9) and (2.10) is the following. Only the Bessel functions J𝑣l (kr) with positive indexes 𝑣l ≥ 0 are finite at the edge r = 0, and hence they can be used in the region r ≤ r0 . Hankel functions are appropriate in the region r ≥ r0 since they satisfy Sommerfeld’s radiation condition at infinity: lim

) √ ( du r − iku = 0 dr

with r → ∞.

(2.12)

52

WEDGE DIFFRACTION: EXACT SOLUTION AND ASYMPTOTICS

Hence, the solutions of Equation (2.4) can be written as ∞ ⎧∑ ⎪ al J𝑣l (kr)H𝑣(1) (kr0 ) sin 𝑣l 𝜑0 sin 𝑣l 𝜑 l ⎪ l=1 us = ⎨ ∞ ∑ ⎪ al J𝑣l (kr0 )H𝑣(1) (kr) sin 𝑣l 𝜑0 sin 𝑣l 𝜑 l ⎪ l=1 ⎩

with r ≤ r0 , (2.13) with r ≥ r0 ,

∞ ⎧ ∑ ⎪ bl J𝑣l (kr)H𝑣(1) (kr0 ) cos 𝑣l 𝜑0 cos 𝑣l l ⎪ l=0 uh = ⎨ ∞ ∑ ⎪ bl J𝑣l (kr0 )H𝑣(1) (kr) cos 𝑣l 𝜑0 cos 𝑣l 𝜑 l ⎪ ⎩ l=0

with r ≤ r0 , (2.14) with r ≥ r0 .

These expressions satisfy the boundary conditions as well as the reciprocity principle; that is, they do not change after interchanging r and r0 , 𝜑 and 𝜑0 . The unknown coefficients al and bl can be found by applying Green’s theorem, 𝜕u dl = Δuds, ∮L 𝜕n ∫S

ds = rdrd𝜑

(2.15)

to the fields us and uh in the region S bounded by the contour L shown in Figure 2.2. This contour consists of two arcs, r = r0 − 𝜀 and r = r0 + 𝜀, and two radial sides, 𝜑 = 𝜑0 − 𝜓 and 𝜑 = 𝜑0 + 𝜓. Substitute us,h into (2.15), taking into account that according to Equation (2.1), Δu = −k2 u + I0 𝛿(r − r0 , 𝜑 − 𝜑0 ),

(2.16)

nˆ

φ0 + ψ

L

r0 + ε

S r0 – ε

r0

φ0

φ0 – ψ

φ =0

φ =α Figure 2.2 Integration contour L in Green’s theorem (2.15).

53

CLASSICAL SOLUTIONS

and let 𝜀 tend to zero. It is clear that ∫S

us,h ds → 0

when

S → 0,

(2.17)

as the Bessel and Hankel functions are finite at r = r0 > 0. As a result, Green’s formula for us,h transforms into (

𝜑0 +𝜓

∫𝜑0 −𝜓

𝜕us,h || − | 𝜕r ||r=r +0 0

𝜑0 +𝜓

= I0

∫𝜑0 −𝜓

𝜕us,h || | 𝜕r ||r=r −0 0

r0 +𝜀

d𝜑 lim

∫r0 −𝜀

) r0 d𝜑

𝛿(r − r0 , 𝜑 − 𝜑0 )rdr

with 𝜀 → 0.

(2.18)

The two-dimensional delta function in polar coordinates equals 1 𝛿(r − r0 , 𝜑 − 𝜑0 ) = 𝛿(r − r0 ) 𝛿(𝜑 − 𝜑0 ); r

(2.19)

therefore, 𝜑0 +𝜓

(

∫𝜑0 −𝜓

𝜕us,h || − | 𝜕r || r=r0 +0

𝜕us,h || | 𝜕r ||r=r −0 0

)

𝜑0 +𝜓

r0 d𝜑 = I0

∫𝜑0 −𝜓

𝛿(𝜑 − 𝜑0 )d𝜑. (2.20)

Equation (2.20) is valid for arbitrary limits of integration. This is possible if the integrands on the left and right sides equal each other: 𝜕us,h || 𝜕us,h || 1 − = I 𝛿(𝜑 − 𝜑0 ). | | 𝜕r ||r=r +0 𝜕r ||r=r −0 r0 0 0 0

(2.21)

This equation can be used to determine the unknown coefficients al and bl in expressions (2.13) and (2.14). To do this, we substitute us of Equation (2.13) into Equation (2.21), multiply both sides by sin 𝑣t 𝜑, where 𝑣t = t𝜋∕𝛼, and integrate them over 𝜑 from 0 to 𝛼. Note that 𝛼

{1

𝛼 0

with l = t t = 1, 2, 3, … with l ≠ t

(2.22)

] [ d d 𝛼 kr0 al J𝑣l (kr0 ) H𝑣(1) (kr0 ) − H𝑣(1) (kr0 ) J𝑣l (kr0 ) = I0 . l 2 dkr0 l dkr0

(2.23)

∫0

sin 𝑣l 𝜑 sin 𝑣t 𝜑d𝜑 =

2

and obtain

54

WEDGE DIFFRACTION: EXACT SOLUTION AND ASYMPTOTICS

The expression inside brackets is the Wronskian, [ ] d d 2i . W J𝑣 (x), H𝑣(1) (x) = J𝑣 (x) H𝑣(1) (x) − H𝑣(1) (x) J𝑣 (x) = dx dx 𝜋x

(2.24)

From Equations (2.23) and (2.24) it follows that al =

𝜋 I . i𝛼 0

(2.25)

We carry out similar manipulations for hard boundary conditions. Substitute uh of Equation (2.14) into Equation (2.21), multiply both sides by cos 𝑣t 𝜑, and integrate over 𝜑 from 0 to 𝛼. As a result, we obtain bl = 𝜀l

𝜋 I i𝛼 0

with 𝜀0 =

1 , 2

𝜀1 = 𝜀2 = 𝜀3 = ⋯ = 1.

(2.26)

Thus, the coefficients al and bl are found, and the functions us,h are determined completely. Now we can write the final expressions for the total field excited by the external filamentary source. In the case of soft boundary conditions, the field us is described by the following expressions:

∞ ⎧𝜋 ∑ ⎪ I0 J𝑣 (kr)H𝑣(1) (kr0 ) sin 𝑣l 𝜑0 sin 𝑣l 𝜑 l ⎪ i𝛼 l=1 l us = ⎨ ∞ ∑ ⎪ 𝜋I J𝑣l (kr0 )H𝑣(1) (kr) sin 𝑣l 𝜑0 sin 𝑣l 𝜑 l ⎪ i𝛼 0 l=1 ⎩

with r ≤ r0 , (2.27) with r ≥ r0 ,

and in the case of hard boundary conditions, the field uh is determined by

∞ ⎧𝜋 ∑ ⎪ I0 𝜀l J𝑣l (kr)H𝑣(1) (kr0 ) cos 𝑣l 𝜑0 cos 𝑣l 𝜑 l ⎪ i𝛼 l=0 uh = ⎨ ∞ ∑ ⎪𝜋I 𝜀l J𝑣l (kr0 )H𝑣(1) (kr) cos 𝑣l 𝜑0 cos 𝑣l 𝜑 l ⎪ i𝛼 0 l=0 ⎩

with r ≤ r0 , (2.28) with r ≥ r0 .

These expressions relate to the excitation of the field by a cylindrical wave, with a source term I0 𝛿(r − r0 , 𝜑 − 𝜑0 ), around the wedge in the region 0 ≤ 𝜑 ≤ 𝛼, 0 ≤ r ≤ ∞. They can be modified for excitation by a plane wave.

TRANSITION TO PLANE WAVE EXCITATION

2.2

55

TRANSITION TO PLANE WAVE EXCITATION

For Hankel functions with large arguments (kr0 → ∞), one can use the asymptotic expression √ H𝑣(1) (kr0 ) l

∼

2 i[kr0 −(𝜋∕2)𝑣l −(𝜋∕4)] e ≈ H0(1) (kr0 )e−i(𝜋∕2)𝑣l . 𝜋kr0

(2.29)

Then the field Equations (2.27) and (2.28) can be rewritten for the region r < r0 as us =

∞ ∑ 𝜋 e−i(𝜋∕2)𝑣l J𝑣l (kr) sin 𝑣l 𝜑0 sin 𝑣l 𝜑, I0 H0(1) (kr0 ) i𝛼 l=1

(2.30)

uh =

∞ ∑ 𝜋 𝜀l e−i(𝜋∕2)𝑣l J𝑣l (kr) cos 𝑣l 𝜑0 cos n𝑣l 𝜑 I0 H0(1) (kr0 ) i𝛼 l=0

(2.31)

or us =

𝜋 I H (1) (kr0 ) 2i𝛼 0 0 ×

∞ ∑

e−i(𝜋∕2)𝑣l J𝑣l (kr)[cos 𝑣l (𝜑 − 𝜑0 ) − cos 𝑣l (𝜑 + 𝜑0 )],

(2.32)

l=0

uh =

𝜋 I H (1) (kr0 ) 2i𝛼 0 0 ×

∞ ∑

𝜀l e−i(𝜋∕2)𝑣l J𝑣l (kr)[cos 𝑣l (𝜑 − 𝜑0 ) + cos 𝑣l (𝜑 + 𝜑0 )].

(2.33)

l=0

To clarify these expressions, we consider the solution of the wave Equation (2.1) in a free infinite homogeneous medium, without the scattering wedge. For this problem it is convenient to use the new polar coordinates (𝜌, 𝜙) with the origin at the radiating source (r0 , 𝜑0 ). It is clear that due to the azimuthal symmetry of the problem, its solution is a function of only one variable, u = u(𝜌). In addition, as the wave equation is a differential equation of the second order, its solution, in general, is the sum of two fundamental solutions of the related homogeneous equation u(𝜌) = c1 H0(1) (k𝜌) + c2 H0(2) (k𝜌)

(2.34)

with constants c1 and c2 . We retain only the first term, u(𝜌) = c1 H0(1) (k𝜌),

(2.35)

56

WEDGE DIFFRACTION: EXACT SOLUTION AND ASYMPTOTICS

nˆ = ρˆ

ρ=ε

Figure 2.3 Circular region (0 ≤ 𝜌 ≤ 𝜀, 0 ≤ 𝜙 ≤ 2𝜋) with a radiating source at the center.

because the second term, with H0(2) (k𝜌), does not satisfy the Sommerfeld radiation condition (2.12) and represents a nonphysical wave incoming from infinity. The constant c1 is found again with Green’s theorem (2.15) applied to the circular region S of a small radius 𝜀 with the center at 𝜌 = 0 (Fig. 2.3). For small values k𝜌 ≪ 1, the function u(k𝜌) and its normal derivative du∕dn = du∕d𝜌 (at the boundary of the region S) are described by the asymptotic approximations u(𝜌) ≈ c1

i2 ln(k𝜌), 𝜋

du(𝜌) i2 ≈ c1 d𝜌 𝜋𝜌

with k𝜌 ≪ 1.

(2.36)

By substitution of these quantities into Green’s theorem and taking the limit with 𝜀 → 0, we find that c1 = I0 ∕i4 and u(𝜌) =

1 I H (1) (k𝜌). i4 0 0

(2.37)

This solution explains the physical meaning of the factor in front of the series in Equations (2.32) and (2.33). It actually represents the field of the incident wave on the edge of the wedge u0 =

1 I H (1) (kr0 ). 4i 0 0

(2.38)

When r0 → ∞ and I0 → ∞, this field can be interpreted as a plane wave traveling to the wedge from the direction 𝜑 = 𝜑0 : uinc = u0 e−ikr cos(𝜑−𝜑0 ) .

(2.39)

As a result, we can rewrite Equations (2.32) and (2.33) in the classical Sommerfeld form (Sommerfeld, 1935): us = u0 ⋅ [u(kr, 𝜑 − 𝜑0 ) − u(kr, 𝜑 + 𝜑0 )],

(2.40)

for uinc = u0 eikr cos(𝜑−𝜑0 ) uh = u0 ⋅ [u(kr, 𝜑 − 𝜑0 ) + u(kr, 𝜑 + 𝜑0 )],

(2.41)

CONVERSION OF THE SERIES SOLUTION TO THE SOMMERFELD INTEGRALS

57

where u(kr, 𝜓) =

∞ 2𝜋 ∑ −i(𝜋∕2)𝑣l 𝜀e J𝑣l (kr) cos 𝑣l 𝜓. 𝛼 l=0 l

(2.42)

Equations (2.40) to (2.42) finally determine the total field generated by the incident plane wave in the presence of a perfectly reflecting wedge.

2.3 CONVERSION OF THE SERIES SOLUTION TO THE SOMMERFELD INTEGRALS Sommerfeld (1935) presented a solution of the wedge diffraction problem in integral form and then transformed it into an infinite series. Here we perform a reverse procedure and convert the infinite series (2.40) and (2.41) into integrals. To do this we use the following expression of the Bessel function (Sommerfeld, 1935): 1 2𝜋 ∫I

J𝑣s (kr) =

III

ei[kr cos 𝛽+𝑣l (𝛽−𝜋∕2)] d𝛽,

(2.43)

where the integration contour is as shown in Figure 2.4. Then the function u(kr𝜓) can be represented as 1 u(kr, 𝜓) = 2𝛼 ∫I

III

[ ikr cos 𝛽

e

1+

∞ ∑

i𝑣l (𝛽−𝜋+𝜓)

e

l=1

+

∞ ∑

] i𝑣l (𝛽−𝜋−𝜓)

e

d𝛽

l=1

Im β

I

–π

III

0

π

2π

Figure 2.4 Integration contour in Equation (2.43).

Re β

(2.44)

58

WEDGE DIFFRACTION: EXACT SOLUTION AND ASYMPTOTICS

with 𝑣l = l𝜋∕𝛼. Here the series are geometrical progressions. Their summation leads to u(kr, 𝜓) =

1 2𝛼 ∫I

III

eikr cos 𝛽

[

1 1 − ei(𝜋∕𝛼)(𝛽−𝜋+𝜓)

−

]

1 1 − e−i(𝜋∕𝛼)(𝛽−𝜋−𝜓)

d𝛽. (2.45)

With a new variable 𝛽 ′ = 𝛽 − 𝜋, this becomes 1 u(kr, 𝜓) = 2𝛼 ∫I′

III′

−ikr cos 𝛽 ′

e

[

1 1 − ei(𝜋∕𝛼)(𝛽 ′ +𝜓)

−

]

1 1 − e−i(𝜋∕𝛼)(𝛽 ′ −𝜓)

d𝛽 ′ . (2.46)

Here the integration contour is shifted by −𝜋 compared to the contour shown in Figure 2.4. According to the difference inside brackets, the function u(kr, 𝜓) can be represented as the sum of two integrals. In the integral related to the first term inside brackets, we replace 𝛽 ′ by 𝛽. In the integral related to the second term inside brackets, we change 𝛽 ′ to −𝛽. As a result, we arrive at the Sommerfeld integral (Sommerfeld, 1935): e−ikr cos 𝛽 1 d𝛽. 2𝛼 ∫C 1 − ei(𝜋∕𝛼)(𝛽+𝜓)

u(kr, 𝜓) =

(2.47)

The integration contour C, consisting of two branches, is shown in Figure 2.5.

(β )

C

D –2π

–π

π

0

D

2π

C

Figure 2.5 Integration contours in Equations (2.47) and (2.50).

CONVERSION OF THE SERIES SOLUTION TO THE SOMMERFELD INTEGRALS

59

Im ζ

D0 –π /2

Re ζ 0

π /2 D0

Figure 2.6 Integration counter in Equation (2.51).

The integrand of u(kr, 𝜓) possesses the first-order poles 𝛽m = 2𝛼m − 𝜓

with m = 0, ±1 = ±2, ±3, … .

(2.48)

Application of the Cauchy theorem to the integral over the closed contour C–D (Fig. 2.5) results in ⎧ 𝑣(kr, 𝜓) + e−ikr cos 𝜓 ⎪ u(kr, 𝜓) = ⎨ 𝑣(kr, 𝜓) ⎪ 𝑣(kr, 𝜓) + e−ikr cos(2𝛼−𝜓) ⎩

with −𝜋 < 𝜓 < 𝜋, with 𝜋 < 𝜓 < 2𝛼 − 𝜋, with 2𝛼 − 𝜋 < 𝜓 < 2𝛼 + 𝜋,

(2.49)

where 𝑣(kr, 𝜓) =

e−ikr cos 𝛽 1 d𝛽. 2𝛼 ∫D 1 − ei(𝜋∕𝛼)(𝛽+𝜓)

(2.50)

The integration contour D consists of two branches (Fig. 2.5). In the integral over the left branch, we change the variable 𝛽 to 𝜁 − 𝜋, and in the integral over the right branch we put 𝛽 = 𝜁 + 𝜋. Then the function 𝑣(kr, 𝜓) transforms into the integral over the contour D0 (Fig. 2.6): 𝑣(kr, 𝜓) = i

sin(𝜋∕n) eikr cos 𝜁 d𝜁 , 2𝜋n ∫D0 cos(𝜋∕n) − cos[(𝜁 + 𝜓)∕n]

(2.51)

where n = 𝛼∕𝜋. Note that 𝑣(kr, 𝜑 − 𝜑0 ) = 𝑣(kr, 𝜑0 − 𝜑). This equality is easily proved by introducing a new variable, 𝜉 = −𝜁 . The physical interpretation of Equation (2.49) is the following. The function 𝑣(kr, 𝜓) describes the diffracted part of the field, and the residues relate to the geometrical optics. This interpretation becomes clear if we consider the functions u(kr, 𝜑 − 𝜑0 ) and u(kr, 𝜑 + 𝜑0 ).

60

WEDGE DIFFRACTION: EXACT SOLUTION AND ASYMPTOTICS

φ = π – φ0 φ0

φ=0

φ = π + φ0

Figure 2.7 The incident plane wave propagates from the direction 𝜑 = 𝜑0 . The line 𝜑 = 𝜋 − 𝜑0 is the boundary of the reflected wave, and the line 𝜑 = 𝜋 + 𝜑0 is the shadow boundary.

In the case 0 < 𝜑0 < 𝛼 − 𝜋, when only one face (𝜑 = 0) of the wedge is illuminated (Fig. 2.7), these functions are determined by u(kr, 𝜑 − 𝜑0 ) = 𝑣(kr, 𝜑 − 𝜑0 ) + e−ikr cos(𝜑−𝜑0 ) u(kr, 𝜑 + 𝜑0 ) = 𝑣(kr, 𝜑 + 𝜑0 ) + e−ikr cos(𝜑+𝜑0 ) u(kr, 𝜑 − 𝜑0 ) = 𝑣(kr, 𝜑 − 𝜑0 ) + e−ikr cos(𝜑−𝜑0 ) u(kr, 𝜑 + 𝜑0 ) = 𝑣(kr, 𝜑 + 𝜑0 )

} with 0 < 𝜑 < 𝜋 − 𝜑0 (2.52) } with 𝜋 − 𝜑0 < 𝜑 < 𝜋 + 𝜑0 (2.53)

u(kr, 𝜑 − 𝜑0 ) = 𝑣(kr, 𝜑 − 𝜑0 ) u(kr, 𝜑 + 𝜑0 ) = 𝑣(kr, 𝜑 + 𝜑0 )

} with 𝜋 + 𝜑0 < 𝜑 < 𝛼.

(2.54)

In Equations (2.52) and (2.53), the term e−ikr cos(𝜑−𝜑0 ) determines the incident plane wave, which exists only in the illuminated region, 0 < 𝜑 < 𝜋 + 𝜑0 , and the term e−ikr cos(𝜑+𝜑0 ) relates to the reflected plane wave existing in the region 0 < 𝜑 < 𝜋 − 𝜑0 . In agreement with the geometrical optics, Equation (2.54) does not contain either the incident or reflected plane waves, because the region 𝜋 + 𝜑0 < 𝜑 < 𝛼 is shadowed by the wedge. The boundaries of the incident and reflected plane waves are shown in Figure 2.7. Figure 2.8 illustrates the situation when both faces of the wedge are illuminated. In this case 𝛼 − 𝜋 < 𝜑0 < 𝜋, and the functions u(kr, 𝜑 − 𝜑0 ) and u(kr, 𝜑 + 𝜑0 ) are determined by } u(kr, 𝜑 − 𝜑0 ) = 𝑣(kr, 𝜑 − 𝜑0 ) + e−ikr cos(𝜑−𝜑0 ) with 0 < 𝜑 < 𝜋 − 𝜑0 (2.55) u(kr, 𝜑 + 𝜑0 ) = 𝑣(kr, 𝜑 + 𝜑0 ) + e−ikr cos(𝜑+𝜑0 ) } u(kr, 𝜑 − 𝜑0 ) = 𝑣(kr, 𝜑 − 𝜑0 ) + e−ikr cos(𝜑−𝜑0 ) with 𝜋 − 𝜑0 < 𝜑 < 2𝛼 − 𝜋 − 𝜑0 u(kr, 𝜑 + 𝜑0 ) = 𝑣(kr, 𝜑 + 𝜑0 ) (2.56) } u(kr, 𝜑 − 𝜑0 ) = 𝑣(kr, 𝜑 − 𝜑0 ) + e−ikr cos(𝜑−𝜑0 ) with 2𝛼 − 𝜋 − 𝜑0 < 𝜑 < 𝛼. u(kr, 𝜑 + 𝜑0 ) = 𝑣(kr, 𝜑 + 𝜑0 ) + e−ikr cos(2𝛼−𝜑−𝜑0 ) (2.57)

61

THE SOMMERFELD RAY ASYMPTOTICS

φ = π – φ0 φ=0

φ = 2α – π – φ 0

φ=α

Figure 2.8 The incident plane wave illuminates both faces of a wedge. The line 𝜑 = 𝜋 − 𝜑0 is the boundary of the wave reflected from the face 𝜑 = 0, and the line 𝜑 = 2𝛼 − 𝜋 − 𝜑0 is the boundary of the wave reflected from the face 𝜑 = 𝛼.

The term e−ikr cos(2𝛼−𝜑−𝜑0 ) describes the plane wave reflected from the face 𝜑 = 𝛼.

2.4

THE SOMMERFELD RAY ASYMPTOTICS

The relationships us = Ez and uh = Hz exist between the acoustic and electromagnetic edge diffracted rays.

A simple asymptotic expression for the function 𝑣(kr, 𝜓) with kr ≫ 1 can be found by the steepest descent method (Copson, 1965; Murray, 1984). With this purpose we replace the integration variable in Equation (2.51) by s=

√

𝜁 2ei𝜋∕4 sin . 2

(2.58)

Then s2 = i(1 − cos 𝜁 ) and 𝑣(kr, 𝜓) =

2 sin(𝜋∕n) i(kr+𝜋∕4) ∞ e−krs ds, (2.59) √ e ∫−∞ {cos(𝜋∕n) − cos[(𝜓 + 𝜁 )∕n]} cos(𝜁 ∕2) n𝜋 2

where n = 𝛼∕𝜋. Here s = 0 is the saddle point. Indeed, when point s moves from the saddle point along the imaginary axis, the function exp(−krs2 ) increases most rapidly. In contrast, this function decreases most rapidly when the point s moves away from the saddle point along the real axis. Because of that, the vicinity of the saddle point provides the major contribution to the integral when kr ≫ 1. According to the steepest descent method, the slowly varying factor of the integrand is expanded into a Taylor power series near the saddle point and is then integrated term by term. If the integrand expansion is convergent only in the vicinity of the saddle point, the series obtained

62

WEDGE DIFFRACTION: EXACT SOLUTION AND ASYMPTOTICS

after integration will be semiconvergent: that is, asymptotic. Retaining the first term in this series for the function 𝑣(kr, 𝜓), we obtain 𝑣(kr, 𝜓) ∼ =

∞ sin(𝜋∕n) 2 ei(kr+𝜋∕4) e−krs ds √ ∫ cos(𝜋∕n) − cos(𝜓∕n) −∞ n𝜋 2

(1∕n) sin(𝜋∕n) ei(kr+𝜋∕4) . √ cos(𝜋∕n) − cos(𝜓∕n) 2𝜋kr

(2.60)

The next terms of the asymptotic series for the function 𝑣(kr, 𝜓) are small quantities on the order of√(kr)−3∕2 and higher. The asymptotic expression (2.60) is valid under the condition kr |cos(𝜓∕2)| ≫ 1 and describes cylindrical waves diverging from the edge: that is, edge waves. According to Equations (2.40), (2.49), and (2.60), the wave diffracted at the edge of the acoustically soft wedge is determined as

ei(kr+𝜋∕4) , uds = u0 [𝑣(kr, 𝜑 − 𝜑0 ) − 𝑣(kr, 𝜑 + 𝜑0 )] ∼ u0 f (𝜑, 𝜑0 , 𝛼) √ 2𝜋kr

(2.61)

where f (𝜑, 𝜑0 , 𝛼) =

( ) sin(𝜋∕n) 1 1 − . n cos(𝜋∕n) − cos[(𝜑 − 𝜑0 )∕n] cos(𝜋∕n) − cos[(𝜑 + 𝜑0 )∕n]

(2.62) Equations (2.41), (2.49), and (2.60) determine the wave arising at the edge of the acoustically hard wedge:

ei(kr+𝜋∕4) , (2.63) udh = u0 [𝑣(kr, 𝜑 − 𝜑0 ) + 𝑣(kr, 𝜑 + 𝜑0 )] ∼ u0 g(𝜑, 𝜑0 , 𝛼) √ 2𝜋kr

where

g(𝜑, 𝜑0 , 𝛼) =

sin(𝜋∕n) n

(

1 1 + cos(𝜋∕n) − cos[(𝜑 − 𝜑0 )∕n] cos(𝜋∕n) − cos[(𝜑 + 𝜑0 )∕n]

) .

(2.64) Functions f and g describe the directivity patterns of the edge waves. The asymptotic expressions (2.60) to (2.64) were introduced by Sommerfeld (1935) and are well known. It is easy to verify that they satisfy the boundary conditions (2.2) and (2.3).

THE PAULI ASYMPTOTICS

63

One should mention that the two-dimensional edge waves (2.61) and (2.63) can be interpreted as continuous sets of edge-diffracted rays. They arise due to diffraction but propagate from the edge in the first asymptotic approximation as ordinary rays in accordance with geometrical optics laws for two-dimensional fields. As shown in the work of Pelosi et al. (1998), the edge-diffracted rays had already been observed visually by Newton, although he did not use such terminology. The term diffracted ray was introduced by Kalashnikov (1912), who also was the first person to present objective experimental proof of the existence of edge-diffracted rays by recording them on a photographic plate. Theoretically, their existence was established first by Rubinowicz (1924) and later by many other researchers. Keller (1962) formulated the concept of diffracted rays in a general form. Here it is also pertinent to remind one about Sommerfeld’s warning against the too formal ray interpretation of diffraction phenomena. He wrote that shining diffraction points on edges do not exist in reality and are just optical illusions: “Das ist naturlich eine optische Tauschung” (Sommerfeld, 1896, p. 369). He explained that such seemingly shining edge points are the result of our perception, or in Sommerfeld’s words, the result of “analytical continuation” of diffracted rays by our eyes. Because of the ray structure of edge waves, the asymptotic expressions (2.61) to (2.64) can be called the ray asymptotics, as emphasized in the title of the present section. These asymptotics have an essential drawback. They are not valid near the shadow boundary (𝜑 ≈ 𝜋 + 𝜑0 ) or near the boundaries of reflected plane waves (𝜑 ≈ 𝜋 − 𝜑0 , 𝜑 ≈ 2𝛼 − 𝜋 − 𝜑0 ). The mathematical reason for this drawback is the following. Two poles, s1 =

√ 𝜋−𝜓 2ei𝜋∕4 sin 2

and

s2 =

( √ 𝜋+𝜓) , 2ei𝜋∕4 sin 𝛼 − 2

(2.65)

of the integrand in Equation (2.59) approach the saddle point s = 0 when 𝜓 = 𝜑 ± 𝜑0 → 𝜋 and 𝜓 = 𝜑 + 𝜑0) → 2𝛼 − 𝜋. In this case, the Taylor expansion for the integrand becomes meaningless because its terms tend to infinity. There is a physical background behind this mathematics. The vicinity of boundaries of incident and reflected waves is the region of the effective transverse diffusion where the field cannot be described in terms of diffracted rays and has a more complex structure. This phenomenon has been considered in detail by Ufimtsev (2003, 2009; Sec. 5.5).

2.5

THE PAULI ASYMPTOTICS

In 1938, Pauli suggested the asymptotic expansion for the function 𝑣(kr, 𝜓), which is valid at the geometrical optics boundaries 𝜑 = 𝜋 ± 𝜑0 and transforms to the Sommerfeld asymptotics away from these boundaries (Pauli, 1938). In this section we provide the derivation for the first term of the Pauli expansion. Usually, for engineering analysis only the first terms in asymptotic expansions are of practical value. Higher-order terms commonly are not utilized, because they are smaller in magnitude and are quite complicated to evaluate. Besides, the high-order terms can occur beyond the frames

64

WEDGE DIFFRACTION: EXACT SOLUTION AND ASYMPTOTICS

of validity of idealized mathematical models used for a description of real physical phenomena. That is why we focus here on the first asymptotic term. According to Equation (2.59), 𝑣(kr, 𝜓) =

sin(𝜋∕n) i(kr+𝜋∕4) ∞ −krs2 ds e , √ e ∫ {cos(𝜋∕n) − cos[(𝜓 + 𝜁 )∕n]} cos(𝜁 ∕2) −∞ n𝜋 2 (2.66)

where s=

√

𝜁 2ei𝜋∕4 sin , 2

n=

𝛼 . 𝜋

(2.67)

with s20 = 2i cos2

𝜓 . 2

(2.68)

s2 = i(1 − cos 𝜁 ),

Let us multiply and divide the integrand by cos 𝜓 + cos 𝜁 = i(s2 − s20 ) Then 𝑣(kr, 𝜓) =

2 sin(𝜋∕n) i(kr−𝜋∕4) ∞ e−krs f (s, 𝜓) ds, √ e ∫−∞ s2 − s20 n𝜋 2

(2.69)

√ where the poles s = ±s0 = ± 2ei(𝜋∕4) cos(𝜓∕2) are outside the integration contour and approach it at the saddle point s = 0 when 𝜓 → 𝜋. The function f (s, 𝜓) =

cos 𝜓 + cos 𝜁 {cos(𝜋∕n) − cos[(𝜓 + 𝜁 )∕n]} cos(𝜁 ∕2)

(2.70)

does not have a pole at the saddle point s = 0 (𝜁 = 0) when 𝜓 = 𝜑 ± 𝜑0 → 𝜋. Therefore, it can be expanded into a regular Taylor series. By integrating this series term by term, Pauli obtained the asymptotic expansion of the function 𝑣(kr, 𝜓) for the large argument kr. The first term of this expansion is determined by √ 𝑣(kr, 𝜓) =

∞ −krs 2 sin(𝜋∕n) 1 + cos 𝜓 e ds. ei(kr−𝜋∕4) ∫0 s2 − s2 𝜋 n cos(𝜋∕n) − cos(𝜓∕n) 0 2

(2.71)

Here the integral can be represented as ∞

∫0

2

2 e−krs ds = e−krs0 2 2 ∫0 s − s0

∞

∞

ds

∫kr

2 −s2 )t 0

e−(s

dt.

(2.72)

65

THE PAULI ASYMPTOTICS

By changing the order of integration, we obtain ∞

∫0

√ 2 ∞ ∞ 𝜋 −krs2 ∞ i|||s0 |||2 t dt e−krs −krs20 s20 t dt −x2 ds = e e e dx = e e 0 √ √ ∫kr ∫kr 2 s2 − s20 t ∫0 t √ 𝜋 −ikr||s ||2 ∞ 2 = (2.73) e | 0| √ eiq dq. |s0 | ∫ kr|s0 | | |

As a result, 𝑣(kr, 𝜓) =

∞ 2 2 sin(𝜋∕n) |cos(𝜓∕2)| −ikr cos 𝜓 e−i(𝜋∕4) eiq dq (2.74) e √ √ n cos(𝜋∕n) − cos(𝜓∕n) 𝜋 ∫ 2kr|cos(𝜓∕2)|

or

∞ cos(𝜓∕2) 2 2 sin(𝜋∕n) cos(𝜓∕2) −ikr cos 𝜓 e−i(𝜋∕4) eiq dq. e √ √ n cos(𝜋∕n) − cos(𝜓∕n) 𝜋 ∫ 2kr cos(𝜓∕2)

𝑣(kr, 𝜓) =

(2.75) This expression represents the slightly modified first term in the Pauli asymptotic expansion. The next term is on the order of (kr)−1∕2 near the boundaries 𝜑 = 𝜋 ± 𝜑0 and is on the order of (kr)−3∕2 away from them. The upper limit of the Fresnel integral in Equation (2.75) should be read as sgn[cos(𝜓∕2)]∞. It always equals infinity but changes its sign when the observation point intersects the geometrical optics boundaries (𝜓 = 𝜑 ± 𝜑0 = 𝜋). Here the function 𝑣(kr, 𝜓) undergoes the discontinuity and in this way ensures the continuity of the function u(kr, 𝜓) and therefore the continuity of the total field. Indeed, by using the formula √ ∞ 𝜋 i(𝜋∕4) iq2 e dq = , (2.76) e ∫0 2 one can show that 𝑣(kr, 𝜋 + 0) = 12 eikr ,

𝑣(kr, 𝜋 − 0) = − 12 eikr ,

u(kr, 𝜋 ± 0) = 12 eikr .

(2.77) (2.78)

Also, with the help of the asymptotic approximations p

∫∞

2

eiq dq ∼

2

−p

2

2 eip eip eiq dq ∼ − , 2ip ∫−∞ 2ip

with p ≫ 1,

(2.79)

66

WEDGE DIFFRACTION: EXACT SOLUTION AND ASYMPTOTICS

it is easy to verify that the Pauli expression (2.75) converts to the Sommerfeld √ asymptotics (2.60) under the condition kr |cos(𝜓∕2)| ≫ 1. As shown by Ufimtsev (2003, 2009; Sec. 5.5), the Pauli asymptotics (2.75) can be considered to be a “stenographic form” of the more physically meaningful expression [√ 𝑣(kr, 𝜓) = V

] [ ] i(kr+(𝜋∕4)) (1∕n) sin(𝜋∕n) kr 1 e (𝜓 − 𝜋) eikr + − √ 2 cos(𝜋∕n) − cos(𝜓∕n) 𝜓 − 𝜋 2𝜋kr (2.80)

with V(𝜏) = e−i𝜏

2

e−i(𝜋∕4) √ 𝜋 ∫𝜏

∞sgn𝜏

2

eiq dq,

(2.81)

which √ follows from the solution of the parabolic equation. Here, the first term, V[ kr∕2(𝜓 − 𝜋)]eikr , describes the transverse diffusion of the wave field in the vicinity of the geometrical optics boundaries and does not depend on the reflective properties of the wedge faces. The second term in Equation (2.80) can be interpreted as the diffraction background. It is of interest that in the particular case when the angle 𝛼 = 2𝜋 and the wedge transforms into the half-plane, the Pauli asymptotic (2.75) transforms to the function √ 2kr cos(𝜓∕2)

e−i𝜋∕4 𝑣hp (kr, 𝜓) = e−ikr cos 𝜓 √ 𝜋 ∫∞ cos(𝜓∕2)

2

eiq dq

(2.82)

and provides an exact (!) solution to the half-plane diffraction problem. Indeed, in this case, n = 2 and Equation (2.51) becomes 𝑣(kr, 𝜓) = −

eikr cos 𝜁 i d𝜁 , 4𝜋 ∫D0 cos[(𝜓 + 𝜁)∕2]

(2.83)

which can be converted to the Fresnel integral. To do this, let us separate the contour D0 (see Fig. 2.6) into two parts at the point 𝜁 = 0. Summation of the integrals over these parts of the integration contour leads to the expression 𝑣(kr, 𝜓) = − =−

i 4𝜋 ∫0

(𝜋∕2)−i∞

eikr cos 𝜁

[

] 1 1 + d𝜁 cos(𝜓 + 𝜁 )∕2 cos(𝜓 − 𝜁 )∕2

(𝜋∕2)−i∞ cos(𝜁 ∕2) 𝜓 i eikr cos 𝜁 cos d𝜁 . 𝜋 2 ∫0 cos 𝜓 + cos 𝜁

(2.84)

THE PAULI ASYMPTOTICS

67

√ We then introduce the integration variable s = 2 ei(𝜋∕4) sin(𝜁 ∕2) and apply the procedure outlined in Equations (2.68) to (2.71). As a result, we obtain √ ∞ −krs2 2 𝜓 e ds. cos ei(kr−𝜋∕4) 𝑣(kr, 𝜓) = − ∫0 s2 − s2 𝜋 2 0

(2.85)

With the help of Equation (2.73), this expression transforms to Equation (2.82). The latter, together with Equations (2.39) to (2.41) and (2.52) to (2.54), provides an exact solution to the half-plane diffraction problem: ∞ cos(𝜓∕2)

2 e−i(𝜋∕4) eiq dq. 𝑣hp (kr, 𝜓) = −e−ikr cos 𝜓 √ √ ∫ 2kr cos(𝜓∕2) 𝜋

(2.86)

Thus, the Pauli asymptotic (2.75) possesses valuable properties. It is simple and provides an exact solution to the half-plane diffraction problem. It describes both the transverse diffusion of the wave field near the geometrical optics boundaries and the diffracted rays away from these boundaries. However, it is not free of certain drawbacks:

r The total field us,h determined with the Pauli asymptotics (2.75) exactly sat-

isfies the boundary conditions (2.2) and (2.3) on the face 𝜑 = 0. However, on the√ face 𝜑 = 𝛼, these boundary conditions are satisfied only asymptotically, when kr |cos(𝜓∕2)| ≫ 1 and the Pauli asymptotic converts to the Sommerfeld expression (2.60). r The Pauli asymptotic (2.75) provides correct values for the wave field in the direction of the shadow boundary (𝜑 = 𝜋 + 𝜑0 ) and in the direction 𝜑 = 𝜋 − 𝜑0 of the plane wave reflected from the face 𝜑 = 0. However, it fails at the direction 𝜑 = 2𝛼 − 𝜋 − 𝜑0 of the plane wave reflected from the face 𝜑 = 𝛼 (Fig. 2.8). It predicts an incorrect infinite value for the field in this √ direction. The reason for this singularity is the existence of the pole s2 = 2 ei𝜋∕4 sin[𝛼 − (𝜋 + 𝜓)∕2] approaching the saddle point when 𝜑 → 2𝛼 − 𝜋 − 𝜑0 . However, this pole was missed by Pauli, a drawback that is corrected in the next section. One can suggest the following remedy to diminish these drawbacks to some extent. The asymptotic (2.75) should be used only in the region 0 ≤ 𝜑 ≤ 𝛼∕2. To calculate the field in the rest of region 𝛼∕2 ≤ 𝜑 ≤ 𝛼, it is necessary to introduce new polar coordinates with the angle 𝜑′ measured from the face 𝜑 = 𝛼 and then to apply the expression (2.75) in the region 0 < 𝜑′ ≤ 𝛼∕2. In this way one can obtain correct values for the field at the boundary of the plane wave reflected from the face 𝜑 = 𝛼 and satisfy the boundary conditions on this face, but at the expense of the field discontinuity in the direction 𝜑 = 𝛼∕2.

68

WEDGE DIFFRACTION: EXACT SOLUTION AND ASYMPTOTICS

The discontinuity of the field at 𝜑 = 𝛼∕2 is a manifestation of the fact that the asymptotic (2.75) does not satisfy the fundamental physical principle. It is not invariant with respect to the choice of coordinate system. Indeed, if we choose the polar coordinates 𝜑′ and 𝜑′0 measured from the face 𝜑 = 𝛼, the Pauli asymptotic leads to the relationships 𝑣Pauli (kr, 𝜑′ − 𝜑′0 ) = 𝑣Pauli [kr, 𝛼 − 𝜑 − (𝛼 − 𝜑0 )] = 𝑣Pauli (kr, 𝜑 − 𝜑0 ),

(2.87)

𝑣Pauli (kr, 𝜑′ + 𝜑′0 ) = 𝑣Pauli (kr, 2𝛼 − 𝜑 − 𝜑0 ) ≠ 𝑣Pauli (kr, 𝜑 + 𝜑0 ).

(2.88)

The last inequality indicates that, strictly speaking, the Pauli asymptotic does not satisfy the invariance principle. However, it satisfies this principle approximately when it transforms into the Sommerfeld ray asymptotics. Next we derive uniform asymptotics applicable in all scattering directions (0 < 𝜑 < 𝛼).

2.6 UNIFORM ASYMPTOTICS: EXTENSION OF THE PAULI TECHNIQUE Here we derive asymptotic expressions under the condition that the incident wave does not undergo double and higher-order multiple reflections at faces of the wedge. This condition is always realized for convex wedges (𝜋 < 𝛼 ≤ 2𝜋) and also for concave wedges and horns (𝜋∕2 < 𝛼 < 𝜋) but only for certain directions of the incident wave. However, the theory developed below can easily be extended for any narrow horns (0 < 𝛼 < 𝜋∕2) with multiple reflections. Now we return to Equation (2.66) and we observe that two poles, s1 =

√ 𝜓 2 ei𝜋∕4 cos 2

and

( √ 𝜓) , s2 = − 2 ei𝜋∕4 cos 𝛼 − 2

(2.89)

can approach the saddle point s = 0 when 𝜓 = 𝜑 ± 𝜑0 → 𝜋 or 𝜓 = 𝜑 + 𝜑0 → 2𝛼 − 𝜋. The pole s1 approaches the saddle point when the direction of observation 𝜑 tends to the shadow boundary 𝜑 = 𝜋 + 𝜑0 or to the boundary 𝜑 = 𝜋 − 𝜑0 of the wave reflected from the face 𝜑 = 0 (Fig. 2.7). The pole s2 approaches the saddle point when the direction 𝜑 tends to the boundary 𝜑 = 2𝛼 − 𝜋 − 𝜑0 of the wave reflected from the face 𝜑 = 𝛼 (Fig. 2.8). All other poles in (2.66) can be ignored, as they are outside the integration contour and never reach the saddle point in the absence of multiple reflections. Taking these observations into account, we multiply and divide the integrand in (2.66) by the factor )( ) ( (cos 𝜁 + cos 𝜓)[cos 𝜁 + cos(2𝛼 − 𝜓)] = − s2 − s21 s2 − s22

(2.90)

UNIFORM ASYMPTOTICS: EXTENSION OF THE PAULI TECHNIQUE

69

and obtain

𝑣(kr, 𝜓) = −

2 sin(𝜋∕n) i(kr+𝜋∕4) ∞ e−krs f (s, 𝜓) ( ) ( ) ds, √ e ∫−∞ s2 − s21 s2 − s22 n𝜋 2

(2.91)

where f (s, 𝜓) =

(cos 𝜁 + cos 𝜓)[cos 𝜁 + cos(2𝛼 − 𝜓)] . {cos(𝜋∕n) − cos[(𝜓 + 𝜁)n]} cos(𝜁 ∕2)

(2.92)

This function is finite and continuous in the vicinity of the saddle point s = 0 and therefore can be expanded into the Taylor series. The integration of this series in Equation (2.91) leads to the uniform asymptotic expansion for 𝑣(kr, 𝜓) valid for any angles 𝜑 and 𝜑0 . One should note that only even terms with factors s2m (m = 1, 2, 3, …) give nonzero contributions to the integral in (2.91). We retain and calculate here the first two terms of this asymptotic expansion. The second term is retained because for 𝛼 → 2𝜋 it partially contains a quantity of the same order as the first term. The related asymptotic expression for the function 𝑣(kr, 𝜓) is determined by 𝑣(kr, 𝜓) = 𝑣1 (kr, 𝜓) + 𝑣2 (kr, 𝜓),

(2.93)

where 2 ∞ sin(𝜋∕n) i(kr+𝜋∕4) e−krs ds f (0, 𝜓) e ( ) ( ), √ ∫−∞ s2 − s2 s2 − s2 n𝜋 2 1 2

(2.94)

2 sin(𝜋∕n) i(kr+𝜋∕4) d2 f (0, 𝜓) ∞ e−krs s2 ds ( ) ( ), √ e ∫−∞ s2 − s2 s2 − s2 ds2 n𝜋2 2 1 2

(2.95)

𝑣1 (kr, 𝜓) = −

𝑣2 (kr, 𝜓) = −

and ∞

∫−∞

[ ] 2 ∞ −krs2 ∞ −krs2 2 e−krs e ds e ds − , ( )( ) ds = 2 s1 − s22 ∫0 s2 − s21 ∫0 s2 − s22 s2 − s21 s2 − s22 ∞

∫0

2

e−krs s2 ds = ∫0 s2 − s21,2

∞

∞

2

e−krs ds + s21,2

∫0

(2.96)

2

e−krs ds. s2 − s21,2

(2.97)

70

WEDGE DIFFRACTION: EXACT SOLUTION AND ASYMPTOTICS

According to Equation (2.73), these integrals are reduced to Fresnel integrals. As a result, we obtain the following asymptotic expression:

(1∕n) sin(𝜋∕n) 1 e−i(𝜋∕4) √ cos(𝜋∕n) − cos(𝜓∕n) sin 𝛼 sin(𝛼 − 𝜓) 𝜋 [ ∞ cos(𝜓∕2) 𝜓 2 × P(𝛼, 𝜓) cos3 e−ikr cos 𝜓 √ eiq dq ∫ 2kr cos(𝜓∕2) 2

𝑣(kr, 𝜓) =

−Q(𝛼, 𝜓) cos

3

] 𝜓 ) −ikr cos(2𝛼−𝜓) ∞ cos(𝛼−𝜓∕2) iq2 e 𝛼− e dq , ∫√2kr cos(𝛼−𝜓∕2) 2

(

(2.98) where {

𝜓 )]2 P(𝛼, 𝜓) = 2 − cos 𝛼 − 2 [

(

}

2 4 1 + sin (𝜓∕n) − cos(𝜋∕n) cos(𝜓∕n) 1+ 2 n [cos(𝜋∕n) − cos(𝜓∕n)]2

, (2.99)

(

𝜓 )2 Q(𝛼, 𝜓) = 2 − cos 2

{

2 4 1 + sin (𝜓∕n) − cos(𝜋∕n) cos(𝜓∕n) 1+ 2 n [cos(𝜋∕n) − cos(𝜓∕n)]2

} . (2.100)

√ One can show that away √ from the boundaries of reflected plane waves, where kr |cos(𝜓∕2)| ≫ 1 and kr |cos[𝛼 − (𝜓∕2)]| ≫ 1, the asymptotic (2.98) transforms into the ray-type asymptotic

(1∕n) sin(𝜋∕n) (1∕n) sin(𝜋∕n) ei(kr+𝜋∕4) + √ cos(𝜋∕n) − cos(𝜓∕n) cos(𝜋∕n) − cos)𝜓∕n) 2𝜋kr { } 2 4 1 + sin (𝜓∕n) − cos(𝜋∕n) cos(𝜓∕n) ei(kr−𝜋∕4) × 1+ 2 . √ n [cos(𝜋∕n) − cos(𝜓∕n)]2 4 𝜋(2kr)3∕2

𝑣(kr, 𝜓) ∼

(2.101)

We emphasize that only the first term here can be interpreted in terms of diffracted rays; the second term, as well as all high-order asymptotic terms not presented in Equation (2.101), have a pronounced wave nature due to the factor (kr)−3∕2 .

UNIFORM ASYMPTOTICS: EXTENSION OF THE PAULI TECHNIQUE

71

At the boundaries of the reflected waves, the function (2.98) is discontinuous, 𝑣(kr, 𝜋 ± 0) = 𝑣(kr, 2𝛼 − 𝜋 ∓ 0) ∞ sin 𝛼

2 1 1 e−i𝜋∕4 𝜋 = ± eikr − cot sin 𝛼eikr cos 2𝛼 √ eiq dq √ ∫ 2 n n 2kr sin 𝛼 𝜋

(2.102)

and compensates for the discontinuities in the geometrical optics part of the total √ field. Under the condition 2kr |sin 𝛼| ≫ 1, it follows from Equation (2.102) that 1 1 𝜋 ei(kr+𝜋∕4) . 𝑣(kr, 𝜋 ± 0) = 𝑣(kr, 2𝛼 − 𝜋 ∓ 0) = ± eikr − cot √ 2 2n n 2𝜋kr

(2.103)

It is easy to check that the asymptotic (2.98) provides the correct result 𝑣(kr, 𝜓) = 0 in the limiting case 𝛼 = 𝜋 when the wedge transforms into the infinite plane and the diffracted field vanishes. Taking into account that P(2𝜋, 𝜓) = Q(2𝜋, 𝜓) = 0 and applying L’Hospital’s rule, one can show that in the other limiting case when 𝛼 → 2𝜋, the asymptotic (2.98) converts into the function (2.86) related to an exact solution of the half-plane diffraction problem. Now let us estimate the accuracy of asymptotic (2.98). This expression represents the sum of the two first terms in the asymptotic series resulting from term-by-term integration of the Taylor series for the integrand in Equation (2.91). Therefore, the error of (2.98) is a magnitude on the order of the neglected third asymptotic term. It is determined by the integral ∞

∫−∞

2

e−krs s4 ds = s2 − s21,2

{

O[(kr)−3∕2 ] O[(kr)−5∕2 ]

with s√ 1,2 = 0 with kr|s1,2 | ≫ 1,

(2.104)

where the values s1,2 = 0 relate to the geometrical optics boundaries of the incident and reflected waves. Note also that the asymptotic expressions for the total field based on Equation (2.98) do not satisfy the boundary conditions (2.2) and (2.3) rigorously. These conditions are satisfied only asymptotically under the condition √ | | kr |s1,2 | ≫ 1. In Equation (2.104) we have used the symbol O[(kr)−m ]to show the behavior of the integral under the condition kr ≫ 1. This is an ordinary definition accepted in the asymptotic theory. The expression f (x) = O(x−m ) means that lim[f (x) ⋅ xm ] = const with x → ∞. As mentioned in Section 2.5, the Pauli asymptotic (2.75) is not invariant with respect to the choice of coordinate system. A similar situation happens with asymptotic expressions (2.98): 𝑣[kr, 𝛼 − 𝜑 − (𝛼 − 𝜑0 )] ≠ 𝑣(kr, 𝜑 − 𝜑0 ),

(2.105)

𝑣[kr, 𝛼 − 𝜑 + (𝛼 − 𝜑0 )] = 𝑣(kr, 𝜑 + 𝜑0 ).

(2.106)

72

WEDGE DIFFRACTION: EXACT SOLUTION AND ASYMPTOTICS

In addition, the asymptotic (2.98) does not satisfy the reciprocity principle: 𝑣(kr, 𝜑 − 𝜑0 ) ≠ 𝑣(kr, 𝜑0 − 𝜑).

(2.107)

However, these shortcomings are admissible for asymptotic expressions, which satisfy the rigorous laws only approximately. Deviations from these laws are asymptotically small and are beyond the accuracy of these asymptotics. For example, asymptotics (2.102) and (2.103) for the field at the geometrical optics boundaries are indeed invariant with respect to the choice of polar coordinates, and according to the estimations (2.104), their error is a small value, on the order of (kr)−3∕2 . In addition, the ray-type asymptotic (2.101) is invariant in the same sense and rigorously satisfies the reciprocity principle. The found asymptotic (2.98) is convenient for field analysis. In the particular case when 𝛼 = 2𝜋 and the wedge transforms into the half-plane, it provides an exact solution. This asymptotic is simple and applicable for all observation angles 0 ≤ 𝜑 ≤ 𝛼. However, because of its asymptotic origination, it does not satisfy the boundary conditions or reciprocity and invariance principles rigorously, only approximately. Numerical comparison of Pauli’s asymptotic (2.75) and its extension (2.98) with the exact solution has been presented by Hacivelioglu et al. (2011, Figs. 9 and 10). Note that Pauli’s idea of asymptotic expansions in terms of confluent hypergeometric functions was extended by Clemmow (1950) for a more general type of singularity which can approach the saddle point, but he did not apply his theory to the wedge diffraction problem. Instead, he illustrated it using the example of diffraction at a black half-plane. However, in this problem the original Pauli method is entirely applicable and provides a proper approximation (Ufimtsev, 2003, 2009; Sec. 1.6). 2.7 FAST CONVERGENT INTEGRALS AND UNIFORM ASYMPTOTICS: THE “MAGIC ZERO” PROCEDURE Return to the integral (2.66): 𝑣(kr, 𝜓) =

∞ (1∕n) sin(𝜋∕n) 2 ds 1 i(kr+𝜋∕4) e−krs . (2.108) √ e ∫ cos(𝜋∕n) − cos[(𝜓 + 𝜁)∕n] cos(𝜁 ∕2) −∞ 𝜋 2

Here the function 𝜁 (s) is defined as follows. According to Equation (2.67), we obtain cos 𝜁 = 1 + is2 . Next, the Euler formula for the cosine function leads to the equation ei𝜁 + e−i𝜁 = 2(1 + is2 ).

(2.109)

Introducing a new variable, t = ei𝜁 , we arrive at the quadratic equation t2 − 2(1 + is2 )t + 1 = 0

(2.110)

√ t = 1 + is2 ± i s2 (s2 − 2i).

(2.111)

with its solution,

FAST CONVERGENT INTEGRALS AND UNIFORM ASYMPTOTICS: THE “MAGIC ZERO” PROCEDURE

73

Thus, √ 𝜁 (s) = −i ln[t(s)] = −i ln[1 + is2 ± i s2 (s2 − 2i)].

(2.112)

Note that setting t = e−i𝜁 in Equation (2.109) also leads to the same quadratic Equation (2.110) and to the solution √ 𝜁 (s) = i ln[1 + is2 ± i s2 (s2 − 2i)].

(2.113)

To define the unique solution 𝜁 (s), one should impose the conditions 𝜁 (−∞) = −

𝜋 + i∞, 2

𝜁 (∞) =

𝜋 − i∞. 2

(2.114)

They allow one to determine the solution as 𝜁 (s) = −sgn(s) i ln[1 + is2 + i |s|

√ s2 − 2i)],

(2.115)

where one should choose the following branches of the logarithm and square-root functions: ln[f (s)] = ln[|f (s)| ei𝛿 ] = ln[|f (s)|] + i𝛿,

−𝜋 ≤ 𝛿 ≤ 𝜋,

(2.116)

𝜋 ≤ 𝜂 ≤ 0, 4

(2.117)

with ln(1) = 0, ln(i) = 𝜋∕2, and √ |√ | s2 − 2i = || s2 − 2i|| ei𝜂 , | |

1 2 𝜂 = − arctan 2 , 2 s

−

with √

{ s2

− 2i →

|s| √ 2 e−i(𝜋∕4)

if if

s → ±∞ . s → 0,

(2.118)

The integral (2.108) is taken along the steepest descent path and converges rapidly under the condition kr ≫ 1 when the series in Equation (2.42) becomes slowly convergent. However, its integration is not easy, because the integrand is singular exactly at the saddle point, s = 𝜁 = 0, when 𝜓 = 𝜑 ± 𝜑0 = 𝜋 or 𝜓 = 𝜑 + 𝜑0 = 2𝛼 − 𝜋. These singularities happen at the boundaries of the incident (𝜑 = 𝜋 + 𝜑0 ) and reflected (𝜑 = 𝜋 − 𝜑0 or 𝜑 = 2𝛼 − 𝜋 − 𝜑0 ) plane waves. To treat this problem, we undertake the following steps. First, we represent the integrand factor as (1∕n) sin(𝜋∕n) 1 = cos(𝜋∕n) − cos[(𝜓 + 𝜁 )∕n] 2n

( cot

𝜁 +𝜓 −𝜋 𝜁 +𝜓 +𝜋 − cot 2n 2n

) .

(2.119)

74

WEDGE DIFFRACTION: EXACT SOLUTION AND ASYMPTOTICS

Then we apply the magic zero procedure: We add to this function zero as a sum of two special identical terms with opposite signs, and transform Equation (2.119) as 𝜁 +𝜓 −𝜋 1 𝜁 +𝜓 −𝜋 𝜁 +𝜓 +𝜋 1 𝜁 +𝜓 −𝜋 1 1 cot − csc − cot + csc . 2n 2n 2 2 2n 2n 2 2 (2.120) It is now seen that the second term cancels the singularity of the first term completely. We then substitute Equation (2.120) into Equation (2.108). Integrating the last term in Equation (2.120) leads to the Fresnel function, ∞

2

e−krs ds 1 i(kr+𝜋∕4) √ e ∫ sin[(𝜁 + 𝜓 − 𝜋)∕2] cos(𝜁 ∕2) −∞ 2𝜋 2 ∞ sin[(𝜓−𝜋)∕2]

=

2 e−i(𝜋∕4) −ikr cos 𝜓 eix dx. e √ √ ∫ 2krsins[(𝜓−𝜋)∕2] 𝜋

(2.121)

As a result, we arrive at a new form of Equation (2.108): 𝑣(kr, 𝜓) =

∞ [ 𝜍+𝜓 −𝜋 1 𝜍+𝜓 −𝜋 2 1 1 i(kr+𝜋∕4) cot − csc e−krs √ e ∫ 2n 2n 2 2 −∞ 𝜋 2 ∞ sin[(𝜓−𝜋)∕2] 𝜍 +𝜓 +𝜋] 2 ds e−i(𝜋∕4) 1 cot + e−ikr cos 𝜓 √ − eix dx. √ ∫ 2n 2n cos(𝜍∕2) 2kr sin[(𝜓−𝜋)∕2] 𝜋

(2.122) Here the integral over the variable s is free of the singularity and convenient for numerical calculation when kr ≫ 1. This form of function 𝑣(kr, 𝜓) is valid under the condition 0 < 𝜑0 < 𝛼 − 𝜋, when only one face (𝜑 = 0) is illuminated. A similar regularization procedure can be accomplished in the case 𝛼 − 𝜋 < 𝜑0 < 𝜋, when both faces of the wedge are illuminated (see Fig. 2.8). The following exact forms are valid for this case: 𝑣(kr, 𝜑 − 𝜑0 ) =

[ ∞ 𝜍 + 𝜑 − 𝜑0 − 𝜋 2 1 1 i(kr+𝜋∕4) e−krs cot √ e ∫ 2n 2n −∞ 𝜋 2 𝜍 + 𝜑 − 𝜑0 − 𝜋 𝜍 + 𝜑 − 𝜑0 + 𝜋 1 1 − csc − cot 2 2 2n 2n

]

∞ sin[(𝜑−𝜑 −𝜋)∕2]

×

0 2 ds e−i(𝜋∕4) eix dx, + e−ikr cos(𝜑−𝜑0 ) √ √ ∫ cos(𝜍∕2) 2kr sin[(𝜑−𝜑0 −𝜋)∕2] 𝜋

(2.123)

FAST CONVERGENT INTEGRALS AND UNIFORM ASYMPTOTICS: THE “MAGIC ZERO” PROCEDURE

𝑣(kr, 𝜑 + 𝜑0 ) =

75

[ ∞ 𝜍 + 𝜑 + 𝜑0 − 𝜋 2 1 1 i(kr+𝜋∕4) e−krs cot √ e ∫ 2n 2n −∞ 𝜋 2 𝜍 + 𝜑 + 𝜑0 − 𝜋 𝜍 + 𝜙 + 𝜙0 − 𝜋 1 1 csc + cot 2 2 2n 2n ] 𝜍 + 𝜙 + 𝜙 − 𝜋 1 ds 0 − csc 2 2 cos(𝜍∕2)

−

∞ sin[(𝜑+𝜑 −𝜋)∕2]

0 2 e−i(𝜋∕4) + e−ikr cos(𝜑+𝜑0 ) √ eix dx √ ∫ 2kr sin[(𝜑+𝜑0 −𝜋)∕2] 𝜋

∞ sin[(𝜙+𝜙 −𝜋)∕2]

0 2 e−i(𝜋∕4) eix dx, + e−ikr cos(𝜙+𝜙0 ) √ √ ∫ 2kr sin[(𝜙+𝜙0 −𝜋)∕2] 𝜋

(2.124) where 𝜙 = 𝛼 − 𝜑 and 𝜙0 = 𝛼 − 𝜑0 . The first two terms in brackets in Equation (2.124) cancel each other in the direction 𝜑 = 𝜋 − 𝜑0 related to the specular reflection from the face 𝜑 = 0. The last two terms in brackets remove the singularity in the direction of the specular reflection (𝜑 = 2𝛼 − 𝜋 − 𝜑0 ) from the face 𝜑 = 𝛼. The integral forms found are convenient both for direct numerical calculation and for asymptotic evaluation. The Fresnel integrals also require numerical integration. For small and large arguments, we can use its known asymptotics. For moderate arguments, the integration of the Fresnel integrals can be accomplished with Equation (2.121). The integrals in Equations (2.122) to (2.124) over the variable s are free of the singularity and can be evaluated by a regular saddle point technique when kr ≫ 1. The related first-order uniform approximations of 𝑣(kr, 𝜓) are determined as ei(kr+𝜋∕4) 𝑣(kr, 𝜓) = √ 2𝜋kr

(

𝜓 −𝜋 𝜓 −𝜋 𝜓 + 𝜋) 1 1 1 cot − csc − cot 2n 2n 2 2 2n 2n ∞ sin[(𝜓−𝜋)∕2]

2 e−i(𝜋∕4) + e−ikr cos 𝜓 √ eix dx √ 𝜋 ∫ 2kr sin[(𝜓−𝜋)∕2]

(2.125)

with 0 < 𝜑0 < 𝛼 − 𝜋 and (

𝜑 − 𝜑0 − 𝜋 1 𝜑 − 𝜑0 − 𝜋 1 cot − csc 2n 2n 2 2 ) 𝜑 − 𝜑0 + 𝜋 1 e−i(𝜋∕4) − cot + e−ikr cos(𝜑−𝜑0 ) √ 2n 2n 𝜋

ei(kr+𝜋∕4) 𝑣(kr, 𝜑 − 𝜑0 ) = √ 2𝜋kr

∞ sin[(𝜑−𝜑0 −𝜋)∕2]

×

∫

√

2kr sin[(𝜑−𝜑0 −𝜋)∕2]

2

eix dx,

(2.126)

76

WEDGE DIFFRACTION: EXACT SOLUTION AND ASYMPTOTICS

(

𝜑 + 𝜑0 − 𝜋 1 𝜑 + 𝜑0 − 𝜋 1 cot − csc 2n 2n 2 2 ) 𝜙 + 𝜙0 − 𝜋 1 𝜙 + 𝜙0 − 𝜋 1 + cot − csc 2n 2n 2 2

ei(kr+𝜋∕4) 𝑣(kr, 𝜑 + 𝜑0 ) = √ 2𝜋kr

∞ sin[(𝜑+𝜑 −𝜋)∕2]

0 2 e−i(𝜋∕4) + e−ikr cos(𝜑+𝜑0 ) √ eix dx √ 𝜋 ∫ 2kr sin[(𝜑+𝜑0 −𝜋)∕2]

∞ sin[(𝜙+𝜙 −𝜋)∕2]

0 2 e−i(𝜋∕4) eix dx + e−ikr cos(𝜙+𝜙0 ) √ √ 𝜋 ∫ 2kr sin[(𝜙+𝜙0 −𝜋)∕2]

(2.127)

with 𝛼 − 𝜋 < 𝜑0 < 𝜋 and 𝜙 = 𝛼 − 𝜑, 𝜙0 = 𝛼 − 𝜑0 . The second asymptotic terms for the integrals over the variable s are on the order of (kr)−3∕2 . Calculation of higherorder asymptotic terms for the functions 𝑣(kr, 𝜑 ± 𝜑0 ) is straightforward, but we do not derive them because of their small practical value. They can be found in a book by Bowman et al. (1987). √ ) ]| [( | Under the conditions 2kr |cos 𝜑 ± 𝜑0 ∕2 | ≫ 1, the uniform asymptotics | | above transform into the Sommerfeld asymptotics (2.61) and (2.63) with the alternative cotangent forms of the functions f (𝜑, 𝜑0 , 𝛼)and g(𝜑, 𝜑0 , 𝛼): ( 𝜋 − 𝜑 − 𝜑0 𝜋 + 𝜑 + 𝜑0 1 f (𝜑, 𝜑0 , 𝛼) = cot + cot 2n 2n 2n ) 𝜋 − 𝜑 + 𝜑0 𝜋 + 𝜑 − 𝜑0 − cot − cot (2.128) 2n 2n ( 𝜋 − 𝜑 − 𝜑0 𝜋 + 𝜑 + 𝜑0 1 g(𝜑, 𝜑0 , 𝛼) = − cot + cot 2n 2n 2n ) 𝜋 − 𝜑 + 𝜑0 𝜋 + 𝜑 − 𝜑0 + cot + cot . (2.129) 2n 2n The material of this section was utilized earlier in a tutorial paper by Hacivelioglu et al. (2011) that contains extensive numerical illustrations of the wedge-diffracted field. In this chapter we studied the edge waves scattered from a wedge with semiinfinite planar faces. The edge waves arising in two-dimensional structures with curved concave and convex faces have been investigated by Molinet (2005; Molinet et al., 2005). PROBLEMS 2.1

)] [ ( The incident wave uinc = u0 exp −ikr cos 𝜑 − 𝜑0 generates the field inside a two-dimensional soft corner (a wedge with the angle 𝛼 < 𝜋). Derive an exact expression for the total field inside the corner with the angle 𝛼 = 𝜋∕2 (Fig. 2.1). Does this field contain the edge wave? Explain why.

PROBLEMS

77

Solution The total field is given by Equations (2.40) and (2.47). For the corner with 𝛼 = 𝜋∕2 it takes the form utot s

1 = 𝜋 ∫C

(

) 1 1 − e−ikr cos 𝛽 d𝛽. 1 − ei2(𝛽+𝜑−𝜑0 ) 1 − ei2(𝛽+𝜑+𝜑0 )

The integrand here contains the poles (2.48)—the zeros of the denominators. Let us find their contribution to the total field. To do this, we evaluate the integral over the closed contour C–D shown in Figure 2.5. According to the Cauchy theorem, poles

us

= −2𝜋i ⋅

1∑ [Res(𝛽m+ ) − [Res(𝛽m− )], 𝜋 m

where Res(𝛽m± ) =

| ± e−ikr cos 𝛽 i | = e−ikr cos 𝛽m | i2(𝛽+𝜑∓𝜑 ) | 0 ]∕d𝛽 d[1 − e |𝛽=𝛽 ± 2 m

is the residue at the pole 𝛽m± . Here 𝛽m+ = m𝜋 − 𝜑 + 𝜑0 are the poles of the first term of the integrand, and 𝛽m− = m𝜋 − 𝜑 − 𝜑0 are the poles of the second term. poles Only those poles contribute to us , which are inside the contour C–D, that is, ± inside the interval −𝜋 < 𝛽m < 𝜋. One can verify that only the following five poles satisfy these conditions when 0 ≤ 𝜑 ≤ 𝜋∕2 for the corner with 𝛼 = 𝜋∕2: 𝛽0+ = −𝜑 + 𝜑0 ,

𝛽1+ = 𝜋 − 𝜑 + 𝜑0 ,

𝛽0− = −𝜑 − 𝜑0 ,

𝛽1− = 𝜋 − 𝜑 − 𝜑0 .

+ 𝛽−1 = −𝜋 − 𝜑 + 𝜑0 ,

poles

reveals that they generate Substitution of the residues for these poles into us the following five plane waves: = uinc = u0 e−ikr cos(𝜑−𝜑0 ) 1. The pole 𝛽0+ generates the incident wave u+ 0 which exists in the entire corner (0 ≤ 𝜑 ≤ 𝜋∕2). 2. The pole 𝛽0− generates the wave u− = −u0 e−ikr cos(𝜑+𝜑0 ) reflected from the 0 face 𝜑 = 0. The corner is soft. Because of that, the reflection coefficient equals −1. This wave exists in the entire corner (0 ≤ 𝜑 ≤ 𝜋∕2) and hits the face 𝜑 = 𝜋∕2. After reflection from that face, it is transformed into the wave generated by the pole 𝛽1+ . 3. The pole 𝛽1+ generates the wave u+ = u0 eikr cos(𝜑−𝜑0 ) reflected from the 1 face 𝜑 = 𝜋∕2 and coming out from the corner. It exists only in the region 𝜑0 ≤ 𝜑 ≤ 𝜋∕2. This wave is the result of two reflections of the incident wave from the corner faces; therefore, its amplitude equals (−1)2 u0 = u0 . = −u0 eikr cos(𝜑+𝜑0 ) reflected from the 4. The pole 𝛽1− generates the wave u− 1 face 𝜑 = 𝜋∕2. It exists in the entire corner (0 ≤ 𝜑 ≤ 𝜋∕2) and hits the face

78

WEDGE DIFFRACTION: EXACT SOLUTION AND ASYMPTOTICS

𝜑 = 0. After reflection from that face it transforms into the wave generated + . by the pole 𝛽−1 + 5. The pole 𝛽−1 generates the wave u+ = u0 eikr cos(𝜑−𝜑0 ) reflected from the −1 face 𝜑 = 0 and coming out from the corner. It exists only in the region 0 ≤ 𝜑 ≤ 𝜑0 . This wave is the result of two reflections of the incident wave from the corner; therefore, its amplitude equals (−1)2 u0 = u0 . The following important features of this rectangular corner are worth noting: r The field generated by the poles represents the geometrical optics (GO) poles waves, us ≡ uGO s . r The waves u+ and u+ merge with each other on the line 𝜑 = 𝜑0 and fill in 1 −1 the entire corner without a gap between them. Together they actually form + u+ = u0 eikr cos(𝜑−𝜑0 ) . the single wave u+ 1 −1 r The incident wave is reflected back exactly by this corner. r The geometrical optics part of the field fills in the whole corner continuously. This GO field satisfies both the wave equation and the boundary conditions. Therefore, it represents the exact solution of the corner problem. r Indeed, the edge-diffracted field does not exist inside the corner. This follows from Equation (2.51) when n = 𝛼∕𝜋 = 12 since sin (𝜋∕n) = sin (2𝜋) = 0. Thus, the total field inside the corner consists only of the plane waves: = u+ + u+ + u+ + u− + u− . We have derived this result utilizing the exact utot s 0 1 0 1 −1 solution. It can be verified by application of the GO reflection law. 2.2

Solve a problem similar to Problem 2.1 but for the hard corner with 𝛼 = 𝜋∕2. Solution The difference from Problem 2.1 lies only in the reflection coefficient, which is now equal to +1. However, this difference does not affect the backscattered plane wave that results from two double reflections. The total field inside the hard corner equals + + + − − utot h = u0 + u1 + u−1 − u0 − u1

whose terms are as defined in Problem 2.1. Notice that utot = 4u0 exactly on h the edge (r = 0). The edge-diffracted wave does not exist, as explained in Problem 2.1. ) ( 2.3 The incident wave Ezinc = E0z exp[−ikr cos 𝜑 − 𝜑0 ] generates the field inside a two-dimensiional perfectly conducting corner (a wedge with the angle 𝛼 < 𝜋 between its faces). Derive an exact expression for the total field inside the corner with the angle 𝛼 = 𝜋∕2. Does this field contain an edge wave? Explain why. Solution This problem is two-dimensional. Therefore, we can use the equivalency relationships between acoustic and electromagnetic problems, which

PROBLEMS

79

are shown at the beginning of this chapter. According to this equivalence, inc inc tot Eztot = utot s if Ez = u . The field us was found in Problem 2.1. + + + − tot Thus, Ez = u0 + u1 + u−1 + u0 + u− where u0 = E0z . An edge-diffracted 1 wave does not exist, as explained in Problem 2.1. 2.4 Solve a problem similar to Problem 2.3 but with the incident wave Hzinc = H0z exp[−ikr cos(𝜑 − 𝜑0 )]. Solution This problem is two-dimensional. Therefore, we can use the equivalency relationships between acoustic and electromagnetic problems. Accordif Hzinc = uinc . The field utot was found ing to this equivalence, Hztot = utot h h + + − − u− , where u = H . An tot in Problem 2.2. Thus, Hz = u0 + u+ + u − u 0 0z 0 1 1 −1 edge-diffracted wave does not exist, as explained in Problem 2.1. 2.5 Use the exact solution (2.40) for a soft wedge and derive asymptotic approximations for the scattering sources jS = 𝜕u∕𝜕n induced on both faces of the wedge, close to the edge (kr ≪ 1). (a) Apply these asymptotics of the Bessel functions: J0 (x) ∼ 1 −

( )2 x , 2

J𝑣 (x) ∼

( )𝑣 x 1 Γ (𝑣 + 1) 2

with x → 0.

(b) Derive the asymptotic expression for the scattering sources jS (kr, 𝛼). Realize the different behavior of this function for the angle 𝛼 > 𝜋 and 𝛼 < 𝜋. Solution Consider the two first terms in Equation (2.42) and substitute the asymptotic approximations for the Bessel functions. The surface scattering sources are defined as js =

𝜕us 1 𝜕us = 𝜕n r r𝜕𝜑

js =

𝜕us 1 𝜕us =− 𝜕n r 𝜕𝜑

on the face 𝜑 = 0, on the face 𝜑 = 𝛼.

Verify that on both faces, js ∼ (kr)(𝜋∕𝛼)−1 with kr → 0. Thus, js → 0 for a concave wedge (𝛼 < 𝜋). But js → ∞ for a convex wedge (𝛼 < 𝜋) when r → 0. 2.6 Analyze a problem similar to Problem 2.5 but for a hard wedge where jh = uh and uh is given by Equation (2.41). Find the behavior of jh for k ≪ 1. Solution Consider the first two terms in Equation (2.42) and substitute there the asymptotic approximations for the Bessel functions. Verify that jh = u0

] [ ] [ 2𝜋 + O (kr)2 + O (kr)𝜋∕𝛼 𝛼

with kr → 0.

Exactly on the edge (r = 0) of the corner with 𝛼 = 𝜋∕2, jh = uh = 4u0 , in agreement with the solution of Problem 2.2.

80

2.7

WEDGE DIFFRACTION: EXACT SOLUTION AND ASYMPTOTICS

Explore the electromagnetic version )] of Problem 2.5 for the incident wave [ ( Ezinc = E0z exp −ikr cos 𝜑 − 𝜑0 exciting a perfectly conducting wedge. Analyze the surface currents close to the edge (kr ≪ 1). ⃗ On the Solution The surface electric current is defined as ⃗je = n̂ × H. face 𝜑 = 0 (𝜑 = 𝛼), it contains only the z-component jez = −Hr (jez = Hr ). ⃗ = ikZ0 H ⃗ and equals The field Hr is found from the Maxwell equation ∇ × E Hr = Y0 (1∕ikr)(𝜕Ez ∕𝜕𝜑), where Y0 = 1∕Z0 . Then we apply the equivalency , and obtain principle, Ez = us with Ezinc = uinc 0 jez = −Hr = −Y0 jez = Hr = Y0

1 𝜕us 1 = −Y0 js ikr 𝜕𝜑 ik

on the face 𝜑 = 0,

1 𝜕us 1 = −Y0 js on the face 𝜑 = 𝛼. ikr 𝜕𝜑 ik

In Problem 2.5 it was shown that js ∼ (kr)𝜋∕𝛼−1 with r → 0. Thus, jez → 0 for a concave wedge (𝛼 < 𝜋) and jez → ∞ for a convex wedge (𝛼 < 𝜋). 2.8

Explore a problem similar to Problem 2.7 but for the incident wave Hzinc = ) ( H0z exp[−ikr cos 𝜑 − 𝜑0 ] exciting a perfectly conducting wedge. Analyze ⃗ the surface currents ⃗je = n̂ × H. Solution According to the definition, jer = Hz on the face 𝜑 = 0 and jer = −Hz on the face 𝜑 = 𝛼. Utilize the equivalence relationship Hz = uh with Hzinc = uinc . The field uh was found in Problem 2.6. As a result, 2𝜋 + O[(kr)2 ] + O[(kr)𝜋∕𝛼 ] with kr → 0 on the face 𝜑 = 0, 𝛼 2𝜋 jer = −H0z + O[(kr)2 ] + O[(kr)𝜋∕𝛼 ] on the face 𝜑 = 𝛼. 𝛼 jer = H0z

The different signs for the current on the faces 𝜑 = 0 and 𝜑 = 𝛼 indicate that the electric current flows to the edge along one face and then flows from the edge along the other face. The current is continuous on the edge, in agreement with the electric charge conservation law. 2.9

Use the Sommerfeld ray asymptotics (2.61) and (2,63). Calculate the surface sources js and jh far from the edge (kr ≫ 1). They are the waves running from the edge. At which wedge (soft or hard) do these waves decrease faster? It is sufficient to consider the waves only on the face 𝜑 = 0. Explain the origination of the phase factor exp (i𝜋∕4). Due to Equations (1.11), (2.61), and (2,63), ( ) ( ) ei(kr+𝜋∕4) 1 𝜕f 0, 𝜑0 , 𝛼 ei(kr+𝜋∕4) , jh ∼ u0 g 0, 𝜑0 , 𝛼 √ js ∼ u0 √ r 𝜕𝜑 2𝜋kr 2𝜋kr

Solution

js = O[(kr)−3∕2 ],

jh = O[(kr)−1∕2 ]

with kr → ∞.

PROBLEMS

81

Thus, the edge waves decrease faster on the soft edge. The phase factor exp (i𝜋∕4) shows that the edge is the caustic of the edge waves/rays. 2.10

Use the Sommerfeld ray asymptotics (2.61) and (2.63) for electromagnetic waves with E- and H-polarization, respectively. Calculate the surface electric currents far from the edge (kr ≫ 1). They are the waves running from the edge. It is sufficient to consider the waves only on the face 𝜑 = 0. Which waves (with component jz or jr ) decrease faster? Explain the origination of the phase factor exp (i𝜋∕4). ⃗ On the Solution The surface electric current is defined as ⃗je = n̂ × H. face 𝜑 = 0, jer = Hz and jez = −Hr . As shown in Problem 2.7, jez = −Hr = −Y0 (1∕ikr)(𝜕Ez ∕𝜕𝜑). Then we use the equivalence between acoustic and electromagnetic waves and in accordance with Equations (2.61) and (2.63) obtain jez = E0z

1 𝜕f (0, 𝜑0 , 𝛼) ei(kr+𝜋∕4) √ r 𝜕𝜑 2𝜋kr

( ) ei(kr+𝜋∕4) jer = H0z g 0, 𝜑0 , 𝛼 √ 2𝜋kr

and with kr ≫ 1.

Thus, the wave with component jz decreases faster. The phase factor exp (i𝜋∕4) shows that the edge is the caustic of the edge waves/rays. 2.11

Find the first two terms of the asymptotic expansion (with p → ∞) for the integrals ∞

∫p

∫p

∫p

eix

2

dx xn

with n ≥ 1.

Set x2 = t and employ integration by parts:

Solution ∞

∞

2

eix dx,

2

( ) ∞ ∞ 1 1 1 it −3∕2 eit t−1∕2 |∞ eit t−1∕2 dt = + e t dt p2 2 ∫p2 2i 2 ∫p2 ( ) ∞ i ip2 1 |∞ 3 e − eit t−3∕2 | 2 + eit t−5∕2 dt = |p 2p 4 2 ∫p2

eix dx =

( ) 2 1 i ip2 e + 3 eip + O p−5 , 2p 4p ( ) ∞ ∞ ∞ 2 dx 1 1 |∞ n + 1 eit t−(n+1)∕2 | 2 + eix n = eit t(n+1)∕2 dt = eit t−(n+3)∕2 dt |p ∫p x 2 ∫p2 2i 2 ∫p2 =

( ) n + 1 eip i eip + + O p−n−5 . n+1 2p 4 pn+3 2

=

2

82

WEDGE DIFFRACTION: EXACT SOLUTION AND ASYMPTOTICS

2.12

An acoustic wave hits a wedge barrier (Fig. 2.7). Behind which barrier (soft or hard) is the intensity of the diffracted acoustic wave higher? Explore this problem utilizing the (2.61) and (2.63). Compute and ) (ray asymptotics ) ( Sommerfeld plot the ratio f 𝜑, 𝜑0 , 𝛼 ∕g 𝜑, 𝜑0 , 𝛼 . Explore the following two examples: (a) 𝛼 = 315◦ , 𝜑0 = 45◦ , 225◦ ≤ 𝜑 ≤ 𝛼. (b) 𝛼 = 355◦ , ∕𝜑0 = 45◦ , 225◦ ≤ 𝜑 ≤ 𝛼. Prove analytically that this ratio equals unity at the shadow boundary 𝜑 = 180◦ + 𝜑0 . Solution ) It( follows) from Equations (2.61) and (2.63) that us ∕uh = ( f 𝜑, 𝜑0 , 𝛼 ∕g 𝜑, 𝜑0 , 𝛼 . Introduce the denotations x = cos

𝜑 − 𝜑0 𝜋 − cos , n n

y = cos

𝜑 + 𝜑0 𝜋 − cos . n n

Then us 1 − x∕y f = = . uh g 1 + x∕y Notice that x → 0 with 𝜑 → 𝜋 + 𝜑0 . Thus f ∕g = 1 with 𝜑 = 𝜋 + 𝜑0 . Accomplish the numerical part of the problem. 2.13

An electromagnetic wave hits a perfectly conducting wedge (Fig. 2.7). Compare the intensity of edge-diffracted waves with Ez - and Hz -polarization. Which of them is more intensive in the shadow region? Use the Sommerfeld ray asymptotics (2.61) and (2.63). Compute and plot the ratio Ez ∕Hz . Explore the following two examples: (a) 𝛼 = 300◦ , 𝜑0 = 45◦ , 225◦ ≤ 𝜑 ≤ 𝛼. (b) 𝛼 = 360◦ , 𝜑0 = 45◦ , 225◦ ≤ 𝜑 ≤ 𝛼. Solution According to the equivalence between the two-dimensional acoustic and electromagnetic waves, ) ( f 𝜑, 𝜑0 , 𝛼 Ez us = = ( ). Hz uh g 𝜑, 𝜑0 , 𝛼 Apply the denotations x and y introduced in Problem 2.12 . Then Ez 1 − x∕y = . Hz 1 + x∕y Accomplish the numerical calculations and plot the ratio Ez ∕Hz .

PROBLEMS

2.14

83

Find the scattering sources jh = uh induced on the face 𝜑 = 0 of a hard wedge by the incident wave uinc = u0 exp[−ikr cos(𝜑 − 𝜑0 )] under the grazing incidence, 𝜑0 = 𝜋 (Fig. 2.8). (a) Derive the asymptotic form with kr ≫ 1. (b) Provide the electromagnetic version of this solution in the case of the grazing incidence of the wave Hz = H0z exp (ikr cos 𝜑). Solution (a) According to Equations (2.41) and (2.49), jh = uh = u0 [eikr + 𝑣(kr, −𝜋) + 𝑣(kr, 𝜋)]. Utilizing Equation (2.51), one can verify that the function 𝑣 (kr, 𝜓) is even, 𝑣 (kr, −𝜓) = 𝑣 (kr, 𝜓). Hence, jh = u0 [eikr + 2𝑣(kr, 𝜋)] is the exact solution. The asymptotic solution can be found using the saddle point technique. The saddle point of function (2.51) is found from the condition d cos 𝜁 ∕d𝜁 = − sin 𝜁 = 0 and equals 𝜁 = 0. Then we find an approximation for the integrand in Equation (2.51) in the vicinity of the saddle point. Simple operations with trigonometric functions show that [ ( )] sin(𝜋∕n) 𝜁 𝜁 1 𝜋 = cot − cot + . cos(𝜋∕n) − cos[(𝜋 + 𝜁 )∕n] 2 2n n 2n In the vicinity of 𝜁 = 0, this function equals (n∕𝜁 ) − (1∕2) cot(𝜋∕n) + O (𝜁 ). Due to the symmetry of the integration contour and function exp (ikr cos 𝜁 ), 𝜋∕2−i∞

∫−𝜋∕2+i∞

eikr cos 𝜁

d𝜁 = 0, 𝜁

𝜋∕2−i∞

∫−𝜋∕2+i∞

eikr cos 𝜁 𝜁 d𝜁 = 0.

Then 𝑣 (kr, 𝜋) ∼ −

i 𝜋 eikr cos 𝜁 d𝜁 , cot 4𝜋n n ∫D0

or in view of Equation (3.9), 𝑣 (kr, 𝜋) ∼ −

𝜋 i cot H0(1) (kr) . 4n n

For the Hankel function with kr ≫ 1, one can use the asymptotic form (2.29), which leads to ( jh ∼ u0

1 𝜋 ei(kr+𝜋∕4) eikr − cot √ n n 2𝜋kr

) .

84

WEDGE DIFFRACTION: EXACT SOLUTION AND ASYMPTOTICS

(b) The electric current equals jer = Hz . Due to the equivalence relationship between acoustic and electromagnetic problems, Hz = uh with u0 = H0z . Thus, the electric current equals [ ] jer = Hoz eikr + 2𝑣(kr, 𝜋) , and its asymptotic approximation is ( jer

2.15

= H0z

e

ikr

1 𝜋 ei(kr+𝜋∕4) − cot √ n n 2𝜋kr

) with kr ≫ 1.

𝜕u

Find the scattering sources js = 𝜕ns = (1∕r)(𝜕us ∕𝜕𝜑) on the face 𝜑 = 0 of a soft wedge. They are induced by the incident wave uinc = u0 exp[−ikr cos(𝜑 − 𝜑0 )] under the grazing incidence, 𝜑0 = 𝜋 (Fig. 2.8). (a) Derive the asymptotic approximation of js with kr ≫ 1. (b) Provide the electromagnetic version of this solution in the case of grazing incidence of the wave Ez = E0z exp(ikr cos 𝜑) on a perfectly conducting wedge. Solution (a) The derivative 𝜕us ∕𝜕𝜑 is a continuous function of variables 𝜑 and 𝜑0 . To find its value for 𝜑 = 0 and 𝜑0 = 𝜋, we calculate it first in the region 𝜑 = 𝜋 − 𝜑0 + 𝛿 when 𝜑0 = 𝜋 − 𝛿 and 0 < 𝛿 ≪ 1, and then take the limit with 𝛿 → 0. According to Equations (2.40) and (2.53), us = e−ikr cos(𝜑−𝜑0 ) + 𝑣(kr, 𝜑 − 𝜑0 ) − 𝑣(kr, 𝜑 + 𝜑0 ). u0 Notice that 𝜕 −ikr cos(𝜑−𝜑0 ) = ikr sin(𝜑 − 𝜑0 )e−ikr cos(𝜑−𝜑0 ) = 0 when e 𝜕𝜑

𝜑0 = 𝜋

and 𝜑 = 0. The problem is to calculate the derivative of function (2.51): sin(𝜋∕n) 𝜕 𝜕 eikr cos 𝜁 𝑣(kr, 𝜓) = i 𝜕𝜓 2𝜋n 𝜕𝜓 ∫D0

(

𝜁 +𝜓 𝜋 cos − cos n n

)−1 d𝜁 .

PROBLEMS

85

It is clear that the operator 𝜕∕𝜕𝜓 can be replaced by d∕d𝜁 . Next we apply integration by parts, ( )−1 sin(𝜋∕n) 𝜁 +𝜓 d 𝜋 𝜕 eikr cos 𝜁 d𝜁 𝑣(kr, 𝜓) = i cos − cos 𝜕𝜓 2𝜋n ∫D0 d𝜁 n n [ |𝜋∕2−i∞ sin(𝜋∕n) eikr cos 𝜁 | =i 2𝜋n cos(𝜋∕n) − cos[(𝜁 + 𝜓)∕n] ||−𝜋∕2+i∞ ] sin 𝜁 +ikr eikr cos 𝜁 d𝜁 . ∫D0 cos(𝜋∕n) − cos[(𝜁 + 𝜓)∕n] The first term in brackets equals zero. We actually need this derivative at the limiting case when 𝜓 = ±𝜋, sin(𝜋∕n) sin 𝜁 𝜕 eikr cos 𝜁 𝑣(kr, ±𝜋) = −kr d𝜁 . ∫ 𝜕𝜓 2𝜋n cos(𝜋∕n) − cos(𝜁 ± 𝜋)∕n D0 This integral is calculated asymptotically with kr ≫ 1. The main contribution to the integral is given by the vicinity of the saddle point 𝜁 = 0. At this point sin 𝜁 cos 𝜁 = lim cos(𝜋∕n) − cos[(𝜁 ± 𝜋)∕n] 𝜁 →0 (1∕n) sin[(𝜁 ± 𝜋)∕n] n =± . sin(𝜋∕n)

lim

𝜁 →0

As a result, in the first-order approximation, | k 𝜕 ∼∓ eikr cos 𝜁 d𝜁 𝑣(kr, 𝜑 ± 𝜋)|| r 𝜕𝜑 2𝜋 ∫D0 |𝜑=0

with kr → ∞

or in view of Equation (3.9), | k 𝜕 ∼ ∓ H0(1) (kr). 𝑣(kr, 𝜑 ± 𝜋)|| r 𝜕𝜑 2 |𝜑=0 For the Hankel function we use its asymptotic form (2.29) and obtain | ei(kr−𝜋∕4) 𝜕 . ∼ ∓k √ 𝑣(kr, 𝜑 ± 𝜋)|| r 𝜕𝜑 |𝜑=0 2𝜋kr

86

WEDGE DIFFRACTION: EXACT SOLUTION AND ASYMPTOTICS

Thus, js =

| 1 𝜕us || 1 𝜕 = u0 [𝑣(kr, 𝜑 − 𝜋) − 𝑣(kr, 𝜑 + 𝜋)]|| r 𝜕𝜑 ||𝜑=0 r 𝜕𝜑 |𝜑=0 ei(lr−(𝜋∕4)) ∼ u0 2k √ 2𝜋kr

with kr → ∞.

(b) The electric current induced on the face 𝜑 = 0 of a perfectly conduct⃗ is found from the ing wedge is defined as jez = −Hr . The vector H ⃗ = ikZo H, ⃗ and its r-component equals Hr = Maxwell equation ∇ × E Y0 (1∕ikr)(𝜕Ez ∕𝜕𝜑) where Y0 = 1∕Z0 . Thus, jez = −Y0

1 𝜕Ez . ikr 𝜕𝜑

Due to the equivalence between acoustic and electromagnetic waves, Ez = us with u0 = E0z . Hence, 𝜕Ez ∕𝜕𝜑 = 𝜕us ∕𝜕𝜑 and jez = −Y0

1 𝜕us i ei(kr+𝜋∕4) = Y0 js ∼ 2Y0 E0z √ ikr 𝜕𝜑 k 2𝜋kr

with kr → ∞.

3 Wedge Diffraction: The Physical Optics Field

(0) (0) (0) The relationships u(0) s = Ez and uh = Hz exist between the PO acoustic and electromagnetic fields. An exception is for an oblique incidence (see Section 4.3).

Exact expressions for the scattered field were derived in Chapter 2. In the present chapter we calculate the physical optics part of the scattered field, that is, the field generated by the uniform component of the induced surface scattering sources. It will be used in Chapter 4 to examine the field radiated by the nonuniform sources as the difference between the exact and physical optics fields.

3.1

ORIGINAL PO INTEGRALS

The geometry of the problem is shown in Figure 3.1. A perfectly reflecting wedge located in a homogeneous medium is excited by a plane wave, uinc = u0 e−ikr cos(𝜑−𝜑0 ) = u0 e−ik(x cos 𝜑0 +y sin 𝜑0 ) ,

(3.1)

Fundamentals of the Physical Theory of Diffraction, Second Edition. Pyotr Ya. Ufimtsev. © 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

87

88

WEDGE DIFFRACTION: THE PHYSICAL OPTICS FIELD

P

φ = π – φ0

y r φ

φ =α – π

nˆ

φ0

φ=0

φ =π φ = π + φ0

nˆ

x

φ=α

Figure 3.1 Wedge and related coordinates.

where we assume that 0 ≤ 𝜑0 ≤ 𝜋. In the PO approximation, the surface sources (induced on the face 𝜑 = 0) are determined according to Equation (1.31): −ikx cos 𝜑0 , j(0) s = −u0 2ik sin 𝜑0 e

j(0) = u0 2e−ikx cos 𝜑0 . h

(3.2)

Equation (1.32) determines the field radiated by these sources: √

u(0) s

2 2 2 ∞ ik sin 𝜑0 ∞ −ik𝜉 cos 𝜑 eik (x−𝜉) +y +𝜁 0 d𝜉 = u0 e d𝜁 , √ ∫−∞ (x − 𝜉)2 + y2 + 𝜁 2 2𝜋 ∫0

u(0) h

1 = u0 2𝜋 ∫0

∞

−ik𝜉 cos 𝜑0

e

∞

d𝜉

∫−∞

√

2

2

(3.3)

2

𝜕 eik (x−𝜉) +y +𝜁 d𝜁 . √ 𝜕n (x − 𝜉)2 + y2 + 𝜁 2

(3.4)

Here we use the denotations 𝜕 𝜕 f (r) = ∇′ f (r) ⋅ n̂ = −∇f (r) ⋅ n̂ = − f (r), 𝜕n 𝜕y

(3.5)

where ∇′ and ∇ are the gradient operators applied to coordinates of the integration and observation points, respectively. In view of Equation (3.5), the field u(0) can be h written as u(0) h

1 𝜕 = −u0 2𝜋 𝜕y ∫0

∞

−ik𝜉 cos 𝜑0

e

∞

d𝜉

∫−∞

eik

√ (x−𝜉)2 +y2 +𝜁 2

d𝜁 . √ (x − 𝜉)2 + y2 + 𝜁 2

(3.6)

In Equations (3.3) and (3.6), the integral over the variable 𝜁 can be expressed through the Hankel function. We utilize two integral forms for this function. The first form, H0(1) (kd)

∞

√

2

2

1 eik d +𝜁 = d𝜁 , √ i𝜋 ∫−∞ d2 + 𝜁 2

(3.7)

ORIGINAL PO INTEGRALS

89

follows from formula 8.421.11 of Gradshteyn and Ryzhik (1994). The second form, ( √ ) ∞ i(𝑣d−𝑤z) 1 e d𝑤 H0(1) k d2 + z2 = 𝜋 ∫−∞ 𝑣

(3.8)

√ (where 𝑣 = k2 − 𝑤2 , Im 𝑣 > 0, and d > 0), can be verified by its conversion to the Sommerfeld formula (Sommerfeld, 1935): H0(1) (𝜌) =

𝛿−i∞

1 ei𝜌 cos 𝛽 d𝛽, 𝜋 ∫−𝛿+i∞

0≤𝛿≤𝜋

(3.9)

√ by setting 𝑤 = k sin t, 𝑣 = k cos t, and k d2 + z2 = 𝜌. Application of Equation (3.7) leads to ] k sin 𝜑0 ∞ −ik𝜉 cos 𝜑 (1) [ √ 2 + y2 d𝜉, 0H k e (x − 𝜉) 0 ∫0 2 [ √ ] ∞ i 𝜕 = −u0 e−ik𝜉 cos 𝜑0 H0(1) k (x − 𝜉)2 + y2 d𝜉. 2 𝜕y ∫0

u(0) s = −u0

(3.10)

u(0) h

(3.11)

Then we use Equation (3.8) and find that u(0) s = −u0

∞ k sin 𝜑0 ∞ ei𝑣|y| −i𝑤x d𝑤 ei(𝑤−k cos 𝜑0 )𝜉 d𝜉, e ∫0 2𝜋 ∫−∞ 𝑣

= sgn(y)u0 u(0) h

∞

1 ei(𝑣|y|−𝑤x) d𝑤 ∫0 2𝜋 ∫−∞

∞

ei(𝑤−k cos 𝜑0 )𝜉 d𝜉.

(3.12) (3.13)

To ensure convergence of the internal integrals, we impose the condition Im (𝑤 − k cos 𝜑0 ) > 0 and obtain u(0) s = u0

i sin 𝜑0 I, 2𝜋 s

u(0) = sign(y)u0 h

1 I , i ⋅ 2𝜋 h

(3.14)

where ∞

Is = k

∫−∞

ei(𝑣|y|−𝑤x) d𝑤, 𝑣(k cos 𝜑0 − 𝑤)

∞

Ih =

∫−∞

ei(𝑣|y|−𝑤x) d𝑤, k cos 𝜑0 − 𝑤

and the integration contour skirts above the pole, 𝑤 = k cos 𝜑0 .

(3.15)

90

WEDGE DIFFRACTION: THE PHYSICAL OPTICS FIELD

3.2 CONVERSION OF PO INTEGRALS TO THE CANONICAL FORM In the integrals Is and Ih , we introduce the polar coordinates by the relationships { x = r cos 𝜑,

|y| =

r sin 𝜑 −r sin 𝜑

with 𝜙 < 𝜋, with 𝜙 > 𝜋

(3.16)

and change the integration variable 𝑤 by 𝜉, setting 𝑤 = −k cos 𝜉,

𝑣 = k sin 𝜉,

Im 𝑣 > 0.

(3.17)

with 𝜙 < 𝜋, with 𝜙 > 𝜋.

(3.18)

Then { 𝑣 |y| − 𝑤x =

kr cos(𝜉 − 𝜙) kr cos(𝜉 + 𝜙)

The equations 𝑤 = −k cos 𝜉 ′ cosh 𝜉 ′′ + ik sin 𝜉 ′ sinh 𝜉 ′′ , 𝑣 = k sin 𝜉 ′ cosh 𝜉 ′′ + ik cos 𝜉 ′ sinh 𝜉 ′′

(3.19)

Im 𝑣 = k cos 𝜉 ′ sinh 𝜉 ′′ > 0

(3.20)

and the condition

determine the integration path F in the complex plane 𝜉 = 𝜉 ′ + i𝜉 ′′ , as shown in Figure 3.2. ξ ′′ = + ∞ (ξ ) F F

0

• π – φ0

π

ξ′ F

ξ ′′ = – ∞ Figure 3.2 Integration contour F in the complex plane 𝜉 = 𝜉 ′ + i𝜉 ′′ .

CONVERSION OF PO INTEGRALS TO THE CANONICAL FORM

91

After these manipulations, we obtain the following expressions: Is =

eikr cos(𝜉−𝜑) d𝜉 ∫F cos 𝜉 + cos 𝜑0

with 𝜑 < 𝜋,

(3.21)

Is =

eikr cos(𝜉+𝜑) d𝜉 ∫F cos 𝜉 + cos 𝜑0

with 𝜑 > 𝜋

(3.22)

and Ih =

eikr cos(𝜉−𝜑) sin 𝜉 d𝜉 ∫F cos 𝜉 + cos 𝜑0

with 𝜑 < 𝜋,

(3.23)

Ih =

eikr cos(𝜉+𝜑) sin 𝜉 d𝜉 ∫F cos 𝜉 + cos 𝜑0

with 𝜑 > 𝜋,

(3.24)

where, as shown in Figure 3.2, the integration contour F skirts above the pole, 𝜉 = 𝜋 − 𝜑0 . In Figures 3.3 and 3.4, we introduce two additional contours, G1 and G2 , related to the cases 𝜑 < 𝜋 and 𝜑 > 𝜋, respectively. In the dashed regions of these figures, the relations Im cos(𝜉 − 𝜑) > 0 and Im cos(𝜉 + 𝜑) > 0 are valid and guarantee the convergence of the integrals Is and Ih . We then apply the Cauchy residue theorem to Im ξ

φ–

π 2

G1

(ξ ) F

π – φ0

–π

π Re ξ

0

F G1

–π + φ

φ

φ+

π 2

π+φ

Figure 3.3 Integration contours F and G1 for the case 𝜑 < 𝜋 . In the shaded regions, Im cos(𝜉 − 𝜑) > 0.

92

WEDGE DIFFRACTION: THE PHYSICAL OPTICS FIELD

Im ξ 3π 2

–φ (ξ )

G2 F

π – φ0

–π

π Re ξ

0

F G2

2π – φ

π–φ

5π 2

–φ

3π + φ

Figure 3.4 Integration contours F and G2 for the case 𝜑 > 𝜋. In the shaded regions, Im cos(𝜉 + 𝜑) > 0.

the integrals over the closed contours F–G1,2 and obtain

⎧ ⎪∫ ⎪ G1 ⎪ ⎪ ⎪ ⎪ ⎪ ∫G Is = ⎨ 1 ⎪ ⎪∫ ⎪ G2 ⎪ ⎪ ⎪ ∫G2 ⎪ ⎩

2𝜋 i −ikr cos(𝜙+𝜙0 ) eikr cos(𝜉−𝜙) d𝜉 + e cos 𝜉 + cos 𝜙0 sin 𝜙0 with 0 ≤ 𝜑 ≤ 𝜋 − 𝜑0 ,

(3.25)

d𝜉

with 𝜋 − 𝜑0 < 𝜑 < 𝜋,

(3.26)

eikr cos(𝜉+𝜑) d𝜉 cos 𝜉 + cos 𝜑0

with 𝜋 < 𝜑 < 𝜋 + 𝜑0 ,

(3.27)

eikr cos(𝜉−𝜑) cos 𝜉 + cos 𝜑0

2𝜋i −ikr cos(𝜑−𝜑0 ) eikr cos(𝜉+𝜑) d𝜉 + e cos 𝜉 + cos 𝜑0 sin 𝜑0 with 𝜋 + 𝜙0 < 𝜙 ≤ 2𝜋,

(3.28)

In these equations we change the integration variable 𝜉 by 𝜁 + 𝜑 in the region 0 ≤ 𝜑 < 𝜋 and by 𝜁 + 2𝜋 − 𝜑 in the region 𝜋 < 𝜑 < 2𝜋. As a result, the integrals in Equations (3.25) to (3.28) are transformed into the integrals over the contour D0

CONVERSION OF PO INTEGRALS TO THE CANONICAL FORM

93

shown in Figure 2.6: ∫G1

eikr cos 𝜁 d𝜁 eikr cos(𝜉−𝜑) d𝜉 = ∫D0 cos 𝜑0 + cos(𝜁 + 𝜑) cos 𝜉 + cos 𝜑0

with 0 ≤ 𝜑 < 𝜋,

∫G2

eikr cos 𝜁 d𝜁 eikr cos(𝜉+𝜑) d𝜉 = ∫D0 cos 𝜑0 + cos(𝜁 − 𝜑) cos 𝜉 + cos 𝜑0

with 𝜋 ≤ 𝜑 < 2𝜋. (3.30)

(3.29)

We omit similar transformations for integral Ih and show only their results: ⎧ ⎪ ∫D ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ∫D0 Ih = ⎨ ⎪ ⎪∫ ⎪ D0 ⎪ ⎪ ⎪ ∫D0 ⎪ ⎩

eikr cos 𝜁 sin(𝜁 + 𝜑) d𝜁 + 2𝜋 i e−ikr cos(𝜑+𝜑0 ) cos 𝜑0 + cos(𝜁 + 𝜑) with 0 ≤ 𝜑 ≤ 𝜋 − 𝜑0 ,

(3.31)

eikr cos 𝜁 sin(𝜁 + 𝜑) d𝜁 cos 𝜑0 + cos(𝜁 + 𝜑)

with 𝜋 − 𝜑0 ≤ 𝜑 < 𝜋,

(3.32)

eikr cos 𝜁 sin(𝜁 − 𝜑) d𝜁 cos 𝜑0 + cos(𝜁 − 𝜑)

with 𝜋 ≤ 𝜑 ≤ 𝜋 + 𝜑0 ,

(3.33)

eikr cos 𝜁 sin(𝜁 − 𝜑) d𝜁 + 2𝜋 i e−ikr cos(𝜑−𝜑0 ) cos 𝜑0 + cos(𝜁 − 𝜑) with 𝜋 + 𝜑0 ≤ 𝜑 ≤ 2𝜋.

(3.34)

Taking into account the symmetry of contour D0 , where 𝜁 runs from −𝜋∕2 + i∞ to 𝜋∕2 − i∞, and changing the variable 𝜁 by −𝜁 in the integrals related to the region 𝜋 < 𝜑 < 2𝜋, one can show that eikr cos 𝜁 d𝜁 eikr cos 𝜁 d𝜁 = , cos 𝜑0 + cos(𝜁 − 𝜑) ∫D0 cos 𝜑0 + cos(𝜁 + 𝜑)

(3.35)

eikr cos 𝜁 sin(𝜁 − 𝜑) eikr cos 𝜁 sin(𝜁 + 𝜑) d𝜁 = − d𝜁 . ∫D0 cos 𝜙0 + cos(𝜁 + 𝜑) cos 𝜙0 + cos(𝜁 − 𝜑)

(3.36)

∫D0 ∫D0

Finally, one can represent the scattered field in the following form:

u(0) s

u(0) h

⎧ −u e−ikr cos(𝜑+𝜑0 ) 0

=

⎪ u0 𝑣(0) s (kr, 𝜑, 𝜑0 ) + ⎨ 0

with 0 ≤ 𝜑 ≤ 𝜋 − 𝜑0 , with 𝜋 − 𝜑0 < 𝜑 < 𝜋 + 𝜑0 , (3.37) with 𝜋 + 𝜑0 < 𝜑 < 2𝜋,

=

⎧ u e−ikr cos(𝜑+𝜑0 ) ⎪ 0 (0) u0 𝑣h (kr, 𝜑, 𝜑0 ) + ⎨ 0 ⎪ −u0 e−ikr cos(𝜙−𝜙0 ) ⎩

with 0 ≤ 𝜑 ≤ 𝜋 − 𝜑0 , with 𝜋 − 𝜑0 < 𝜑 < 𝜋 + 𝜑0 , (3.38) with 𝜋 + 𝜑0 < 𝜑 < 2𝜋,

⎪ −u0 e−ikr cos(𝜙−𝜙0 ) ⎩

94

WEDGE DIFFRACTION: THE PHYSICAL OPTICS FIELD

where 𝑣(0) s (kr, 𝜑, 𝜑0 ) =

i sin 𝜑0 eikr cos 𝜁 d𝜁 , 2𝜋 ∫D0 cos 𝜑0 + cos(𝜁 + 𝜑)

(3.39)

(kr, 𝜑, 𝜑0 ) = 𝑣(0) h

eikr cos 𝜁 sin(𝜁 + 𝜑) 1 d𝜁. i2𝜋 ∫D0 cos 𝜑0 + cos(𝜁 + 𝜑)

(3.40)

These expressions determine the PO field scattered by the face 𝜑 = 0 of the wedge. They are valid under the condition 0 ≤ 𝜑0 < 𝛼 − 𝜋. However, when 𝛼 − 𝜋 < 𝜑0 < 𝜋, the other face (𝜑 = 𝛼) is also illuminated and generates the additional scattered field. This additional field is also determined by Equations (3.37) and (3.38), where one should replace 𝜑 by 𝛼 − 𝜑 and 𝜑0 by 𝛼 − 𝜑0 . scattered in this case by both faces of the wedge is described Thus, the PO field u(0) s,h by the following equations: (0) (0) u(0) s = u0 [𝑣s (kr, 𝜑, 𝜑0 ) + 𝑣s (kr, 𝛼 − 𝜑, 𝛼 − 𝜑0 )]

⎧ −u0 e−ikr cos(𝜑−𝜑0 ) ⎪ ⎪0 +⎨ −ikr cos(2𝛼−𝜑−𝜑0 ) ⎪ −u0 e ⎪ −u e−ikr cos(𝜑−𝜑0 ) ⎩ 0

with 0 ≤ 𝜑 ≤ 𝜋 − 𝜑0 , with 𝜋−𝜑0 < 𝜑 < 2𝛼 − 𝜋 − 𝜑0 , with 2𝛼 − 𝜋 − 𝜑0 ≤ 𝜑 ≤ 𝛼,

(3.41)

with 𝛼 < 𝜑 < 2𝜋,

u(0) = u0 [𝑣(0) (kr, 𝜑, 𝜑0 ) + 𝑣(0) (kr, 𝛼 − 𝜑, 𝛼 − 𝜑0 )] h h h ⎧ u0 e−ikr cos(𝜑−𝜑0 ) ⎪ ⎪0 +⎨ −ikr cos(2𝛼−𝜑−𝜑0 ) ⎪ u0 e ⎪ −u e−ikr cos(𝜑−𝜑0 ) ⎩ 0

with 0 ≤ 𝜑 ≤ 𝜋 − 𝜑0 , with 𝜋 − 𝜑0 < 𝜑 < 2𝛼 − 𝜋 − 𝜑0 , with 2𝛼 − 𝜋 − 𝜑0 ≤ 𝜑 ≤ 𝛼,

(3.42)

with 𝛼 < 𝜑 < 2𝜋.

As shown in Equations (3.37), (3.38) and (3.41), (3.42), the PO scattered field consists of the plane waves and the diffracted part described by functions 𝑣(0) s and (0) 𝑣h . Fast convergent integrals and asymptotic expressions for these functions are derived in the next section. 3.3 FAST CONVERGENT INTEGRALS AND ASYMPTOTICS FOR THE PO DIFFRACTED FIELD

The relationships us = Ez and uh = Hz exist between the acoustic and electromagnetic fields studied in this section. In this section we derive exact forms of the functions 𝑣(0) (kr, 𝜑, 𝜑0 ) in terms of the s,h fast convergent integrals, which are free of singularities. Introducing a new variable

FAST CONVERGENT INTEGRALS AND ASYMPTOTICS FOR THE PO DIFFRACTED FIELD

s=

95

√ 2ei(𝜋∕4) sin(𝜁 ∕2), one can rewrite Equations (3.39) and (3.40) as sin 𝜑0 2 1 i(kr+𝜋∕4) ds e−krs , √ e ∫ cos 𝜑 + cos(𝜁 + 𝜑) cos(𝜁 ∕2) −∞ 0 𝜋 2 ∞

𝑣(0) s (kr, 𝜑, 𝜑0 ) =

(3.43)

∞ sin(𝜁 + 𝜑) 2 1 i(kr+𝜋∕4) ds e e−krs (kr, 𝜑, 𝜑 ) = − , (3.44) 𝑣(0) √ 0 h ∫ cos 𝜑 + cos(𝜁 + 𝜑) cos(𝜁 ∕2) −∞ 0 𝜋 2

where the inverse function 𝜁 (s) is determined by Equations (2.115) to (2.118). In view of the identities ) 𝜁 + 𝜑 − 𝜑0 − 𝜋 𝜁 + 𝜑 + 𝜑0 − 𝜋 cot − cot , (3.45) 2 2 ( ) 𝜁 + 𝜑 − 𝜑0 − 𝜋 𝜁 + 𝜑 + 𝜑0 − 𝜋 sin(𝜁 + 𝜑) 1 =− cot + cot , (3.46) cos 𝜑0 + cos(𝜁 + 𝜑) 2 2 2 sin 𝜑0 1 = cos 𝜑0 + cos(𝜁 + 𝜑) 2

(

Equations (3.43) and (3.44) take the form 𝑣(0) s (kr, 𝜑, 𝜑0 ) = ∞

×

∫−∞

e

−krs2

(

𝜁 + 𝜑 + 𝜑0 − 𝜋 𝜁 + 𝜑 − 𝜑0 − 𝜋 − cot cot 2 2

𝑣(0) (kr, 𝜑, 𝜑0 ) = h ∞

×

∫−∞

e−krs

2

1 i(kr+𝜋∕4) √ e 2𝜋 2

(

1 i(kr+𝜋∕4) √ e 2𝜋 2

𝜁 + 𝜑 + 𝜑0 − 𝜋 𝜁 + 𝜑 − 𝜑0 − 𝜋 + cot cot 2 2

)

)

ds , cos(𝜁 ∕2)

(3.47)

ds . cos(𝜁 ∕2)

(3.48)

Here, the integrands are singular at the saddle point s = 𝜁 = 0 when 𝜑 → 𝜋 ± 𝜑0 (i.e., on the GO boundaries). Then we apply the magic zero procedure introduced in Section 2.7, utilize Equation (2.121), and obtain 𝑣(0) s (kr, 𝜑, 𝜑0 ) =

(

𝜁 + 𝜑 − 𝜑0 − 𝜋 𝜁 + 𝜑 − 𝜑0 − 𝜋 − csc 2 2 ) 𝜁 + 𝜑 + 𝜑0 − 𝜋 𝜁 + 𝜑 + 𝜑0 − 𝜋 ds − cot + csc 2 2 cos(𝜁 ∕2)

ei(kr+𝜋∕4) ∞ −krs2 e √ 2𝜋 2 ∫−∞

cot

∞ sin[(𝜑−𝜑 −𝜋)∕2]

0 2 e−i(𝜋∕4) eix dx + e−ikr cos(𝜑−𝜑0 ) √ √ ∫ 2kr sin[(𝜑−𝜑0 −𝜋)∕2] 𝜋

e−i(𝜋∕4) ∞ sin(𝜑+𝜑0 −𝜋)∕2 ix2 e dx, − e−ikr cos(𝜑+𝜑0 ) √ √ 𝜋 ∫ 2kr sin(𝜑+𝜑0 −𝜋)∕2

(3.49)

96

WEDGE DIFFRACTION: THE PHYSICAL OPTICS FIELD

(

𝜁 + 𝜑 − 𝜑0 − 𝜋 𝜁 + 𝜑 − 𝜑0 − 𝜋 − csc 2 2 ) 𝜁 + 𝜑 + 𝜑0 − 𝜋 𝜁 + 𝜑 + 𝜑0 − 𝜋 ds + cot − csc 2 2 cos(𝜁 ∕2)

ei(kr+𝜋∕4) ∞ −krs2 e √ 2𝜋 2 ∫−∞

𝑣(0) (kr, 𝜑, 𝜑0 ) = h

cot

∞sins[(𝜑−𝜑 −𝜋)∕2]

0 2 e−i(𝜋∕4) eix dx + e−ikr cos(𝜑−𝜑0 ) √ √ 𝜋 ∫ 2kr sin[(𝜑−𝜑0 −𝜋)∕2]

e−i(𝜋∕4) ∞ sin[(𝜑+𝜑0 −𝜋)∕2] ix2 e dx. + e−ikr cos(𝜑+𝜑0 ) √ √ 𝜋 ∫ 2kr sin[(𝜑+𝜑0 −𝜋)∕2]

(3.50)

It is clear that the integrals over variable s are free of singularities at the saddle point s = 𝜁 = 0 when the observation point approaches the GO boundaries 𝜑 = 𝜋 ± 𝜑0 . These fast convergent integrals (with kr ≫ 1) are convenient for both numerical and asymptotic evaluations. Indeed, integrating the Taylor series of the integrands in the vicinity of the saddle point, one can obtain the asymptotic expansions valid with kr ≫ 1. The first-order are determined as asymptotics found in this way for functions 𝑣(0) s,h 𝑣(0) s (kr, 𝜑, 𝜑0 ) =

( 𝜑 − 𝜑0 − 𝜋 𝜑 − 𝜑0 − 𝜋 1 cot − csc 2 2 2 ) 𝜑 + 𝜑0 − 𝜋 ei(kr+𝜋∕4) 𝜑 + 𝜑0 − 𝜋 + csc − cot √ 2 2 2𝜋kr ∞ sin[(𝜑−𝜑 −𝜋)∕2]

0 2 e−i(𝜋∕4) eix dx + e−ikr cos(𝜑−𝜑0 ) √ √ ∫ 2kr sin[(𝜑−𝜑0 −𝜋)∕2] 𝜋

e−i(𝜋∕4) ∞ sin[(𝜑+𝜑0 −𝜋)∕2] ix2 e dx, (3.51) − e−ikr cos(𝜑+𝜑0 ) √ √ 𝜋 ∫ 2kr sin[(𝜑+𝜑0 −𝜋)∕2] ( 𝜑 − 𝜑0 − 𝜋 𝜑 − 𝜑0 − 𝜋 1 𝑣(0) (kr, 𝜑, 𝜑 ) = cot − csc 0 h 2 2 2 ) 𝜑 + 𝜑0 − 𝜋 ei(kr+𝜋∕4) 𝜑 + 𝜑0 − 𝜋 − csc + cot √ 2 2 2𝜋kr e−i(𝜋∕4) ∞ sin[(𝜑−𝜑0 −𝜋)∕2] ix2 e dx + e−ikr cos(𝜑−𝜑0 ) √ √ 𝜋 ∫ 2kr sin[(𝜑−𝜑0 −𝜋)∕2] ∞ sin[(𝜑+𝜑 −𝜋)∕2]

0 2 e−i(𝜋∕4) eix dx. (3.52) + e−ikr cos(𝜑+𝜑0 ) √ √ ∫ 2kr sin[(𝜑+𝜑0 −𝜋)∕2] 𝜋

The second and third terms of the Taylor series lead, respectively, to the integrals ∞

∫−∞

2

e−krs s ds = 0

∞

and

∫−∞

2

e−krs s2 ds = O[(kr)−3.2 ].

97

FAST CONVERGENT INTEGRALS AND ASYMPTOTICS FOR THE PO DIFFRACTED FIELD

Hence, the error of the asymptotic approximations (3.51) and (3.52) is on the order of (kr)−3∕2 . Examine these asymptotics for 0 < 𝜑0 < 𝛼 − 𝜋 when only one face (𝜑 = 0) is illuminated. They are discontinuous on the GO boundaries 𝜑 = 𝜋 ± 𝜑0 and ensure continuity of the total PO field there. The directions 𝜑 = 𝜋 ± 𝜑0 − 0 and 𝜑 = 𝜋 ± 𝜑0 + 0 relate to opposite sides of the GO boundaries. It follows from Equations (3.51) and (3.52) that 𝜑0 ei(kr+𝜋∕4) 1 ikr 1 𝑣(0) s (kr, 𝜋 + 𝜑0 ± 0, 𝜑0 ) = ± 2 e + 2 tan 2 √ 2𝜋kr 2 e−i(𝜋∕4) ∞ eix dx, − eikr cos 2𝜑0 √ √ ∫ 2kr sin 𝜑0 𝜋

(3.53)

𝜑 ei(kr+𝜋∕4) 1 1 𝑣(0) (kr, 𝜋 + 𝜑0 ± 0, 𝜑0 ) = ± eikr − tan 0 √ h 2 2 2 2𝜋kr ∞

2 e−i(𝜋∕4) eix dx. + eikr cos 2𝜑0 √ √ 𝜋 ∫ 2kr sin 𝜑0

(3.54)

The first terms here coincide with the Pauli asymptotics (2.77) for the exact solution of the wedge diffraction problem. Thus, PO provides a correct asymptotic approximation of the diffracted field in the vicinity of the GO shadow boundary. The PO asymptotics valid in the vicinity of the GO boundary of reflected rays (𝜑 = 𝜋 − 𝜑0 ) are derived in Problem 3.1. Away from the GO boundaries, Equations (3.51) and (3,52) reduce to 𝑣(0) s (kr, 𝜑, 𝜑0 ) =

sin 𝜑0 ei(kr+𝜋∕4) , √ cos 𝜑 + cos 𝜑0 2𝜋kr

𝑣(0) (kr, 𝜑, 𝜑0 ) = − h

sin 𝜑 ei(kr+𝜋∕4) . √ cos 𝜑 + cos 𝜑0 2𝜋kr

(3.55)

(3.56)

As a result, the ray asymptotics for the diffracted part of the PO field (3.37), (3.38) and (3.41), (3.42) can be written as

ei(kr+𝜋∕4) , u(0)d = u0 f (0) (𝜑, 𝜑0 ) √ s 2𝜋kr

(3.57)

ei(kr+𝜋∕4) u(0)d . = u0 g(0) (𝜑, 𝜑0 ) √ h 2𝜋kr

(3.58)

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WEDGE DIFFRACTION: THE PHYSICAL OPTICS FIELD

Here f (0) (𝜑, 𝜑0 ) =

sin 𝜑0 , cos 𝜑 + cos 𝜑0

g(0) (𝜑, 𝜑0 ) = −

sin 𝜑 cos 𝜑 + cos 𝜑0

(3.59)

for 0 < 𝜑0 < 𝛼 − 𝜋, when only the face 𝜑 = 0 is illuminated by the incident wave, and f (0) (𝜑, 𝜑0 ) =

sin 𝜑0 sin(𝛼 − 𝜑0 ) + , cos 𝜑 + cos 𝜑0 cos(𝛼 − 𝜑) + cos(𝛼 − 𝜑0 )

g(0) (𝜑, 𝜑0 ) = −

(3.60)

sin 𝜑 sin(𝛼 − 𝜑) − , cos 𝜑 + cos 𝜑0 cos(𝛼 − 𝜑) + cos(𝛼 − 𝜑0 )

(3.61)

when both faces are illuminated (𝛼 − 𝜋 < 𝜑0 < 𝜋). within the same level of Next we derive other asymptotics for the functions 𝑣(0) s,h approximation but in a more compact form. Utilizing the identities sin 𝜑0 1 = cos 𝜑0 + cos (𝜁 + 𝜑) 2 sin (𝜁 + 𝜑) 1 = cos 𝜑0 + cos (𝜁 + 𝜑) 2

( tan (

𝜁 + 𝜑 − 𝜑0 𝜁 + 𝜑 + 𝜑0 − tan 2 2

𝜁 + 𝜑 + 𝜑0 𝜁 + 𝜑 − 𝜑0 tan + tan 2 2

) ,

(3.62)

,

(3.63)

)

one can represent Equations (3.39) and (3.40) as 𝑣(0) s (kr, 𝜑, 𝜑0 )

i eikr cos 𝜁 = 4𝜋 ∫D0

𝑣(0) (kr, 𝜑, 𝜑0 ) = h

1 eikr cos 𝜁 i4𝜋 ∫D0

(

𝜁 + 𝜑 − 𝜑0 𝜁 + 𝜑 + 𝜑0 − tan tan 2 2

( tan

)

𝜁 + 𝜑 − 𝜑0 𝜁 + 𝜑 + 𝜑0 + tan 2 2

d𝜁 )

(3.64) d𝜁 . (3.65)

A main contribution to these integrals is provided by the vicinity of the saddle point 𝜁 = 0. Note that the function sin[(𝜁 + 𝜓)∕2] contained in tan[(𝜁 + 𝜓)∕2] is regular in the vicinity of this point and can be expanded into the Taylor series. We take the first term in this series, preserve cos[(𝜁 + 𝜓)∕2] in the denominator, utilize the identity √

2kr cos(𝜓∕2) eikr cos 𝜁 d𝜁 2 e−i(𝜋∕4) i eix dx, (3.66) = e−ikr cos 𝜓 √ − ∫ 4𝜋 ∫D0 cos[(𝜁 + 𝜓)∕2] ∞ cos(𝜓∕2) 𝜋

FAST CONVERGENT INTEGRALS AND ASYMPTOTICS FOR THE PO DIFFRACTED FIELD

99

which follows from Equations (2.82) and (2.83), and arrive at the following asymptotics: √

𝑣(0) s (kr, 𝜑, 𝜑0 )

2kr cos[(𝜑−𝜑0 )∕2] 𝜑 − 𝜑0 −ikr cos(𝜑−𝜑 ) e−i𝜋∕4 2 0 e eix dx = sin √ ∫ 2 ∞ cos[(𝜑−𝜑 )∕2] 𝜋 0 √

2kr cos[(𝜑+𝜑0 )∕2] 𝜑 + 𝜑0 −ikr cos(𝜑+𝜑 ) e−i𝜋∕4 2 0 e eix dx, (3.67) − sin √ ∫ 2 𝜋 ∞ cos[(𝜑+𝜑0 )∕2] √

(kr, 𝜑, 𝜑0 ) 𝑣(0) h

2kr cos[(𝜑−𝜑0 )∕2] 𝜑 − 𝜑0 −ikr cos(𝜑−𝜑 ) e−i𝜋∕4 2 0 e eix dx = sin √ ∫ 2 𝜋 ∞ cos[(𝜑−𝜑0 )∕2] √

2kr cos[(𝜑+𝜑0 )∕2] 𝜑 + 𝜑0 −ikr cos(𝜑+𝜑 ) e−i𝜋∕4 2 0 e eix dx. (3.68) + sin √ ∫ 2 𝜋 ∞ cos[(𝜑+𝜑0 )∕2]

As shown in Problem 3.2, the next terms of the Taylor series produce corrections on the order of (kr)−3∕2 . Thus, within this approximation these asymptotics are equivalent to asymptotics (3.51) and (3.52). At the GO boundary 𝜑 = 𝜋 + 𝜑0 ± 0, these asymptotics reduce to ∞ sin 𝜑

0 2 e−i(𝜋∕4) 1 ikr e − cos 𝜑0 eikr cos 2𝜑0 √ eix dx, 𝑣(0) √ s (kr, 𝜋 + 𝜑0 ± 0, 𝜑0 ) = ± 2 ∫ 2kr sin 𝜑0 𝜋 (3.69)

∞ sin 𝜑

−i(𝜋∕4) 0 2 1 ikr ikr cos 2𝜑0 e e e eix dx, 𝑣(0 (kr, 𝜋 + 𝜑 ± 0, 𝜑 ) = ± + cos 𝜑 √ √ 0 0 0 h ∫ 2 2kr sin 𝜑0 𝜋 (3.70)

and look different from Equations (3.53) and (3.54). However, they are asymptotically equivalent within their level of approximation. Indeed, under the condition √ 2kr sin 𝜑0 ≫ 1, they provide the same approximation as Equations (3.53) and (3.54), ei(kr+𝜋∕4) 1 ikr 1 e − cot 𝜑0 √ + O[(kr)−3∕2 ], 𝑣(0) s (kr, 𝜋 + 𝜑0 ± 0, 𝜑0 ) = ± 2 2 2𝜋kr

(3.71)

ei(kr+𝜋∕4) 1 1 + O[(kr)−3∕2 ]. 𝑣(0) (kr, 𝜋 + 𝜑0 ± 0, 𝜑0 ) = ± eikr + cot 𝜑0 √ h 2 2 2𝜋kr

(3.72)

Although PO does not satisfy the reciprocity principle and boundary conditions rigorously, it describes some features of the scattered field within approximations

100

WEDGE DIFFRACTION: THE PHYSICAL OPTICS FIELD

acceptable in engineering practice. In particular, it correctly determines the total scattering cross-section of large opaque objects and predicts the field behavior in the vicinity of GO boundaries. The related results of a numerical comparison of the PO approximation against the exact solution for the wedge diffraction problem have been presented by Hacivelioglu et al. (2011). In the present book, PO is considered as a constitutive part of PTD and is utilized in the next chapter to calculate the field generated by the nonuniform component of surface scattering sources.

PROBLEMS 3.1 Derive the asymptotic approximations for the functions (3.51) and (3.52) at the GO boundary of reflected rays (𝜑 = 𝜋 − 𝜑0 ± 0). Solution Substitute 𝜑 = 𝜋 − 𝜑0 ± 0 into Equations (3.51) and (3.52). Verify that r cot 𝜑 + 𝜑0 − 𝜋 − csc 𝜑 + 𝜑0 − 𝜋 = 0. 2 2 ∞ sin[(𝜑+𝜑0 −𝜋)∕2] −i(𝜋∕4) 2 e e−i(𝜋∕4) r e−ikr cos(𝜑+𝜑0 ) √ eix dx = eikr √ √ 𝜋 ∫ 2kr sin[(𝜑+𝜑0 −𝜋)∕2] 𝜋 ±∞

×

∫0

2 1 eix dx = ± eikr . 2

r e−ikr cos(𝜑−𝜑0 ) e−i(𝜋∕4) √ −∞ sin 𝜑0

×

∫−√2kr sin 𝜑0

𝜋

ix2

e dx

∞ sin[(𝜑−𝜑0 −𝜋)∕2] 2 e−i(𝜋∕4) eix dx = eikr cos 2𝜑0 √ √ ∫ 2kr sin[(𝜑−𝜑0 −𝜋)∕2] 𝜋 ∞ sin 𝜑0 −i(𝜋∕4) 2 e eix dx. = −eikr cos 2𝜑0 √ √ 𝜋 ∫ 2kr sin 𝜑0

r cot 𝜑 − 𝜑0 − 𝜋 − csc 𝜑 − 𝜑0 − 𝜋 = − cot 𝜑 + csc 𝜑 = tan 𝜑0 . 0 0 2

2

2

Therefore, 𝜑0 ei(kr+𝜋∕4) 1 ikr 1 e − eikr cos 2𝜑0 𝑣(0) (kr, 𝜋 − 𝜑 ± 0, 𝜑 ) = ∓ + tan √ 0 0 s 2 2 2 2𝜋kr e−i(𝜋∕4) ∞ sin 𝜑0 ix2 e dx, × √ √ 𝜋 ∫ 3kr sin 𝜑0 𝜑0 ei(kr+𝜋∕4) 1 ikr 1 e 𝑣(0) − eikr cos 2𝜑0 (kr, 𝜋 − 𝜑 ± 0, 𝜑 ) = ± + tan √ 0 0 h 2 2 2 2𝜋kr ∞ sin 𝜑

0 2 e−i(𝜋∕4) eix dx. × √ √ ∫ 3kr sin 𝜑0 𝜋

PROBLEMS

3.2

101

Evaluate the error of approximations (3.67) and (3.68). Solution

Consider the integral

∫D0

eikr cos 𝜁 tan

sin[(𝜁 + 𝜓)∕2] 𝜁 +𝜓 eikr cos 𝜁 d𝜁 = d𝜁 . ∫D0 2 cos[(𝜁 + 𝜓)∕2)

Expand sin[(𝜁 + 𝜓)∕2] into the Taylor series in the vicinity of the saddle point 𝜁 = 0, sin

𝜁 +𝜓 𝜓 𝜓 𝜓 1 1 = sin + 𝜁 cos − 𝜁 2 sin + O(𝜁 3 ). 2 2 2 2 8 2

The first term here generates the asymptotics (3.67) and (3.68). Evaluate the contribution generated by the next terms: I(𝜓) = =

∫D0 ∫D0

eikr cos 𝜁

] [ 𝜓 𝜓 1 1 1 𝜁 cos − 𝜁 2 sin + O(𝜁 3 ) d𝜁 cos[(𝜁 + 𝜓)∕2] 2 2 8 2

eikr cos 𝜁

𝜁 (1∕2) cos(𝜓∕2) − 𝜁 2 (1∕8) sin(𝜓∕2) + ⋯ d𝜁 . cos(𝜓∕2) − 𝜁 (1∕2) sin(𝜓∕2) − 𝜁 2 (1∕8) cos(𝜓∕2) + ⋯

When 𝜓 ≠ 𝜋, I(𝜓) =

∫D0

=

∫D0

eikr cos 𝜁

𝜁 (1∕2) − 𝜁 2 (1∕8) tan(𝜓∕2) + O(𝜁 3 ) d𝜁 1 − 𝜁 (1∕2) tan(𝜓∕2) − 𝜁 2 (1∕8) + ⋯

eikr cos 𝜁 [(1∕2)𝜁 + O(𝜁 2 )]d𝜁 .

Here

∫D0

eikr cos 𝜁 𝜁 d𝜁 =

0

∫−(𝜋∕3)+i∞

eikr cos 𝜁 𝜁 d𝜁 +

(𝜋∕2)−i∞

∫0

0

=

∫D0

(𝜋∕2)−i∞

∫(𝜋∕2)−i∞

eikr cos 𝜁 𝜁 2 d𝜁 ∼ eikr

eikr cos x xdx +

2

∫D0

∫0

eikr cos 𝜁 𝜁 d𝜁 = 0,

e−i(kr∕2)𝜁 𝜁 2 d𝜁 = O[(kr)−3∕2 ]

Thus, I(𝜓) = O[(kr)−3∕2 ] when 𝜓 ≠ 𝜋.

eikr cos 𝜁 𝜁 d𝜁

with kr ≫ 1.

102

WEDGE DIFFRACTION: THE PHYSICAL OPTICS FIELD

When 𝜓 = 𝜋∕2, the integral I(𝜓) can be written as I(𝜓) =

𝜁2 1 1 eikr cos 𝜁 eikr cos 𝜁 [𝜁 + O(𝜁 3 )] d𝜁 . [1 + O(𝜁 2 )] d𝜁 = 8 ∫D0 sin(𝜁 ∕2) 4 ∫D0

It equals zero because it contains only the odd powers of the variable 𝜁 . Thus, I(𝜓) = O[(kr)−3∕2 ] when 𝜓 ≠ 𝜋 and I(𝜋) = 0. Next take into account the fact that functions (3.64) and (3.65) contain the terms with parameters 𝜓− = 𝜑 − 𝜑0 and 𝜓+ = 𝜑 + 𝜑0 . When one of them equals 𝜋, the other is not 𝜋. Therefore, the corrections to approximations (3.67) and (3.68) generated by the neglected terms in the Taylor series are on the order of (kr)−3∕2 .

4 Wedge Diffraction: Radiation by Fringe Components of Surface Sources (1) (1) (1) The relationships u(1) s = Ez and uh = Hz exist between the acoustic and electromagnetic fields generated by the nonuniform/fringe sources j(1) . An exception exists for an oblique incidence (see Section 4.3.2).

We have now reached the moment when we can construct the integral and asymptotic expressions for a field radiated by the nonuniform/fringe component of the surface sources which are induced at the wedge by the incident wave. The exact expressions for the total field generated around the wedge have been derived in Chapter 2. The physical optics (PO) part of this field (which is generated by the uniform component of the surface sources) has been studied in Chapter 3. The contribution to the diffracted field by the nonuniform/fringe component is the difference between the exact total field and its PO part. This contribution is investigated in this chapter.

Fundamentals of the Physical Theory of Diffraction, Second Edition. Pyotr Ya. Ufimtsev. © 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

103

104

WEDGE DIFFRACTION: RADIATION BY FRINGE COMPONENTS OF SURFACE SOURCES

4.1 INTEGRALS AND ASYMPTOTICS According to the exact solution [see Equations (2.40), (2.41), and (2.52) to (2.57)], the total field around the wedge consists of the diffracted and geometrical optics parts, go

uts,h = uds,h + us,h ,

(4.1) go

where uds,h is described by the functions 𝑣(kr, 𝜓), and us,h is the sum of the incident and reflected plane waves. Equations (3.37) to (3.42) represent the scattered field in the PO approximation. By summation with the incident wave (3.1), these equations determine the PO part of the total field: go

= u(0)d + us,h u(0)t s,h s,h

(4.2)

where u(0)d is the diffracted part of the field, which is described by the functions s,h go

𝑣(0) . The geometrical optics part us,h of the PO field is the same quantity as that in s,h Equation (4.1). The field (4.1) is generated by total surface source js,h = j(0) + j(1) , consisting of s,h s,h the uniform and nonuniform components, and the PO field (4.2) is radiated only by . Therefore, the fringe field created by the nonuniform the uniform component j(0) s,h component is the difference = uts,h − u(0)t = uds,h − u(0)d . u(1) s,h s,h s,h

(4.3)

For 0 < 𝜑0 < 𝛼 − 𝜋, when only one face (𝜑 = 0) is illuminated, this field is determined according to Chapters 2 and 3 as u(1) s = 𝑣(kr, 𝜑 − 𝜑0 ) − 𝑣(kr, 𝜑 + 𝜑0 ) − 𝑣(0) s (kr, 𝜑, 𝜑0 ), u0 u(1) h u0

= 𝑣(kr, 𝜑 − 𝜑0 ) + 𝑣(kr, 𝜑 + 𝜑00 ) − 𝑣(0) (kr, 𝜑, 𝜑0 ). h

(4.4)

(4.5)

For 𝛼 − 𝜋 < 𝜑0 < 𝜋, when both faces are illuminated, the field u(1) is determined as s,h u(1) s (0) = 𝑣(kr, 𝜙 − 𝜙0 ) − 𝑣(kr, 𝜙 + 𝜙0 ) − 𝑣(0) s (kr, 𝜑, 𝜑0 ) − 𝑣s (kr, 𝛼 − 𝜑, 𝛼 − 𝜑0 ), (4.6) u0 u(1) h u0

= 𝑣(kr, 𝜑 − 𝜑0 ) + 𝑣(kr, 𝜑 + 𝜑0 ) − 𝑣(0) (kr, 𝜑, 𝜑0 ) − 𝑣(0) (kr, 𝛼 − 𝜑, 𝛼 − 𝜑0 ). (4.7) h h

(kr, 𝜙, 𝜙0 ) The function 𝑣(kr, 𝜓) is defined in Equation (2.59) and the functions 𝑣(0) s,h are determined in Equations (3.43) and (3.44) by the integrals over the same steepest

INTEGRALS AND ASYMPTOTICS

105

descent path along the real axis (−∞ ≤ s ≤ ∞). Therefore, the functions u(1) can be s,h also represented by the integrals over this axis. , we transform the integrands Before presenting the final integrals for functions u(1) s,h of functions uds,h and u(0)d as follows: s,h (1∕n) sin(𝜋∕n) (1∕n) sin(𝜋∕n) ∓ cos(𝜋∕n) − cos[(𝜁 + 𝜓1 )∕n] cos(𝜋∕n) − cos[(𝜁 + 𝜓2 )∕n] ( ) 𝜁 + 𝜓1 − 𝜋 𝜁 + 𝜓1 + 𝜋 𝜁 + 𝜓2 − 𝜋 𝜁 + 𝜓2 + 𝜋 cot − cot ∓ cot ± cot , 2n 2n 2n 2n

1 = 2n

(4.8) sin 𝜑0 1 = cos 𝜑0 + cos (𝜁 + 𝜑) 2

(

sin (𝜁 + 𝜑) 1 =− cos 𝜑0 + cos (𝜁 + 𝜑) 2

cot

𝜁 + 𝜓1 − 𝜋 𝜁 + 𝜓2 − 𝜋 − cot 2 2

( cot

)

𝜁 + 𝜓1 − 𝜋 𝜁 + 𝜓2 − 𝜋 + cot 2 2

,

(4.9)

) ,

(4.10)

where 𝜓1 = 𝜑 − 𝜑0 ,

𝜓2 = 𝜑 + 𝜑0 .

(4.11)

We present similar equations for 𝛼 − 𝜋 < 𝜑0 < 𝜋 when both faces of the wedge are illuminated: sin(𝛼 − 𝜑0 ) sin 𝜑0 + cos 𝜑0 + cos (𝜁 + 𝜑) cos(𝛼 − 𝜑0 ) + cos (𝜁 + 𝛼 − 𝜑) 1 = 2

( ) 𝜁 + 𝜓1 − 𝜋 𝜁 + 𝜓2 − 𝜋 𝜁 + 𝜙1 − 𝜋 𝜁 + 𝜙2 − 𝜋 cot − cot + cot − cot , 2 2 2 2 (4.12)

sin(𝜁 + 𝛼 − 𝜑) sin(𝜁 + 𝜑) + cos 𝜑0 + cos(𝜁 + 𝜑) cos(𝛼 − 𝜑0 ) + cos(𝜁 + 𝛼 − 𝜑) =−

1 2

( ) 𝜁 + 𝜓1 − 𝜋 𝜁 + 𝜓2 − 𝜋 𝜁 + 𝜙1 − 𝜋 𝜁 + 𝜙2 − 𝜋 cot + cot + cot + cot , 2 2 2 2 (4.13)

where 𝜙1 = (𝛼 − 𝜑) − (𝛼 − 𝜑0 ) = 𝜑0 − 𝜑 = −𝜓1 , 𝜙2 = (𝛼 − 𝜑) + (𝛼 − 𝜑0 ) = 2𝛼 − 𝜑 − 𝜑0 = 2𝛼 − 𝜓2 .

(4.14)

106

WEDGE DIFFRACTION: RADIATION BY FRINGE COMPONENTS OF SURFACE SOURCES

The relationships

cot

𝜁 + 𝜓1 + 𝜋 𝜁 − 𝜙1 + 𝜋 = cot , 2n 2n

cot

( ) 𝜁 + 𝜓2 + 𝜋 𝜁 + 2𝛼 − 𝜙2 + 𝜋 𝜁 − 𝜙2 + 𝜋 = cot = cot 𝜋 + 2n 2n 2n

with

𝛼 =𝜋 n (4.15)

are also useful in this case. In the case 0 < 𝜑0 < 𝛼 − 𝜋, when only the face 𝜑 = 0 is illuminated, the equations above allow one to represent the fringe field as

u(1) s (kr, 𝜑, 𝜑0 ) = u0

∞

2 ei(kr+𝜋∕4) ds e−krs √ ∫ cos(𝜁 ∕2) −∞ 𝜋 2

( ×

𝜍 + 𝜓1 − 𝜋 1 𝜍 + 𝜓1 − 𝜋 1 cot − cot 2n 2n 2 2

−

𝜍 + 𝜓2 − 𝜋 1 𝜍 + 𝜓2 − 𝜋 1 cot + cot 2n 2n 2 2

𝜍 + 𝜓1 + 𝜋 𝜍 + 𝜓2 + 𝜋 1 1 − cot + cot 2n 2n 2n 2n

u(1) (kr, 𝜑, 𝜑0 ) = u0 h

,

(4.16)

.

(4.17)

∞

2 ei(kr+𝜋∕4) ds e−krs √ ∫ cos(𝜁 ∕2) −∞ 𝜋 2

( ×

)

𝜍 + 𝜓1 − 𝜋 𝜍 + 𝜓1 − 𝜋 1 1 cot − cot 2n 2n 2 2

+

𝜍 + 𝜓2 − 𝜋 1 𝜍 + 𝜓2 − 𝜋 1 cot − cot 2n 2n 2 2

𝜍 + 𝜓1 + 𝜋 𝜍 + 𝜓2 + 𝜋 1 1 − cot − cot 2n 2n 2n 2n

)

The quantity 𝜁 = 𝜁 (s) is defined according to Equations (2.115) to (2.118).

INTEGRALS AND ASYMPTOTICS

107

For 𝛼 − 𝜋 < 𝜙0 < 𝜋, when both faces are illuminated, the fringe field is determined by u(1) s (kr, 𝜑, 𝜑0 ) = u0

∞

2 ei(kr+𝜋∕4) ds e−krs √ ∫ cos(𝜁 ∕2) −∞ 𝜋 2

(

𝜍 + 𝜓1 − 𝜋 1 𝜍 + 𝜓1 − 𝜋 1 cot − cot 2n 2n 2 2

×

−

𝜍 + 𝜓2 − 𝜋 1 𝜍 + 𝜓2 − 𝜋 1 cot + cot 2n 2n 2 2

−

𝜍 − 𝜙1 + 𝜋 1 𝜍 + 𝜙1 − 𝜋 1 cot − cot 2n 2n 2 2

𝜍 − 𝜙2 + 𝜋 1 𝜍 + 𝜙2 − 𝜋 1 cot + cot + 2n 2n 2 2 (kr, 𝜑, 𝜑0 ) = u0 u(1) h

,

(4.18)

.

(4.19)

∞

2 ei(kr+𝜋∕4) ds e−krs √ ∫ cos(𝜁 ∕2) −∞ 𝜋 2

( ×

)

𝜍 + 𝜓1 − 𝜋 𝜍 + 𝜓1 − 𝜋 1 1 cot − cot 2n 2n 2 2

+

𝜍 + 𝜓2 − 𝜋 1 𝜍 + 𝜓2 − 𝜋 1 cot − cot 2n 2n 2 2

−

𝜁 − 𝜙1 + 𝜋 1 𝜁 + 𝜙1 − 𝜋 1 cot − cot 2n 2n 2 2

−

𝜁 − 𝜙2 + 𝜋 1 𝜁 + 𝜙2 − 𝜋 1 cot − cot 2n 2n 2 2

)

The terms in parentheses in Equations (4.14) to (4.19) are arranged in pairs to ensure cancellation of their possible singularities at the saddle point 𝜁 = s = 0, which appear at the GO boundaries where 𝜓1,2 = 𝜋 and 𝜙1,2 = 𝜋. Only the singularities related to 𝜓2 = 𝜋 and 𝜙2 = 𝜋 exist outside the wedge at the boundaries of the reflected plane waves. The singularities associated with 𝜓1 = 𝜋 and 𝜙1 = 𝜋 relate to the shadow boundary which is inside the wedge, that is, outside the physical space. Nevertheless, they affect the field in the physical space when the incidence directions are close to the grazing directions (𝜑0 = 𝜋 or 𝜑0 = 𝛼). The integrals derived for the fringe field are taken along the steepest descent path and are very convenient for numerical and analytical analysis. For finite values of kr, they are fast convergent. Far away from the edge when kr ≫ 1, they can

108

WEDGE DIFFRACTION: RADIATION BY FRINGE COMPONENTS OF SURFACE SOURCES

be evaluated asymptotically using the saddle point method (Copson, 1965; Murray, 1984). According to this method, the slowly varying integrand is expanded into the Taylor power series in the vicinity of the saddle point s = 0 and is then integrated term by term. The first term does not depend on s and is removed √ from the integral 2 sign. The remaining integral from the function exp(−krs ) equals 𝜋∕kr. The second term of the Taylor expansion is proportional to s and leads to the integral ∞

∫−∞

2

e−krs s ds = 0.

The third term generates the integral ∞

∫−∞

2

e−krs s2 ds = O[(kr)−3∕2 ]

with kr ≫ 1.

Finally, we obtain the following asymptotics for the fringe waves:

ei(kr+𝜋∕4) (1) + O[(kr)−3∕2 ], u(1) s = u0 f (𝜑, 𝜑0 , 𝛼) √ 2𝜋kr

(4.20)

ei(kr+𝜋∕4) + O[(kr)−3∕2 ], u(1) = u0 g(1) (𝜑, 𝜑0 , 𝛼) √ h 2𝜋kr

(4.21)

where f (1) (𝜑, 𝜑0 , 𝛼) = f (𝜑, 𝜑0 , 𝛼) − f (0) (𝜑, 𝜑0 ),

(4.22)

g(1) (𝜑, 𝜑0 , 𝛼) = g(𝜑, 𝜑0 , 𝛼) − g(0) (𝜑, 𝜑0 ).

(4.23)

Two equivalent forms of functions f (1) and g(1) are shown below when 0 < 𝜑0 < 𝛼 − 𝜋 and only the face 𝜑 = 0 is illuminated. The cotangent form follows straightforwardly from Equations (4.16) and (4.17) as 𝜓 −𝜋 1 𝜓 −𝜋 𝜓 +𝜋 1 1 cot 1 − cot 1 − cot 1 2n 2n 2 2 2n 2n 𝜓 𝜓 𝜓 − 𝜋 − 𝜋 +𝜋 1 1 1 − cot 2 + cot 2 + cot 2 , 2n 2n 2 2 2n 2n 𝜓 −𝜋 1 𝜓 −𝜋 𝜓 +𝜋 1 1 cot 1 − cot 1 − cot 1 g(1) (𝜑, 𝜑0 , 𝛼) = 2n 2n 2 2 2n 2n 𝜓 𝜓 𝜓 − 𝜋 − 𝜋 +𝜋 1 1 1 + cot 2 − cot 2 − cot 2 . 2n 2n 2 2 2n 2n f (1) (𝜑, 𝜑0 , 𝛼) =

(4.24)

(4.25)

INTEGRALS AND ASYMPTOTICS

109

Another form follows from Equations (4.24) and (4.25) if we take into account Equations (4.8) to (4.10) and set 𝜁 = 0 there: f

(1)

[ ] 𝜋 1 1 1 − (𝜑, 𝜑0 , 𝛼) = sin n n cos(𝜋∕n) − cos[(𝜁 + 𝜓1 )∕n] cos(𝜋∕n) − cos[(𝜁 + 𝜓2 )∕n]

sin 𝜑0 , (4.26) cos 𝜑0 + cos 𝜑 [ ] 𝜋 1 1 1 (1) + g (𝜑, 𝜑0 , 𝛼) = sin n n cos(𝜋∕n) − cos[(𝜁 + 𝜓1 )∕n] cos(𝜋∕n) − cos[(𝜁 + 𝜓2 )∕n] −

+

sin 𝜑 . cos 𝜑0 + cos 𝜑

(4.27)

For 𝛼 − 𝜋 < 𝜑0 < 𝜋 when both faces are illuminated, the related functions f (1) and g(1) are obtained by adding the term −

sin(𝛼 − 𝜑0 ) to the right-hand side of Equation (4.26) cos(𝛼 − 𝜑0 ) + cos(𝛼 − 𝜑)

and adding the term sin(𝛼 − 𝜑) to the right-hand side of Equation (4.27). cos(𝛼 − 𝜑0 ) + cos(𝛼 − 𝜑) The cotangent forms are preferable for numerical calculations, whereas Equations (4.26) and (4.27) are more compact and have clear origination. Indeed, they can be written in terms of the functions f , g and f (0) , g(0) introduced in Chapters 2 and 3, respectively: f (1) = f − f (0) ,

g(1) = g − g(0) .

(4.28)

itself represents the Thus, the field generated by the nonuniform component j(1) s,h cylindrical wave diverging from the edge of the wedge. This form of the field is a concentrates near the consequence of the fact that the nonuniform component j(1) s,h is sometimes called the fringe component. edge. For this reason the quantity j(1) s,h Approximations (4.20) and (4.21) reveal a ray structure of this part of the diffracted field, and because of that they can be called ray asymptotics. The directivity patterns of the fields (4.20) and (4.21) possess a wonderful property. In contrast to the functions f , g, f (0) , and g(0) , which are singular at the geometrical optics boundaries, the functions f (1) and g(1) are finite there. It turns out that the singularities of functions f and g are totally canceled by the singularities of the functions f (0) and g(0) , respectively. The following equations determine the finite values of functions f (1) and g(1) for these special directions.

110

WEDGE DIFFRACTION: RADIATION BY FRINGE COMPONENTS OF SURFACE SOURCES

For the direction 𝜑 = 𝜋 − 𝜑0 [which is the boundary of the plane wave reflected from the face 𝜑 = 0 (Fig. 2.7)], the functions f (1) and g(1) have the values {

f (1) g(1)

} =

(1∕n) sin(𝜋∕n) 1 1 𝜋 + cot 𝜑0 ± cot cos(𝜋∕n) − cos[(𝜋 − 2𝜑0 )∕n] 2 2n n

(4.29)

when 0 < 𝜑0 < 𝛼 − 𝜋, and {

f (1) g(1)

} =

(1∕n) sin(𝜋∕n) 1 1 𝜋 + cot 𝜑0 ± cot cos(𝜋∕n) − cos[(𝜋 − 2𝜑0 )∕n] 2 2n n ⎧ ⎫ sin(𝛼 − 𝜑0 ) ⎪ ⎪ ⎪ cos(𝛼 − 𝜑0 ) − cos(𝛼 + 𝜑0 ) ⎪ −⎨ ⎬ sin(𝛼 + 𝜑0 ) ⎪ ⎪ ⎪ cos(𝛼 − 𝜑 ) − cos(𝛼 + 𝜑 ) ⎪ 0 0 ⎭ ⎩

(4.30)

when 𝛼 − 𝜋 < 𝜑0 < 𝜋. For the direction 𝜑 = 𝜋 + 𝜑0 [which is the shadow boundary of the incident wave (Fig. 2.7)], the values of functions f (1) and g(1) are determined by {

f (1) g(1)

} =∓

(1∕n) sin(𝜋∕n) 1 1 𝜋 ± cot 𝜑0 − cot cos(𝜋∕n) − cos[(𝜋 + 2𝜑0 )∕n] 2 2n n

(4.31)

when 0 < 𝜑0 < 𝜋. The direction 𝜑 = 𝜋 + 𝜑0 under the condition 𝛼 − 𝜋 < 𝜑0 < 𝜋 is inside the wedge and is not of interest. For 𝛼 − 𝜋 < 𝜑0 < 𝜋, when both faces of the wedge are illuminated (Fig. 2.8), the functions f (1) and g(1) have the following values at the direction 𝜑 = 2𝛼 − 𝜋 − 𝜑0 (which is the boundary of the plane waves reflected from the face 𝜑 = 𝛼): {

f (1) g(1)

} =

(1∕n) sin(𝜋∕n) 1 1 𝜋 + cot(𝛼 − 𝜑0 ) ± cot cos(𝜋∕n) − cos[(𝜑 − 𝜑0 )∕n] 2 2n n ⎧ sin 𝜑0 ⎪− ⎪ cos 𝜑 + cos 𝜑0 +⎨ sin 𝜑 ⎪ ⎪ cos 𝜑 + cos 𝜑 0 ⎩

⎫ ⎪ ⎪ ⎬. ⎪ ⎪ ⎭

(4.32)

One should note that the functions f (1) and g(1) are singular in the two special directions 𝜑=0

when

𝜑0 = 𝜋

(4.33)

INTEGRALS AND ASYMPTOTICS

111

and 𝜑=𝛼

when

𝜑0 = 𝛼 − 𝜋,

(4.34)

which relate to the grazing reflections from the faces under the grazing incidence. This is a special case when the integrands in (4.16) to (4.19) cannot be expanded into the Taylor series because its terms become infinite. In Section 7.9 we develop a special version of PTD that is free of the grazing singularity. Figures 4.1 and 4.2 illustrate the behavior and beauty of the functions f (1) and g(1) . The Appendix to Chapter 4 (at the back of the book) contains MATLAB codes for numerical calculations of the exact integrals (4.16) to (4.19) and their asymptotics (4.20) and (4.21) for the edge waves radiated by the nonuniform/fringe sources j(1) s,h as well as for the directivity patterns of these waves, that is, for functions f (1) and g(1) . The appendix also includes additional illustrative figures for the fringe waves. Figures A4.1a and A4.1b show that the asymptotics (4.20) and (4.21) provide good approximations for the fringe waves at a distance r = 𝜆 from the edge when kr = 2𝜋 ≈ 6.28. Figures A4.3a and A4.3b are of special interest. They demonstrate the origination of the grazing singularity when 𝜑0 → 𝛼 − 𝜋 = 120◦ , as predicted by Equation (4.34). These figures also show that in contrast to the asymptotics, the exact integrals for the fringe waves are free from the grazing singularity.

Figure 4.1 Directivity patterns of edge waves radiated by nonuniform/fringe components of the surface sources. The function f (1) (g(1) ) corresponds to the case of an acoustically soft (hard) wedge; they also describe the Ez - (Hz )-component of the electromagnetic wave scattered at a perfectly conducting wedge. [From Ufimtsev (1957) by permission of Zhurnal Tekhnicheskoi Fiziki.]

112

WEDGE DIFFRACTION: RADIATION BY FRINGE COMPONENTS OF SURFACE SOURCES

0

0.2

0.4

0.6

0.8

1.0

φ = φ0 α = 360° f1 < 0 g1 < 0 Figure 4.2 Directivity patterns of edge waves radiated by nonuniform/fringe components of the surface sources. The function f (1) (g(1) ) corresponds to the case of an acoustically soft (hard) half-plane; they also describe the Ez - (Hz )-component of the electromagnetic wave scattered at the perfectly conducting half-plane. [From Ufimtsev (1957) by permission of Zhurnal Tekhnicheskoi Fiziki.]

4.2 INTEGRAL FORMS OF FUNCTIONS f (1) AND g(1) It is a well-known fact in the antenna and scattering theory that the directivity pattern of a far field can be considered to be a conformal Fourier transform of radiating and scattering sources distributed over antennas and scatterers. This is shown clearly in Equation (1.19). In this section we establish this type of relationship (1) between the directivity patterns f (1) and g(1) and their sources j(1) s and jh at the wedge. The geometry of the problem is shown in Figure 3.1, and the incident wave is given by Equation (3.1). The nonuniform components of the surface sources (1) j(1) s = u0 Js ,

j(1) = u0 Jh(1) h

(4.35)

radiate the field defined by Equation (1.10). We consider first radiation from the wedge face 𝜑 = 0: √

u(1) s

2 2 2 ∞ ∞ u eik (x−𝜉) +y +𝜁 =− 0 Js(1) (k𝜉, 𝜑0 ) d𝜉 d𝜁 , √ ∫−∞ (x − 𝜉)2 + y2 + 𝜁 2 4𝜋 ∫0

u(1) h

2 2 2 ∞ ∞ u 𝜕 eik (x−𝜉) +y +𝜁 =− 0 Jh(1) (k𝜉, 𝜑0 ) d𝜉 d𝜁 . √ ∫−∞ (x − 𝜉)2 + y2 + 𝜁 2 4𝜋 𝜕y ∫0

(4.36)

√

(4.37)

INTEGRAL FORMS OF FUNCTIONS f (1) AND g(1)

113

In view of Equation (3.7), ∞

√ Js(1) (k𝜉, 𝜑0 )H0(1) [k (x − 𝜉)2 + y2 ] d𝜁 ,

u(1) s = −u0

i 4 ∫0

= −u0 u(1) h

i 𝜕 4 𝜕y ∫0

∞

√ Jh(1) (k𝜉, 𝜑0 )H0(1) [k (x − 𝜉)2 + y2 ] d𝜁 .

(4.38)

(4.39)

According to Equations (1.11), (2.61) and (2.63), the functions Js(1) and Jh(1) decrease as (k𝜉)−3∕2 and (k𝜉)−1∕2 , respectively, with increasing distance 𝜉 from the edge. At a certain distance 𝜉 = 𝜉effective , these functions are sufficiently small and can 2 , the Hankel be approximated by zero for 𝜉 ≥ 𝜉eff . In the far zone, where r ≫ k𝜉eff function in Equations (4.38) and (4.39) can be replaced by its asymptotics (2.29). This leads to the following approximations: ei(kr+𝜋∕4) = −u u(1) √ 0 s 2 2𝜋kr ∫0

𝜉eff

Js(1) (k𝜉, 𝜑0 )e−ik𝜉 cos 𝜑 d𝜉,

ei(kr+𝜋∕4) u(1) = −u0 ik sin 𝜑 √ h 2 2𝜋kr ∫0

𝜉eff

Jh(1) (k𝜉, 𝜑0 )e−ik𝜉 cos 𝜑 d𝜉.

(4.40)

(4.41)

These expressions describe the field radiated from the face 𝜑 = 0. By replacement of 𝜑 by 𝛼 − 𝜑 and 𝜑0 by 𝛼 − 𝜑0 , we can find the field radiated from the face 𝜑 = 𝛼. The total field created by both faces must be equal to that of Equations (4.20) and (4.21). By equating them, we obtain the useful relationships

f

(1)

[ 𝜉eff 1 (𝜑, 𝜑0 , 𝛼) = − Js(1) (k𝜉, 𝜑0 ) e−ik𝜉 cos 𝜑 d𝜉 2 ∫0 𝜉eff

+ g(1) (𝜑, 𝜑0 , 𝛼) = −

∫0

] Js(1) (k𝜉, 𝛼 − 𝜑0 ) e−ik𝜉 cos(𝛼−𝜑) d𝜉 ,

(4.42)

[ 𝜉eff ik Jh(1) (k𝜉, 𝜑0 ) e−ik𝜉 cos 𝜑 d𝜉 sin 𝜑 ∫0 2

+ sin(𝛼 − 𝜑)

𝜉eff

∫0

] Jh(1) (k𝜉, 𝛼 − 𝜑0 ) e−ik𝜉 cos(𝛼−𝜑) d𝜉 .

(4.43)

These expressions show that the functions f (1) and g(1) can also be interpreted as the directivity patterns of elementary diffracted waves generated (in the plane normal to the edge) by sources distributed at the wedge along lines normal to the edge.

114

WEDGE DIFFRACTION: RADIATION BY FRINGE COMPONENTS OF SURFACE SOURCES

4.3 OBLIQUE INCIDENCE OF A PLANE WAVE AT A WEDGE

(t) For an oblique incidence, the relationship u(t) s = Ez exists for diffracted rays is not generated by the total surface currents, j(t) = j(0) + j(1) . However, u(0,1) s equal to Ez(0,1) due to the polarization coupling in the electromagnetic PO field. = Hz(0,1,t) exists between the acoustic and The complete equivalence u(0,1,t) h electromagnetic diffracted waves.

4.3.1

Acoustic Waves

Here we extend the results of Section 4.1 to the general case when the incident wave propagates under the oblique direction to the edge (Fig. 4.3). It is given by the equation ̃ cos 𝛽+z cos 𝛾) uinc = u0 eik(x cos 𝛼+y

(4.44)

with 0 < 𝛾 ≤ 𝜋. We use the “tilde hat” for the angle 𝛼̃ to avoid possible confusion with the external angle 𝛼 of the wedge. The boundary conditions on the wedge faces are shown in Equations (2.2) and (2.3). To satisfy these conditions, the diffracted field must have the same dependence on the coordinate z as the incident wave (4.44): ud = u(r, 𝜑) eikz cos 𝛾 .

(4.45)

The substitution of this function ud into the wave Equation (2.4) leads to an equation for the function u(r,𝜑), Δu(r,𝜑) + k12 u(r,𝜑) = 0

with k1 = k sin 𝛾,

y

0

x

γ

Figure 4.3 Oblique incidence of a plane wave at a wedge.

(4.46)

OBLIQUE INCIDENCE OF A PLANE WAVE AT A WEDGE

115

where the Laplacian operator Δ is defined by Equation (2.5). It is also expedient to represent the incident wave (4.40) in the form of Equation (3.1): uinc = u0 eikz cos 𝛾 e−ik1 (x cos 𝜑0 +y sin 𝜑0 ) ,

(4.47)

where sin 𝛾 cos 𝜑0 = − cos 𝛼, ̃

sin 𝛾 sin 𝜑0 = − cos 𝛽

(4.48)

and tan 𝜑0 =

cos 𝛽 cos 𝛼̃

with 0 ≤ 𝜑0 < 𝜋.

(4.49)

Thus, we reduced the three-dimensional diffraction problem for the oblique incidence to a two-dimensional problem for the normal incidence (𝛾 = 𝜋∕2) considered in Chapter 2. The solution for the oblique incidence can be found automatically using simple replacements in the solution for the normal incidence: namely, the quantity u0 should be replaced by u0 eikz cos 𝛾 , the wave number k by k1 = k sin 𝛾, and the angle ̃ 𝜑0 by 𝜑0 = arctan(cos 𝛽∕ cos 𝛼). This rule has been established here for the exact solution of the wedge diffraction problem and for its asymptotics. One can show that it is also valid for the PO part of the field. First, by substitution of the incident wave (4.47) into Equation (1.31), we find the PO surface sources: ikz cos 𝛾 ⋅ 2ik1 sin 𝜑0 e−ik1 x cos 𝜑0 , j(0) s = −u0 e

(4.50)

j(0) = u0 eikz cos 𝛾 ⋅ 2e−ik1 x cos 𝜑0 . h

(4.51)

Comparison with Equation (3.2) shows that the sources (4.50) and (4.51) satisfy the rule above for the transition from normal to oblique incidence. If the sources of the field satisfy this rule, one can expect that the field generated does also. To verify that, we substitute (4.50) and (4.51) into the original expressions (1.32) for the PO field: √

u(0) s

2 2 2 ∞ ik sin 𝜑0 ∞ −ik 𝜉 cos 𝜑 eik (x−𝜉) +y +(z−𝜁 ) 0 d𝜉 = u0 1 e 1 eik𝜁 cos 𝛾 √ d𝜁 , ∫−∞ ∫0 2𝜋 (x − 𝜉)2 + y2 + (z − 𝜁 )2

(4.52) u(0) h

1 𝜕 = −u0 2𝜋 𝜕y ∫0

∞

−ik1 𝜉 cos 𝜑0

e

∞

d𝜉

∫−∞

ik𝜁 cos 𝛾

e

eik

√ (x−𝜉)2 +y2 +(z−𝜁 )2

d𝜁 , √ (x − 𝜉)2 + y2 + (z − 𝜁 )2 (4.53)

116

WEDGE DIFFRACTION: RADIATION BY FRINGE COMPONENTS OF SURFACE SOURCES

or √

u(0) s

ikz cos 𝛾

= u0 e

2 2 2 ∞ ik1 sin 𝜑0 ∞ −ik 𝜉 cos 𝜑 eik (x−𝜉) +y +s 0 d𝜉 e 1 eiks cos 𝛾 √ ds, ∫−∞ ∫0 2𝜋 (x − 𝜉)2 + y2 + s2

(4.54) 1 𝜕 2𝜋 𝜕y ∫0

= −u0 eikz cos 𝛾 u(0) h

∞

e−ik1 𝜉 cos 𝜑0 d𝜉

∞

∫−∞

eiks cos 𝛾 √

eik

√

(x−𝜉)2 +y2 +s2

(x − 𝜉)2 + y2 + s2

ds. (4.55)

The integrals over the variable s still contain the wave number k for the normal incidence and need further investigation. Let us rewrite them as ∞

∫−∞

√

iks cos 𝛾

e

2

2

eik D +s ds √ D2 + s2

with D =

√

(x − 𝜉)2 + y2 .

(4.56)

Then we return to the integral (3.8) for the Hankel√function and make the following changes: 𝑤 = s, z = t, d = −ip, k = −iD, and q = p2 − t2 with D > 0 and Imq > 0. After these manipulations it follows from (3.8) that H0(1) (qD)

√

∞

2

∞

2

√

2

2

1 1 eip D +s eip D +s = e−ist √ ds = eist √ ds. i𝜋 ∫−∞ i𝜋 ∫−∞ D2 + s2 D2 + s2

By setting t = k cos 𝛾, p = k, and q = ∞

∫−∞

√ p2 − t2 = k sin 𝛾 = k1 here, we find that

√

iks cos 𝛾

e

(4.57)

2

2

eik D +s ds = i𝜋H0(1) (k1 D). √ 2 2 D +s

(4.58)

This relationship allows one to rewrite the PO fields (4.54) and (4.55) in the form k1 sin 𝜑0 ∞ −ik 𝜉 cos 𝜑 (1) √ 0H e 1 (k1 (x − 𝜉)2 + y2 ) d𝜉, 0 ∫0 2 ∞ √ i 𝜕 = −u0 eikz cos 𝛾 e−ik1 𝜉 cos 𝜑0 H0(1) (k1 (x − 𝜉)2 + y2 ) d𝜉. 2 𝜕y ∫0

ikz cos 𝛾 u(0) s = −u0 e

(4.59)

u(0) h

(4.60)

Comparison with expressions (3.10) and (3.11) finally confirms that the PO fields really satisfy the rule above for the transition to oblique incidence. Thus, it has been proved that this rule is applicable both to the exact solution and to the PO approximation. Hence, this rule is also applicable to their difference, which generated by the nonuniform component j(1) of the surface sources of is the field u(1) s,h s,h the diffracted field. By application of this rule to Equations (4.20) and (4.21), one can

OBLIQUE INCIDENCE OF A PLANE WAVE AT A WEDGE

117

easily find the high-frequency approximations for the far field u(1) generated under s,h oblique incidence: ei(k1 r+𝜋∕4) ikz cos 𝛾 (1) , u(1) e s = u0 f (𝜑, 𝜑0 , 𝛼) √ 2𝜋k1 r

(4.61)

ei(k1 r+𝜋∕4) ikz cos 𝛾 (1) u(1) = u g (𝜑, 𝜑 , 𝛼) , e √ 0 0 h 2𝜋k1 r

(4.62)

where the functions f (1) and g(1) are as defined in Section 4.1 and the angle 𝜑0 is determined by Equation (4.49). The waves (4.61) and (4.62) have the form of conic waves and can be interpreted in terms of diffracted rays. Indeed, their eikonal S = z cos 𝛾 + r sin 𝛾

(4.63)

describes the phase fronts in the form of conic surfaces where S = const. The gradient of the eikonal ∇S = ẑ cos 𝛾 + r̂ sin 𝛾

(4.64)

indicates the directions of the edge-diffracted rays. They are distributed over the cone surface shown in Figure 4.4. The axis of this cone is directed along the edge. All rays form the same angle 𝛾 with the edge as that of the incident ray. One should note that the diffraction cone was discovered theoretically by Rubinowicz (1924) for arbitrary curved edges. For curved edges, the axis of z eφ er eγ

γ Figure 4.4 Cone of diffracted rays.

118

WEDGE DIFFRACTION: RADIATION BY FRINGE COMPONENTS OF SURFACE SOURCES

the diffraction cone is directed along the tangent to the edge at the diffraction point. Rubinowicz made this discovery by asymptotic evaluation of the Kirchhoff diffraction integral. He also established that edge-diffracted rays satisfy Fermat’s principle. Later, this concept of edge-diffracted rays was included in the geometrical theory of diffraction (Keller, 1962). The ray interpretation of edge-diffracted waves (4.61) and (4.62) was also suggested in PTD (Ufimtsev, 1962b). Senior and Uslenghi (1972) proved the existence of these rays in experiments with the diffraction of laser radiation. 4.3.2

Electromagnetic Waves

For electromagnetic waves Ezinc = E0z eikz cos 𝛾 e−ik1 r cos(𝜑−𝜑0 ) ,

(4.65)

Hzinc = H0z eikz cos 𝛾 e−ik1 r cos(𝜑−𝜑0 ) ,

(4.66)

incident at a perfectly conducting wedge, the following approximations describe the edge-diffracted field (Ufimtsev, 1975, 2003, 2008b, 2009): { } eik1 r+i𝜋∕4 ikz cos 𝛾 , Ez(0) = E0z f (0) (𝜑, 𝜑0 ) − Z0 H0z [𝜀(𝜑0 ) − 𝜀(𝛼 − 𝜑0 )] cos 𝛾 × √ e 2𝜋k1 r (4.67) eik1 r+i𝜋∕4 ikz cos 𝛾 Hz(0) = H0z g(0) (𝜑, 𝜑0 ) √ , e 2𝜋k1 r

(4.68)

{ } eik1 r+i𝜋∕4 ikz cos 𝛾 e Ez(1) = E0z f (1) (𝜑, 𝜑0 , 𝛼) + Z0 H0z [𝜀(𝜑0 ) − 𝜀(𝛼 − 𝜑0 )] cos 𝛾 × √ , 2𝜋k1 r (4.69) eik1 r+i𝜋∕4 ikz cos 𝛾 Hz(1) = H0z g(1) (𝜑, 𝜑0 , 𝛼) √ . e 2𝜋k1 r Here Z0 = 1∕Y0 =

(4.70)

√ 𝜇0 ∕𝜀0 = 120𝜋 ohms is the impedance of free space, and { 𝜀(x) =

1 0

with with

0 < x < 𝜋, 𝜋 < x.

(4.71)

In Equations (4.67) and (4.69), terms with the factor Z0 H0z show polarization coupling. This effect is a consequence of the discontinuity of the PO current component, which is normal to the edge. Such a discontinuity happens at the edge when only

OBLIQUE INCIDENCE OF A PLANE WAVE AT A WEDGE

119

one of its faces is illuminated. It is worthwhile to clarify this phenomenon (Ufimtsev, 2008b). The PO current (0) inc ikz cos 𝛾 j(0) r = jx = 2𝜀(𝜑0 )H0z e

(4.72)

exists on the entire face 𝜑 = 0, that is, in the region 0 ≤ x ≤ ∞ (Fig. 4.3). Outside the face (y = 0, x < 0) this current equals zero. Thus, it undergoes discontinuity at the point x = 0. According to the continuity equation ∇ ⋅ ⃗j(0) = i𝜔𝜌(0) ,

(4.73)

2 𝜀(𝜑0 )H0z eikz cos 𝛾 𝛿(x) i𝜔

(4.74)

the electric charge filament 𝜌(0) =

appears directly on the edge (x = 0). This charge creates the electric field ⃗ = −∇Φ E

(4.75)

with the potential Φ=

1 𝜀(𝜑0 )Z0 H0z eikz cos 𝛾 H0(1) (k1 r). 2k

(4.76)

Away from the edge (k1 r ≫ 1) this field contains the component eik1 r+i𝜋∕4 Ez = −𝜀(𝜑0 )Z0 H0z eikz cos 𝛾 cos 𝛾 √ . 2𝜋k1 r

(4.77)

This term is exactly that present in Equation (4.67). The other term with 𝜀(𝛼 − 𝜑0 ) is produced by the edge discontinuity of the PO current: ikz cos 𝛾 −ik1 r cos(𝛼−𝜑0 ) j(0) e r = −2𝜀(𝛼 − 𝜑0 )H0z e

(4.78)

induced on the wedge face 𝜑 = 𝛼. Polarization coupling disappears when both wedge faces are illuminated. Indeed, in this case, 𝜀(𝜑0 ) = 𝜀(𝛼 − 𝜑0 ) = 1 and the coupling terms cancel each other. This is the mathematical explanation. The physics is also clear. In this case the current j(0) r flows around the edge freely, without discontinuity, and therefore the electric charge 𝜌(0) is not accumulated at the edge.

120

WEDGE DIFFRACTION: RADIATION BY FRINGE COMPONENTS OF SURFACE SOURCES

According to Equations (4.67) and (4.69), polarization coupling also disappears under the normal incidence (𝛾 = 𝜋∕2). In this case, according to Equation (4.76), Ez = −𝜕Φ∕𝜕z = 0 since cos 𝛾 = 0. The physics behind this mathematics is as follows. The physical quantities of the electric charges (4.74) are determined as 𝜌 = Re[𝜌(0) ] =

2 𝜀(𝜑0 )H0z sin(kz cos 𝛾 − 𝜔t)𝛿(x). 𝜔

(4.79)

It is seen that at each moment in time, charges of different magnitude exist at different points of the edge. Because of that, they create the total nonzero field component Ez . However, under the normal incidence (𝛾 = 𝜋∕2) the edge is filled in uniformly by the charges. Such a homogeneous charge filament can create an electric field only with radial component Er . Notice also that the PO edge charges (4.74) do not create a magnetic field at all. This point is well understood from a mathematical point of view, because according to the Maxwell equations, the potential electric field does not induce the magnetic field. The physics is also clear: These edge charges oscillate in time, but they do not move along the edge and do not produce an electric current that could generate a magnetic field. Finally, notice that the field generated by the total surface current ⃗j(t) = ⃗j(0) + ⃗j(1) is free of this polarization coupling, eik1 r+i𝜋∕4 ikz cos 𝛾 , Ez(t) = E0z f (𝜑, 𝜑0 , 𝛼) √ e 2𝜋k1 r

(4.80)

eik1 r+i𝜋∕4 ikz cos 𝛾 e Hz(t) = H0z g(𝜑, 𝜑0 , 𝛼) √ . 2𝜋k1 r

(4.81)

PROBLEMS The functions f (𝜑, 𝜑0 , 𝛼) and g(𝜑, 𝜑0 , 𝛼) as well as the functions f (0) (𝜑, 𝜑0 ) and g(0) (𝜑, 𝜑0 ) are singular at the geometrical optics boundaries of the incident and reflected rays. Verify that their differences, f (1) − f (0) and g(1) − g(0) , are finite there. Prove Equations (4.29) to (4.32). 4.1 Prove Equation (4.29). Proof Equation (4.29) determines the field in the direction 𝜑 = 𝜋 − 𝜑0 of the specular reflection from the face 𝜑 = 0 (Fig. 2.7). According to Equations (2.62) and (2.64), ( ) { } sin(𝜋∕n) f 1 1 ∓ . = g n cos(𝜋∕n) − cos[(𝜑 − 𝜑0 )∕n] cos(𝜋∕n) − cos[(𝜑 + 𝜑0 )∕n]

PROBLEMS

121

Also, in view of Equations (2.119) and (3.45), (3.46), ( (1∕n) sin(𝜋∕n) 𝜓 −𝜋 𝜓 + 𝜋) 1 cot , = − cot cos(𝜋∕n) − cos(𝜓∕n) 2n 2n 2n sin 𝜑0 𝜑 + 𝜑0 − 𝜋 1 𝜑 − 𝜑0 − 𝜋 1 = − cot + cot , cos 𝜑0 + cos 𝜑 2 2 2 2 𝜑 + 𝜑0 − 𝜋 1 𝜑 − 𝜑0 − 𝜋 sin 𝜑 1 = − cot − cot . cos 𝜑0 + cos 𝜑 2 2 2 2 When 𝜑 → 𝜋 − 𝜑0 , the quantity 𝜑 + 𝜑0 − 𝜋 tends to zero and 𝜑 − 𝜑0 − 𝜋 → −2𝜑0 . In this case, the related cotangent terms take the forms 𝜑 + 𝜑0 − 𝜋 1 1 cot = + O(𝜑 + 𝜑0 − 𝜋), 2n 2n 𝜑 + 𝜑0 − 𝜋 𝜑 + 𝜑0 + 𝜋 1 𝜋 1 cot → cot , 2n 2n 2n n 𝜑 + 𝜑0 − 𝜋 1 1 cot = + O(𝜑 + 𝜑0 − 𝜋), 2 2 𝜑 + 𝜑0 − 𝜋 𝜑 − 𝜑0 − 𝜋 1 1 cot → − cot 𝜑0 . 2 2 2 Utilizing these estimations, one can easily verify that singularities of the functions f and g are really canceled by singularities of the functions −f (0) and −g(0) . As a result, {

4.2

f (1) g(1)

} =

(1∕n) sin(𝜋∕n) 1 1 𝜋 + cot 𝜑0 ± cot . cos(𝜋∕n) − cos[(𝜋 − 2𝜑0 )∕n] 2 2n n

Prove Equation (4.31). Proof Equation (4.31) determines the field in the direction of the shadow boundary (𝜑 = 𝜋 + 𝜑0 , Fig. 2.7). In this case, 𝜑 → 𝜋 + 𝜑0 , 𝜑 − 𝜑0 − 𝜋 → 0, and 𝜑 + 𝜑0 − 𝜋 → 2𝜑0 . With the replacement of 𝜑0 by −𝜑0 in the estimations shown in Problem 4.1, one obtains 𝜑 − 𝜑0 − 𝜋 1 1 cot = + O(𝜑 − 𝜑0 − 𝜋), 2n 2n 𝜑 − 𝜑0 − 𝜋 𝜑 − 𝜑0 + 𝜋 1 1 𝜋 cot → cot , 2n 2n 2n n

122

WEDGE DIFFRACTION: RADIATION BY FRINGE COMPONENTS OF SURFACE SOURCES

𝜑 − 𝜑0 − 𝜋 1 1 cot = + O(𝜑 − 𝜑0 − 𝜋), 2 2 𝜑 − 𝜑0 − 𝜋 𝜑 + 𝜑0 − 𝜋 1 1 cot → cot 𝜑0 . 2 2 2 As a result, {

f (1) g(1)

} =∓

(1∕n) sin(𝜋∕n) 1 1 𝜋 ± cot 𝜑0 − cot . cos(𝜋∕n) − cos[(𝜋 + 2𝜑0 )∕n] 2 2n n

In the same way, one can prove Equations (4.30) and (4.32).

5 First-Order Diffraction at Strips and Polygonal Cylinders The relationships us = Ez and uh = Hz exist between the acoustic and electromagnetic fields for these problems.

In Chapters 3 and 4 we have built a foundation for the solution of two-dimensional diffraction problems. General asymptotic expressions have been derived for first-order edge-diffracted waves generated by both uniform and nonuniform components of the surface sources. In this chapter this general theory is applied to high-frequency diffraction at strips and cylinders with triangular cross-sections. These specific diffraction problems have been studied comprehensively and reported in the literature. In particular, the uniform asymptotic expressions (with arbitrary high asymptotic precision) for the directivity pattern and for the surface field at the strips have been derived by Ufimtsev (1969, 1970, 2003, 2009). In these publications one can also find many other references related to the strip diffraction problem. Among them we should note the first and classical solution by Schwarzschild (1902). High-frequency diffraction at polygonal cylinders was investigated by Morse (1964) and Borovikov (1966). We consider these problems again here, to demonstrate the first applications of PTD.

Fundamentals of the Physical Theory of Diffraction, Second Edition. Pyotr Ya. Ufimtsev. © 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

123

124

FIRST-ORDER DIFFRACTION AT STRIPS AND POLYGONAL CYLINDERS

(r, ϕ ) y y =α

1

φ0

x

y = –α 2 Figure 5.1 Cross-section of the strip by the plane z = const.

5.1 DIFFRACTION AT A STRIP The geometry of the problem is shown in Figure 5.1. Soft (1.5) or hard (1.6) boundary conditions are imposed at the strip. The incident wave is given by uinc = u0 eik(x cos 𝜙0 +y sin 𝜙0 )

with

− 𝜋∕2 < 𝜙0 < 𝜋∕2.

(5.1)

The diffracted field is investigated around the strip in the directions −𝜋∕2 < 𝜙 < 3𝜋∕2. For a description of edge waves, we utilize the local coordinates r1 , 𝜑1 , 𝜑01 and r2 , 𝜑2 , 𝜑02 , where 𝜑1 and 𝜑2 are measured from the illuminated side of the strip (Fig. 5.2). 5.1.1

Physical Optics Part of a Scattered Field

The physical optics (PO) part of a scattered field generated by the uniform components (1.31) of the surface sources is determined by the integrals (1.32). However, this

ϕ0

ϕ

r1

φ1

1

φ01 r2

φ2 ϕ0

φ02 2

Figure 5.2 Local coordinates for edge waves.

DIFFRACTION AT A STRIP

125

integration process can be avoided. It turns out that the far field can be calculated immediately as the sum of two edge waves described in general form by Equations (3.57) and (3.58): ei(kr1 +𝜋∕4) ei(kr2 +𝜋∕4) (0) + u0 (2)f (0) (𝜑2 , 𝜑02 ) √ , u(0) s = u0 (1)f (𝜑1 , 𝜑01 ) √ 2𝜋kr1 2𝜋kr2

(5.2a)

ei(kr1 +𝜋∕4) ei(kr2 +𝜋∕4) = u0 (1)g(0) (𝜑1 , 𝜑01 ) √ + u0 (2)g(0) (𝜑2 , 𝜑02 ) √ u(0) h 2𝜋kr1 2𝜋kr2

(5.2b)

with u0 (1) = u0 eika sin 𝜙0 ,

u0 (2) = u0 e−ika sin 𝜙0 .

(5.3)

For the far zone (r ≫ ka2 ), these expressions can be simplified: [ (0) ] ei(kr + 𝜋∕4) ika(sin 𝜙0 −sin 𝜙) , + f (0) (2) e−ika(sin 𝜙0 −sin 𝜙) √ u(0) s = u0 f (1)e 2𝜋kr

(5.4)

[ ] ei(kr + 𝜋∕4) , (5.5) = u0 g(0) (1)eika(sin 𝜙0 −sin 𝜙) + g(0) (2) e−ika(sin 𝜙0 −sin 𝜙) √ u(0) h 2𝜋kr where f (0) (1) ≡ f (0) (𝜑1 , 𝜑01 ),

f (0) (2) ≡ f (0) (𝜑2 , 𝜑02 ),

(5.6)

g (1) ≡ g (𝜑1 , 𝜑01 ),

g (2) ≡ g (𝜑2 , 𝜑02 ).

(5.7)

(0)

(0)

(0)

(0)

In accordance with Equations (3.59), functions f (0) and g(0) are defined in terms of the basic coordinates 𝜙 and 𝜙0 as f (0) (1) = −f (0) (2) =

cos 𝜙0 , sin 𝜙0 − sin 𝜙

(5.8)

g(0) (1) = −g(0) (2) =

cos 𝜙 , sin 𝜙0 − sin 𝜙

(5.9)

with −𝜋∕2 < 𝜙 < 3𝜋∕2 and −𝜋∕2 < 𝜙0 < 𝜋∕2. The field expressions (5.4) and (5.5) possess a wonderful property. Although all functions (5.8) and (5.9) are singular at the directions 𝜙 = 𝜙0 and 𝜙 = 𝜋 − 𝜙0 , their combinations in Equations (5.4) and (5.5) are always finite, due to the relationships

126

FIRST-ORDER DIFFRACTION AT STRIPS AND POLYGONAL CYLINDERS

f (0) (1) = −f (0) (2) and g(0) (1) = −g(0) (2). This property of expressions (5.4) and (5.5) becomes obvious when they are written in the explicit form ei(kr+𝜋∕4) (0) , u(0) s = u0 Φs (𝜙, 𝜙0 ) √ 2𝜋kr

(5.10)

ei(kr+𝜋∕4) (0) u(0) , = u Φ (𝜙, 𝜙 ) √ 0 0 h h 2𝜋kr

(5.11)

where

Φ(0) s (𝜙, 𝜙0 ) = i2 cos 𝜙0 Φ(0) (𝜙, 𝜙0 ) = i2 cos 𝜙 h

sin[ka(sin 𝜙 − sin 𝜙0 )] , sin 𝜙 − sin 𝜙0

sin[ka(sin 𝜙 − sin 𝜙0 )] . sin 𝜙 − sin 𝜙0

(5.12) (5.13)

Now it is clear that (0) Φ(0) s (𝜙0 , 𝜙0 ) = Φh (𝜙0 , 𝜙0 ) = i2ka cos 𝜙0

(5.14)

for the forward direction 𝜙 = 𝜙0 , and (0) Φ(0) s (𝜋 − 𝜙0 , 𝜙0 ) = −Φh (𝜋 − 𝜙0 , 𝜙0 ) = i2ka cos 𝜙0

(5.15)

for the specular direction 𝜙 = 𝜋 − 𝜙0 . In accordance with the two-dimensional form of the optical theorem (Ufimtsev, 2003, 2009), the total scattering cross-section is defined as 𝜎 tot =

2 ImΦ(𝜙0 , 𝜙0 ), k

(5.16)

which, following Equation (5.14), equals 𝜎s(0) tot = 𝜎h(0)tot = 2A

(5.17)

where A = 2a cos 𝜙0 is the “width” of the incident wave part intercepted by the strip (Fig. 5.3). Equation (5.17) can be interpreted as the normalized total scattered power per unit length of the strip in the z-axis direction.

DIFFRACTION AT A STRIP

127

A

ϕ0

Figure 5.3 Cross-section A of the incident wave intercepted by the strip.

It is expedient to introduce the two-dimensional bistatic scattering cross-section 𝜎 by a relationship similar to (1.24): Psc av =

𝜎Pinc av 2𝜋r

,

(5.18)

where 1 2 | |2 Pinc av = 2 k Z |u0 | ,

1 2 | sc | 2 Psc av = 2 k Z |u | .

(5.19)

Therefore, in view of Equations (5.10) and (5.11), the bistatic scattering crosssection is defined by

(0) 𝜎s,h =

1 | (0) |2 |Φs,h (𝜙, 𝜙0 )| . | | k

(5.20)

This is a general definition of the bistatic scattering cross-section 𝜎 for any twodimensional scattered fields represented in the form of Equations (5.10) and (5.11). For backscattering when 𝜙 = 𝜋 + 𝜙0 , the directivity patterns are given by the simple expressions (0) Φ(0) s = −Φh = i cot 𝜙0 sin(2ka sin 𝜙0 ),

(5.21)

which are consistent with the general relationship (1.37). Finally, one should notice another peculiarity of the directivity patterns (5.12) and (5.13) for the field generated by the uniform components of the surface sources. They have exact zeros in the directions where ka(sin 𝜙0 − sin 𝜙) = ±n𝜋

with n = 1, 2, 3, … .

(5.22)

128

FIRST-ORDER DIFFRACTION AT STRIPS AND POLYGONAL CYLINDERS

This property is the consequence of the relationships f (0) (1) = −f (0) (2) and g(0) (1) = −g(0) (2). 5.1.2

Total Scattered Field

The nonuniform/fringe components of scattering sources concentrate near the edges of the strip and generate the two edge waves described by Equations (4.20) and (4.21). Their sum equals ei(kr1 +𝜋∕4) ei(kr2 +𝜋∕4) (1) + u0 (2)f (1) (𝜑2 , 𝜑02 , 2𝜋) √ , u(1) s = u0 (1)f (𝜑1 , 𝜑01 , 2𝜋) √ 2𝜋kr1 2𝜋kr2

(5.23)

ei(kr1 +𝜋∕4) ei(kr2 +𝜋∕4) (1) (1) u(1) = u (1)g (𝜑 , 𝜑 , 2𝜋) + u (2)g (𝜑 , 𝜑 , 2𝜋) . √ √ 0 1 01 0 2 02 h 2𝜋kr1 2𝜋kr2

(5.24)

where the quantities u0 (1) and u0 (2) are defined by Equation (5.3). Due to the relationships (4.22) and (4.23), these expressions can be written in the form (0) u(1) s = us − us ,

u(1) = uh − u(0) . h h

(5.25)

Therefore, the total first-order scattered field equals (1) us = u(0) s + us

ei(kr1 +𝜋∕4) ei(kr2 +𝜋∕4) = u0 (1)f (𝜙1 , 𝜙01 , 2𝜋) √ + u0 (2)f (𝜙2 , 𝜙02 , 2𝜋) √ , (5.26) 2𝜋kr1 2𝜋kr2 + u(1) uh = u(0) h h ei(kr1 +𝜋∕4) ei(kr2 +𝜋∕4) = u0 (1)g(𝜙1 , 𝜙01 , 2𝜋) √ + u0 (2)g(𝜙2 , 𝜙02 , 2𝜋) √ , (5.27) 2𝜋kr1 2𝜋kr2 with functions f and g defined in Equations (2.62) and (2.64). For the far zone these field expressions are simplified to

[ ] ei(kr+𝜋∕4) , us = u0 f (1)eika(sin 𝜙0 −sin 𝜙) + f (2) e−ika(sin 𝜙0 −sin 𝜙) √ 2𝜋kr

(5.28)

[ ] ei(kr+𝜋∕4) uh = u0 g(1)eika(sin 𝜙0 −sin 𝜙) + g(2) e−ika(sin 𝜙0 −sin 𝜙) √ , 2𝜋kr

(5.29)

DIFFRACTION AT A STRIP

129

where f (1) ≡ f (𝜑1 , 𝜑01 , 2𝜋),

f (2) ≡ f (𝜑2 , 𝜑02 , 2𝜋),

g(1) ≡ g(𝜑1 , 𝜑01 , 2𝜋),

g(2) ≡ g(𝜑2 , 𝜑02 , 2𝜋).

(5.30)

In terms of basic coordinates 𝜙 and 𝜙0 , these functions are determined by {

f (1) g(1)

}

1 = 2

( −

1 1 ∓ sin[(𝜙 − 𝜙0 )∕2] cos[(𝜙 + 𝜙0 )∕2]

) with − 𝜋∕2 ≤ 𝜙 ≤ 3𝜋∕2, (5.31)

{

f (2) g(2)

} =

1 2

(

1 1 ∓ sin[(𝜙 − 𝜙0 )∕2] cos[(𝜙 + 𝜙0 )∕2]

) with − 𝜋∕2 ≤ 𝜙 ≤ 𝜋∕2, (5.32)

but {

f (2) g(2)

} =

1 2

( −

1 1 ± sin[(𝜙 − 𝜙0 )∕2] cos[(𝜙 + 𝜙0 )∕2]

) with 𝜋∕2 ≤ 𝜙 ≤ 3𝜋∕2. (5.33)

As can be seen, the functions g(2) and 𝜕f (2)∕𝜕𝜙 are discontinuous in the direction 𝜙 = 𝜋∕2. This is a consequence of the fact that they relate to the field generated by the scattering sources distributed over the entire half-plane −a < y < ∞. The functions g(1) and 𝜕f (1)∕𝜕𝜙 are also discontinuous. They have different values in the directions 𝜙 = −𝜋∕2 and 𝜙 = 3𝜋∕2 related to the different sides of the half-plane −∞ < y < a containing the sources of the field. This discontinuity in the field of the first-order edge waves (5.28) and (5.29) can be eliminated in two ways. First, in calculation of the field generated by the nonuniform , one should restrict the integration region by the actual surface of component j(1) s,h the strip. In other words, one should truncate (outside the strip) the component j(1) s,h related to the semi-infinite half-plane. This approach is presented in Section 5.1.4 [see also Michaeli (1987) and Johansen (1996)]. Another remedy is calculation of the multiple edge diffraction investigated in Chapters 9 and 10 and also in works of Ufimtsev (1969, 1970, 2009). In view of Equations (5.31) to (5.33), the field expressions (5.28) and (5.29) can be written in the form ei(kr+𝜋∕4) , us,h = u0 Φs,h (𝜙, 𝜙0 ) √ 2𝜋kr

(5.34)

130

FIRST-ORDER DIFFRACTION AT STRIPS AND POLYGONAL CYLINDERS

where Φs (𝜙, 𝜙0 ) = −

cos[ka(sin 𝜙0 − sin 𝜙)] sin[ka(sin 𝜙0 − sin 𝜙)] +i cos[(𝜙0 + 𝜙)∕2] sin[(𝜙0 − 𝜙)∕2] with

Φs (𝜙, 𝜙0 ) =

− 𝜋∕2 ≤ 𝜙 ≤ 𝜋∕2,

(5.35)

cos[ka(sin 𝜙0 − sin 𝜙)] sin[ka(sin 𝜙0 − sin 𝜙)] −i sin[(𝜙0 − 𝜙)∕2] cos[(𝜙0 + 𝜙)∕2] with

𝜋∕2 ≤ 𝜙 ≤ 3𝜋∕2,

(5.36)

and Φh (𝜙, 𝜙0 ) =

cos[ka(sin 𝜙0 − sin 𝜙)] sin[ka(sin 𝜙0 − sin 𝜙)] +i cos[(𝜙0 + 𝜙)∕2] sin[(𝜙0 − 𝜙)∕2] with

Φh (𝜙, 𝜙0 ) =

− 𝜋∕2 ≤ 𝜙 ≤ 𝜋∕2,

(5.37)

cos[ka(sin 𝜙0 − sin 𝜙)] sin[ka(sin 𝜙0 − sin 𝜙)] +i sin[(𝜙0 − 𝜙)∕2] cos[(𝜙0 + 𝜙)∕2] with 𝜋∕2 ≤ 𝜙 ≤ 3𝜋∕2.

(5.38)

(𝜙, 𝜑0 )] do not have As can be seen, these functions [in contrast to the functions Φ(0) s,h exact zeros in the directions determined by Equation (5.22). Note another property of Equations (5.35) to (5.38). According to them, ||Φs (𝜙, 𝜙0 || = ||Φh (𝜙, 𝜙0 )||. That is why in the first-order PTD approximation, the bistatic scattering cross-section is the same for the soft and hard strips and does not depend on polarization of the incident electromagnetic wave, as is confirmed in Figures 5.5 and 5.6. In the following we provide the explicit expressions of functions Φs,h for certain specific directions of observation. For the forward direction 𝜙 = 𝜙0 , 1 Φs (𝜙0 , 𝜙0 ) = i2ka cos 𝜙0 − , (5.39) cos 𝜙0 1 , (5.40) Φh (𝜙0 , 𝜙0 ) = i2ka cos 𝜙0 + cos 𝜙0 and for the specular direction 𝜙 = 𝜋 − 𝜙0 , 1 , cos 𝜙0 1 . Φh (𝜋 − 𝜙0 , 𝜙0 ) = −i2ka cos 𝜙0 − cos 𝜙0 Φs (𝜋 − 𝜙0 , 𝜙0 ) = i2ka cos 𝜙0 −

(5.41) (5.42)

In addition, for the backscattering direction 𝜙 = 𝜋 + 𝜙0 , sin(2ka sin 𝜙0 ) , sin 𝜙0 sin(2ka sin 𝜙0 ) . Φh (𝜋 + 𝜙0 , 𝜙0 ) = − cos(2ka sin 𝜙0 ) − i sin 𝜙0 Φs (𝜋 + 𝜙0 , 𝜙0 ) = − cos(2ka sin 𝜙0 ) + i

(5.43) (5.44)

DIFFRACTION AT A STRIP

131

In conformity with Equations (4.33) and (4.34), functions (5.39) to (5.42) are singular under the grazing incidence (𝜙0 = ±𝜋∕2). A comparison with Equation (5.14) shows that the first term in Equations (5.39) and (5.40) is caused by the uniform component and the second term by the nonuniform component of the surface sources. In view of Equation (5.16), this observation gives the impression that the field generated by the nonuniform component does not contribute to the total scattered power. An interesting question arises: How does it happen that the nonuniform component j(1) generates the field (5.23) and (5.24) but does not contribute to the total scattered power? The answer is the following. The field (5.23) and (5.24) does contribute to the total scattered field (5.16), but through high-order edge waves generated due to multiple diffraction of the primary edge waves (5.34). An additional comment is necessary regarding the field Equations (5.28) and (5.29) and functions f and g involved in these equations. The functions f and g are singular in the forward (𝜙 = 𝜙0 ) and specular (𝜙 = 𝜋 − 𝜙0 ) directions related to the geometrical optics boundaries of the incident and reflected rays. However, these singularities cancel each other in the field Equations (5.28) and (5.29) and provide finite values in these special directions, as shown in Equations (5.39) to (5.42). This cancellation is due to the fact that the incident wave is a plane wave. Because of this, the geometrical optics boundaries related to the different edges of the strip are parallel to each other (Fig. 5.3) and merge in the far zone from the strip. This case represents an exception compared to a more general situation when the source of the incident wave can be at a finite distance from the scattering object. For example, in the case of the cylindrical incident wave (Fig. 5.4), the shadow boundaries caused by the strip edges are not parallel, and therefore the related singularities of functions f and g are separated in space and cannot cancel each other. In such a case, the traditional PTD procedure consists of calculation of the fields generated separately by the uniform and nonuniform components of the scattering sources with the subsequent summation of these fields. Each of these fields is finite, and hence their sum is finite as well.

Figure 5.4 Shadow boundaries caused by the strip edges in the field of the incident cylindrical wave. At these boundaries, the functions f and g are singular: f = ∞, g = ∞.

132

FIRST-ORDER DIFFRACTION AT STRIPS AND POLYGONAL CYLINDERS

5.1.3

Numerical Analysis of a Scattered Field

The relationships us = Ez and uh = Hz exist between the acoustic and electromagnetic fields for this problem.

In this section we present the numerical analysis of a scattered field by utilizing different approaches. Note that the geometry of the problem is shown in Figure 5.1. The soft (1.5) or hard (1.6) boundary conditions are imposed on the strip. The incident wave is the plane wave (5.1). The relationships us = Ez and uh = Hz show the equivalence between the acoustic and electromagnetic fields studied in this section. Equation (5.20) determines the bistatic scattering cross-section. The quantity calculated is the normalized scattering cross-section | Φs,h (𝜙, 𝜙0 ) |2 | | = where l = 2a, (5.45) | | | | kl kl2 | | which is plotted on the decibel scale. For parameters ka and 𝜙0 we take the values ka = 3𝜋 and 𝜙0 = 45◦ , that is, 𝜑0 = 𝜋∕4 rad. In this case, the strip width l = 2a equals 3𝜆 and application of the asymptotic theory is justified. The scattering cross-section is investigated in the directions 𝜋∕2 ≤ 𝜙 ≤ 3𝜋∕2 around the strip. Note that the function Φs is symmetrical with respect to the strip plane [Φs (𝜋 − 𝜙, 𝜙0 ) = Φs (𝜙, 𝜙0 )], and the function Φh is antisymmetrical [Φh (𝜋 − 𝜙, 𝜙0 ) = −Φh (𝜙, 𝜙0 )]. That is why it is sufficient to calculate these functions only in the interval 𝜋∕2 ≤ 𝜙 ≤ 3𝜋∕2. Calculations are performed using the following approaches: 𝜎s,h

r The PO approximation with the functions Φ(0) given in Section 5.1.1. s,h r The first-order PTD approximation with the functions Φ presented in Section s,h 5.1.2.

r The first-order TED approximation (Ufimtsev, 1969, 1970, 2003, 2009) with the functions ̃ 1 (𝛼, 𝛼0 ) + Ψ ̃ 1 (−𝛼, −𝛼0 ), Φs (𝜙, 𝜙0 ) = Ψ ̃ ̃ 1 (−𝛼, −𝛼0 ), Φh (𝜙, 𝜙0 ) = Φ1 (𝛼, 𝛼0 ) + Φ where ̃ 1 (𝛼, 𝛼0 ) = Ψ

√ √ 1 − 𝛼 1 − 𝛼0

√ i𝜒(𝛼+𝛼0 )

(5.46) (5.47)

√ 1 − 𝛼 1 + 𝛼0

e + 𝛼 + 𝛼0 𝛼 + 𝛼0 [ ] 1 + 𝛼0 1−𝛼 × 𝜓(q, 𝛼) − 𝜓(q, −𝛼0 ) eiq+i𝜒(𝛼−𝛼0 ) 2 2 √ 1 − 𝛼 i⋅2q [√ + e 1 − 𝛼0 𝜓(q, 𝛼0 )ei𝜒𝛼0 2Ds √ ] − 1 + 𝛼0 𝜓(q, −𝛼0 )𝜓(q, 1) eiq−i𝜒𝛼0 𝜓(q, 𝛼)ei𝜒𝛼 ,

(5.48)

133

DIFFRACTION AT A STRIP

√ ̃ 1 (𝛼, 𝛼0 ) = − Φ

√ 1 + 𝛼 1 + 𝛼0 [

𝛼 + 𝛼0

√

× (1 + 𝛼0 )

ei𝜒(𝛼+𝛼0 ) +

1 𝛼 + 𝛼0

1+𝛼 𝜑(q, −𝛼0 ) − (1 − 𝛼) 2

√

1 − 𝛼0 𝜑(q, 𝛼) 2

1 i2q [ 𝜑(q, 𝛼0 )ei𝜒𝛼0 e Dh ] − 𝜑(q, −𝛼0 )𝜑(q, 1) eiq−i𝜒𝛼0 𝜑(q, 𝛼)ei𝜒𝛼 .

]

× eiq+i𝜒(𝛼−𝛼0 ) −

(5.49)

In these expressions, 𝛼 = sin 𝜙,

𝛼0 = − sin 𝜙0 ,

𝜒 = ka,

q = 2𝜒 = 2ka,

(5.50)

∞

2 2 eit dt, 𝜑(q, 𝛼) = √ e−i𝜋∕4 e−iq(1+𝛼) √ ∫ q(1+𝛼) 𝜋 √ 𝜕𝜑(q, 𝛼) i 1+𝛼 1 ei𝜋∕4 , 𝜓(q, 𝛼) = √ = 𝜑(q, 𝛼) − √ 𝜕q 2 2𝜋q 2(1 + 𝛼)

Ds = 1 − 𝜓 2 (q, 1)ei2q , Dh = 1 − 𝜑2 (q, 1)ei2q .

(5.51)

(5.52) (5.53)

The absolute error of the TED approximations (5.46) and (5.47) is equal to ̃ s,h (𝛼, 𝛼0 ) + Q ̃ s,h (−𝛼, −𝛼0 ), Qs,h (𝛼, 𝛼0 ) = Q

(5.54)

where ̃ s (𝛼, 𝛼0 ) = Q ̃ h (𝛼, 𝛼0 ) = Q

O(q−1∕2 ) , [1 + q(1 + 𝛼)][1 + q(1 − 𝛼0 )] √ √ √ 1 + 𝛼 1 + 𝛼0 O( q) [1 + q(1 + 𝛼)][1 + q(1 + 𝛼0 )]

(5.55)

(5.56)

under the conditions q(1 ± 𝛼) ≫ 1 and q(1 ± 𝛼0 ) ≫ 1. Here the symbol O(qm ) is ̃ used to show the asymptotic behavior of quantities Q(𝛼, 𝛼0 , q) when q → ∞. The definition of this symbol has been given above in conjunction with Equation (2.104). Also, √ lim

1 − 𝛼2

𝜕 Q (𝛼, 𝛼0 ) = 0 𝜕𝛼 s

with 𝛼 → ±1

(5.57)

and Qh (𝛼, ±1) = Qh (±1, 𝛼0 ) = 0.

(5.58)

134

FIRST-ORDER DIFFRACTION AT STRIPS AND POLYGONAL CYLINDERS

Figure 5.5 Scattering at an acoustically soft strip (Ez -polarization of the electromagnetic field).

Equations (5.57) and (5.58) are consequences of the fact that the scattered field (5.46) and (5.47) and its normal derivatives at the plane x = 0 are continuous outside the strip. In other words, according to approximations (5.46) and (5.47), no scattering sources exist outside the strip surface. We have calculated functions (5.46) and (5.47) with the well-established estimations (5.55) and (5.56) to demonstrate the accuracy of the PO and PTD approximations. The results are plotted in Figures 5.5 and 5.6 on the decibel scale as 10 log(𝜎s,h ∕kl2 ). These figures show that the accuracy of the first-order PTD approximation is higher for the soft boundary condition. Only a small discrepancy with the exact TED curve is observed in the vicinity of the direction 𝜙 = 90◦ , and the PTD and TED curves actually merge in the region 100◦ < 𝜙 ≤ 270◦ . The reason for the better approximation provided by the PTD for the soft boundary condition is the faster attenuation of the primary edge waves of the scattering sources j(1) s compared to the similar waves of j(1) . In this case, the amplitude of these waves (at the location of h −3∕2 for the soft strip and on the the opposite edge of the strip) is on the order of (2ka) order of (2ka)−1∕2 for the hard strip. It is also seen in Figures 5.5 and 5.6 that PO cannot provide a reasonable approximation for the scattered field in the vicinity of the minima of the directivity patterns, where PO predicts wrong pure zeros. The incorrect values of the PTD approximation for the function 𝜎h in the vicinity of the directions 𝜙 = 90◦ and 𝜙 = 270◦ are caused by the fictitious scattering sources j(1) h

DIFFRACTION AT A STRIP

135

Figure 5.6 Scattering at an acoustically hard strip (Hz -polarization of the electromagnetic field).

distributed outside the strip surface, as discussed in Section 5.1.2. This shortcoming of the first-order PTD is eliminated in the next section.

5.1.4

First-Order PTD with Truncated Scattering Sources jh(1)

The relationship uh = Hz exists between the acoustic and electromagnetic fields for this problem.

The geometry of the problem is shown in Figure 5.1 and the incident wave is given by Equation (5.1). To calculate the scattered field uh , we apply Equation (1.10) and utilize the following observations.

r The symbol r was used in (1.10) for the distance between the integration and √

observation points. Now we replace it by 𝜌√= x2 + (y − 𝜂)2 + 𝜁 2 and retain the symbol r for the polar coordinate r = x2 + y2 of the observation point, assuming that x = r cos 𝜙 and y = r sin 𝜙. r The surface of integration in Equation (1.10) covers both faces of the strip: S = S− + S+ (Fig. 5.7).

136

FIRST-ORDER DIFFRACTION AT STRIPS AND POLYGONAL CYLINDERS

y

nˆ–

nˆ+ S– S+

x

uinc Figure 5.7 The integration surface S = S− + S+ in Equation (1.10) applied to the strip problem.

r The operator 𝜕∕𝜕n in Equation (1.10) acts on coordinates of the integration/ source point. According to Equations (1.14) and (3.5), it can be replaced by the operators acting on coordinates of the observation point. Namely, 𝜕∕𝜕n− = 𝜕∕𝜕x for the illuminated face S− of the strip and 𝜕∕𝜕n+ = −𝜕∕𝜕x for the shadowed face S+ , where 𝜕∕𝜕x is the differentiation with respect to the observation point (outside the strip). r The integral in (1.10) over the variable 𝜁 applied to the strip problem can [ √ ] be expressed through the Hankel function H0(1) k x2 + (y − 𝜂)2 as shown in Equation (3.7). r We calculate the scattered field in the far zone where r ≫ ka2 . For this zone the Hankel function can be approximated by its asymptotic expression H0(1)

[ √ ] √ 2 ik(r−𝜂 sin 𝜙)−i𝜋∕4 2 2 e k x + (y − 𝜂) ∼ . 𝜋kr

(5.59)

r We denote the scattering sources as j(0) = u0 J (0) and j(1) = u0 J (1) . h h h h r Then the scattered field in the far zone can be represented in the following form: a

ei(𝜋∕4) 𝜕 eikr = u Jh(0) (𝜂, x = −0)e−ik𝜂 sin 𝜙 d𝜂, u(0) √ √ 0 h 2 2𝜋k 𝜕x r ∫−a [ a ei(𝜋∕4) 𝜕 eikr (1) Jh(1) (𝜂, x = −0)e−ik𝜂 sin 𝜙 d𝜂 uh = u0 √ √ ∫ 𝜕x r −a 2 2𝜋k a

−

∫−a

Jh(1) (𝜂, x = +0)e−ik𝜂 sin 𝜙 d𝜂

(5.60)

] (5.61)

exist only on the illuminated Remember that the uniform scattering sources j(0) h face of the strip where x = −0.

DIFFRACTION AT A STRIP

137

r In the far-zone approximation one can set eikr 𝜕 eikr √ = ik cos 𝜙 √ . 𝜕x r r

(5.62)

r According to definition (1.31) of the uniform scattering source, one has Jh(0) (𝜂, x = −0) = 2eik𝜂 sin 𝜙0 .

r The nonuniform scattering source j(1) is found by utilizing the exact solution for h

the half-plane diffraction problem presented in Section 2.5. According to this solution, u(1) ∕u0 = 𝑣(kr, 𝜑 − 𝜑0 ) + 𝑣(kr, 𝜑 + 𝜑0 ) where 𝑣(kr, 𝜓) = 𝑣hp (kr, 𝜓) h is given by Equation (2.86). Due to this solution: b The function j(1) is antisymmetrical, j(1) (𝜂, x = +0) = −j(1) (𝜂, x = −0) and, h

h

consequently, Jh(1) (𝜂, x = +0) = −Jh(1) (𝜂, x = −0).

h

b The function J (1) (𝜂, x = −0) is determined as h

] { [ 𝜋 Jh(1) (𝜂, x = −0) = 2 𝑣 k(a − 𝜂), − 𝜙0 eika sin 𝜙0 2 ] } [ 𝜋 + 𝑣 k(a + 𝜂), + 𝜙0 e−ika sin 𝜙0 . 2

(5.63)

r In view of the observations above, the functions (5.60) and (5.61) can be approximated by u(0) = u0 i2 cos 𝜙 h

sin[ka(sin 𝜙0 − sin 𝜙)] ei(kr+𝜋∕4) , √ sin 𝜙0 − sin 𝜙 2𝜋kr

(5.64)

] a{ [ ei(kr+𝜋∕4) 𝜋 u(1) 𝑣 k(a − 𝜂), − 𝜙0 eika sin 𝜙0 = u0 2ik cos 𝜙 √ h 2 2𝜋kr ∫−a ] } [ 𝜋 + 𝑣 k(a + 𝜂), + 𝜙0 e−ik𝜂 sin 𝜙 d𝜂. 2

(5.65)

r The asymptotic expression (5.64) was found here by direct integration of the uniform component of the scattering sources. It is identical to the PO expression (5.11) derived by the summation of two edge waves. r The field (5.65) is calculated by integration by parts. r We provide the final results in terms of the directivity patterns: ei(kr+𝜋∕4) u(0,1) = u0 Φ(0,1) (𝜙, 𝜙0 ) √ h h 2𝜋kr

(5.66)

138

FIRST-ORDER DIFFRACTION AT STRIPS AND POLYGONAL CYLINDERS

where Φ(0) is given by Equation (5.13) and h (𝜙, 𝜙0 ) = Φ(1) h

cos 𝜙 𝜉

{ − i4 sin(ka𝜉)

[ √ ei⋅3𝜋∕4 −ika𝜉 √2ka(1+sin 𝜙0 ) it2 2ka(1−sin 𝜙0 ) it2 e e dt − eika𝜉 e dt +2 √ ∫ ∫ 0 0 𝜋 √ 1 + sin 𝜙0 ika𝜉 √2ka(1+sin 𝜙) it2 e e dt − ∫0 1 + sin 𝜙 √ ]} 1 − sin 𝜙0 −ika𝜉 √2ka(1−sin 𝜙) it2 e e dt , + (5.67) ∫0 1 − sin 𝜙 where 𝜉 = sin 𝜙0 − sin 𝜙. + u(1) (found by trunThe directivity pattern of the total scattered field utrh = u(0) h h

+ Φ(1) : cation of the nonuniform sources) equals Φtrh = Φ(0) h h Φtrh (𝜙, 𝜙0 ) =

cos 𝜙 𝜉

{ − i2 sin(ka𝜉)

[ √ ei⋅3𝜋∕4 −ika𝜉 √2ka(1+sin 𝜙0 ) it2 2ka(1−sin 𝜙0 ) it2 e e dt − eika𝜉 e dt +2 √ ∫ ∫ 0 0 𝜋 √ 1 + sin 𝜙0 ika𝜉 √2ka(1+sin 𝜙) it2 e e dt − ∫0 1 + sin 𝜙 √ ]} 1 − sin 𝜙0 −ika𝜉 √2ka(1−sin 𝜙) it2 e e dt + . (5.68) ∫0 1 − sin 𝜙 In the ray region [where ka(1 ± sin 𝜙0 ) ≫ 1 and ka(1 ± sin 𝜙) ≫ 1], the asymptotic approximation of this function is given by Φtrh (𝜙, 𝜙0 ) = g(1)eika(sin 𝜙0 −sin 𝜙) + g(2)e−ika(sin 𝜙0 −sin 𝜙) [ ] eika(sin 𝜙0 +sin 𝜙) e−ika(sin 𝜙0 +sin 𝜙) − cos 𝜙 + √ √ (1 + sin 𝜙) 1 + sin 𝜙0 (1 − sin 𝜙) 1 − sin 𝜙0 ei(2ka+𝜋∕4) , × √ 2𝜋ka

(5.69)

where the functions g(1) and g(2) are as determined in Equations (5.31) to (5.33). These functions relate to the primary edge waves arising at the ends of the strip.

139

DIFFRACTION AT A STRIP

The terms in the second line of Equation (5.69) represent the radiation of the primary edge waves when they reach the opposite end of the strip. These terms can be interpreted as the part of the true edge waves arising due to the secondary diffraction. The true secondary edge waves are described by the TED approximation (5.47), which leads to the following asymptotic expression: (𝜙, 𝜙0 ) = g(1)eika(sin 𝜙0 −sin 𝜙) + g(2)e−ika(sin 𝜙0 −sin 𝜙) ΦTED h [ ] eika(sin 𝜙0 +sin 𝜙) e−ika(sin 𝜙0 +sin 𝜙) + √ + √ √ √ 1 + sin 𝜙) 1 + sin 𝜙0 1 − sin 𝜙 1 − sin 𝜙0 ×

ei(2ka+𝜋∕4) . √ 𝜋ka

(5.70)

Here the second line represents the exact asymptotic form of the secondary edge waves. The function Φtrh (𝜙, 𝜙0 ) contains terms singular in the specular direction 𝜙 = 𝜋 − 𝜙0 . However, such singular terms cancel each other and generate the finite quantity for this function: Φtrh (𝜋 − 𝜙0 , 𝜙0 ) = i2ka cos 𝜙0 {

[ √ ei𝜋∕4 2ka(1+sin 𝜙0 ) eit2 dt 4ka √ 𝜋 ∫0 ] √ 2ka(1−sin 𝜙0 ) it2 e dt

− cos 𝜙0 +

∫0 √

[ ] ei2ka(1−sin 𝜙0 ) 2ka i3𝜋∕4 ei2ka(1+sin 𝜙0 ) e + + √ √ 𝜋 1 + sin 𝜙0 1 − sin 𝜙0 [ √ ei3𝜋∕4 1 2ka(1+sin 𝜙0 ) it2 e dt − √ 𝜋 1 + sin 𝜙0 ∫0 ]} √ 1 2ka(1−sin 𝜙0 ) eit2 + dt . 1 − sin 𝜙0 ∫0

(5.71)

Expressions (5.68) and (5.71) were used to calculate the normalized scattering crosssection (5.45) in the region 𝜋∕2 ≤ 𝜙 ≤ 3𝜋∕2. The results are plotted in Figure 5.8. It is seen that the truncated PTD is in good agreement with the exact asymptotic theory identified here as the TED (Ufimtsev, 1969, 1970, 2003, 2009). Significant improvement compared to the nontruncated version of PTD (Fig. 5.6) has been achieved in the vicinity of the directions 𝜙 = 90◦ and 𝜙 = 270◦ .

140

FIRST-ORDER DIFFRACTION AT STRIPS AND POLYGONAL CYLINDERS

Figure 5.8 Scattering at a hard strip predicted by the truncated version of PTD (Hz -polarization of the electromagnetic field).

5.2 DIFFRACTION AT A TRIANGULAR CYLINDER

The relationships us = Ez and uh = Hz exist between the acoustic and electromagnetic fields for this problem.

For simplicity we consider here the diffraction at a cylinder with the cross-section in the form of an equilateral triangle (Fig. 5.9). Two special cases will be investigated: y r

2 l

ϕ l

1

x

l 3 Figure 5.9 Cross-section of the scattering cylinder. The numbers 1, 2, and 3 denote the edges; r and 𝜙 are polar coordinates of the field point.

DIFFRACTION AT A TRIANGULAR CYLINDER

141

(1) The symmetric case, when the incident plane wave propagates in the direction parallel to the bisector of the triangle; and (2) Backscattering, when the scattered field is evaluated for the direction from which the incident wave comes. First, we study these problems utilizing the PO approximation, and after that it will be corrected by taking into account the first-order edge waves generated by the nonuniform/fringe . scattering sources j(1) s,h

5.2.1

Symmetric Scattering: PO Approximation

The incident wave, uinc = u0 eikx ,

(5.72)

generates the identical scattering sources j(0) at faces 1–2 and 1–3 of the cylinder s,h (Fig. 5.9), which are symmetric with respect to the x-axis. Soft (1.5) or hard (1.6) boundary conditions are imposed on the faces. The scattering cylinder is equilateral: The width of each face is equal to l, and each internal angle between faces equals 60◦ . The Cartesian coordinates of edges 1, 2, and 3 are equal to (0,0), (h,a), and (h,−a), respectively, where h = l cos(𝜋∕6) and a = l∕2. Due to the symmetry of the problem, it is sufficient to calculate the scattered field in the directions 0 ≤ 𝜙 ≤ 𝜋. The traditional integration technique for calculation of the PO scattered field can be avoided here. Indeed, in this approximation, the scattering cylinder can be considered as the combination of the strips 1–2 and 1–3. As shown in Section 5.1.1, the far field scattered by a strip consists of two edge waves determined by Equations (3.57) and (3.58). We omit all routine calculations related to the transition from the local polar coordinates (r1,2,3 , 𝜑1,2,3 ) (used for description of individual edge waves) to the basic coordinates (r, 𝜙), and provide the final equations for the scattered field in the region r ≫ kl2 , 0 ≤ 𝜙 ≤ 𝜋: ei(kr+𝜋∕4) , u(0) = u0 Φ(0) (𝜙, 0) √ s,h s,h 2𝜋kr

(5.73)

(0) (0) i𝜓2 Φ(0) + f (0) (3)ei𝜓3 , s = f (1) + f (2)e

(5.74)

Φ(0) = g(0) (1) + g(0) (2)ei𝜓2 + g(0) (3)ei𝜓3 , h

(5.75)

where

with 𝜓2 = kh(1 − cos 𝜙) − ka sin 𝜙,

𝜓3 = kh(1 − cos 𝜙) + ka sin 𝜙.

(5.76)

142

FIRST-ORDER DIFFRACTION AT STRIPS AND POLYGONAL CYLINDERS

It is understood here that the uniform sources j(0) induced on each face (1–2 s,h and 1–3) radiate the field in the entire surrounding space (0 ≤ 𝜙 ≤ 2𝜋).

The functions f (0) and g(0) determine the directivity patterns of the edge waves identified by the corresponding numbers in the argument of these functions. They are given as f (0) (1) =

sin(𝜋∕6) sin(𝜋∕6) + , cos(𝜋∕6 − 𝜙) − cos(𝜋∕6) cos(𝜋∕6 + 𝜙) − cos(𝜋∕6)

(5.77)

g(0) (1) =

sin(𝜋∕6 − 𝜙) sin(𝜋∕6 + 𝜙) + , cos(𝜋∕6 − 𝜙) − cos(𝜋∕6) cos(𝜋∕6 + 𝜙) − cos(𝜋∕6)

(5.78)

f (0) (2) =

sin(𝜋∕6) , cos(𝜋∕6) − cos(𝜋∕6 − 𝜙)

g(0) (2) =

sin(𝜋∕6 − 𝜙) , (5.79) cos(𝜋∕6) − cos(𝜋∕6 − 𝜙)

f (0) (3) =

sin(𝜋∕6) , cos(𝜋∕6) − cos(𝜋∕6 + 𝜙)

g(0) (3) =

sin(𝜋∕6 + 𝜙) . (5.80) cos(𝜋∕6) − cos(𝜋∕6 + 𝜙)

All these functions have singularities in the forward direction 𝜙 = 0, which cancel each other, being substituted into the functions Φ(0) . This results in the expression s,h (0) Φ(0) s (0,0) = Φh (0,0) = ikl.

(5.81)

A similar situation with the compensation of the singularities occurs for the specular direction 𝜙 = 𝜋∕3 when Φ(0) s

(

) 𝜋 l 𝜋 , 0 = ik + tan ei𝜓3 , 3 2 6

Φ(0) h

(

) 𝜋 l 1 ei𝜓3 . , 0 = −ik + 3 2 cos(𝜋∕6)

(5.82)

Due to the symmetry of the problem, the same field is scattered in the direction of the specular reflection from the face 1–3: ( ( ) ) 𝜋 (0) 𝜋 − , 0 = Φ , 0 , Φ(0) s s 3 3

( ( ) ) 𝜋 (0) 𝜋 − Φ(0) , 0 = Φ , 0 . h h 3 3

(5.83)

It is worth noting that the sum of the dominant terms in the reflected fields (5.82) and (5.83) equals ±ikl and conforms to the field (5.81) scattered in the forward direction. This result is in agreement with the fundamental law that the total power of the reflected field (1.81) is asymptotically (with kl ≫ 1) equal to the total power of the shadow radiation (1.85).

DIFFRACTION AT A TRIANGULAR CYLINDER

5.2.2

143

Backscattering: PO Approximation

In this problem it is appropriate to consider the incident wave in the form uinc = u0 e−ik(x cos 𝜙0 +y sin 𝜙0 )

with 0 ≤ 𝜙0 ≤ 𝜋.

(5.84)

The scattered field is evaluated in the direction 𝜙 = 𝜙0 . The basic feature of the PO approximation for the backscattering follows from the properties of functions f (0) and g(0) defined in Equations (3.59) to (3.61) through the local polar angles 𝜑 and 𝜑0 . These angles determine the directions to the field point and to the source of the incident wave, respectively. For the backscattering direction 𝜑 = 𝜑0 , it turns out that f (0) (𝜑0 , 𝜑0 ) = −g(0) (𝜑0 , 𝜑0 ). This means that the fields scattered back to the source by the acoustically soft and hard cylinders differ only in sign, and therefore their directivity patterns differ in this way as well: (0) Φ(0) s = −Φh .

(5.85)

This result is in agreement with the general property (1.37) of the PO approximation. It is clear from Section 5.2.1 that the scattered field consists of the sum of the edge waves. We again omit simple routine calculations of these waves and present the final expressions for the directivity patterns of the total scattered field. We have different expressions for different intervals of observation because of the different number of contributions to the scattered field. In the interval 0 ≤ 𝜙 ≤ 𝜋∕6, only two diffracted waves exist incoming from edges 2 and 3 (Fig. 5.9). In this interval, (0) i𝜓2 + f (0) (3)ei𝜓3 , Φ(0) s (𝜙) = f (2)e

(5.86)

where f (0) (2) = −f (0) (3) = − 12 cot 𝜙, 𝜓2 = −2k(h cos 𝜙 + a sin 𝜙),

(5.87) 𝜓3 = −2k(h cos 𝜙 − a sin 𝜙)

(5.88)

with h = l cos(𝜋∕6) and a = l∕2. Equation (5.86) can be rewritten in the form Φ(0) s (𝜙) = i cos 𝜙

sin(2ka sin 𝜙) −i⋅2kh cos 𝜙 e , sin 𝜙

(5.89)

which predicts the value −i⋅2kh = ikl e−i⋅2kh . Φ(0) s (0) = i2ka e

(5.90)

144

FIRST-ORDER DIFFRACTION AT STRIPS AND POLYGONAL CYLINDERS

In the interval 𝜋∕6 < 𝜙 < 𝜋∕2, the scattered field consists of the three edge waves and, respectively, (0) (0) i𝜓2 + f (0) (3)ei𝜓3 , Φ(0) s (𝜙) = f (1) + f (2)e

(5.91)

where ( ) 1 𝜋 −𝜙 , f (0) (1) = − tan 2 6 [ ( ) ] 1 𝜋 tan − 𝜙 − cot 𝜙 , f (0) (2) = 2 6 f (0) (3) =

1 2

cot 𝜙.

(5.92) (5.93) (5.94)

In the interval 𝜋∕2 < 𝜙 < 5𝜋∕6, only two edge waves contribute to the total scattered field, and (0) (0) i𝜓2 Φ(0) s (𝜙) = f (1) + f (2)e

(5.95)

with function f (0) (1) as defined in Equation (5.92). The function f (0) (2) in this interval differs from Equation (5.93) and equals f (0) (2) =

( ) 1 𝜋 tan −𝜙 2 6

(5.96)

because the face 2–3 is not illuminated by the incident wave and does not generate the scattered field (in the framework of the PO approximation). In the specular direction 𝜙 = 2𝜋∕3, Equation (5.95) determines the value Φ(0) s

(

2𝜋 3

) = ikl,

(5.97)

which is consistent with Equation (5.90). In the interval 5𝜋∕6 < 𝜙 ≤ 𝜋, again all three edges generate the scattered field, and the directivity pattern is determined by Equation (5.91), but with the different functions f (0) (1) and f (0) (3): [ ( ) ( )] 𝜋 1 𝜋 tan − 𝜙 + tan +𝜙 , 2 6 6 ( ) 1 𝜋 +𝜙 . f (0) (3) = tan 2 6 f (0) (1) = −

(5.98) (5.99)

However, the function f (0) (2) is still defined by Equation (5.96). The function (5.98) differs from (5.92) because in this interval of observation, two faces, 1–2 and 1–3, are

DIFFRACTION AT A TRIANGULAR CYLINDER

145

illuminated. The function (5.99) differs from Equation (5.94) because the different faces of edge 3 are illuminated in the intervals 0 ≤ 𝜙 < 𝜋∕2 and 5𝜋∕6 < 𝜙 ≤ 𝜋.

5.2.3

Symmetric Scattering: First-Order PTD Approximation

Here the acoustic quantity Φs (Φh ) is equivalent to the directivity pattern of the Ez (Hz )-component of the electromagnetic field.

To improve the PO approximation presented in Section 5.2.1, we include in the scattered field the contributions generated by the nonuniform scattering sources j(1) . s,h In the first-order approximation, they have the form of edge waves (4.20) and (4.21). When these waves are added to the PO edge waves (3.57) and (3.58), one can see that [due to the relationships (4.22) and (4.23)] the resulting edge waves are defined by the Sommerfeld asymptotics (2.61) and (2.63). Because the acoustically soft and hard cylinders are perfectly reflecting (nontransparent), only those edge waves contribute to the scattered field, which come from the edges visible from the observation point. We also note that in this section we consider the symmetric case when the incident wave is given by Equation (5.72) and propagates along the bisector of the cylinder (Fig. 5.9). The basic polar coordinates r and 𝜙 are used in the following for the description of the scattered field and the angle 𝜙 changes in the interval 0 ≤ 𝜙 ≤ 𝜋. In accordance with these comments, the directivity patterns of the scattered field can be written as follows. In the interval 0 ≤ 𝜙 < 𝜋∕6, Φs (𝜙, 0) = f (2)ei𝜓2 + f (3)ei𝜓3 , Φh (𝜙, 0) = g(2)e

i𝜓2

i𝜓3

+ g(3)e

,

(5.100) (5.101)

where {

} [ ] sin(𝜋∕n) f (2) 1 1 = ∓ g(2) n cos(𝜋∕n) − cos(𝜋∕n − 𝜙∕n) cos(𝜋∕n) + cos(𝜋∕5 + 𝜙∕n)

{

}

f (3) g(3)

with 0 ≤ 𝜙 ≤ 𝜋, (5.102) [ ] sin(𝜋∕n) 1 1 = ∓ n cos(𝜋∕n) − cos(𝜋∕n + 𝜙∕n) cos(𝜋∕n) + cos(𝜋∕5 − 𝜙∕n) with 0 ≤ 𝜙 ≤ 𝜋∕2,

(5.103)

where n = 𝛼∕𝜋 = 5∕3 (here 𝛼 = 5𝜋∕3 is the external angle between the faces of the edge) and the phase functions 𝜓2,3 are as defined in Equation (5.76). The first terms in these functions are singular for the forward direction (𝜙 = 0). However, the

146

FIRST-ORDER DIFFRACTION AT STRIPS AND POLYGONAL CYLINDERS

singularities of the functions related to edges 2 and 3 cancel each other, resulting in the following expressions: Φs (0) = ikl −

(2∕n) sin(𝜋∕n) 1 𝜋 cot − , n n cos(𝜋∕n) + cos(𝜋∕5)

(5.104)

Φh (0) = ikl −

(2∕n) sin(𝜋∕n) 1 𝜋 cot + . n n cos(𝜋∕n) + cos(𝜋∕5)

(5.105)

Comparison of these Equations with Equation (5.81) shows that the first term here relates to the PO field, and the last two terms represent contributions from the nonuniform/fringe component j(1) . s,h In the interval 𝜋∕6 < 𝜙 < 𝜋∕2, three edge waves form the total scattered field: Φs (𝜙) = f (1) + f (2)ei𝜓2 + f (3)ei𝜓3 ,

(5.106)

Φh (𝜙) = g(1) + g(2)ei𝜓2 + g(3)ei𝜓3 ,

(5.107)

where [ ] { } sin(𝜋∕n) f (1) 1 1 ∓ . = g(1) n cos(𝜋∕n) − cos(𝜋∕n − 𝜙∕n) cos(𝜋∕n) + cos(𝜋∕n − 𝜙∕n) with 𝜋∕6 ≤ 𝜙 ≤ 𝜋

(5.108)

There is the specular direction (𝜙 = 𝜋∕3) in this sector of observation. In this direction, the second terms in the functions f (1), g(1) and f (2), g(2) are singular. But these singular terms cancel each other in Equations (5.106) and (5.107) and the scattered field remains finite: ( ) ( ) l 1 𝜋 𝜋 𝜋 = ik + tan + cot + f (3)ei𝜓3 , 3 2 n n n ( ) ( ) l 1 𝜋 𝜋 𝜋 = −ik + tan − cot + g(3)ei𝜓3 . Φh 3 2 n n n Φs

(5.109) (5.110)

Due to the symmetry of the problem, these equations also describe the field in the direction of specular reflection from the face 1–3. Here the comments presented at the end of Section 5.2.1 are also pertinent. The power of the total reflected field is asymptotically equal to the power of the shadow radiation determined by the first term in (5.104) and (5.105). It is also clear that the dominant term ±ikl∕2 in (5.109) and (5.110) relates to the PO contribution. In the interval 𝜋∕2 < 𝜙 < 5𝜋∕6, edge 3 is invisible and the scattered field consists of two edge waves: Φs (𝜙) = f (1) + f (2)ei𝜓2 ,

(5.111)

Φh (𝜙) = g(1) + g(2)ei𝜓2 .

(5.112)

DIFFRACTION AT A TRIANGULAR CYLINDER

147

In the interval 5𝜋∕6 < 𝜙 ≤ 𝜋, all three edges are visible and the scattered field is determined again by Equations (5.106) and (5.107) with the functions [ ] { } sin(𝜋∕n) f (3) 1 1 ∓ = g(3) n cos(𝜋∕n) − cos(𝜋∕n − 𝜙∕n) cos(𝜋∕n) + cos(𝜋∕n + 𝜙∕n) with 5𝜋∕6 ≤ 𝜙 ≤ 𝜋.

(5.113)

This expression differs from Equation (5.103), although both are exact forms of the same generic expressions, (2.62) and (2.64). This difference is due to the fact that the local polar coordinate 𝜑3 [used in the generic definition of functions f (3) and g(3)] cannot be described in the regions 𝜑3 < 𝜋∕6 and 𝜑3 > 𝜋∕6 (Fig. 5.10) by a single expression in terms of the basic coordinate 𝜙 under the restriction 0 ≤ 𝜙 ≤ 𝜋. Notice also that the asymptotic expressions found above for the quantities uh and dus ∕d𝜙 are discontinuous in the directions 𝜙 = 𝜋∕6, 𝜙 = 𝜋∕2, and 𝜙 = 5𝜋∕6, which are the geometrical optics boundaries for the edge waves. These discontinuities can be diminished when the higher-order edge waves (arising due to multiple edge diffraction) are taken into account. However, as shown in Section 5.2.5, such discontinuities are already not significant when l ≥ 3𝜆. It is easy to construct similar asymptotics for the bistatic scattering for arbitrary directions of the incident wave. However, their shortcoming is the grazing singularity (4.33) and (4.34), which appears in the case of the grazing directions of the incident wave. This singularity can be removed with application of the uniform theory developed in Section 7.9. An alternative procedure free of the grazing singularity is based on the truncation of elementary strips (Michaeli, 1987; Breinbjerg, 1992; Johansen, 1996).

φ3 = 5π /3 y

φ3 = 0

γ

2 l

ϕ=π

ϕ l

1

φ3 = π /6

φ3

ϕ=0 x

3

φ3 = 2π /3 Figure 5.10 Sector 0 ≤ 𝜑3 ≤ 5𝜋∕3 is the domain of the functions f (3) and g(3). The field point (r, 𝜙) is located in the region 0 ≤ 𝜙 ≤ 𝜋.

148

FIRST-ORDER DIFFRACTION AT STRIPS AND POLYGONAL CYLINDERS

5.2.4

Backscattering: First-Order PTD Approximation

Here the acoustic quantity Φs (Φh ) is equivalent to the directivity pattern of the Ez (Hz )-component of the electromagnetic field.

In Section 5.2.3 it was explained that in the first-order PTD approximation, the field scattered by a triangular cylinder is a linear combination of the edge waves defined in the general form by Equations (2.61) and (2.63). Now we apply this PTD theory for the investigation of the backscattering, assuming that the incident wave is given by Equation (5.84) with 0 ≤ 𝜙0 ≤ 𝜋. The scattered field is found in the direction 𝜙 = 𝜙0 in the sector 0 ≤ 𝜙 ≤ 𝜋. The geometry of the problem is shown in Figure 5.9. We omit the simple but tedious calculations of the individual edge waves in terms of the basic coordinates r and 𝜙 and present the final expressions for the directivity patterns of the total scattered field. In the interval 0 ≤ 𝜙 ≤ 𝜋∕6, Φs (𝜙) = f (2)eik𝜓2 + f (3)ei𝜓3 ,

(5.114)

Φh (𝜙) = g(2)eik𝜓2 + g(3)ei𝜓3 ,

(5.115)

where {

{

f (2) g(2)

f (3) g(3)

}

}

[ ] sin(𝜋∕n) 1 1 = ∓ n cos(𝜋∕n) − 1 cos(𝜋∕n) − cos(𝜋∕n + 2𝜙∕n) with 0 ≤ 𝜙 ≤ 𝜋, (5.116) [ ] sin(𝜋∕n) 1 1 = ∓ n cos(𝜋∕n) − 1 cos(𝜋∕n) − cos(𝜋∕n − 2𝜙∕n) with 0 ≤ 𝜙 ≤ 𝜋∕2.

(5.117)

Here the edge parameter equals n = 𝛼∕𝜋 = 5∕3, and the quantities 𝜓2,3 are defined in Equation (5.88). It follows from these equations that for the direction 𝜙 = 0, [ ] (2∕n) sin(𝜋∕n) −i⋅2kh 1 𝜋 e , Φs (0) = ikl + cot + n n cos(𝜋∕n) − 1 [ ] (2∕n) sin(𝜋∕n) −i⋅2kh 1 𝜋 e Φh (0) = −ikl − cot + , n n cos(𝜋∕n) − 1 where h = l cos(𝜋∕6).

(5.118) (5.119)

149

DIFFRACTION AT A TRIANGULAR CYLINDER

In the interval 𝜋∕6 < 𝜙 ≤ 𝜋∕2, the additional wave appears incoming from the edge 1. Hence, in this interval, Φs (𝜙) = f (1) + f (2)eik𝜓2 + f (3)ei𝜓3 , ik𝜓2

Φh (𝜙) = g(1) + g(2)e

i𝜓3

+ g(3)e

,

(5.120) (5.121)

where functions f (2), f (3) and g(2), g(3) are defined in Equations (5.116) and (5.117) and {

f (1) g(1)

}

[ ] sin(𝜋∕n) 1 1 ∓ = n cos(𝜋∕n) − 1 cos(𝜋∕n) − cos(𝜋∕5 − 2𝜙∕n) with 𝜋∕6 < 𝜙 ≤ 𝜋.

(5.122)

In the interval 𝜋∕2 < 𝜙 < 5𝜋∕6, the diffracted wave from edge 3 disappears, and Φs (𝜙) = f (1) + f (2)eik𝜓2 ,

(5.123)

.

(5.124)

ik𝜓2

Φh (𝜙) = g(1) + g(2)e

There is the specular direction 𝜙 = 2𝜋∕3 in this interval, where ( Φs ( Φh

2𝜋 3 2𝜋 3

) = ikl +

1 𝜋 (2∕n) sin(𝜋∕n) cot + , n n cos(𝜋∕n) − 1

) = −ikl −

1 𝜋 (2∕n) sin(𝜋∕n) cot + . n n cos(𝜋∕n) − 1

(5.125) (5.126)

Due to the symmetry of the problem, these expressions differ from Equations (5.118) and (5.119) only by the absence of the phase factor e−i2kh [caused by the choice of the incident wave in the form of Equation (5.84) and by the choice of the coordinates’ origin at edge 1]. In the interval 5𝜋∕6 < 𝜙 ≤ 𝜋, all three edges generate the scattered field, which again is determined by Equations (5.120) and (5.121), where the functions f (3) and g(3) are defined by the expressions {

f (3) g(3)

}

[ ] sin(𝜋∕n) 1 1 ∓ , (5.127) = n cos(𝜋∕n) − 1 cos(𝜋∕n) + cos(2𝜙∕n)

different from Equation (5.117). These expressions complete the description of the backscattered field.

150

FIRST-ORDER DIFFRACTION AT STRIPS AND POLYGONAL CYLINDERS

5.2.5

Numerical Analysis of a Scattered Field

Here the scattering cross-section 𝜎s (𝜎h ) for acoustic waves is equal to the scattering cross-section for electromagnetic waves with the component Ez (Hz ).

Numerical calculations were performed for the normalized scattering cross-section (5.20), | Φs,h (𝜙) |2 | | =| | , | kl | kl2 | |

𝜎s,h

(5.128)

on the decibel scale for the equilateral cylinder with the parameter kl = 6𝜋 when l = 3𝜆. The results are plotted in Figures 5.11 and 5.12 for symmetric scattering (when the incident wave propagates in the direction parallel to the cylinder bisector; Fig. 5.9) and in Figures 5.13 and 5.14 for backscattering. As shown in Figures 5.11 and 5.12, PTD improves the PO approximation significantly at the minima of the scattering cross-section. The difference between the PTD and PO data is also appreciable at maxima. This difference is pronounced for an acoustically hard cylinder and reaches about 6 to 9 dB in the sector 160–180◦ . A similar situation is observed for the backscattering in Figures 5.13 and 5.14. In

Figure 5.11 Bistatic scattering at an acoustically soft cylinder (Ez -polarization of the electromagnetic wave).

DIFFRACTION AT A TRIANGULAR CYLINDER

151

Figure 5.12 Bistatic scattering at an acoustically hard cylinder (Hz -polarization of the electromagnetic wave).

Figure 5.13

Backscattering at an acoustically soft cylinder (Ez -polarization of the electromagnetic wave).

152

FIRST-ORDER DIFFRACTION AT STRIPS AND POLYGONAL CYLINDERS

Figure 5.14 Backscattering at an acoustically hard cylinder (Hz -polarization of the electromagnetic wave).

particular, the difference between the PTD and PO curves for the acoustically hard cylinder is about 5 to 9 dB in the directions 50–70◦ and 170–180◦ . More accurate PTD results for triangular cylinders are presented in a paper by are set to zero (truncated) Johansen (1996), where the primary scattering sources j(1) s,h on extensions of the cylinder faces. The PTD approximation with higher asymptotic accuracy for this scattering problem can be constructed by application of the improved PTD version derived in Section 7.9 and utilizing the multiple diffraction technique developed in Chapters 9 and 10.

PROBLEMS 5.1 Derive the PO approximation for the far field scattered at a soft strip shown in Figure 5.1. The incident plane wave is given by Equation (5.1). Calculate the field by integration of the surface scattering sources and verify that the result is identical to the edge wave approximation (5.4). For the illuminated side of the soft strip at its point y = 𝜂,

Solution n̂ = −̂x,

𝜕u j(0) s = 2 𝜕n

inc

uinc = u0 eik𝜂 sin 𝜙0 , = −u0 2ik cos 𝜙0 eik𝜂 sin 𝜙0 .

𝜕uinc 𝜕n

= − 𝜕u𝜕x = −u0 ik cos 𝜙0 eik𝜂 sin 𝜙0 , inc

153

PROBLEMS

According to Equation (1.32), a

∞

eikr eikr 1 ik eik𝜂 sin 𝜙0 d𝜂 j(0) ds = u0 cos 𝜙0 d𝜁 , s ∫−∞ r ∫−a 4𝜋 ∫ r 2𝜋 √ r = x2 + (y − 𝜂)2 + 𝜁 2 .

u(0) s =−

In view of Equation (3.7), ∞

∫−∞

[ √ ] eikr d𝜁 = i𝜋H0(1) k x2 + (y − 𝜂)2 . r

√ We calculate the field in the far zone where 𝜌 = x2 + y2 ≫ ka2 . In this region one can use the asymptotic form (2.29) for the Hankel function, H0(1)

√ [ √ ] k x2 + (y − 𝜂)2 ∼

2 eik √ 2 2 𝜋k x + (y − 𝜂)

√ x2 +(y−𝜂)2 −i(𝜋∕4)

e

.

In the far zone √ √ y x2 + (y − 𝜂)2 ≈ 𝜌 − 𝜌 𝜂 = 𝜌 − 𝜂 sin 𝜙, 𝜌 = x2 + y2 , √ √ 2 ik(𝜌−𝜂 sin 𝜙) −i(𝜋∕4) e e , H0(1) (k x2 + (y − 𝜂)2 ) ≈ 𝜋k𝜌 and the field approximation is given by a

ei(k𝜌+𝜋∕4) eik(sin 𝜙0 −sin 𝜙)𝜂 d𝜂, u(0) ik cos 𝜙0 s = u0 √ ∫−a 2𝜋k𝜌 that is, uPO ≡ u(0) s = u0 2i cos 𝜙0

sin[ka(sin 𝜙0 − sin 𝜙)] ei(k𝜌+𝜋∕4) , √ sin 𝜙0 − sin 𝜙 2𝜋k𝜌

√ which coincides with Equation (5.10), where r = x2 + y2 ≡ 𝜌. Use the Euler formula for the sine function and verify that this uPO transforms exactly into Equation (5.4). 5.2

Derive the PO approximation for the far field scattered at the hard strip shown in Figure 5.1. The incident plane wave is given by Equation (5.1). Calculate the field by integration of the surface scattering sources and verify that the result is identical to the edge wave approximation (5.5).

154

FIRST-ORDER DIFFRACTION AT STRIPS AND POLYGONAL CYLINDERS

Solution The geometry of the problem is shown in Figure 5.1. The incident wave is given as uinc = u0 exp[ik(x cos 𝜙0 + y sin 𝜙0 )]. On the illuminated side = 2u0 eik𝜂 sin 𝜙0 . The scattered field is described by Equation of the strip, j(0) h (1.32): a

∞

1 𝜕 eikr 𝜕 eikr 1 ik𝜂 sin 𝜙0 e d𝜂 j(0) ds = d𝜁, ∫−∞ 𝜕n r 4𝜋 ∫ h 𝜕n r 2𝜋 ∫−a √ r = x2 + (y − 𝜂)2 + 𝜁 2 .

≡ u(0) = uPO h h

It follows from Problem 1.5 that ∞

∫−∞

∞

eikr 𝜕 eikr 𝜕 d𝜁 = d𝜁. 𝜕n r 𝜕x ∫−∞ r

As shown in Problem 5.1 for the far zone, ∞

∫−∞

eikr d𝜁 ≈ r

√

2𝜋 i(k𝜌+𝜋∕4) −ik𝜂 sin 𝜙 e e , k𝜌

𝜌=

√

x2 + y2 .

Hence, √ ∞ ikr e 2𝜋 i(k𝜌+𝜋∕4) −ik𝜂 sin 𝜙 𝜕 e e . d𝜁 ≈ ik cos 𝜙 𝜕x ∫−∞ r k𝜌 and integrating over the variable 𝜂, Substituting this approximation into u(0) h one obtains uPO ≡ u(0) = u0 i2 cos 𝜙 h h

sin[ka(sin 𝜙0 − sin 𝜙)] ei(k𝜌+𝜋∕4) , √ sin 𝜙0 − sin 𝜙 2𝜋k𝜌

√ which coincides with Equation (5.11), where r = x2 + y2 ≡ 𝜌. Use the Euler formula for the sine function and verify that this uPO transforms exactly into the edge wave approximation (5.5). 5.3 Derive the PO approximation for the far field scattered at the perfectly conducting strip shown in Figure 5.1. The incident wave is given as Ezinc = E0z eik(x cos 𝜑0 +y sin 𝜑0 ) . Verify that that the result is equivalent to the edge wave approximation (5.4) under the condition E0z = u0 .

PROBLEMS

155

Solution According to Equation (1.128), EzPO = Ezrefl + Ezsh . The reflected and shadow parts of the PO field have been found in Problems 1.8 and 1.9. Their sum equals EzPO = E0z 2i cos 𝜙0

sin[ka(sin 𝜙0 − sin 𝜙)] ei(kr+𝜋∕4) , √ sin 𝜙0 − sin 𝜙 2𝜋kr

√ where r = x2 + y2 . Use the Euler formula for the sine function and verify that EzPO is equivalent to Equation (5.4) under the condition E0z = u0 . 5.4

Derive the PO approximation for the far field scattered at the perfectly conducting strip shown in Figure 5.1. The incident plane wave is given as Hzinc = H0z eik(x cos 𝜑0 +y sin 𝜑0 ) . Calculate the field by integration of the surface currents and verify that the result is equivalent to the edge wave approximation (5.5) under the condition H0z = u0 . Solution The scattered field is determined by Equations (1.104) and (1.105). The PO surface electric current is determined by Equation (1.113). It contains ik𝜂 sin 𝜙9 at the surface point y = 𝜂. The magonly the component j(0) y = 2H0z e netic surface current does not exists on a perfectly conducting surface, ⃗jm = 0. It follows from Equations (1.104) and (1.105) that a

HzPO =

∞

eik𝜌 𝜕 e 1 𝜕 eik𝜂 sin 𝜙0 d𝜂 Ay = H0z d𝜁 , ∫−∞ 𝜌 𝜕x 2𝜋 𝜕x ∫−a

√ where 𝜌 = x2 + (y − 𝜂)2 + 𝜁 2 . Then we apply Equations (3.7) and (2.29), utilize the far-field approximation √ √ √ 2𝜋 i(kr−𝜋∕4) −ik𝜂 sin 𝜙 e e , r = x2 + y2 , H0(1) (k x2 + (y − 𝜂)2 ) ≈ kr calculate the integral a

HzPO = H0z

𝜕 ei(kr+𝜋∕4) eik𝜂(sin 𝜙9 −sin 𝜙) d𝜂, √ 𝜕x 2𝜋kr ∫−a

and finally obtain HzPO = H0z 2i cos 𝜙

sin[ka(sin 𝜙0 − sin 𝜙)] ei(kr+𝜋∕4) . √ sin 𝜙0 − sin 𝜙 2𝜋kr

Applying the Euler formula for the sine function, one finds that HzPO is equivalent given by Equation (5.5) under the condition u0 = H0z . to uPO h

6 Axially Symmetric Scattering of Acoustic Waves at Bodies of Revolution A similar problem for electromagnetic waves is considered in Chapter 2 of the book by Ufimtsev (2003, 2009).

In this chapter we develop the first-order PTD for acoustic waves scattered at bodies of revolution with sharp edges. Axially symmetric scattering is studied. This situation occurs when an incident plane wave propagates in the direction along the symmetry axis of a body of revolution. An edge of a body of revolution is a circle. When its diameter is much greater than a wavelength, the nonuniform scattering induced near the edge are asymptotically identical to those near the edge sources j(1) s,h of a tangential conic surface consisting of two parts (Fig. 6.1). Diffraction at this surface is an appropriate canonical problem that is studied in Section 6.1. Its solution is used in the next sections to determine the field scattered at certain bodies of revolution.

Fundamentals of the Physical Theory of Diffraction, Second Edition. Pyotr Ya. Ufimtsev. © 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

157

158

AXIALLY SYMMETRIC SCATTERING OF ACOUSTIC WAVES AT BODIES OF REVOLUTION

uinc z

Figure 6.1 Body of revolution (solid lines) with a circular edge and a conic surface (dashed lines) tangential to a body at the edge points.

6.1 DIFFRACTION AT A CANONICAL CONIC SURFACE The geometry of the problem is illustrated in Figures 6.1 and 6.2. The solid lines in Figure 6.1 show a general view of a body of revolution with a circular edge. The dashed tangent lines belong to the tangential conic surface. The cross-section of this surface by the meridian plane and some related denotations are presented in ρ

r′

r′ uinc

ϑ′

ω

φ0

(R, ϑ , ψ )

ξ

1

ξ

Ω

2

α – φ0

Figure 6.2 Cross-section of the canonical conic surface.

z

DIFFRACTION AT A CANONICAL CONIC SURFACE

159

Figure 6.2. Here 𝜉 is the distance from the edge along the generatrix; r′ , 𝜗′ , and 𝜓 ′ are the spherical coordinates and 𝜌 and 𝜓 ′ are the polar coordinates of the point on the conic surface; R, 𝜗, and 𝜓 are the coordinates of the observation point; 𝜑0 is the angle of incidence measured from the illuminated side of the conic surface; and 𝛼 − 𝜑0 is the angle of incidence measured from the shadowed side; the meaning of the angles 𝜔 and Ω is clear; the edge points 1 and 2 are symmetrical. 6.1.1

Integrals for the Scattered Field

It is supposed that the incident plane wave uinc = u0 eikz

(6.1)

propagates along the symmetry axis of the conic surface. The incident wave undergoes diffraction at this surface and creates there the nonuniform/fringe scattering sources (1) j(1) s = u0 Js ,

j(1) = u0 Jh(1) . h

(6.2)

It is obvious that due to the symmetry of the problem, these sources do not depend on the polar angle 𝜓 ′ . The scattered field is described by general Equations (1.16) and (1.17). In the particular case of a conic surface, the quantities involved in these equations are determined by the following expressions. The quantities ds− = (a − 𝜉 sin 𝜔) d𝜉 d𝜓 ′ , ds+ = (a − 𝜉 sin Ω) d𝜉 d𝜓 ′

(6.3)

are the differential elements of the conic surface on its illuminated (z < 0) and shadowed (z > 0) sides, respectively: r′ sin 𝜗′ = a − 𝜉 sin 𝜔 r′ cos 𝜗′ = −𝜉 cos 𝜔 r′ sin 𝜗′ = a − 𝜉 sin Ω r′ cos 𝜗′ = 𝜉 cos Ω

for points with z < 0,

(6.4)

for points with z > 0;

(6.5)

̂ ⋅ n̂ )− = sin 𝜗 cos 𝜔 cos(𝜓 ′ − 𝜓) − sin 𝜔 cos 𝜗 (m

for points with z < 0,

(6.6)

̂ ⋅ n̂ )+ = sin 𝜗 cos Ω cos(𝜓 ′ − 𝜓) + sin Ω cos 𝜗 (m

for points with z > 0.

(6.7)

Note that 𝜓 ′ and 𝜓 are the spherical coordinates of the integration and observation points, respectively. In this section we also use the symbol Ω for the angle shown in Figure 6.2. In Equations (1.16) and (1.17), the same symbol was used for another angle (Fig. 1.2). We note this to avoid possible confusion.

160

AXIALLY SYMMETRIC SCATTERING OF ACOUSTIC WAVES AT BODIES OF REVOLUTION

In view of the comments above, the scattered fields (1.16) and (1.17) can be represented in the form u(1) s = −u0 [ ×

1 eikR 4𝜋 R 𝜉eff

∫0 𝜉eff

+

∫0

Js(1) (k𝜉, 𝜑0 )eik𝜉 cos 𝜔 cos 𝜗 (a − 𝜉 sin 𝜔) d𝜉

Js(1) (k𝜉, 𝛼

−ik𝜉 cos Ω cos 𝜗

− 𝜑0 )e

2𝜋

∫0

(a − 𝜉 sin Ω) d𝜉

− (𝜓 ′ ,𝜓)

e−ikΨ 2𝜋

−ikΨ+ (𝜓 ′ ,𝜓)

e

∫0

d𝜓 ′ ] d𝜓

′

(6.8) = −u0 u(1) h [ ×

ik eikR 4𝜋 R 𝜉eff

∫0 𝜉eff

+

∫0 2𝜋

×

∫0

Jh(1) (k𝜉, 𝜑0 )eik𝜉 cos 𝜔 cos 𝜗 (a − 𝜉 sin 𝜔) d𝜉

2𝜋

∫0

− (𝜓 ′ ,𝜓)

e−ikΨ

̂ ⋅ n̂ )− d𝜓 ′ (m

Jh(1) (k𝜉, 𝛼 − 𝜑0 )e−ik𝜉 cos Ω cos 𝜗 (a − 𝜉 sin Ω) d𝜉 + (𝜓 ′ ,𝜓)

e−ikΨ

] ̂ ⋅ n̂ )+ d𝜓 ′ , (m

(6.9)

where Ψ− (𝜓 ′ , 𝜓) = (a − 𝜉 sin 𝜔) sin 𝜗 cos(𝜓 ′ − 𝜓), Ψ+ (𝜓 ′ , 𝜓) = (a − 𝜉 sin Ω) sin 𝜗 cos(𝜓 ′ − 𝜓).

(6.10)

The first integrals in brackets in Equations (6.8) and (6.9) relate to the illuminated surface (z < 0), and the second integrals relate to the shadowed surface (z > 0). These integrals are calculated over the interval 0 ≤ 𝜉 ≤ 𝜉eff as the nonuniform sources (1) Js,h (k𝜉) decrease with increasing 𝜉 and can be neglected at a certain distance 𝜉 > 𝜉eff from the edge. In the next sections we present the asymptotic estimates for the scattered field. 6.1.2

Ray Asymptotics

First we investigate the field in the observation points, which are visible from all edge points (0 ≤ 𝜓 ′ ≤ 2𝜋). This happens for the two intervals of observation directions: 0 ≤ 𝜗 ≤ Ω and 𝜋 − 𝜔 ≤ 𝜗 ≤ 𝜋. We assume that k(a − 𝜉eff sin 𝜔) sin 𝜗 ≫ 1, k(a − 𝜉eff sin Ω) sin 𝜗 ≫ 1

(6.11)

DIFFRACTION AT A CANONICAL CONIC SURFACE

161

and apply the stationary-phase technique (Copson, 1965; Murray, 1984) to the integrals over the variable 𝜓 ′ . The stationary points 𝜓1,2 are found from the condition dΨ± (𝜓 ′ , 𝜓) = 0, d𝜓 ′

(6.12)

which leads to the equation d cos(𝜓 ′ − 𝜓) = − sin(𝜓 ′ − 𝜓) = 0 at d𝜓 ′

𝜓 ′ = 𝜓st′ .

(6.13)

In the interval 0 ≤ 𝜓 ′ ≤ 2𝜋, two stationary points exist: 𝜓1 = 𝜓

and

𝜓2 = 𝜋 + 𝜓.

(6.14)

With respect to the observation point P(R, 𝜗, 𝜓), the stationary points 𝜓1 and 𝜓2 are the nearest and farthest edge points, respectively. In accordance with this asymptotic method, the function cos(𝜓 ′ − 𝜓) contained in Ψ± is approximated by cos(𝜓 ′ − 𝜓) ≈ 1 − 12 (𝜓 ′ − 𝜓1 )2 in the vicinity of the point 𝜓 ′ = 𝜓1 ,

(6.15)

cos(𝜓 ′ − 𝜓) ≈ −1 + 12 (𝜓 ′ − 𝜓2 )2 in the vicinity of the point 𝜓 ′ = 𝜓2 .

(6.16)

̂ ⋅ n̂ )± is approximated by its value at the stationary The slowly varying factor (m points. The initial integral over the variable 𝜓 ′ asymptotically equals the sum of two integrals calculated in the vicinity of each stationary point. The intervals of integration in these integrals are extended from −∞ to +∞. These standard manipulations lead to the asymptotic expression 2𝜋

∫0

−ikΨ± (𝜓 ′ ,𝜓)

e

√ √ √ 2𝜋 ̂ ⋅ n̂ ) d𝜓 ∼ √ (m ( )] √ [ √ Ω √ k a − 𝜉 sin sin 𝜗 𝜔 ±

′

{ ×

{

̂ ⋅ n̂ )± ||𝜓 ′ =𝜓 (m 1

exp

[ −ik a − 𝜉 sin

{ [ ̂ ⋅ n̂ )± ||𝜓 ′ =𝜓 exp + (m 2

(

ik a − 𝜉 sin

(

Ω 𝜔

Ω

𝜔 )]

)]

} sin 𝜗

ei(𝜋∕4)

} sin 𝜗

} e−i(𝜋∕4)

,

(6.17)

162

AXIALLY SYMMETRIC SCATTERING OF ACOUSTIC WAVES AT BODIES OF REVOLUTION

φ1

ϑ α – φ1 1

φ01 = ω

ω

Ω

z

Figure 6.3 Local polar coordinates at the stationary point 1 (𝜓st′ = 𝜓).

where | ̂ ⋅ n̂ )+ |𝜓 ′ =𝜓1 = sin(𝜗 + Ω), (m | | ̂ ⋅ n̂ )+ |𝜓 ′ =𝜓2 = − sin(𝜗 − Ω), (m |

| ̂ ⋅ n̂ )− |𝜓 ′ =𝜓1 = sin(𝜗 − 𝜔), (m | (6.18) | ̂ ⋅ n̂ )− |𝜓 ′ =𝜓2 = − sin(𝜗 + 𝜔). (m |

Equation (6.17) allows one to reduce Equations (6.8) and integrals over √ (6.9) to single√ the variable 𝜉. These integrals will contain the factors a − 𝜉 sin 𝜔 and a − 𝜉 sin Ω, √ which can be approximated to a under the condition a ≫ 𝜉eff . Before we present the resulting expressions for Equations (6.8) and (6.9), we introduce the local polar coordinates 𝜑1 and 𝜑2 at the stationary points 𝜓1 and 𝜓2 . In Figure 6.2 these points are denoted as 1 and 2. The local coordinates are shown in Figure 6.3 for point 1 and in Figures 6.4 and 6.5 for point 2. Considering the relationships above among coordinates 𝜑1 ,𝜑2 , and 𝜗, and utilizing Equation (6.17), one can obtain the following approximations for Equations (6.8)

Ω

ω

z

ϑ

φ

02

=ω

φ2

2

α – φ2 Figure 6.4 Local coordinates at the stationary point 2 (𝜓st′ = 𝜋 + 𝜓) for the observation directions in the interval 𝜋 − 𝜔 ≤ 𝜗 ≤ 𝜋.

DIFFRACTION AT A CANONICAL CONIC SURFACE

z

Ω

ω

163

α

–φ2

φ02 = ω 2

ϑ

φ2

Figure 6.5 Local coordinates at the stationary point 2 (𝜓st′ = 𝜋 + 𝜓) for the observation directions in the interval 0 ≤ 𝜗 ≤ Ω.

and (6.9): eikR a u(1) s = −u0 √ 2 2𝜋ka sin 𝜗 R [ 𝜉eff { Js(1) (k𝜉, 𝜑0 )e−ik𝜉 cos 𝜑1 d𝜉 × e−ika sin 𝜗+i(𝜋∕4) ∫0 ] 𝜉eff (1) −ik𝜉 cos(𝛼−𝜑1 ) Js (k𝜉, 𝛼 − 𝜑0 )e d𝜉 + ∫0 [ 𝜉eff ika sin 𝜗−i(𝜋∕4) +e Js(1) (k𝜉, 𝜑0 )e−ik𝜉 cos 𝜑2 d𝜉 ∫0 ]} 𝜉eff (1) −ik𝜉 cos(𝛼−𝜑2 ) Js (k𝜉, 𝛼 − 𝜑0 )e d𝜉 , + ∫0 eikR ika = −u0 √ u(1) h 2 2𝜋ka sin 𝜗 R [ { −ika sin 𝜗+i(𝜋∕4) sin 𝜑1 × e + sin(𝛼 − 𝜑1 )

𝜉eff

∫0 [

𝜉eff

∫0

∫0

Jh(1) (k𝜉, 𝜑0 )e−ik𝜉 cos 𝜑1 d𝜉 ]

Jh(1) (k𝜉, 𝛼 − 𝜑0 )e−ik𝜉 cos(𝛼−𝜑1 ) d𝜉

+ eika sin 𝜗−i(𝜋∕4) sin 𝜑2 + sin(𝛼 − 𝜑2 )

𝜉eff

(6.19)

𝜉eff

∫0

Jh(1) (k𝜉, 𝜑0 )e−ik𝜉 cos 𝜑2 d𝜉

Jh(1) (k𝜉, 𝛼 − 𝜑0 )e−ik𝜉 cos(𝛼−𝜑2 ) d𝜉

]} .

(6.20)

164

AXIALLY SYMMETRIC SCATTERING OF ACOUSTIC WAVES AT BODIES OF REVOLUTION

(1) Under the condition ka ≫ 1, the scattering sources Js,h near the edge of the conic surface are asymptotically equivalent to those near the edge of the tangential wedge. Hence, the expressions inside the brackets in (6.19) and (6.20) are also asymptotically equivalent to the similar expressions in Equations (4.42) and (4.43) which relate to the wedge diffraction problem. Utilizing this observation, one can rewrite Equations (6.19) and (6.20) as

] eikR [ (1) u0 a f (1)e−ika sin 𝜗+i(𝜋∕4) + f (1) (2)eika sin 𝜗−i(𝜋∕4) u(1) , s = √ R 2𝜋ka sin 𝜗 (6.21) ] ikR [ (1) u0 a −ika sin 𝜗+i(𝜋∕4) (1) ika sin 𝜗−i(𝜋∕4) e g u(1) = (1)e + g (2)e . √ h R 2𝜋ka sin 𝜗 (6.22) These expressions can be interpreted as the ray asymptotics for the field u(1) . They s,h show that under the condition ka sin 𝜗 ≫ 1, this field consists of two diffracted rays incoming from stationary points 1 and 2 at the circular edge. As is also seen there, the ray from point 2 undergoes the additional phase shift −𝜋∕2 while crossing the focal line along the z-axis. As noted in connection with Equation (6.14) and demonstrated in Figures 6.3 and 6.4, points 1 and 2 are the nearest and farthest edge points, respectively, to the observation point P(R, 𝜗, 𝜓). In this meaning, these points are understood everywhere in the present chapter. The approximations above were derived for the regions 0 < 𝜗 ≤ Ω and 𝜋 − 𝜔 ≤ 𝜗 < 𝜋 when both stationary points are visible from the observation point. The same technique is used for calculation of the diffracted field in the region Ω ≤ 𝜗 ≤ 𝜋 − 𝜔 when stationary point 2 is not visible and therefore does not contribute to the firstorder edge waves. In this case, only the vicinity of stationary point 1 participates in the calculation, which results in u(1) s = √ u(1) h

u0 a

f (1) (1)e−ika sin 𝜗+i(𝜋∕4)

eikR , R

2𝜋ka sin 𝜗 u a eikR g(1) (1)e−ika sin 𝜗+i(𝜋∕4) =√ 0 . R 2𝜋ka sin 𝜗

(6.23) (6.24)

Functions f (1) and g(1) are defined by Equations (4.22) and (4.23) with Equations (3.59) to (3.61) and (2.62) and (2.64). According to these equations, ( ) sin(𝜋∕n) 1 1 f (1) (1) = − n cos(𝜋∕n) − cos[(𝜋 − 𝜗)∕n] cos(𝜋∕n) − cos[(𝜋 + 2𝜔 − 𝜗)∕n] −

sin 𝜔 , cos 𝜔 − cos(𝜔 − 𝜗)

(6.25)

DIFFRACTION AT A CANONICAL CONIC SURFACE

g(1) (1) =

sin(𝜋∕n) n

−

(

165

1 1 + cos(𝜋∕n) − cos[(𝜋 − 𝜗)∕n] cos(𝜋∕n) − cos[(𝜋 + 2𝜔 − 𝜗)∕n]

sin(𝜔 − 𝜗) , cos 𝜔 − cos(𝜔 − 𝜗)

)

(6.26)

where n=

𝜔+Ω 𝛼 =1+ . 𝜋 𝜋

(6.27)

These expressions for functions f (1) and g(1) are valid in the entire region 0 ≤ 𝜗 ≤ 𝜋, although we have two different expressions for functions f (1) (2) and g(1) (2). For the region 0 ≤ 𝜗 ≤ Ω, f (1) (2) =

sin(𝜋∕n) n

−

g(1) (2) =

1 1 − cos(𝜋∕n) − cos[(𝜋 + 𝜗)∕n] cos(𝜋∕n) − cos[(𝜋 + 2𝜔 + 𝜗)∕n]

sin 𝜔 , cos 𝜔 − cos(𝜔 + 𝜗)

sin(𝜋∕n) n

−

(

(

(6.28)

1 1 + cos(𝜋∕n) − cos[(𝜋 + 𝜗)∕n] cos(𝜋∕n) − cos[(𝜋 + 2𝜔 + 𝜗)∕n]

sin(𝜔 + 𝜗) , cos 𝜔 − cos(𝜔 + 𝜗)

)

)

(6.29)

and for the region 𝜋 − 𝜔 ≤ 𝜗 ≤ 𝜋, the corresponding expressions are f

(1)

sin(𝜋∕n) (2) = n

−

(

sin 𝜔 , cos 𝜔 − cos(𝜔 + 𝜗)

sin(𝜋∕n) g (2) = n

(

(1)

−

1 1 − cos(𝜋∕n) − cos[(𝜋 − 𝜗)∕n] cos(𝜋∕n) − cos[(𝜋 − 2𝜔 − 𝜗)∕n]

(6.30)

1 1 + cos(𝜋∕n) − cos[(𝜋 − 𝜗)∕n] cos(𝜋∕n) − cos[(𝜋 − 2𝜔 − 𝜗)∕n]

sin(𝜔 + 𝜗) . cos 𝜔 − cos(𝜔 + 𝜗)

)

)

(6.31)

Some terms in functions f (1) and g(1) are singular in certain directions 𝜗; however, such singularities cancel each other and these functions are always finite. For the forward direction 𝜗 = 0 (i.e., the shadow boundary behind the body), (1∕n) sin(𝜋∕n) 1 𝜋 1 − cot + cot 𝜔, cos(𝜋∕n) − cos[(𝜋 + 2𝜔)∕n] 2n n 2

(6.32)

(1∕n) sin(𝜋∕n) 1 𝜋 1 − cot − cot 𝜔. cos(𝜋∕n) − cos[(𝜋 + 2𝜔)∕n] 2n n 2

(6.33)

f (1) (1) = f (1) (2) = − g(1) (1) = g(1) (2) =

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AXIALLY SYMMETRIC SCATTERING OF ACOUSTIC WAVES AT BODIES OF REVOLUTION

For the specular direction 𝜗 = 2𝜔 (corresponding to the ray reflected from the conic surface), f (1) (1) =

(1∕n) sin(𝜋∕n) 1 𝜋 1 + cot + cot 𝜔, cos(𝜋∕n) − cos[(𝜋 − 2𝜔)∕n] 2n n 2

(6.34)

g(1) (1) =

(1∕n) sin(𝜋∕n) 1 𝜋 1 − cot + cot 𝜔. cos(𝜋∕n) − cos[(𝜋 − 2𝜔)∕n] 2n n 2

(6.35)

Notice also that for the directions 𝜗 = 0 and 𝜗 = 𝜋, the relationships f (1) (1) = f (1) (2)

and

g(1) (1) = g(1) (2)

(6.36)

are valid, due to the symmetry of the problem. 6.1.3

Focal Fields

Focal fields for acoustic and electromagnetic waves generated by the nonuniform sources j(1) are different, due to the vector nature of electromagnetic fields.

Given the incident wave (6.1), every point at the z-axis is a focal point for diffracted edge waves (Figs. 6.1 and 6.2). In the far zone, the position of any focal points with z > 0 is determined by the coordinate 𝜗 = 0, and the points with z < 0 are characterized by the coordinate 𝜗 = 𝜋. For the focal points, the general field expressions (6.8) and (6.9) are simplified. Under the condition a ≫ 𝜉eff , they can be written as u(1) s = −u0 [ ×

a eikR 2 R 𝜉eff

∫0

Js(1) (k𝜉, 𝜑0 )e±ik𝜉 cos 𝜔 d𝜉 +

𝜉eff

∫0

] Js(1) (k𝜉, 𝛼 − 𝜑0 )e∓ik𝜉 cos Ω d𝜉 , (6.37)

u(1) = −u0 h

ika eikR 2 R

[ × ∓ sin 𝜔

𝜉eff

∫0 𝜉eff

± sin Ω

∫0

Jh(1) (k𝜉, 𝜑0 )e±ik𝜉 cos 𝜔 d𝜉

] Jh(1) (k𝜉, 𝛼 − 𝜑0 )e∓ik𝜉 cos Ω d𝜉 .

(6.38)

167

DIFFRACTION AT A CANONICAL CONIC SURFACE

The upper (lower) sign here relates to 𝜗 = 0 (𝜗 = 𝜋). In terms of the local coordinates, they look the same for both directions: u(1) s = −u0

a eikR 2 R

[ ×

𝜉eff

∫0

Js(1) (k𝜉, 𝜑0 )e−ik𝜉 cos 𝜑1

𝜉eff

d𝜉 +

∫0

] Js(1) (k𝜉, 𝛼

− 𝜑0 )e

−ik𝜉 cos(𝛼−𝜑1 )

d𝜉 ,

(6.39)

u(1) = −u0 h

ika eikR 2 R

[ × sin 𝜑1

𝜉eff

∫0

+ sin(𝛼 − 𝜑1 )

Jh(1) (k𝜉, 𝜑0 )e−ik𝜉 cos 𝜑1 d𝜉 𝜉eff

∫0

] Jh(1) (k𝜉,

𝛼 − 𝜑0 )e

−ik𝜉 cos(𝛼−𝜑1 )

d𝜉

(6.40)

with 𝜑1 = 𝜔 (𝜑1 = 𝜋 + 𝜔) when 𝜗 = 𝜋 (𝜗 = 0). Then we use Equations (4.42) and (4.43) and find the following expressions for the focal field: (1) u(1) s = u0 af (1)

eikR , R

(6.41)

u(1) = u0 ag(1) (1) h

eikR . R

(6.42)

√ It is seen that the focal field is ka times higher in magnitude than the ray fields (6.21) and (6.22). In conjunction with (6.41) and (6.42), we recall the relationships (6.36), which are valid for the focal points.

6.1.4

Bessel Interpolations for the Field u(1) s,h

In previous sections we derived the ray and focal asymptotic expressions for the field u(1) . The ray asymptotics (6.21) to (6.24) are valid under the condition ka sin 𝜗 ≫ s,h 1 (i.e., away from the focal line), and asymptotic expressions (6.41) and (6.42) determine the field directly on the focal line (where sin 𝜗 = 0). Now our goal is to construct such approximations, that would describe the diffracted field continuously in both the ray and focal regions. This can be done utilizing the Bessel functions J0 (ka sin 𝜗) and J1 (ka sin 𝜗).

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AXIALLY SYMMETRIC SCATTERING OF ACOUSTIC WAVES AT BODIES OF REVOLUTION

For the large argument (ka sin 𝜗 ≫ 1), they can be approximated by (Gradshteyn and Ryzhik, 1994) 1 (eika sin 𝜗−i(𝜋∕4) + e−ika sin 𝜗+i(𝜋∕4) ), J0 (ka sin 𝜗) ∼ √ 2𝜋ka sin 𝜗

(6.43)

1 J1 (ka sin 𝜗) ∼ √ (eika sin 𝜗−i(3𝜋∕4) + e−ika sin 𝜗+i(3𝜋∕4) ). 2𝜋ka sin 𝜗

(6.44)

It follows from these expressions that e−ika sin 𝜗+i(𝜋∕4) 1 ≈ [J0 (ka sin 𝜗) − iJ1 (ka sin 𝜗)], √ 2 2𝜋ka sin 𝜗

(6.45)

eika sin 𝜗−i(𝜋∕4) 1 ≈ [J0 (ka sin 𝜗) + iJ1 (ka sin 𝜗)]. √ 2 2𝜋ka sin 𝜗

(6.46)

With these observations, the ray asymptotics (6.21) and (6.22) can be written as u(1) s = u0

a eikR (1) {f (1)[J0 (ka sin 𝜗) − iJ1 (ka sin 𝜗)] 2 R

+ f (1) (2)[J0 (ka sin 𝜗) + iJ1 (ka sin 𝜗)]}, = u0 u(1) h

(6.47)

a eikR (1) {g (1)[J0 (ka sin 𝜗) − iJ1 (ka sin 𝜗)] 2 R

+ g(1) (2)[J0 (ka sin 𝜗) + iJ1 (ka sin 𝜗)]}.

(6.48)

Now we continue these expressions analytically into the focal region. If we take Equation (6.36) into account as well as the relationships J0 (0) = 1 and J1 (0) = 0, we can see that the expressions (6.47) and (6.48) transform exactly into the focal asymptotics (6.41) and (6.42) when 𝜗 → 0 and 𝜗 → 𝜋. This means that formulas (6.47) and (6.48) can be considered as the appropriate approximations for the diffracted field in the regions 0 ≤ 𝜗 ≤ Ω and 𝜋 − 𝜔 ≤ 𝜗 ≤ 𝜋. In the region Ω ≤ 𝜗 ≤ 𝜋 − 𝜔, where stationary point 2 is not visible, the modified versions of (6.47) and (6.48), eikR a (1) u(1) s = u0 2 f (1)[J0 (ka sin 𝜗) − iJ1 (ka sin 𝜗)] R ,

(6.49)

eikR a = u0 g(1) (1)[J0 (ka sin 𝜗) − iJ1 (ka sin 𝜗)] , u(1) h 2 R

(6.50)

SCATTERING AT A DISK

169

represent analytical continuation of the ray asymptotics (6.23) and (6.24) into the entire region Ω ≤ 𝜗 ≤ 𝜋 − 𝜔. In the following sections, the results found above for the canonical problem are utilized in calculating the field scattered at certain bodies of revolution.

6.2

SCATTERING AT A DISK

The geometry of the problem is shown in Figure 6.6. The incident plane wave is given by Equation (6.1) and propagates in the positive direction of the z-axis. Because of that and due to the axial symmetry of the disk, the scattered field is the same in all meridian planes 𝜑 = const(0 ≤ 𝜑 ≤ 2𝜋). In addition, the scattered field is symmetric with respect to the disk plane: us (−z) = us (z) and uh (−z) = −uh (z). Therefore, it is sufficient to investigate the field only in the plane 𝜑 = 𝜋∕2 for the directions 0 ≤ 𝜗 ≤ 𝜋∕2. 6.2.1

Physical Optics Approximation

(0) (0) (0) The relationships u(0) s = E𝜑 and uh = H𝜑 in the PO approximation are valid inc inc for the disk diffraction problem under the conditions uinc s = E𝜑 and uh = inc H𝜑 .

Acoustic Waves In this approximation, the scattered field is considered as the radiation generated by the uniform component (1.31) of the scattering sources, which are induced on the illuminated side (z = −0) of the disk. This field is determined by y

(R, ϑ , π2 )

1

x

r′

φ′

uinc

z

a nˆ 2

Figure 6.6 An incident wave propagates in the direction along the z-axis and undergoes diffraction at a circular disk of radius a. Edge points 1 and 2 are in the plane x = 0.

170

AXIALLY SYMMETRIC SCATTERING OF ACOUSTIC WAVES AT BODIES OF REVOLUTION

Equations (1.16) and (1.17), where one should set 𝜗′ =

𝜋 , 2

𝜑=

𝜋 , 2

z ̂ ⋅ n̂ = − ≈ − cos 𝜗, m r

cos Ω = sin 𝜗 sin 𝜑′ , jh = j(0) = 2u0 . h

js = j(0) s = −2iku0 ,

(6.51) (6.52)

With these settings, the field is described by a

2𝜋

u(0) s = u0

ik eikR r′ dr′ ∫0 2𝜋 R ∫0

u(0) = u0 h

ik eikR r′ dr′ cos 𝜗 ∫0 2𝜋 R ∫0

e−ikr

a

′ sin 𝜗 sin 𝜑′

2𝜋

d𝜑′ ,

′ sin 𝜗 sin 𝜑′

(6.53) d𝜑′ .

(6.54)

J0 (x)x dx = xJ1 (x),

(6.55)

e−ikr

According to Gradshteyn and Ryzhik (1994), 1 2𝜋 ∫0

2𝜋

e−ix sin 𝜑 d𝜑′ = J0 (x), ′

∫

where J0 and J1 are the Bessel functions. With these relationships, one can represent Equations (6.53) and (6.54) as u(0) s = u0

eikR ia J1 (ka sin 𝜗) , sin 𝜗 R

(6.56)

u(0) = u0 h

eikR ia cos 𝜗J1 (ka sin 𝜗) . sin 𝜗 R

(6.57)

For small arguments (ka sin 𝜗 ≪ 1), one can replace J1 (ka sin 𝜗) by (ka sin 𝜗)∕2. Then we set 𝜗 = 0, 𝜋 and find the field on the focal line: u(0) s = u0

ika2 eikR 2 R

= ±u0 u(0) h

ika2 eikR 2 R

for for

𝜗 = 0 and 𝜗 = 𝜋, ( ) 0 𝜗= . 𝜋

(6.58) (6.59)

In the directions away from the focal line where ka sin 𝜗 ≫ 1, the Bessel function in Equations (6.56) and (6.57) can be replaced by its asymptotic expression (6.44). Equations we obtain in this way reveal a ray structure of the field. It consists of two diffracted rays coming from the stationary points 1 and 2: u0 a eikR [f (0) (1)e−ika sin 𝜗+i(𝜋∕4) + f (0) (2)eika sin 𝜗−i(𝜋∕4) ] , u(0) s = √ R 2𝜋ka sin 𝜗

(6.60)

ikR u0 a (0) −ika sin 𝜗+i(𝜋∕4) (0) ika sin 𝜗−i(𝜋∕4) e [g = (1)e + g (2)e ] , u(0) √ h R 2𝜋ka sin 𝜗

(6.61)

SCATTERING AT A DISK

171

where the functions f (0) (1) = −f (0) (2) = −

1 , sin 𝜗

(6.62)

g(0) (1) = −g(0) (2) = −

cos 𝜗 sin 𝜗

(6.63)

determine the directivity patterns of diffracted rays in the PO approximation. 6.2.2

Relationships Between Acoustic and Electromagnetic PO Fields

To show the relationship between the acoustic and electromagnetic diffracted fields, we present here the PO approximation for the field scattered at a perfectly conducting disk (Ufimtsev, 1962): E𝜗(0) = Z0 H𝜑(0) = −iaZ0 H0x sin 𝜑 E𝜑(0)

=

−Z0 H𝜗(0)

cos 𝜗 eikR J1 (ka sin 𝜗) , sin 𝜗 R

(6.64)

1 eikR = −iaZ0 H0x cos 𝜑 J1 (ka sin 𝜗) . sin 𝜗 R

This field is due to diffraction of the incident wave, Eyinc = −Z0 Hxinc = −Z0 H0x eikz = E0y eikz , where Z0 = equations

√

(6.65)

𝜇0 ∕𝜀0 = 120𝜋 ohms is the impedance of free space. In view of the

E0𝜑 = E0y cos 𝜑 = −Z0 H0x cos 𝜑,

H9𝜑 = −H0x sin 𝜑,

(6.66)

the expressions (6.64) can be rewritten as E𝜑(0) = E0𝜑 H𝜑(0)

eikR ia J1 (ka sin 𝜗) , sin 𝜗 R

(6.67)

eikR ia = H0𝜑 cos 𝜗J1 (ka sin 𝜗) . sin 𝜗 R

Their comparison with Equations (6.56) and (6.57) reveals the following equivalence existing in the PO approximation between the acoustic and electromagnetic diffracted fields:

(0) u(0) s = E𝜑

if

u0 = E0𝜑 ,

u(0) h

if

u0 = H0𝜑 .

=

H𝜑(0)

(6.68)

These equations are in an agreement with relationships (1.120) and (1.121).

172

AXIALLY SYMMETRIC SCATTERING OF ACOUSTIC WAVES AT BODIES OF REVOLUTION

6.2.3

Field Generated by Fringe Scattering Sources

The field generated by the nonuniform/fringe scattering sourcesj(1) was investigated s,h in Section 6.1.4. Here we reproduce the related approximations: u(1) s = u0

a eikR (1) {f (1)[J0 (ka sin 𝜗) − iJ1 (ka sin 𝜗)] 2 R

+ f (1) (2)[J0 (ka sin 𝜗) + iJ1 (ka sin 𝜗)]}, u(1) = u0 h

(6.69)

a eikR (1) {g (1)[J0 (ka sin 𝜗) − iJ1 (ka sin 𝜗)] 2 R

+ g(1) (2)[J0 (ka sin 𝜗) + iJ1 (ka sin 𝜗)]},

(6.70)

where 0 ≤ 𝜗 ≤ 𝜋. The ray-type asymptotics of (6.69) and (6.70) are shown in Equations (6.21) and (6.22). The focal asymptotics of (6.69) and (6.70) are determined by Equations (6.41) and (6.42). General expressions for the functions f (1) (1), f (1) (2) and g(1) (1), g(1) (2) are given by Equations (6.25) to (6.31). For the scattering disk, they are written below. The functions f (1) (1) and g(1) (1) are described in the entire region 0 ≤ 𝜗 ≤ 𝜋 by f

(1)

(

1 (1) = − 2

g(1) (1) =

1 2

( −

1 1 + sin(𝜗∕2) cos(𝜗∕2) 1 1 + sin(𝜗∕2) cos(𝜗∕2)

) +

1 , sin 𝜗

(6.71)

+

cos 𝜗 . sin 𝜗

(6.72)

)

Functions f (1) (2) and g(1) (2) are described by f

(1)

(

1 (2) = 2

g(1) (2) =

1 2

1 1 − sin(𝜗∕2) cos(𝜗∕2)

(

)

1 1 + sin(𝜗∕2) cos(𝜗∕2)

−

1 , sin 𝜗

(6.73)

−

cos 𝜗 sin 𝜗

(6.74)

−

1 , sin 𝜗

(6.75)

−

cos 𝜗 sin 𝜗

(6.76)

)

in the region 0 ≤ 𝜗 ≤ 𝜋∕2, and by f (1) (2) =

1 2

g(1) (2) = − in the region 𝜋∕2 ≤ 𝜗 ≤ 𝜋.

( − 1 2

1 1 + sin(𝜗∕2) cos(𝜗∕2)

(

)

1 1 + sin(𝜗∕2) cos(𝜗∕2)

)

SCATTERING AT A DISK

173

Functionsf (1) (1) and f (1) (2) are symmetric with respect to the disk plane, f (1) (1, 𝜋 − 𝜗) = f (1) (1, 𝜗),

f (1) (2, 𝜋 − 𝜗) = f (1) (2, 𝜗),

(6.77)

but functions g(1) (1) and g(1) (2) are antisymmetric: g(1) (1, 𝜋 − 𝜗) = −g(1) (1, 𝜗),

g(1) (2, 𝜋 − 𝜗) = −g(1) (2, 𝜗).

(6.78)

Here, the functions with the argument 𝜋 − 𝜗 correspond to the left half-space (z < 0). It follows from Equations (6.71) to (6.76) that f (1) (1) = f (1) (2) = − 12 g(1) (1) = g(1) (2) = ± 12

𝜗 = 0 and 𝜗 = 𝜋, ( ) 0 for 𝜗 = . 𝜋

for

(6.79) (6.80)

After substitution of these values into Equations (6.41) and (6.42), it follows that u(1) s =−

u0 a eikR 2 R

for

u(1) =± h

u0 a eikR 2 R

for

𝜗 = 0 and 𝜗 = 𝜋, ( ) 0 𝜗= . 𝜋

(6.81) (6.82)

It is worth noting that the amplitude of the focal field generated by the nonuniform sources does not depend on frequency. 6.2.4

Total Scattered Field

The sum us,h = u(0) + u(1) s,h s,h

(6.83)

provides the first-order PTD approximation for the scattered field:

r Quantities u(0) and u(1) are defined by Equations (6.56) and (6.57) and Equations s,h s,h (6.69) and (6.70).

r Their ray-type asymptotics are determined in Equations (6.60), (6.61) and (6.21), (6.22). When they are included in Equation (6.83), the functions f (0) and g(0) contained both in (6.60) and (6.61) and in (6.21) and (6.22) cancel each other. As

174

AXIALLY SYMMETRIC SCATTERING OF ACOUSTIC WAVES AT BODIES OF REVOLUTION

a result, the ray asymptotics for the total field contain only the Sommerfeld functions f and g: u a eikR [f (1)e−ika sin 𝜗+i(𝜋∕4) + f (2)eika sin 𝜗−i(𝜋∕4) ] , us = √ 0 R 2𝜋ka sin 𝜗

(6.84)

eikR . R

(6.85)

uh = √

u0 a 2𝜋ka sin 𝜗

[g(1)e−ika sin 𝜗+i(𝜋∕4) + g(2)eika sin 𝜗−i(𝜋∕4) ]

The functions f (1, 2) and g(1, 2) are described by Equations (6.71) to (6.76), where the last terms (being outside the brackets) should be omitted.

r The focal asymptotics for u(0) and u(1) are presented in Equations (6.58), (6.59) s,h s,h and (6.81), (6.82). Their summation leads to

us = u0

( ) i eikR ika2 1+ 2 ka R

for

( ) i eikR ika2 1− uh = ±u0 2 ka R

for

𝜗 = 0 and 𝜗 = 𝜋,

𝜗=

( ) 0 𝜋

.

(6.86)

(6.87)

Approximation (6.87) for the direction 𝜗 = 𝜋 agrees with Equation (14.114) in the work by Bowman et al. (1987). The results of numerical calculations shown in Figures 6.7 and 6.8 supplement the foregoing analysis of the scattered field. The normalized scattering cross-section is the quantity 𝜎norm =

𝜎s,h 𝜋a2 (ka)2

plotted on the decibel scale. The ordinary scattering cross-section 𝜎 is defined by Equation (1.26). In Figures 6.7 and 6.8, the contribution to the scattered field generated by the nonuniform component of the scattering sources is shown clearly. The field scattered by the hard disk must be equal to zero in the direction 𝜗 = 90◦ . The finite value of the PTD field in this direction (Fig. 6.8) is caused by the fictitious nonuniform sources distributed in the plane outside the disk surface. This shortcoming can be removed by the truncation of these fictitious sources, as shown in Section 5.1.4.

SCATTERING AT A DISK

175

Figure 6.7 Scattering at an acoustically soft disk. The curve “FRINGE” relates to the field generated by the nonuniform (“fringe”) scattering sources. According to Equations (6.68), the PO curve here also demonstrates the scattering of electromagnetic waves at a perfectly conducting disk.

Figure 6.8 Scattering at an acoustically hard disk. The curve “FRINGE” relates to the field generated by the nonuniform (“fringe”) scattering sources. According to Equations (6.68), the PO curve here also demonstrates the scattering of electromagnetic waves at a perfectly conducting disk.

176

AXIALLY SYMMETRIC SCATTERING OF ACOUSTIC WAVES AT BODIES OF REVOLUTION

6.3 SCATTERING AT CONES: FOCAL FIELD

The PO approximations for acoustic and electromagnetic focal fields in this problem are identical. Asymptotics for the total acoustic and electromagnetic fields are different.

6.3.1

Asymptotic Approximations for the Field

The geometry of the problem is shown in Figure 6.9. The incident plane wave is given by Equation (6.1). First, we calculate the focal field radiated by the uniform components j(0) of the induced scattering sources. They are defined by Equation s,h (1.31) and determined on the cone surface by ikz j(0) s = −2u0 ik sin 𝜔e

and

j(0) = 2u0 eikz . h

(6.88)

The scattered field in the far zone is defined by Equations (1.16) and (1.17), where one should set ̂ ⋅ n̂ = − sin 𝜔 cos 𝜗 m

with 𝜗 = 0 or 𝜗 = 𝜋

(6.89)

and ds = 𝜉 sin 𝜔 d𝜉 d𝜑′

with

𝜉 = r′ .

(6.90)

For the observation points on the focal line (𝜗 = 0, 𝜋), the integral over the variable 𝜑′ equals 2𝜋. Hence, the field u(0) can be represented in the form s,h 2 u(0) s = u0 ik sin 𝜔

b

eikR eik𝜉(1−cos 𝜗) cos 𝜔 𝜉 d𝜉, R ∫0

u(0) = u0 ik sin2 𝜔 cos 𝜗 h

b

eikR eik𝜉(1−cos 𝜗) cos 𝜔 𝜉 d𝜉, R ∫0

y

ρ=α Ω

nˆ

uinc

(6.91)

ω

z

l

Figure 6.9 Cross-section of a cone by the plane y0z. A radius of the edge is a.

(6.92)

SCATTERING AT CONES: FOCAL FIELD

177

where b = a∕ sin 𝜔 is the length of the cone generatrix and a is the radius of the edge. It follows from these equations that (0) u(0) s = uh = u0

ika2 eikR 2 R

(6.93)

in the forward direction (𝜗 = 0), and (0) u(0) s = −uh = −u0

( ) ikR i eikR a i e tan2 𝜔 + u0 tan2 𝜔 + tan 𝜔 ei2kl 4k R 4k 2 R

(6.94)

in the backscattering direction (𝜗 = 𝜋). Forward scattering Equation (6.93) is exactly the same as that according to Equations (6.58) and (6.59) for the disk and actually represents the shadow radiation introduced in Section 1.3.4. This result is in complete agreement with the shadow contour theorem presented in Section 1.3.5, since the cone and disk of radius a produce the same shadow and generate the same shadow radiation. Comparison of the acoustic field (6.94) with the electromagnetic PO field scattered by a perfectly conducting cone [Equation (2.4.3) in Ufimtsev (2003, 2009)] reveals the relationships

Ex(0) = u(0) s

if

E0x = u0 ,

Hy(0)

if

H0y = u0

=

u(0) h

(6.95)

for the direction 𝜗 = 𝜋. This result is in complete agreement with the general relationships (1.120) and (1.121). The field radiated by the nonuniform components of the scattering sources j(1) s,h (induced near the circular edge on the cone) was investigated in Section 6.1. According to Equations (6.41) and (6.42), this field equals (1) u(1) s = u0 af

eikR , R

(6.96)

= u0 ag(1) u(1) h

eikR R

(6.97)

(1) u(1) s = u0 af

eikR i2kl e , R

(6.98)

= u0 ag(1) u(1) h

eikR i2kl e R

(6.99)

in the forward direction 𝜗 = 0, and

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AXIALLY SYMMETRIC SCATTERING OF ACOUSTIC WAVES AT BODIES OF REVOLUTION

in the backscattering direction 𝜗 = 𝜋. Here f (1) = − g(1) =

(1∕n) sin(𝜋∕n) 1 𝜋 1 − cot + cot 𝜔, cos(𝜋∕n) − cos[(𝜋 + 2𝜔)∕n] 2n n 2

(1∕n) sin(𝜋∕n) 1 𝜋 1 − cot − cot 𝜔 cos(𝜋∕n) − cos[(𝜋 + 2𝜔)∕n] 2n n 2

for the forward direction 𝜗 = 0, and ( ) sin(𝜋∕n) 1 1 1 (1) − − tan 𝜔, f = n cos(𝜋∕n) − 1 cos(𝜋∕n) − cos(2𝜔∕n) 2 ( ) sin(𝜋∕n) 1 1 1 g(1) = + + tan 𝜔 n cos(𝜋∕n) − 1 cos(𝜋∕n) − cos(2𝜔∕n) 2

(6.100)

(6.101)

(6.102)

(6.103)

for the backscattering direction 𝜗 = 𝜋. In these equations, n = 1 + (𝜔 + Ω)∕𝜋. In contrast to the equivalence of PO acoustic and electromagnetic fields noted generated by nonuniform scattering sources is different above, the acoustic field u(1) s,h from the electromagnetic field radiated by nonuniform electric currents. The electromagnetic field has been given by Ufimtsev [2003, 2009; Equations (2.3.18) and (2.3.19)] and is determined by a linear combination of the functions f (1) and g(1) , (1) and the field u(1) depends but in the acoustic case the field u(1) s depends only on f h only on g(1) . The equations above describe the field u(1) generated only by the nonuniform s,h scattering sources induced near the cone edge. Two other types of nonuniform sources are induced on the cone surface, which we neglected here. The first is caused by the smooth bending of the surface, which is a small quantity inversely proportional to k𝜌 at a distance far from the cone tip. Here 𝜌 is a polar coordinate of the surface (𝜌 = a at the edge points). The second is a nonuniform component concentrated near the cone tip. It is caused both by the sharp tip and by the large curvature of the cone surface due to its smooth bending near the tip. At the far distance 𝜉 from the tip, it is on the order of 1∕k𝜉. In the case of electromagnetic diffraction, it was shown (Ufimtsev, 2003, 2009) that in the backscattering direction (𝜗 = 𝜋) one can neglect the field radiated by these types of nonuniform scattering sources. Asymptotic analysis of an electromagnetic field scattered by a semi-infinite cone also confirms this observation (Felsen, 1955; Bowman et al., 1987). Taking these comments into account, the first-order PTD approximation for the backscattered total field (at the focal line 𝜗 = 𝜋) can be found by summation of Equations (6.94) and (6.98), (6.99): [ i tan2 𝜔(1 − ei2kl ) us = u0 a − 4ka ( ) ] ikR sin(𝜋∕n) 1 1 i2kl e + − e , n cos(𝜋∕n) − 1 cos(𝜋∕n) − cos(2𝜔∕n) R

(6.104)

179

SCATTERING AT CONES: FOCAL FIELD

[

i tan2 𝜔(1 − ei2kl ) 4ka ( ) ] ikR sin(𝜋∕n) 1 1 i2kl e + + e . n cos(𝜋∕n) − 1 cos(𝜋∕n) − cos(2𝜔∕n) R

uh = u0 a

(6.105)

In the limiting case when 𝜔 → 𝜋∕2 and the front part of the object (Fig. 6.9) transforms into the disk, the equations above for the backscattered field are reduced to (

ika (1∕n) sin(𝜋∕n) 1 𝜋 + + cot 2 cos(𝜋∕n) − 1 2n n

)

eikR , R

(6.106)

( ) ika (1∕n) sin(𝜋∕n) 1 𝜋 eikR uh = u0 a − + − cot 2 cos(𝜋∕n) − 1 2n n R

(6.107)

us = u0 a

with n = 3∕2 + Ω∕𝜋. Finally, with Ω → 𝜋∕2, these expressions transform into ( us = u0

ika2 a − 2 2

( uh = −u0

)

ika2 a + 2 2

eikR , R

)

eikR , R

(6.108)

(6.109)

which coincide exactly with Equations (6.86) and (6.87) for the field scattered back from the disk. Additional references to scattering at semi-infinite cones (perfectly reflecting, semitransparent, and impedance cones) can be found in work of Bernard et al. (2008) and Lyalinov and Zhu (2013). 6.3.2

Numerical Analysis of Backscattering

For the field represented in the form us,h = u0 Φs,h

eikR , R

(6.110)

the scattering cross-section is defined according to Equation (1.26) by 2 𝜎s,h = 4𝜋 ||Φs,h || .

(6.111)

We calculated the normalized scattering cross-section norm = 𝜎s,h

𝜎s,h 𝜋a2

(6.112)

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AXIALLY SYMMETRIC SCATTERING OF ACOUSTIC WAVES AT BODIES OF REVOLUTION

norm ). Note that this normalized on the decibel scale, that is, the quantity 10 log(𝜎s,h scattering cross-section is different from that used in Section 6.2.3 for the disk problem. According to Section 6.3.1, the PO predicts the following approximation:

| i |2 tan2 𝜔(1 − ei2kl ) − tan 𝜔 ei2kl | . 𝜎 (0) = 𝜎s(0) = 𝜎h(0) = 𝜋a2 || | | 2ka

(6.113)

In the limiting case when 𝜔 → 𝜋∕2, 𝜎 (0) = 𝜋a2 (ka)2 .

(6.114)

Together with the contribution from the nonuniform scattering sources j(1) , the s,h total backscattering cross-section equals | i 𝜎s = 𝜋a2 || tan2 𝜔(1 − ei2kl ) | 2ka ( ) |2 2 sin(𝜋∕n) 1 1 | − − ei2kl | , | n cos(𝜋∕n) − 1 cos(𝜋∕n) − cos(2𝜔∕n) |

(6.115)

| i tan2 𝜔(1 − ei2kl ) 𝜎h = 𝜋a2 || | 2ka ( ) |2 2 sin(𝜋∕n) 1 1 | + ei2kl | + | n cos(𝜋∕n) − 1 cos(𝜋∕n) − cos(2𝜔∕n) |

(6.116)

with n = 1 + (𝜔 + Ω)∕𝜋. In the limiting case when 𝜔 → 𝜋∕2, | (2∕n) sin(𝜋∕n) 1 𝜋 |2 𝜎s = 𝜋a2 ||ika + + cot || , cos(𝜋∕n) − 1 n n| |

(6.117)

| (2∕n) sin(𝜋∕n) 1 𝜋 |2 − cot || 𝜎h = 𝜋a2 ||−ika + cos(𝜋∕n) − 1 n n| |

(6.118)

with n = 3∕2 + Ω∕𝜋. Finally, when both 𝜔 and Ω are equal to 𝜋∕2 and the cone transforms into the disk, 𝜎s = 𝜋a2 | ika − 1 |2 ,

(6.119)

𝜎h = 𝜋a2 | ika + 1 |2 .

(6.120)

The normalized backscattering cross-section (6.112) of the cone is analyzed numerically as the function of three variables: the length l, the angle 𝜔, and the

SCATTERING AT CONES: FOCAL FIELD

181

Figure 6.10 Backscattering at a cone: dependence on the cone length. According to Equation (6.95), the PO curve also represents the scattering of electromagnetic waves from a perfectly conducting cone.

angle Ω. The MATLAB codes for these calculations are given in the Appendix to Chapter 6. Some results are presented on the decibel scale as follows:

r For the calculation of Equation (6.112) as a function of the length l, the variables

were set as 𝜔 = 45◦ , Ω = 90◦ , a = l tan 𝜔 = l, and 10 ≤ kl < 30. In this case, 1.5𝜆 < l < 4.8𝜆 and 3𝜆 < 2a < 9.6𝜆. The results are demonstrated in Figure 6.10. As is shown here, the data for the hard cone are higher than those for the soft cone. The difference between them is about 15 dB. The PO data are approximately in the middle. r For the calculation of Equation (6.112) as a function of the angle 𝜔, the variables were set as ka = 3𝜋, 10◦ ≤ 𝜔 ≤ 90◦ , and Ω = 90◦ . In this case, 2a = 3𝜆 and 0 ≤ l ≤ 8.5𝜆. The results are plotted in Figure 6.11. A big difference can be observed between the soft and hard data here, at about 40 dB for narrow cones. r For the calculation of Equation (6.112) as a function of the angle Ω (Fig. 6.9), the variables were set as 𝜔 = 10◦ , ka = 3𝜋, kl ≃ 17𝜋, and 0◦ ≤ Ω ≤ 𝜋 − 𝜔 = 170◦ . In this case, 2a = 3𝜆 and l ≃ 8.5𝜆. The results are plotted in Figure 6.12. The PO approximation does not depend on the angle Ω, which is why it is represented here by a straight horizontal line. The difference between the soft and hard data is about 42 to 57 dB. The influence of the cone base shape approaches 11 dB for the soft cone and 16 dB for the hard cone.

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AXIALLY SYMMETRIC SCATTERING OF ACOUSTIC WAVES AT BODIES OF REVOLUTION

Figure 6.11 Backscattering at a cone: dependence on the vertex angle 𝜔. According to Equation (6.95), the PO curve also represents the scattering of electromagnetic waves from a perfectly conducting cone.

Figure 6.12 Backscattering at a cone: dependence on the angle Ω. According to Equation (6.95), the PO curve also represents the scattering of electromagnetic waves from a perfectly conducting cone.

BACKSCATTERED FOCAL FIELDS

183

ρ ω

nˆ

θ ρ θ

ρ=a

Ω

R2

uinc

z l Figure 6.13 Generatrix of the body of revolution.

6.4 BODIES OF REVOLUTION WITH NONZERO GAUSSIAN CURVATURE: BACKSCATTERED FOCAL FIELDS In this section we study symmetrical scattering at bodies of revolution whose illuminated side is an arbitrary smooth convex surface with nonzero Gaussian curvature. A generatrix of such a surface and related geometry are shown in Figure 6.13. The incident plane wave (6.1) propagates in the positive direction of the z-axis, which represents the symmetry axis of a scattering object. We use two systems of coordinates: cylindrical coordinates 𝜌, 𝜑, z and spherical coordinates r, 𝜗, 𝜑. The generatrix is given as a function 𝜌 = 𝜌(z). It is assumed that d2 𝜌∕dz2 ≠ 0 for the illuminated side (0 ≤ z ≤ l) of the object. This condition ensures that the Gaussian curvature of this surface is not zero. We also utilize the following notation related to the edge points (𝜌 = a): d𝜌∕dz = tan 𝜔 for the illuminated side (z = l − 0) and d𝜌∕dz = − tan Ω for the shadowed side (z = l + 0). The shadowed side is an arbitrary smooth surface with 0 ≤ Ω ≤ 𝜋 − 𝜔. In the limiting case Ω = 𝜋 − 𝜔, the scattering object is an infinitely thin (but still perfectly reflecting) screen 𝜌 = 𝜌(z) with 0 ≤ z ≤ l. The principal radii of curvature of the scattering surface are determined according to the differential geometry (Bronshtein and Semendyaev, 1985):

R1 =

{1 + [𝜌′ (z)]2 }3∕2 |𝜌′′ (z)|

and

√ R2 = 𝜌(z) 1 + [𝜌′ (z)]2 ,

(6.121)

where 𝜌′ = d𝜌(z)∕dz and 𝜌′′ = d2 𝜌(z)∕dz2 . The radius R1 relates to the normal section of the surface by the plane (𝜌, z). The radius R2 relates to the orthogonal normal section, which is not the plane z = const, but it is tilted to this plane under the angle 𝜃 (Fig. 6.13). As 𝜌′ = tan 𝜃 with 𝜔 ≤ 𝜃 ≤ 𝜋∕2, the principal radii can be represented as R1 =

1 |𝜌′′ (z)| cos3 𝜃

and

R2 =

𝜌(z) . cos 𝜃

(6.122)

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AXIALLY SYMMETRIC SCATTERING OF ACOUSTIC WAVES AT BODIES OF REVOLUTION

The radius R2 is shown in Figure 6.13. The Gaussian curvature is determined by |𝜌′′ (z)| 1 | cos4 𝜃. =| R1 R2 𝜌(z)

kG =

(6.123)

The expressions above for R1 , R2 , and kG become indefinite at the point z = 0. To disclose these indefinitenesses, one can use the alternative expressions √ 𝜌 1 + [z′ (𝜌)]2 R2 = , z′ (𝜌)

{1 + [z′ (𝜌)]2 }3∕2 , R1 = z′′ (𝜌)

(6.124)

where z = z(𝜌), z′ = dz(𝜌)∕d𝜌, and z′′ = d2 z(𝜌)∕d𝜌2 . It follows from these equations that at the point 𝜌 = z = 0 where z′ (𝜌) = 0, R1 = R2 =

1 . z′′ (𝜌)

(6.125)

The equality of two principal radii means that the vertex point𝜌 = z = 0 of the scattering surface is an umbilic point. 6.4.1

PO Approximation

This is the field radiated by the uniform components (1.31) of the scattering sources ikz j(0) s = iu0 2knz e

and

j(0) = u0 2eikz . h

(6.126)

According to the differential geometry (Bronshtein and Semendyaev, 1985), n̂ = x̂ √

cos 𝜓 1 + [𝜌′ (z)]2

+ ŷ √

sin 𝜓

𝜌′ (z) − ẑ √ 1 + [𝜌′ (z)]2 1 + [𝜌′ (z)]2

(6.127)

or n̂ = x̂ cos 𝜃 cos 𝜓 + ŷ cos 𝜃 sin 𝜓 − ẑ sin 𝜃.

(6.128)

Here, we use the letter 𝜓 for the polar coordinate of the scattering surface and retain the letter 𝜑 for the polar coordinate of the field point. The scattered field in the far zone is determined by Equations (1.16) and (1.17), where one should set √ ds = 𝜌(z) 1 + (𝜌′ )2 dz d𝜓,

r′ sin 𝜗′ = 𝜌(z),

r′ cos 𝜗′ = z,

̂ = ∇r = x̂ sin 𝜗 cos 𝜑 + ŷ sin 𝜗 sin 𝜑 + ẑ cos 𝜗. m

(6.129) (6.130)

BACKSCATTERED FOCAL FIELDS

185

Due to the axial symmetry of the scattered field, it is sufficient to calculate the field only in the meridian plane 𝜑 = 𝜋∕2. Taking this into account, one has ̂ ⋅ n̂ = m

sin 𝜗 sin 𝜓 − 𝜌′ (z) cos 𝜗 , √ 1 + [𝜌′ (z)]2

(6.131)

l

u(0) s = u0

ik eikR eikz(1−cos 𝜗) 𝜌(z)𝜌′ (z) dz ∫0 2𝜋 R ∫0

u(0) = −u0 h

e−ik𝜌(z) sin 𝜗 sin 𝜓 d𝜓,

(6.132)

l

ik eikR eikz(1−cos 𝜗) 𝜌(z) dz 2𝜋 R ∫0 2𝜋

×

2𝜋

∫0

−ik𝜌(z) sin 𝜗 sin 𝜓

e

(6.133)

[sin 𝜗 sin 𝜓 − 𝜌 (z) cos 𝜗] d𝜓. ′

It follows from this equations that in the forward focal direction (𝜗 = 0), (0) u(0) s = uh = u0

ika2 eikR . 2 R

(6.134)

For the backscattering direction (𝜗 = 𝜋), the expressions (6.132) and (6.133) are reduced to (0) u(0) s = −uh = u0 ik

l

eikR ei2kz 𝜌(z)𝜌′ (z) dz. R ∫0

(6.135)

By integrating by parts, one obtains the following asymptotic estimations: (0) u(0) s = −uh = −u0

( )] ikR [ 1 e 1 𝜌(0)𝜌′ (0) − 𝜌(l)𝜌′ (l)ei2kl + O . 2 k R

(6.136)

The indefiniteness for the first term in brackets is disclosed by these manipulations: d𝜌(z) dz 𝜌 1 1 = ′′ . = lim 2 = lim 𝜌→0 dz(𝜌)∕d𝜌 𝜌→0 d z(𝜌)∕d𝜌2 z (0)

𝜌(0)𝜌′ (0) = lim 𝜌(z) z→0

(6.137)

In view of Equation (6.125), this quantity determines the radius of curvature of the scattering surface at the point z = 0. Now Equation (6.136) can be written as

u(0) s

=

−u(0) h

[ ] ikR 1 1 i2kl e = −u0 − a tan 𝜔 e . 2 z′′ (0) R

(6.138)

Here the first term represents the ordinary ray reflected from the vertex of the body, and the second term is the first-order focal field generated by the uniform components

186

AXIALLY SYMMETRIC SCATTERING OF ACOUSTIC WAVES AT BODIES OF REVOLUTION

j(0) of the scattering sources induced near the circular edge (z = l). Indeed, due to s,h Equations (6.125) and (6.138), the backscattering cross-section (1.26) related to the reflection from the body vertex equals 𝜎=𝜋

1 = 𝜋R21,2 [z′′ (0)]2

(6.139)

and agrees totally with Equation (1.27). We note that according to Equations (1.120) and (1.121), Equations (6.138) and (6.139) are also valid for electromagnetic waves scattered from perfectly conducting objects. 6.4.2

Total Backscattered Focal Field: First-Order PTD Asymptotics

In this approximation, the total scattered field in the direction 𝜗 = 𝜋 is found by summation of its components (6.138) and (6.41), (6.42) generated by the uniform and j(1) , respectively. Note that the functions and nonuniform scattering sources j(0) s,h s,h f (1) and g(1) in Equations (6.41) and (6.42) are determined for the case 𝜗 = 𝜋 by Equations (6.102) and (6.103). The summation results in the following asymptotic expressions: [ ( ) ] ikR u0 1 2 sin(𝜋∕n) e 1 1 −a − , us = − ei2kl 2 z′′ (0) n cos(𝜋∕n) − 1 cos(𝜋∕n) − cos(2𝜔∕n) R

(6.140) )] ikR [ ( u0 1 2 sin(𝜋∕n) e 1 1 i2kl +a + e uh = 2 z′′ (0) n cos(𝜋∕n) − 1 cos(𝜋∕n) − cos(2𝜔∕n) R

(6.141) with n = 1 + (𝜔 + Ω)∕𝜋, where 0 ≤ 𝜔 ≤ 𝜋∕2 and 0 ≤ Ω ≤ 𝜋 − 𝜔. In the next two sections we consider the backscattering from two specific bodies of revolution. 6.4.3

Backscattering from Paraboloids

The PO approximations for acoustic and electromagnetic fields in this problem are identical. Focal asymptotics for the acoustic and electromagnetic fields are different.

Asymptotics for the Scattered Field The illuminated surface of the scattering object is a paraboloid with the generatrix given by the equation 𝜌2 (z) = 2pz,

(6.142)

187

BACKSCATTERED FOCAL FIELDS

where 0 ≤ z ≤ l and l = a2 ∕(2p). Differentiation of Equation (6.142) gives 𝜌(z)𝜌′ (z) = p. At the point z = l one has 𝜌(l) = a,𝜌′ (l) = tan 𝜔, and p = a tan 𝜔. Hence, the paraboloid length equals l = 12 a cot 𝜔. The focus of a paraboloid is located at the point z = p∕2. According to Equation (6.125), the focal parameter p equals the radius of the paraboloid curvature at the vertex point z = 0. The shape of the shadowed side of the scattering object and its other geometrical characteristics are shown in Figure 6.13. Due to Equations (6.138) and (6.140), (6.141), the focal backscattered field is described as follows. The PO part of the field equals (0) u(0) s = −uh = −

u0 eikR (p − a tan 𝜔 ei2kl ) . 2 R

(6.143)

The first-order PTD approximations for the total field is determined by [ ( ) ] ikR u0 2 sin(𝜋∕n) 1 1 i2kl e p−a − e , us = − 2 n cos(𝜋∕n) − 1 cos(𝜋∕n) − cos(2𝜔∕n) R (6.144) [ ( ) ] ikR u0 2 sin(𝜋∕n) 1 1 i2kl e uh = p+a + e 2 n cos(𝜋∕n) − 1 cos(𝜋∕n) − cos(2𝜔∕n) R (6.145) with n = 1 + (𝜔 + Ω)∕𝜋. Comparison with the electromagnetic PO field scattered by a perfectly conducting paraboloid [Equation (2.5.2) in Ufimtsev (2003, 2009)] reveals the following relationships:

Ex(0) = u(0) s

if

Exinc = uinc ,

Hy(0) = u(0) h

if

Hyinc = uinc .

(6.146)

This result is in complete agreement with the general relationships (1.120) and (1.121). Taking into account that p = a tan 𝜔, the expressions above can be rewritten as ikR a (0) i2kl e = −u = −u ) tan 𝜔 (1 − e , (6.147) u(0) 0 s h 2 R [ ( ) ] ikR 2 sin(𝜋∕n) 1 1 a i2kl e − e , us = −u0 tan 𝜔 − 2 n cos(𝜋∕n) − 1 cos(𝜋∕n) − cos(2𝜔∕n) R

[

uh = u 0

2 sin(𝜋∕n) a tan 𝜔 + 2 n

(

1 1 + cos(𝜋∕n) − 1 cos(𝜋∕n) − cos(2𝜔∕n)

(6.148)

)

] ei2kl

eikR . R

(6.149)

188

AXIALLY SYMMETRIC SCATTERING OF ACOUSTIC WAVES AT BODIES OF REVOLUTION

These equations are useful for investigation of the continuous transformation of the paraboloid into the flat disk when 𝜔 → 𝜋∕2. Utilizing the relationship l = a2 ∕2p = (a cot 𝜔)∕2, one can show that in the limiting case 𝜔 = 𝜋∕2, (0) u(0) s = −uh = u0

ika2 eikR 2 R

(6.150)

and us = u0

a 2

uh = −u0

( ika +

a 2

1 𝜋 (2∕n) sin(𝜋∕n) cot + n n cos(𝜋∕n) − 1

( ika +

)

1 𝜋 (2∕n) sin(𝜋∕n) cot − n n cos(𝜋∕n) − 1

eikR , R

)

eikR R

(6.151)

(6.152)

with n = 3∕2 + Ω∕𝜋. The distinguishing feature of the PO field (6.147) is its oscillations with the pure zeros corresponding to parameters kl = m𝜋, where m = 1, 2, 3, …. Other properties of the scattered field are illustrated in the next section. Numerical Analysis of Backscattering Here we calculate the normalized scattering cross-section (6.112), which is determined in terms of the scattered field by Equations (6.110) and (6.111). According to the section foregoing “Asymptotics for the Scattered Field,” one can derive the following expressions for the scattering crosssection. The physical optics approximation is given by |2 | 𝜎s(0) = 𝜎h(0) = 𝜋a2 |tan 𝜔(1 − ei2kl )| | |

(6.153)

and the first-order PTD by ( ) | |2 2 sin(𝜋∕n) 1 1 | 𝜎s = 𝜋a |tan 𝜔 − − ei2kl | , | | n cos(𝜋∕n) − 1 cos(𝜋∕n) − cos(2𝜔∕n) | | (6.154) 2|

( ) | |2 2 sin(𝜋∕n) 1 1 | 𝜎h = 𝜋a |tan 𝜔 + + ei2kl | | | n cos(𝜋∕n) − 1 cos(𝜋∕n) − cos(2𝜔∕n) | | (6.155) 2|

with n = 1 + (𝜔 + Ω)∕𝜋.

BACKSCATTERED FOCAL FIELDS

189

When l → 0 and 𝜔 → 𝜋∕2, the expressions above transform into 𝜎 (0) = 𝜋a2 (ka)2 ,

(6.156) |2

| 1 𝜋 (2∕n) sin(𝜋∕n) | 𝜎s = 𝜋a2 ||ika + cot + , n n cos(𝜋∕n) − 1 || |

(6.157)

2 | 1 𝜋 (2∕n) sin(𝜋∕n) || 𝜎h = 𝜋a2 ||ika + cot − n n cos(𝜋∕n) − 1 || |

(6.158)

with n = 3∕2 + Ω∕𝜋. Here we present three types of calculation similar to those in Section 6.3.2 for the cone. The first type consists of calculation for conformal paraboloids, which differ by their length; the second type relates to the transformation of paraboloids into the disk, and the third type reveals the influence of the shadowed base of paraboloids on backscattering. The quantity calculated is the normalized scattering cross-section (6.112). The MATLAB codes for these calculations are given in the Appendix to Chapter 6. Some results are presented on the decibel scale as follows:

r In the study of conformal paraboloids, the focal parameter is set constant (kp = 3𝜋 tan 14◦ ) and the length of paraboloids changes in the interval 6𝜋 ≤ kl ≤ 36. In this case, the radius of the paraboloid base and its length are in the intervals 1.5𝜆 ≤ a ≤ 2𝜆

and

3𝜆 ≤ l ≤ 5.8𝜆.

For a given frequency (k = const), the focal parameter p is constant for all paraboloids with different l. This condition actually means that all these paraboloids are just the different sections of the same semi-infinite paraboloid. This is why they are called conformal. According to the relationship l = a2 ∕2p √ = (a cot 𝜔)∕2, the angle 𝜔 (Fig. 6.13) is determined by the equation cot 𝜔 = 2kl∕ka. For the given values of kl, this angle is in the interval 10.24◦ ≤ 𝜔 ≤ 14◦ . For the angle Ω (Fig. 6.13), we take its value as 90◦ . The results of calculations are plotted in Figure 6.14. The difference between the soft and hard data approaches 16 to 19 dB. Figure 6.14 demonstrates the rough PO data, which do not depend at all on the boundary conditions and are totally incorrect in the vicinity of minima. r The next topic is the transformation of a paraboloid into a disk. In this process each intermediate shape between the initial parabolid and the final disk is a paraboloid whose focal parameter p depends on its length l. It follows from the equation l = a2 ∕2p that p = a2 ∕2l. To find the angle 𝜔 (Fig. 6.13), we use the additional equation p = a tan 𝜔 and obtain tan 𝜔 = a∕2l or tan 𝜔 = ka∕2kl. For the parameter ka, we take the constant value ka = 3𝜋, which does not depend on the paraboloid length. In this case, the diameter of all the intermediate

190

AXIALLY SYMMETRIC SCATTERING OF ACOUSTIC WAVES AT BODIES OF REVOLUTION

Figure 6.14 Backscattering at conformal paraboloids of different lengths (with constant focal parameter p). According to Equation (6.146), the PO curve also represents scattering of electromagnetic waves from a perfectly conducting paraboloids.

paraboloids and the final disk equals 2a = 3𝜆. For the initial paraboloid (which is transformed into the disk), we take kl = 6𝜋, that is. l = 3𝜆. The results are plotted in Figure 6.15. r Now we consider the influence of the paraboloid shadowed base on backscattering. The illuminated part of the object under investigation is a paraboloid with parameters ka = 3𝜋 and kl = 6𝜋 when the base diameter and the length of the paraboloid are equal: 2a = l = 3𝜆. The critical parameter of the base, which affects the edge wave, is the angle Ω. It changes from zero to Ω = 𝜋 − 𝜔, where 𝜔 = tan−1 (ka∕2kl) ≃ 14◦ . In the limiting case Ω = 𝜋 − 𝜔, the scattering object transforms into a perfectly reflecting infinitely thin screen. The results of this investigation are shown in Figure 6.16. The PO approximation does not depend on the shape of the shadowed part of the paraboloid. For the chosen parameter, kl = 6𝜋, it predicts the zero value for the scattered field. On the decibel scale it equals minus infinity and is outside the figure area. As shown in the figure, backscattering from a soft paraboloid depends only a little on the angle Ω (the change of scattering cross-section is about 3 dB), although a strong dependence is observed for the hard paraboloid (about 20 dB).

BACKSCATTERED FOCAL FIELDS

191

Figure 6.15 Transformation of the paraboloid into the disk with continuous maintenance of the paraboloidial shape. According to Equation (6.146), the PO curve also represents the scattering of electromagnetic waves from perfectly conducting paraboloids.

Figure 6.16 Influence of the base shape on backscattering.

192

AXIALLY SYMMETRIC SCATTERING OF ACOUSTIC WAVES AT BODIES OF REVOLUTION

ρ nˆ

ω

ρ=a

Ω a

b

ω

uinc

z

l

Figure 6.17 Generatrix of a spherical segment with an arbitrary shadowed base.

6.4.4

Backscattering from Spherical Segments

The PO approximations for acoustic and electromagnetic fields in this problem are identical. Focal asymptotics for the total acoustic and electromagnetic fields are different.

Asymptotics for the Scattered Field The illuminated surface of the scattering object is a spherical segment whose generatrix is shown in Figure 6.17 and is given by the equation z(𝜌) = b −

√

b2 − 𝜌2

with 0 ≤ z ≤ l,

(6.159)

where b is the radius of the spherical surface. It follows from Equation (6.159) that z′ (𝜌) =

dz(𝜌) 𝜌 = cot 𝜃, =√ d𝜌 b2 − 𝜌2

z′′ (𝜌) =

d2 z(𝜌) b2 = , d𝜌2 (b2 − 𝜌2 )3∕2

(6.160) z′′ (0) =

1 . b

(6.161)

The angle 𝜃(z) in Equation (6.160) is displayed in Figure 6.13. At the point z = l, 𝜌 = a, this angle equals 𝜃(l) = 𝜔. For the given quantities band a, the angle 𝜔 and the segment length l are defined by √ b2 − a2 tan 𝜔 = a

√ and

l=b−

b2 − a2 .

(6.162)

But for the 𝜔 and a given, the segment radius b and length l are determined as b=

a , cos 𝜔

l=a

1 − sin 𝜔 . cos 𝜔

(6.163)

193

BACKSCATTERED FOCAL FIELDS

The last relationships are helpful for investigation of the continuous transformation of the spherical segment into a flat disk when 𝜔 → 𝜋∕2, l → 0, and a = const. is determined According to Equations (6.138), (6.161), and (6.163), the field u(0) s,h by ikR 1 (0) i2kl e ) u(0) s = −uh = −u0 2 (b − a tan 𝜔 e R

(6.164)

or by (0) u(0) s = −uh = −u0

eikR a (1 − sin 𝜔 ei2kl ) . 2 cos 𝜔 R

(6.165)

Comparison with the electromagnetic PO field scattered by a perfectly conducting spherical segment [Equation (2.6.4) in Ufimtsev (2003, 2009)] reveals the following relationships: Ex(0) = u(0) s

if Exinc = uinc

Hy(0) = u(0) h

if Hyinc = uinc .

(6.166)

This result is in complete agreement with the general relationships (1.120) and (1.121). + In view of Equations (6.140), (6.141), (6.161), and (6.163), the field us,h = u(0) s,h u(1) is described by s,h [ ( ) ] ikR 2 sin(𝜋∕n) 1 1 1 i⋅2kl e − e , us = −u0 b − a 2 n cos(𝜋∕n) − 1 cos(𝜋∕n) − cos(2𝜔∕n) R (6.167) [ ( ) ] ikR 2 sin(𝜋∕n) 1 1 1 i⋅2kl e uh = u0 b + a + e , 2 n cos(𝜋∕n) − 1 cos(𝜋∕n) − cos(2𝜔∕n) R (6.168) or by us = −u0

[ ( ) ] ikR 2 sin(𝜋∕n) 1 1 1 a e − − ei⋅2kl , 2 cos 𝜔 n cos(𝜋∕n) − 1 cos(𝜋∕n) − cos(2𝜔∕n) R [

uh = u 0

2 sin(𝜋∕n) 1 a + 2 cos 𝜔 n

(

1 1 + cos(𝜋∕n) − 1 cos(𝜋∕n) − cos(2𝜔∕n)

(6.169)

)

] ei⋅2kl

eikR R

(6.170) with n = 1 + (𝜔 + Ω)∕𝜋 where 0 ≤ 𝜔 ≤ 𝜋∕2 and 0 ≤ Ω ≤ 𝜋 − 𝜔.

194

AXIALLY SYMMETRIC SCATTERING OF ACOUSTIC WAVES AT BODIES OF REVOLUTION

In the limiting case when the spherical segment transforms continuously into the flat disk (𝜔 → 𝜋∕2 and l → 0), Equations (6.164), (6.165) and (6.167), (6.168) are reduced exactly to Equations (6.150), (6.151), and (6.152), respectively. Numerical Analysis of Backscattering In this section we calculate the normalized scattering cross-section (6.112), taking into account Equations (6.110) and (6.111). According to the preceding section, the following expressions for an ordinary scattering cross-section are valid. The PO approximation is given by | 1 |2 − tan 𝜔 ei⋅2kl || 𝜎s(0) = 𝜎h(0) = 𝜋a2 || | cos 𝜔 |

(6.171)

and the first-order PTD by ( ) | 1 |2 2 sin(𝜋∕n) 1 1 i⋅2kl | − − e | , 𝜎s = 𝜋a | | cos 𝜔 | n cos(𝜋∕n) − 1 cos(𝜋∕n) − cos(2𝜔∕n) | | (6.172) 2|

( ) | 1 |2 2 sin(𝜋∕n) 1 1 | | 𝜎h = 𝜋a2 | + + ei⋅2kl | | cos 𝜔 | n cos(𝜋∕n) − 1 cos(𝜋∕n) − cos(2𝜔∕n) | | (6.173) with n = 1 + (𝜔 + Ω)∕𝜋. Two types of calculation are presented below. The MATLAB codes for these calculations are given in the Appendix to Chapter 6. Some results are presented on the decibel scale as follows:

r The first is the continuous transformation of the spherical segment into the flat disk in the limiting case 𝜔 → 𝜋∕2. It is assumed that all transition surfaces are spherical with the curvature radius b = a∕ cos 𝜔. The initial object is given by the parameters ka = 3𝜋, kl0 ≃ 2.5𝜋, 𝜔0 = 10◦ , and Ω = 90◦ . In terms of the wavelength, the base radius and the length of the spherical element are equal to a = 1.5𝜆 and l0 ≃ 1.26𝜆. The numerical results for the normalized scattering cross-section are plotted in Figure 6.18. It shows clearly the influence of the ) concentrated in nonuniform/fringe component of the scattering sources (j(1) s,h the vicinity of the edge. r In the next calculation we investigate the influence on backscattering of the shadowed part of the spherical segment. The critical parameter of this part is the angle Ω shown in Figure 6.17. It changes from zero to 𝜋 − 𝜔. In the limiting case, when Ω = 𝜋 − 𝜔, the scattering object transforms into the perfectly reflecting infinitely thin screen. The illuminated spherical part of the segment is determined by the parameters 𝜔 = 10◦ , ka = 3𝜋, and kl ≃ 2.5𝜋. In terms of the wavelength, a = 1.5𝜆 and l ≃ 1.26𝜆. The results are plotted in Figure 6.19.

195

BACKSCATTERED FOCAL FIELDS

Figure 6.18 Transformation of the spherical segment into the flat disk. According to Equation (6.166), the PO curve also represents the scattering of electromagnetic waves from a perfectly conducting spherical segment.

Normalized Scattering Cross-Section

10 9 8 7

Backscattering at a Spherical Object

6 5 PO PTD SOFT PTD HARD

4 3 2 1 0

0

20

40

60

80

100

120

140

160

Ω (degress)

Figure 6.19 Influence of the shadowed part of the spherical segment on backscattering. According to Equation (6.166), the PO curve also represents the scattering of electromagnetic waves from a perfectly conducting spherical segment. (Produced by F. Hacivelioglu and L. Sevgi.)

196

AXIALLY SYMMETRIC SCATTERING OF ACOUSTIC WAVES AT BODIES OF REVOLUTION

The PO approximation does not depend on the shape of the shadowed part and is represented in Figure 6.19 by the horizontal straight line. However, according to PTD, backscattering increases up to 1.5 dB for an acoustically soft object and up to 7.5 dB for an acoustically hard object. 6.5 BODIES OF REVOLUTION WITH NONZERO GAUSSIAN CURVATURE: AXIALLY SYMMETRIC BISTATIC SCATTERING The geometry of the problem is illustrated in Figure 6.20. The incident plane wave (6.1) propagates in the positive direction of the z-axis, which is the symmetry axis of a scattering body of revolution. The generatrix of the illuminated side of this body is given by the equation 𝜌 = 𝜌(z) with 0 ≤ z ≤ l under the condition d2 𝜌∕dz2 ≠ 0. This condition ensures that the Gaussian curvature of this surface is not zero. The shadowed side is an arbitrary smooth surface with 0 ≤ Ω ≤ 𝜋 − 𝜔. In the limiting case Ω = 𝜋 − 𝜔, the scattering object is an infinitely thin perfectly reflecting screen with 𝜌 = 𝜌(z) and 0 ≤ z ≤ l. The tangent to the generatrix forms the angle 𝜃 with the z-axis. At the edge points z = l − 0, this angle equals 𝜃(l) = 𝜔 = tan−1 [d𝜌(l)∕dz]. The principal radii of curvature (R1 , R2 ) are defined in Equations (6.121) and (6.122), and the Gaussian curvature is given by Equation (6.123). The unit normal n̂ to the illuminated surface is defined in Equation (6.128). Due to the axial symmetry of the problem, it is sufficient to calculate the scattered field only in the meridian plane 𝜑 = 𝜋∕2. 6.5.1

Ray Asymptotics for the PO Field

These asymptotics can be derived from the general integral expressions (6.132) and (6.133) under the condition k𝜌 sin 𝜗 ≫ 1. First we consider the observation points in the region 𝜋 − 𝜔 < 𝜗 ≤ 𝜋, where the entire illuminated surface of the scattering object is visible (Fig. 6.21). The integrals in Equations (6.132) and (6.133) over the variable 𝜓 are calculated using the stationary-phase technique (Copson, 1965; Murray, 1984). The details of this method were explained briefly in Section 6.1.2. The phase function in these ρ ω

θ

nˆ

θ ρ θ

ρ=a

Ω

R2

uinc

z l Figure 6.20

Generatrix of a body of revolution.

AXIALLY SYMMETRIC BISTATIC SCATTERING

197

ϑ = 2ω

y

ρ ω ρ=a 1

Ω

ϑ = π –ω

uinc

ϑ =π

z z=l

Ω

2

ω

Figure 6.21 Cross-section of a body of revolution by the plane y0z.

integrals has two stationary points, 𝜓1 = 𝜋∕2 and 𝜓2 = 3𝜋∕2. Asymptotic evaluation of these integrals leads to the following expressions: u(0) s = u0 √

eikR 2𝜋k sin 𝜗 R ik

[ × ei𝜋∕4

l

∫0

√ eikΦ1 (z) 𝜌(z)𝜌′ (z) dz + e−i𝜋∕4

l

∫0

] √ eikΦ2 (z) 𝜌(z)𝜌′ (z) dz

eikR ik u(0) = −u0 √ h 2𝜋k sin 𝜗 R { l √ eikΦ1 (z) 𝜌(z)[sin 𝜗 − 𝜌′ (z) cos 𝜗] dz × ei𝜋∕4 ∫0 } l √ eikΦ2 (z) 𝜌(z)[sin 𝜗 + 𝜌′ (z) cos 𝜗] dz , − e−i𝜋∕4 ∫0

(6.174)

(6.175)

198

AXIALLY SYMMETRIC SCATTERING OF ACOUSTIC WAVES AT BODIES OF REVOLUTION

where Φ1 (z) = z(1 − cos 𝜗) − 𝜌(z) sin 𝜗,

(6.176)

Φ2 (z) = z(1 − cos 𝜗) + 𝜌(z) sin 𝜗.

(6.177)

Here the integrals with the factor exp[Φ1 (z)] present a field generated by the vicinity of the stationary line 𝜓 = 𝜋∕2, 0 < z ≤ l, and the integrals with exp[Φ2 (z)] describe a field generated by the vicinity of the stationary line 𝜓 = 3𝜋∕2, 0 < z ≤ l (Fig. 6.21). Now we check the functions Φ1 (z) and Φ2 (z) for the presence of stationary points zst . It follows from the equation Φ′1 (z) = 1 − cos 𝜗 − 𝜌′ (zst ) sin 𝜗 = 0

(6.178)

that 𝜌′ (zst ) =

d𝜌 𝜗 = tan 𝜃(zst ) = tan dz 2

(6.179)

𝜗 . 2

(6.180)

and 𝜃(zst ) =

This equation determines the reflection point zst on the scattering surface shown in Figure 6.20. At this point, the tangent to the generatrix 𝜌(z) forms the angle 𝜃 = 𝜗∕2 with the z-axis, which agrees with the reflection law. We then check the function Φ2 (z)for the stationary point. It follows from the equation Φ′2 (z) = 1 − cos 𝜗 + 𝜌′ (zst ) sin 𝜗 = 0

(6.181)

that d𝜌 𝜗 = tan 𝜃(zst ) = − tan dz 2

(6.182)

𝜗 2

(6.183)

𝜌′ (zst ) = and 𝜃(zst ) = −

with

− 𝜋 < 𝜗 < 0.

This stationary point relates to the reflected ray in the meridian plane 𝜑 = 3𝜋∕2. For the same value zst in Equations (6.180) and (6.183), this ray is exactly symmetrical to the reflected ray shown in Figure 6.20. As we consider the scattered field only in the meridian plane 𝜑 = 𝜋∕2, the function Φ2 (z) does not have any stationary points for the scattering directions in this plane. By introducing into Equations (6.174) and (6.175) a new integration variable 𝜉 = z for the integrals with function Φ1 (z), and 𝜉 = −z for the integrals with function Φ2 (z), one can represent their sum as l

I(P) =

∫−l

F(𝜉, P)eikΦ(𝜉,P) d𝜉,

(6.184)

AXIALLY SYMMETRIC BISTATIC SCATTERING

199

where P denotes the location of the observation point. For high frequency of the field (when k ≫ 1), the factor exp[ikΦ(𝜉, P)], being a function of the integration variable 𝜉, undergoes fast oscillations. Because of that, most differential contributions F(𝜉, P)eikΦ(𝜉,P) d𝜉 to the integral I(P) cancel each other asymptotically. Only those in the vicinity of the stationary point 𝜉st and of the endpoints 𝜉 = −l and 𝜉 = l provide substantial contributions to I(P). The contribution of the stationary point is calculated using the stationary-phase technique, and the contributions by the endpoints are found by integrating by parts (Copson, 1965; Murray, 1984). The resulting asymptotic approximation for I(P) is given by √ 2𝜋 F(𝜉st , P)eikΦ(𝜉st ,P)+i𝜋∕4 I(P) ∼ kΦ′′ (𝜉st , P) [ ] F(−l, P) ikΦ(−l,P) 1 F(l, P) ikΦ(l,P) + − ′ e e . (6.185) ik Φ′ (l, P) Φ (−l, P) The first term in Equation (6.185) represents the contribution from the stationary point, and the remainder of the terms provide contributions from the endpoints. Only the dominant asymptotic terms for each contribution are retained here. in the form The procedure outlined was used to represent the scattered field u(0) s,h of three contributions: = u(0) (z ) + u(0) (1) + u(0) (2), u(0) s,h s,h st s,h s,h

(6.186)

where (0) u(0) s (zst ) = −uh (zst ) = −u0

1√ eikR R1 (zst )R2 (zst ) eikΦ1 (zst ) 2 R

(6.187)

and u0 a (0) u(0) s (1) + us (2) = √ 2𝜋ka sin 𝜗 × [f (0) (1)e−ika sin 𝜗+i𝜋∕4 + f (0) (2)eika sin 𝜗−i𝜋∕4 ]

eikR ikl(1−cos 𝜗) , e R (6.188)

u a u(0) (1) + u(0) (2) = √ 0 h h 2𝜋ka sin 𝜗 × [g(0) (1)e−ika sin 𝜗+i𝜋∕4 + g(0) (2)eika sin 𝜗−i𝜋∕4 ]

eikR ikl(1−cos 𝜗) . e R (6.189)

The functions u(0) (z ) describe the ordinary ray reflected at the stationary point zst s,h st determined by Equations (6.179) and (6.180). The quantities R1 and R2 , the principal radii of curvature of the scattering surface, are defined by Equations (6.121) and (6.122).

200

AXIALLY SYMMETRIC SCATTERING OF ACOUSTIC WAVES AT BODIES OF REVOLUTION

Expressions (6.188) and (6.189) determine the sum of two edge-diffracted rays diverging from edge points 1 and 2 shown in Figure 6.21. The directivity patterns of these rays are defined by the functions

6.5.2

f (0) (1) =

sin 𝜔 , cos 𝜔 − cos(𝜔 − 𝜗)

f (0) (2) =

sin 𝜔 , cos 𝜔 − cos(𝜔 + 𝜗)

(6.190)

g(0) (1) =

sin(𝜔 − 𝜗) , cos 𝜔 − cos(𝜔 − 𝜗)

g(0) (2) =

sin(𝜔 + 𝜗) . cos 𝜔 − cos(𝜔 + 𝜗)

(6.191)

Bessel Interpolations for the PO Field in the Region 𝝅 − 𝝎 ≤ 𝝑 ≤ 𝝅

With the application of relationships (6.45) and (6.46), the ray asymptotics above can be written in the form [[ √ 1 − = u R1 (zst )R2 (zst ) eikΦ1 (zst ) u(0) 0 s 2 a + {f (0) (1)[J0 (ka sin 𝜗) − iJ1 (ka sin 𝜗)] 2 ]] eikR , (6.192) + f (0) (2)[J0 (ka sin 𝜗) + iJ1 (ka sin 𝜗)]} eikl(1−cos 𝜗) R [[ √ 1 = u R1 (zst )R2 (zst ) eikΦ1 (zst ) u(0) 0 h 2 a + {g(0) (1)[J0 (ka sin 𝜗) − iJ1 (ka sin 𝜗)] 2 ]] eikR , (6.193) + g(0) (2)[J0 (ka sin 𝜗) + iJ1 (ka sin 𝜗)]} eikl(1−cos 𝜗) R where J0 and J1 are the Bessel functions. These asymptotics are valid away from the focal line (𝜗 = 𝜋) under the condition ka sin 𝜗 ≫ 1. The focal field is described by the asymptotic (6.138), which can be rewritten as [ √ ] ikR 1 a (0) i2kl e − = −u = u R (0)R (0) + tan 𝜔 e , u(0) 0 1 2 s h 2 2 R

(6.194)

where R1 (0) = R2 (0) = 1∕z′′ (0). When 𝜗 → 𝜋, the asymptotics (6.192) and (6.193) transform exactly into the focal asymptotics (6.194). Therefore, the expressions (6.192) and (6.193) can be considered as the appropriate approximations valid in the entire region 𝜋 − 𝜔 ≤ 𝜗 ≤ 𝜋. 6.5.3

Bessel Interpolations for the PTD Field in the Region 𝝅 − 𝝎 ≤ 𝝑 ≤ 𝝅

generated by the The PTD field consists of the sum of the PO field and the field u(1) s,h of the scattering sources. The components j(1) caused nonuniform components j(1) s,h s,h by the smooth bending of the scattering surface generate a far field on the order of k−1 (Schensted, 1955), and those caused by the sharp edge create a field of the order k−1∕2 [as shown in Equations (6.21) and (6.22)]. Therefore, in the first-order

201

AXIALLY SYMMETRIC BISTATIC SCATTERING

PTD approximation, one can retain only the dominant contributions generated by edge-type sources j(1) . The uniform asymptotics for these contributions in the region s,h 𝜋 − 𝜔 ≤ 𝜗 ≤ 𝜋 are given by the expressions (6.47) and (6.48), where one should include the additional factor exp[ikl(1 − cos 𝜗)], due to the shift in the coordinates’ origin. By summation of the modified Equations (6.47) and (6.48) with the PO asymptotics (6.192) and (6.193), we obtain [[ √ 1 = u0 − R1 (zst )R2 (zst ) eikΦ1 (zst ) uPTD s 2 a + {f (1)[J0 (ka sin 𝜗) − iJ1 (ka sin 𝜗)] 2 + f (2)[J0 (ka sin 𝜗) + iJ1 (ka sin 𝜗)]} eikl(1−cos 𝜗) [[ √ 1 R1 (zst )R2 (zst ) eikΦ1 (zst ) 2 a + {g(1)[J0 (ka sin 𝜗) − iJ1 (ka sin 𝜗)] 2

]]

eikR , R

(6.195)

eikR . R

(6.196)

= u0 uPTD h

+ g(2)[J0 (ka sin 𝜗) + iJ1 (ka sin 𝜗)]} eikl(1−cos 𝜗)

]]

The functions f (1, 2) and g(1, 2) are defined by Equations (6.25), (6.26), and (6.30), (6.31), where one should omit the last terms, which are canceled exactly by the terms f (0) (1, 2) and g(0) (1, 2) during the summation of modified Equations (6.47) and (6.48) with Equations (6.192) and (6.193). 6.5.4 Asymptotics for the PTD Field in the Region 2𝝎 < 𝝑 ≤ 𝝅 − 𝝎 Away from the GO Boundary𝝑 = 2𝝎 In this observation region, the stationary edge point 2 (Fig. 6.21) is not visible (when Ω < 2𝜔), and its first-order contribution to the scattered field equals zero. We also assume that the observation directions are far from the last GO ray (reflected at point 1 and shown in Fig. 6.21), where the functions f (1) and g(1)are singular. Under these conditions, we can use the obvious modifications of Equations (6.195) and (6.196) for the scattered field: [[ √ 1 = u R1 (zst )R2 (zst ) eikΦ1 (zst ) uPTD 0 − s 2 ]] ikR e a + f (1)[J0 (ka sin 𝜗) − iJ1 (ka sin 𝜗)] eikl(1−cos 𝜗) , (6.197) 2 R = u0 uPTD h

[[ √ 1 R1 (zst )R2 (zst ) eikΦ1 (zst ) 2

a + g(1)[J0 (ka sin 𝜗) − iJ1 (ka sin 𝜗)] eikl(1−cos 𝜗) 2

]]

eikR . R

(6.198)

202

AXIALLY SYMMETRIC SCATTERING OF ACOUSTIC WAVES AT BODIES OF REVOLUTION

6.5.5 Uniform Approximations for the PO Field in the Ray Region 2𝝎 ≤ 𝝑 ≤ 𝝅 − 𝝎, Including the GO Boundary 𝝑 = 2𝝎 The ray asymptotics (6.197) and (6.198) are not applicable in the vicinity of the geometrical optics boundary 𝜗 = 2𝜔, where the wave field does not have a ray structure. In this “transition region”, the process of the transverse diffusion of the wave field happens to actually give birth to diffracted rays (Ufimtsev, 2003, 2009). The singularities of the functions f (1) and g(1) and the singularity of the factor 1∕Φ′ (l) in Equation (6.185) for the direction 𝜗 → 2𝜔 [in this case, 𝜉st → l and Φ′ (l, P) → 0] provide mathematical evidence of the existence of this process. For calculation of the field integral (6.184) in this region, we should use a more accurate stationary-phase method that allows the approach of the stationary phase to the endpoint (Felsen and Marcuvitz, 1994). Next we present the basics of this technique. First, we modify the canonical integral (6.184). For the sake of simplicity, the symbol P (related to coordinates of the observation point) is omitted. We notice that edge point 2 (Fig. 6.21) is not visible, and therefore its first-order contribution to the scattered field equals zero. In the integral (6.184), this point corresponds to the endpoint 𝜉 = −l. To exclude its contribution, we set 𝜉 = −∞for the lower limit of integration. Then we introduce a new variable t by the equation Φ(𝜉) = Φ(𝜉st ) + t2 .

(6.199)

Notice that according to Equation (6.176), the second derivative, Φ′′ (𝜉st ) = −𝜌′′ (𝜉st ) sin 𝜗, is positive and therefore the quantity Φ(𝜉st ) is the minimum of the function Φ(𝜉). Taking this into account, we define the variable t as the continuous and differentiable function of the old variable 𝜉: {√ t(𝜉) =

Φ(𝜉) − Φ(𝜉st ) √ − Φ(𝜉) − Φ(𝜉st )

for

𝜉 ≥ 𝜉st ,

for

𝜉 ≤ 𝜉st ,

(6.200)

where the radical is understood in the arithmetic sense. In the vicinity of the stationary point, where Φ′ (𝜉st ) = 0, one can use the Taylor approximations 1 ′′ 1 Φ (𝜉st )(𝜉 − 𝜉st )2 + Φ′′′ (𝜉st )(𝜉 − 𝜉st )3 + ⋯ , 2! 3! √ [ ] ′′′ 1 Φ (𝜉st ) 1 ′′ ) Φ (𝜉st ) 1 + (𝜉 − 𝜉 t(𝜉) = (𝜉 − 𝜉st ) st . 2 6 Φ′′ (𝜉st )

Φ(𝜉) = Φ(𝜉st ) +

(6.201) (6.202)

Now the canonical integral (6.184) can be represented in the form t(l)

I = eikΦ(𝜉st )

∫−∞

2

eikt G(t) dt,

(6.203)

AXIALLY SYMMETRIC BISTATIC SCATTERING

203

where d𝜉 2t(𝜉) = F(𝜉) ′ , dt Φ (𝜉) √ 2t(𝜉) 2 F(𝜉st ). G(0) = lim ′ F(𝜉) = 𝜉→𝜉st Φ (𝜉) Φ′′ (𝜉st ) G(t) = F(𝜉)

(6.204) (6.205)

The next idea is to extract (in the explicit form!) the Fresnel integral from Equation (6.203). It is accomplished by use of a simple procedure: { ikΦ(𝜉st )

I=e

t(l)

G(0)

∫−∞

ikt2

e

t(l)

dt +

∫−∞

ikt2

e

} [G(t) − G(0)] dt

(6.206)

or √ kt(l)

1 I = √ eikΦ(𝜉st ) G(0) ∫−∞ k

2

eix dx +

( ) G[t(l)] − G(0) ikΦ(l) 1 + O 2 . (6.207) e i2kt(l) k

√ Under the condition kt(l) ≫ 1, when the observation point is far from the geometrical optics boundary 𝜗 = 2𝜔, this expression is reduced asymptotically to the first two terms in Equation (6.185). When the observation point approaches the boundary (𝜗 = 2𝜔 + 0 and t = +0), one should utilize Equations (6.201) and (6.202) and the additional approximations 1 1 = Φ′ (𝜉) (𝜉 − 𝜉st )Φ′′ (𝜉st ) 2t(𝜉) = Φ′ (𝜉)

√

2 Φ′′ (𝜉st )

} ′′′ 1 Φ (𝜉st ) 2 ) + O[(𝜉 − 𝜉 ) ] , (𝜉 − 𝜉 st st 2 Φ′′ (𝜉st )

(6.208)

} { ′′′ 1 Φ (𝜉st ) 2 ) + O[(𝜉 − 𝜉 ) ] , 1− (𝜉 − 𝜉 st st 3 Φ′′ (𝜉st )

(6.209)

{ 1−

F(𝜉) = F(𝜉st ) + F ′ (𝜉st )(𝜉 − 𝜉st ) + ⋯ , √ G(t) − G(0) = (𝜉 − 𝜉st )

[ ] Φ′′′ (𝜉st ) 1 2 ′ (𝜉 ) − ) F F(𝜉 . st st Φ′′ (𝜉st ) 3 Φ′′ (𝜉st )

(6.210) (6.211)

These relationships lead to the following value of the canonical integral at the boundary 𝜗 = 2𝜔 + 0: √

I=

𝜋 F(l)eikΦ(l)+i𝜋∕4 2kΦ′′ (l) [ ] ( ) Φ′′′ (l) ikΦ(l) 1 1 1 ′ . + (l) − + O F F(l) e ikΦ′′ (l) 3 Φ′′ (l) k2

(6.212)

204

AXIALLY SYMMETRIC SCATTERING OF ACOUSTIC WAVES AT BODIES OF REVOLUTION

This technique is applied further to calculation of the PO field: u(0) s = u0 √

ik 2𝜋k sin 𝜗

ei𝜋∕4 Is

eikR , R

ikR ik i𝜋∕4 e e = −u I , u(0) √ 0 h h R 2𝜋k sin 𝜗

(6.213)

(6.214)

where Is,h are defined by Equation (6.203) with Fs (𝜉) =

√

𝜌(𝜉)𝜌′ (𝜉),

Fh (𝜉) =

Φ(𝜉) = 𝜉(1 − cos 𝜗) − 𝜌(𝜉) sin 𝜗.

√

𝜌(𝜉)[sin 𝜗 − 𝜌′ (𝜉) cos 𝜗],

(6.215) (6.216)

We omit all intermediate routine manipulations and obtain the final approximations (neglecting terms of order k−2 ): √ { k t(l) 2 eikR ei3𝜋∕4 √ (0) ikΦ(zst ) R1 (zst )Rst (zst ) e eix dx us = u0 √ ∫−∞ R 2 𝜋 ] [ } √ R1 (zst )R2 (zst ) ikΦ(l)+i𝜋∕4 a (0) e f (1) − + √ , (6.217) √ 4 𝜋k t(l) 2𝜋ka sin 𝜗 √ { kt(l) ikR i3𝜋∕4 √ 2 e e ikΦ(zst ) = u R (z )R (z ) e eix dx − u(0) √ 0 1 st st st h ∫ R −∞ 2 𝜋 [ } ] √ R1 (zst )R2 (zst ) ikΦ(l)+i𝜋∕4 a (0) g (1) + + √ , (6.218) e √ 4 𝜋k t(l) 2𝜋ka sin 𝜗 √ where t(l) = Φ(l) − Φ(zst ); the functions f (0) (1) and g(0) (1) are defined in Equations (6.190) and (6.191), and R1,2 , the principal radii of curvature of the scattering surface, are defined in Equation (6.121). √ Far from the geometrical optics boundary ( kt(l) ≫ 1), Equations (6.217) and (6.218) transform into [ ] a eikR 1√ (0) ikΦ(zst ) (0) ikΦ(l)+i𝜋∕4 f (1)e R1 (zst )Rst (zst ) e +√ − , us = u0 R 2 2𝜋ka sin 𝜗 (6.219) u(0) h

eikR = u0 R

[

a 1√ g(0) (1)eikΦ(l)+i𝜋∕4 R1 (zst )Rst (zst ) eikΦ(zst ) + √ 2 2𝜋ka sin 𝜗

]

(6.220)

AXIALLY SYMMETRIC BISTATIC SCATTERING

205

These expressions totally agree with Equations (6.186) to (6.189) (keeping in mind that edge point 2 is not visible in the region 2𝜔 ≤ 𝜗 < 𝜋 − 𝜔). The first terms in Equations (6.219) and (6.220) are the ordinary reflected rays, and the second terms are the edge-diffracted rays. Exactly on the geometrical optics boundary (𝜗 = 2𝜔 + 0), Equations (6.217) and (6.218) are reduced to

u(0) h

{

1√ R1 (l)R2 (l) 4 } [ ] ′′′ (l) Φ 1 ei𝜋∕4 1 + √ Fs′ (l) − Fs (l) ′′ eikΦ(l) , ′′ 3 Φ (l) 2𝜋k sin 𝜗 Φ (l) { eikR 1 √ = u0 R1 (l)R2 (l) R 4 } [ ] Φ′′′ (l) ikΦ(l) 1 ei𝜋∕4 1 ′ −√ Fh (l) − Fh (l) ′′ e . ′′ 3 Φ (l) 2𝜋k sin 𝜗) Φ (l)

u(0) s = u0

eikR R

−

(6.221)

(6.222)

Note that all the derivatives in these expressions are taken with respect to the variable 𝜉 in the integral (6.184), and the subscripts s and h indicate the type (soft or hard) of scattering object. The first-order PTD approximation for the scattered field in this region can be generated by found by summation of the PO field (6.217), (6.218), with the field u(1) s,h nonuniform scattering sources j(1) . For the field u(1) , one can use Equations (6.49) s,h s,h and (6.50), which are not singular at the boundary of reflected rays 𝜗 = 2𝜔. Taking into account the different locations of the coordinates’ origin used in Section 6.1 and can be written as here, the field u(1) s,h ikR a (1) ikl(1−cos 𝜗) e u(1) , s = u0 2 f (1)[J0 (ka sin 𝜗) − iJ1 (ka sin 𝜗)]e R

(6.223)

a eikR = u0 g(1) (1)[J0 (ka sin 𝜗) − iJ1 (ka sin 𝜗)]eikl(1−cos 𝜗) u(1) , h 2 R

(6.224)

where the functions f (1) and g(1) are as defined in Section 6.1.2. Here we note again that edge point 2 is not visible in the region 2𝜔 ≤ 𝜗 < 𝜋 − 𝜔 if Ω < 2𝜔. 6.5.6 Approximation of the PO Field in the Shadow Region for Reflected Rays The ordinary rays reflected from the scattering surface do not exist in the shadow region 0 ≤ 𝜗 < 2𝜔 (Fig. 6.21). This circumstance can be used to obtain a helpful approximation for the PO fields (6.174) and (6.175), which otherwise are difficult to calculate. Indeed, according to Equation (1.70), the PO field consists of two

206

AXIALLY SYMMETRIC SCATTERING OF ACOUSTIC WAVES AT BODIES OF REVOLUTION

parts: the reflected field and the shadow radiation. The reflected field [defined by the integral (1.71)] contains all reflected rays and an additional diffracted field that can be neglected in the shadow region, where the basic component of the scattered field is the shadow radiation (1.72). This observation leads directly to the approximation ≈ ush = u(0) s,h

1 4𝜋 ∫Sil

( uinc

𝜕 eikr 𝜕uinc eikr − 𝜕n r 𝜕n r

) ds,

(6.225)

where the integration is performed over the illuminated side of the scattering object. This integral in general is also difficult to calculate. However, the shadow contour theorem developed in Section 1.3.5 greatly simplifies the calculation. According to this theorem, the integral (6.225) is identical to the integral over the black disk located in the plane z = l and having the radius a. The field scattered by the black disk can be represented in the form (1.74): (0) ush = 12 [u(0) s + uh ].

(6.226)

(0) In the far-field approximation, the quantities u(0) s and uh were as calculated in Section 6.2.1. Utilizing Equations (6.56) and (and (6.57) and taking into account the shift of the coordinates’ origin accepted in the present section, the field (6.226) can be written as

ush = u0

eikR ikl(1−cos 𝜗) ia 1 + cos 𝜗 . J1 (ka sin 𝜗) e 2 sin 𝜗 R

(6.227)

It is seen that this field really concentrates in the shadow region. It is zero in the backscattering direction (𝜗 = 𝜋), but it is large and exactly equal to the PO field generated by the perfectly reflecting disk in the shadow direction 𝜗 = 0: ush = u0

ika2 eikR . 2 R

(6.228)

The first-order PTD field in the shadow region can be found by the summation of field generated by the edge sources j(1) and determined the field (6.227) with the u(1) s,h s,h in Section 6.1.4. When utilizing the results of that section, one should include the additional factor exp[ikl(1 − cos 𝜗)] in Equations (6.47) to (6.50) because of the above-mentioned origin shift. This section completes our analysis of the field scattered by the bodies of revolution with nonzero Gaussian curvature. Note that the Appendix to Chapter 6 contains MATLAB codes for numerical calculations of backscattering from bodies of revolution.

PROBLEMS

207

PROBLEMS 6.1

Verify the asymptotic approximation (6.17) for the integral from a function with two stationary points. Solution

Consider the integral 2𝜋

I(𝜓) =

∫0

e−ikΨ(𝜓

′ ,𝜓)

F(𝜓 ′ , 𝜓) d𝜓 ′

with two stationary points (𝜓1 and 𝜓2 ) in the interval (0, 2𝜋). Under the condition k ≫ 1 and according to the stationary-phase technique, this integral asymptotically equals [ ] 2 ik d Ψ(𝜓1 , 𝜓) ′ 2 exp − (𝜓 − 𝜓 ) d𝜓 ′ 1 ∫−∞ 2 d𝜓 ′2 [ ] ∞ 2 ik d Ψ(𝜓2 , 𝜓) ′ −ikΨ(𝜓2 ,𝜓) 2 + F(𝜓2 , 𝜓)e exp − (𝜓 − 𝜓2 ) d𝜓 ′ . ∫−∞ 2 d𝜓 ′2

I(𝜓) ∼ F(𝜓1 , 𝜓)e−ikΨ(𝜓1 ,𝜓)

∞

Taking into account the equality ∞

∫−∞

2

e±ipx dx =

√

𝜋 ±i(𝜋∕4) e p

with p > 0

and specific functions 𝜓 and F, one obtains the approximation (6.17). 6.2

Derive Equations (6.25) and (6.26) for functions f (1) (1) and g(1) (1) related to stationary point 1 (Fig. 6.3 in the canonical problem). Solution

According to Equations (4.22) and (4.23), f (1) (1) = f (𝜑1 , 𝜑01 , 𝛼) − f (0) (𝜑1 , 𝜑01 ), g(1) (1) = g(𝜑1 , 𝜑01 , 𝛼) − g(0) (𝜑1 , 𝜑01 ).

The functions f , g and f (0) , g(0) are defined by Equations (2.62), (2.64), and (3.59), respectively. Utilizing the definition of angles 𝜑1 and 𝜑01 shown in Figure 6.3, one obtains Equations (6.25) and (6.26). 6.3

Derive Equations (6.30) and (6.31) for the functions f (1) (2) and g(1) (2) related to stationary point 2 (Fig. 6.4). Solution The functions f , g and f (0) , g(0) are defined by Equations (2.62), (2.64), and (3.59), respectively. Utilizing the definition of angles 𝜑2 and 𝜑02 shown in Figure 6.4, we obtain Equations (6.30) and (6.31).

208

AXIALLY SYMMETRIC SCATTERING OF ACOUSTIC WAVES AT BODIES OF REVOLUTION

6.4 Verify Equations (6.34) and (6.35) for the functions f (1) (1) and g(1) (1) related to the specular direction 𝜗 = 2𝜔 (in the canonical problem). Solution The functions f , g and f (0) , g(0) are singular in the specular direction (𝜑 = 𝜋 − 𝜑0 ); however, their singularities cancel each other. The resulting expressions for functions f (1) , and g(1) are given in Equations (4.29). According to Figure 6.3, 𝜑01 = 𝜔. Substitution of this angle into Equations (4.29) provides the solution. 6.5 Derive the PO approximation (6.64) for the electromagnetic field scattered from a perfectly conducting disk illuminated by the plane wave (Fig. 6.6): inc = E0y eikz , Eyinc = −Z0 H0x

E0y = −Z0 H0x .

Hxinc = H0x eikz ,

Solution Start with Equations (1.108), (1.109), and (1.112). Take into account ⃗ m = 0. The electric vector-potential contains only one component: that A a

A(0) y =−

1 eikR r′ dr′ H0x ∫0 2𝜋 R ∫0

2𝜋

e−ikr

′ sin 𝜗 sin 𝜑′

d𝜑′ .

According to Equation (6.55), A(0) y = −H0x

a eikR J (ka sin 𝜗). k sin 𝜗 R 1

(0) The components A(0) 𝜑 and A𝜗 are determined as (0) A(0) 𝜑 = Ay cos 𝜑,

A(0) = A(0) y sin 𝜑 cos 𝜗. 𝜗

Subsequent substitution of A(0) into Equations (1.108) and (1.109) leads to 𝜑,𝜗 Equations (6.64). 6.6 Derive the focal asymptotics (6.86) and (6.87) for the disk diffraction problem. Solution us,h = u(0) + u(1) . s,h s,h The quantities u(0) are determined by Equations (6.58) and (6.59) and the s,h quantities u(1) by Equations (6.81) and (6.82). Their sum provides the solution. s,h 6.7 Use Equations (6.104) and (6.105) for the focal field scattered by a cone and prove their limiting forms (6.106) and (6.107) when a cone transforms into a disk. In this case 𝜔 → 𝜋∕2 and l = a cot 𝜔 → 0.

PROBLEMS

Solution

209

Use the following approximations:

2l = 2a cot 𝜔 = 2a tan

(

) 𝜋 − 𝜔 → a(𝜋 − 2𝜔) → 0, 2

1 − ei⋅2kl → −i2ka cot 𝜔 + 2(ka cot 𝜔)2 , ( ) 𝜋 𝜋 − 2𝜔 2 tan 𝜔 = cot − 𝜔 = cot → , 2 2 𝜋 − 2𝜔 1 ika ika 1 i tan2 𝜔(1 − ei⋅2kl ) → tan 𝜔 + → + , 4ka 2 2 2 𝜋 − 2𝜔 (1∕n) sin(𝜋∕n) (1∕n) sin{[(𝜋 + 2𝜔)∕2n] − [(2𝜔 − 𝜋)∕2n]} = cos(𝜋∕n) − cos(2𝜔∕n) 2 sin[(𝜋 + 2𝜔)∕2n] sin[(2𝜔 − 𝜋)∕2n] 𝜋 1 1 cot + , 2n n 2𝜔 − 𝜋 (1∕n) sin(𝜋∕n) 1 𝜋 1 ei2kl → − cot + (1 + i2kl) cos(𝜋∕n) − cos(2𝜔∕n) 2n n 2𝜔 − 𝜋 →−

→−

𝜋 1 1 cot + − ika. 2n n 2𝜔 − 𝜋

Substitution of these approximations into Equations (6.104) and (6.105) provides the solution. 6.8

Equations (6.148) and (6.149) determine the field scattered by a paraboloid. Show that these expressions transform into Equations (6.151) and (6.152) when 𝜔 → 𝜋∕2, l = 12 a cot 𝜔 → 0, and the paraboloid transforms continuously into a disk. Use approximations similar to those in Problem 6.8. Solution (

) 𝜋 2 −𝜔 → , 2 𝜋 − 2𝜔 ( ) 𝜋 1 2l = a cot 𝜔 = a tan − 𝜔 → a (𝜋 − 2𝜔) → 0, 2 2 (2∕n) sin(𝜋∕n) 2 𝜋 2 ei2kl → − cot + (1 + i2kl) cos(𝜋∕n) − cos(2𝜔∕n) 2n n 2𝜔 − 𝜋

tan 𝜔 = cot

𝜋 2 1 − ika. → − cot + n n 2𝜔 − 𝜋 Substitution of these approximations into Equations (6.148) and (6.149) provides the solution.

7 Elementary Acoustic and Electromagnetic Edge Waves This chapter is based on papers by Butorin and Ufimtsev (1986), Butorin et al. (1987), and Ufimtsev (1989, 1991, 2006a,b).

The relationships dus = dEt

if

uinc (𝜁 ) = Etinc (𝜁 ),

duh = dHt

if

uinc (𝜁 ) = Htinc (𝜁 )

exist between acoustic and electromagnetic elementary edge waves (EEWs) propagating in directions that belong to the diffraction cone. Here ̂t is the tangent to the scattering edge at the diffraction point 𝜁 . The relationships du(0,1) = s (0,1) dEt are valid only in the absence of polarization coupling in the PO field.

In previous sections it was demonstrated that edge-diffracted waves provide a significant contribution to a scattered field. By themselves these waves represent the linear superposition of EEWs generated in a certain vicinity of infinitesimal elements of the scattering edge; that is, u=

∫L

du(𝜁 ).

Fundamentals of the Physical Theory of Diffraction, Second Edition. Pyotr Ya. Ufimtsev. © 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

211

212

ELEMENTARY ACOUSTIC AND ELECTROMAGNETIC EDGE WAVES

Here L denotes the edge of the scattering object and 𝜁 is the curvilinear coordinate measured along the edge and associated with its length (d𝜁 = dl). It is supposed that the curvature radius of the edge L is large in terms of the wavelength and that it can change slowly along the edge. The angle between faces of the edge can also change along the edge. The integrand du(𝜁 ) stands for the field of an EEW. Our goal is to derive high-frequency asymptotics for EEWs. Having obtained them, one can calculate the edge waves arising due to diffraction at the large class of objects with arbitrary smoothly curved edges. To achieve this goal, we utilize the asymptotic localization principle. According to this principle, the nonuniform/fringe component j(1) (of the scattering surface sources) induced near the edge is asymptotically (with k → ∞) equivalent to the component j(1) can induced on the canonical wedge tangential to the real edge. To be able to understand the appropriate tangency point and the appropriate vicinity of the point responsible for the radiation of EEWs, we should appeal to the physical structure of edge waves diffracted at the canonical wedge.

7.1 ELEMENTARY STRIPS ON A CANONICAL WEDGE In Chapter 4 it was shown that under the oblique incidence on the wedge, the scattered edge waves have the form of conic waves, which can be interpreted as the edge-diffracted rays distributed over the diffraction cone (Fig. 4.4). Hence, the nonuniform component j(1) can induced on wedge faces is also the ray field running from the edge along the generatrix of this cone. Now it becomes clear how we should choose the appropriate tangency point (of the actual scattering edge L with edge E of the canonical tangential wedge; Fig. 7.1) and how to determine its vicinity responsible for radiation of the EEWs. The appropriate tangency point must be the origin of the diffracted ray coming to the observation point on the tangential wedge. We emphasize that in general the tangency point is not the edge point nearest this observation point! What concerns the appropriate vicinity of the tangency point, it must be an infinitely narrow (elementary) strip oriented along the diffracted ray. In Figure 7.1, that is strip A. This figure also helps us to understand why any other elementary strip (e.g., strip B) is not acceptable. It is seen that the orientation of such a strip is not consistent with

L E

B

A

Figure 7.1 Here L is the edge of the actual scattering object and E is the edge of the canonical tangential wedge. The arrows show the edge-diffracted rays diverging from the edge E. Elementary strips A and B belong to the face of the canonical wedge.

INTEGRALS FOR j(1) ON ELEMENTARY STRIPS s,h

213

incident rays

A

B

C

Figure 7.2 Polygonal facet of a scattering object. Only diffracted rays (dashed lines) coming from the upper edge are shown here and discussed in the text.

the localization principle. Indeed, the field on this strip does not depend on the local properties of the incident wave and the real scattering edge L at the tangency point. Instead, it consists of spurious edge waves and rays coming from fictitious scattering points on auxiliary edge E which do not belong to real scattering edge L. To complete the definition of the elementary strip, we should determine its length. Although the nonuniform/fringe sources j(1) concentrate near the edge, they are distributed over the entire elementary strip up to its infinite end. By integration of j(1) over such a semi-infinite strip, we can find the first-order asymptotics for the EEW. However, sometimes it is reasonable to truncate that part of the elementary strip that is outside the real scattering facet (Michaeli, 1987; Breinbjerg, 1992; Johansen, 1996). A similar truncation procedure was considered in Section 5.1.4. From a physical point of view, the truncation results in the additional edge wave (arising at the truncation points), which can be interpreted as part of the second-order diffraction. In conclusion, we note a special case when the orientation of elementary strips can be arbitrary. Such a situation is possible for truncated strips on polygonal facets illuminated by a plane wave (Fig. 7.2). Sections A and C of the polygonal facet are free of diffracted rays, and section B is filled in continuously by such rays. Two parallel thin lines show the elementary strip. All other elementary strips are parallel to this one and have different lengths because they occupy only section B. The field scattered from the facet (i.e., the field radiated by the nonuniform sources distributed in the region B) is determined by the integral over section B. It is clear that the result of integration does not depend on the shape of subsections of the region B, and therefore does not depend on the orientation of the elementary strips. 7.2

(1) ON ELEMENTARY STRIPS INTEGRALS FOR js,h

First we choose the elementary strips according to the rule formulated in Section 7.1. They are oriented along the edge-diffracted rays and shown in Figure 7.3. The incident plane wave is given by i

uinc = u0 eik𝜙

(7.1)

214

ELEMENTARY ACOUSTIC AND ELECTROMAGNETIC EDGE WAVES

x1

x2

γ0 γ0 β2

R

β1 r

τ2

ζ

φ

ϑ

τ1 γ0

x′

2

t

z

ki 1

x

Figure 7.3 Element of the canonical wedge with two elementary strips oriented along axes x1 and x2 with the origin at point 𝜁 at the edge.

and propagates in the direction k̂ i = ∇𝜙i ≡ grad 𝜙i . In the polar cylindrical coordinates (r, 𝜑, z), this wave is determined by the expression uinc = u0 e−ikz cos 𝛾0 e−ik1 r cos(𝜑−𝜑0 )

(7.2)

with k1 = k sin 𝛾0 , 0 < 𝛾0 < 𝜋, 0 ≤ 𝜑0 ≤ 𝜋, and 0 ≤ 𝜑 ≤ 𝛼. Here 𝛼 is the external angle between faces 1 and 2 of the wedge (𝜋 ≤ 𝛼 ≤ 2𝜋). Notice the relationship 𝛾0 = 𝜋 − 𝛾 between the angle 𝛾0 in Figure 7.3 and the angle 𝛾 in Figure 4.3. For the description of the diffracted field at the observation point, we shall use two additional local coordinate systems: the spherical coordinates R, 𝜗, 𝜑 and the coordinates R, 𝛽1 , 𝛽2 , where the angles 𝛽1,2 are measured from the elementary strips, that is, from the axes x1,2 , and do not exceed 180◦ (0 ≤ 𝛽1,2 ≤ 𝜋). Note that according to the localization principle, the results of the present theory are also applicable in the general case, for an arbitrary incident wave uinc = u0 (z) exp[ik𝜙i (z)] with the angle 𝛾0 defined by the equation k̂ i ⋅ ̂t = cos(𝜋 − 𝛾0 ) = − cos 𝛾0 , where k̂ i = ∇𝜙i .

The local cylindrical coordinates r, 𝜑, z and spherical coordinates R, 𝜗, 𝜑 are introduced at every edge point 𝜁 according to the right-hand rule (with respect to their unit vectors, r̂ × 𝜑̂ = ẑ , R̂ × 𝜗̂ = 𝜑). ̂ Remember that these coordinates are introduced in such a way that the angle 𝜑 is measured from the illuminated face of the edge and the tangent ̂t to the edge is directed along the local polar axis ẑ (̂t = ẑ ). When both faces are illuminated, we can measure the angle 𝜑 from any face, but in this case we should choose the correct direction of the polar axis z and the tangent ̂t (̂z = ̂t = r̂ × 𝜑). ̂ This note is important for correct applications of the theory of EEWs developed here, especially in the case of electromagnetic waves.

INTEGRALS FOR j(1) ON ELEMENTARY STRIPS s,h

215

In accordance with Sections 2.3, 4.1, and 4.3, the incident wave (7.2) generates around the wedge the field us (r, 𝜑) = u0 e−ikz cos 𝛾0 [𝑣(k1 r, 𝜑 − 𝜑0 ) − 𝑣(k1 r, 𝜑 + 𝜑0 )] + us (r, 𝜑),

(7.3)

uh (r, 𝜑) = u0 e−ikz cos 𝛾0 [𝑣(k1 r, 𝜑 − 𝜑0 ) + 𝑣(k1 r, 𝜑 + 𝜑0 )] + uh (r, 𝜑),

(7.4)

go

go

go

where us,h is the geometrical optics field (consisting of the incident and reflected plane waves), and 𝑣(k1 r, 𝜓) = with

1 e−ik1 r cos 𝜂 d𝜂 2𝛼 ∫D 𝑤(𝜂 + 𝜓)

(7.5)

) ( 𝜋 𝑤(x) = 1 − exp i x . 𝛼

(7.6)

The integration counter D is shown in Figure 2.5, where the variable 𝜂 is denoted as 𝛽. induced on the elemenWe are interested in the nonuniform scattering sources j(1) s,h tary strips. In view of Equation (1.11), they are determined as go

j(1) s =

𝜕us (r, 𝜑) 𝜕us (r, 𝜑) − 𝜕n 𝜕n go

= uh (r, 𝜑) − uh (r, 𝜑) j(1) h

at at

𝜑 = 0, 𝛼,

𝜑 = 0, 𝛼.

(7.7) (7.8)

The normal derivatives in Equation (7.7) are defined as 1 𝜕 𝜕 = 𝜕n r 𝜕𝜑

at 𝜑 = 0

and

𝜕 1 𝜕 =− 𝜕n r 𝜕𝜑

at 𝜑 = 𝛼.

(7.9)

The differentiation of the function 𝑣(k1 r, 𝜑 ∓ 𝜑0 ) is carried out by parts: 𝜕 1 𝜕 e−ik1 r cos 𝜂 𝑤−1 (𝜂 + 𝜑 ∓ 𝜑0 ) d𝜂 𝑣(k r, 𝜑 ∓ 𝜑0 ) = 𝜕𝜑 1 2𝛼 ∫D 𝜕𝜑 =

1 𝜕 e−ik1 r cos 𝜂 𝑤−1 (𝜂 + 𝜑 ∓ 𝜑0 ) d𝜂 2𝛼 ∫D 𝜕𝜂

=

−𝜋∕2−i∞ 𝜋∕2+i∞ 1 1 e−ik1 r cos 𝜂 || e−ik1 r cos 𝜂 || + | | 2𝛼 𝑤(𝜂 + 𝜑 ∓ 𝜑0 ) |−3𝜋∕2+i∞ 2𝛼 𝑤(𝜂 + 𝜑 ∓ 𝜑0 ) |3𝜋∕2−i∞

−

ik1 r ik r e−ik1 r cos 𝜂 sin 𝜂 e−ik1 r cos 𝜂 sin 𝜂 d𝜂 = − 1 d𝜂. 2𝛼 ∫D 𝑤(𝜂 + 𝜑 ∓ 𝜑0 ) 2𝛼 ∫D 𝑤(𝜂 + 𝜑 ∓ 𝜑0 ) (7.10)

216

ELEMENTARY ACOUSTIC AND ELECTROMAGNETIC EDGE WAVES

Here the terms related to the ends of the counter D are equal to zero. Note also that according to Figure 7.3, the relationships z = 𝜁 − 𝜉1,2 cos 𝛾0

and

r = 𝜉1,2 sin 𝛾0

(7.11)

are valid for the observation points x1,2 = 𝜉1,2 on the elementary strips along the axes x1,2 . The observations above result in the following integral expressions for j(1) : s,h j(1) s = u0

ik1 −ik(𝜁−𝜉 cos 𝛾 ) cos 𝛾 1 0 0 e−ik1 𝜉1 sin 𝛾0 cos 𝜂 e ∫D 2𝛼

× [𝑤−1 (𝜂 + 𝜑0 ) − 𝑤−1 (𝜂 − 𝜑0 )] sin 𝜂 d𝜂, j(1) = u0 h

(7.12)

1 −ik(𝜁−𝜉1 cos 𝛾0 ) cos 𝛾0 e−ik1 𝜉1 sin 𝛾0 cos 𝜂 e ∫D 2𝛼

× [𝑤−1 (𝜂 + 𝜑0 ) + 𝑤−1 (𝜂 − 𝜑0 )] d𝜂

(7.13)

on the strip 1 (𝜑 = 0), and j(1) s = u0

ik1 −ik(𝜁 −𝜉 cos 𝛾 ) cos 𝛾 2 0 0 e−ik1 𝜉2 sin 𝛾0 cos 𝜂 e ∫D 2𝛼

× [𝑤−1 (𝜂 + 𝛼 − 𝜑0 ) − 𝑤−1 (𝜂 + 𝛼 + 𝜑0 )] sin 𝜂 d𝜂, j(1) = u0 h

(7.14)

1 −ik(𝜁 −𝜉2 cos 𝛾0 ) cos 𝛾0 e−ik1 𝜉2 sin 𝛾0 cos 𝜂 e ∫D 2𝛼

× [𝑤−1 (𝜂 + 𝛼 − 𝜑0 ) + 𝑤−1 (𝜂 + 𝛼 + 𝜑0 )] d𝜂

(7.15)

on the strip 2 (𝜑 = 𝛼). The identity 𝑤(𝜂 + 𝛼 + 𝜑0 ) = 1 − ei(𝜋∕𝛼)(𝜂+𝛼+𝜑0 ) = 1 − ei(𝜋∕𝛼)(2𝛼+𝜂−𝛼+𝜑0 ) = 1 − ei(𝜋∕𝛼)(𝜂−𝛼+𝜑0 ) = 𝑤(𝜂 − 𝛼 + 𝜑0 ) allows one to rewrite Equations (7.14) and (7.15) in the more convenient form j(1) s = u0

ik1 −ik(𝜁 −𝜉 cos 𝛾 ) cos 𝛾 2 0 0 e−ik1 𝜉2 sin 𝛾0 cos 𝜂 e ∫D 2𝛼

× [𝑤−1 (𝜂 + 𝛼 − 𝜑0 ) − 𝑤−1 (𝜂 − 𝛼 + 𝜑0 )] sin 𝜂 d𝜂, j(1) = u0 h

(7.16)

1 −ik(𝜁 −𝜉2 cos 𝛾0 ) cos 𝛾0 e−ik1 𝜉2 sin 𝛾0 cos 𝜂 e ∫D 2𝛼

× [𝑤−1 (𝜂 + 𝛼 − 𝜑0 ) + 𝑤−1 (𝜂 − 𝛼 + 𝜑0 )] d𝜂.

(7.17)

TRIPLE INTEGRALS FOR ELEMENTARY EDGE WAVES

217

Now it is seen that Equations (7.16) and (7.17) related to j(1) on strip 2 follow from s,h (1) Equations (7.12) and (7.13) for js,h on strip 1 after the formal replacement of 𝜑0 by 𝛼 − 𝜑0 and 𝜉1 by 𝜉2 . In the next section, these expressions for j(1) are utilized to s,h calculate the field of the EEWs.

7.3

TRIPLE INTEGRALS FOR ELEMENTARY EDGE WAVES

The field du(1) of the EEW generated by the source j(1) (induced on elementary strip s,h s,h 1, on the face 𝜑 = 0, and on elementary strip 2, on the face 𝜑 = 𝛼) is determined in accordance with Equation (1.10), where the differential element of the surface is defined as ds1,2 = sin 𝛾0 d𝜁 d𝜉1,2 .

In applications of the theory of EEWs, one should remember:

r The differential element of the edge is always positive (d𝜁 > 0). r The coordinate 𝜁 is associated with arc length of the edge. r The tangent ̂t to the edge (which plays the role of a local polar axis ẑ )

is defined as ̂t = d⃗r(𝜁 )∕d𝜁 , where ⃗r(𝜁 ) is the position vector of the edge point 𝜁 .

Thus, (1) (1) du(1) s = du1 + du2

and

du(1) = d𝑣(1) + d𝑣(1) , h 1 2

(7.18)

where sin 𝛾0 ∞ (1) eikr1,2 j d𝜉1,2 , 4𝜋 ∫0 s1,s2 r1,2

(7.19)

sin 𝛾0 ∞ (1) 𝜕 eikr1,2 j d𝜉1,2 . 4𝜋 ∫0 h1,h2 𝜕n r1,2

(7.20)

du(1) = −d𝜁 1,2

d𝑣(1) = d𝜁 1,2

The quantities du(1) , d𝑣(1) and du(1) , d𝑣(1) describe the field 1 1 2 2 √ generated by elementary strips 1 and 2, respectively. The quantities r1 = (x1 − 𝜉1 )2 + h21 and √ r2 = (x2 − 𝜉2 )2 + h22 determine the distances between the observation and integration points (Fig. 7.4).

218

ELEMENTARY ACOUSTIC AND ELECTROMAGNETIC EDGE WAVES

y1 x1

h1

y1

x1

ξ

x1

z1 Figure 7.4 Local Cartesian coordinates (x1 , y1 , z1 ) with axes x1 and z1 placed in wedge face 1 (𝜑 = 0), where the axis y1 is normal to this face (̂y1 = n̂ ). Similar Cartesian coordinates (x2 , y2 , z2 ) are also introduced with axes x2 and z2 in face 2 (𝜑 = 𝛼) and with ŷ 2 = n̂ .

For the Green’s function of the free space, we use Equation (6.616-3) in the book of Gradshteyn and Ryzhik (1994): ∞

i eikr1,2 = eip(x1,2 −𝜉1,2 ) H0(1) (qh1,2 ) dp, r1,2 2 ∫−∞

(7.21)

√ √ where q = k2 − p2 with Im(q) ≥ 0 and h1,2 = y21,2 + z21,2 ≥ 0. The square root q is a two-valued function. To make it a single-valued function we introduce two branch cuts [−∞ ≤ Re(p) ≤ −k and k ≤ Re(p) ≤ ∞] in the complex plane (p). The integration contour in Equation (7.21) is located on the upper side of the left cut and skirts over the branch point p = −k, then follows along the real axis to the right cut and skirts under the branch point p = k. After that, the integration contour follows along the lower side of the right cut. The normal derivatives of the Green’s function are defined by eikr1,2 𝜕 eikr1,2 eikr1,2 𝜕 eikr1,2 ŷ 1,2 = −∇ ŷ 1,2 = − = ∇′ 𝜕n r1,2 r1,2 r1,2 𝜕y1,2 r1,2 =

iy1,2

∞

eip(x1,2 −𝜉1,2 ) qH1(1) (qh1,2 ) dp.

2h1,2 ∫−∞

(7.22)

Note that the differential operator ∇′ (∇) performs the differentiation with respect to the coordinates of the integration (observation) point. The quantities y1,2 are expressed in terms of the spherical coordinates R, 𝜗, 𝜑 as y1 = R sin 𝜗 sin 𝜑

and

y2 = R sin 𝜗 sin(𝛼 − 𝜑),

(7.23)

and the quantities h1,2 are determined in terms of the coordinates R, 𝛽1 , 𝛽2 as h1 = R sin 𝛽1

and

h2 = R sin 𝛽2 .

(7.24)

TRIPLE INTEGRALS FOR ELEMENTARY EDGE WAVES

219

After substituting the sources j(1) and the expressions above into the field formulas s,h (7.19) and (7.20), one obtains = u0 e−ik𝜁 cos 𝛾0 d𝜁 du(1) 1

k sin2 𝛾0 I (x , h , 𝜑 ), 16𝜋𝛼 u 1 1 0

(7.25)

= u0 e−ik𝜁 cos 𝛾0 d𝜁 d𝑣(1) 1

i sin 𝛾0 sin 𝜗 sin 𝜑 I𝑣 (x1 , h1 , 𝜑0 ) 16𝜋𝛼 sin 𝛽1

(7.26)

with Iu (x1 , h1 , 𝜑0 ) =

∞

∫0 ×

I𝑣 (x1 , h1 , 𝜑0 ) =

2𝛾 0

eik𝜉1 cos

∫D

×

∫D

∫−∞

2 𝛾 cos 𝜂 0

e−ik𝜉1 sin

∞

∫0

∞

d𝜉1

2𝛾 0

eik𝜉1 cos

∫−∞

2 𝛾 cos 𝜂 0

e−ik𝜉1 sin

[𝑤−1 (𝜂 + 𝜑0 ) − 𝑤−1 (𝜂 − 𝜑0 )] sin 𝜂 d𝜂, (7.27) ∞

d𝜉1

eip(x1 −𝜉1 ) H0(1) (qh1 ) dp

eip(x1 −𝜉1 ) qH1(1) (qh1 ) dp

[𝑤−1 (𝜂 + 𝜑0 ) + 𝑤−1 (𝜂 − 𝜑0 )] d𝜂

(7.28)

and du(1) = u0 e−ik𝜁 cos 𝛾0 d𝜁 2

k sin2 𝛾0 I (x , h , 𝛼 − 𝜑0 ), 16𝜋𝛼 u 2 2

(7.29)

= u0 e−ik𝜁 cos 𝛾0 d𝜁 d𝑣(1) 2

i sin 𝛾0 sin 𝜗 sin(𝛼 − 𝜑) I𝑣 (x2 , h2 , 𝛼 − 𝜑0 ). 16𝜋𝛼 sin 𝛽2

(7.30)

It may be seen that Equations (7.29) and (7.30) for the fields generated by strip 2 can be obtained from Equations (7.25) and (7.26) for the fields from strip 1 by the formal replacements x1 → x2 ,

h1 → h2 ,

𝛽1 → 𝛽2 .

𝜑0 → 𝛼 − 𝜑0 ,

𝜑 → 𝛼 − 𝜑.

(7.31)

and For this reason, further calculations are carried out only for the fields du(1) 1

; the final expressions for the fields du(1) and d𝑣(1) will be obtained using the d𝑣(1) 1 2 2 relationships (7.31). The next section deals with analytical work on the integrals Iu and I𝑣 .

220

ELEMENTARY ACOUSTIC AND ELECTROMAGNETIC EDGE WAVES

7.4 TRANSFORMATION OF TRIPLE INTEGRALS INTO ONE-DIMENSIONAL INTEGRALS First, we change the order of integration in Equations (7.27) and (7.28): ∞

Iu =

∫−∞

eipx1 H0(1) (qh1 ) dp ∞

×

∫0 ∞

I𝑣 =

∫−∞ ∫0

[𝑤−1 (𝜂 + 𝜑0 ) − 𝑤−1 (𝜂 − 𝜑0 )] sin 𝜂 d𝜂

e−i𝜉1 (p2 +k2 cos 𝜂) d𝜉1 ,

eipx1 qH1(1) (qh1 ) dp ∞

×

∫D

∫D

(7.32) [𝑤−1 (𝜂 + 𝜑0 ) + 𝑤−1 (𝜂 − 𝜑0 )] d𝜂

e−i𝜉1 (p2 +k2 cos 𝜂) d𝜉1 ,

(7.33)

where p2 = p − k cos2 𝛾0

and

k2 = k sin2 𝛾0 .

(7.34)

The integral over the variable 𝜉 is calculated under the condition Im(p2 + k2 cos 𝜂) < 0 to ensure its convergence. Then

Iu =

∞ [𝑤−1 (𝜂 + 𝜑0 ) − 𝑤−1 (𝜂 − 𝜑0 )] sin 𝜂 1 eipx1 H0(1) (qh1 ) dp d𝜂, ∫D i ∫−∞ p2 + k2 cos 𝜂

(7.35)

I𝑣 =

∞ [𝑤−1 (𝜂 + 𝜑0 ) + 𝑤−1 (𝜂 − 𝜑0 )] 1 eipx1 qH1(1) (qh1 ) dp d𝜂. ∫D i ∫−∞ p2 + k2 cos 𝜂

(7.36)

The condition above for the integral convergence is fulfilled for all points on the contour D (with the exception of points 𝜂 = ±𝜋), because of the inequality Im(k2 cos 𝜂) < 0 that is valid there. For the points 𝜂 = ±𝜋 and p ≠ k, the convergence can be achieved by the temporary assumption that the wave number has a small imaginary part, k = k′ + ik′′ with k′ > 0 and 0 < k′′ ≪ 1. This assumption represents a small attenuation of acoustic waves due to a small temporary admitted viscosity of the medium. After the transition to Equations (7.35) and (7.36), this assumption can be omitted. A special situation occurs when 𝜂 = ±𝜋 and point p approaches the point p = k. As explained above, the integration contour in Equation (7.21) skirts under the branch point p = k. This means that under this branch point, the integration point is complex, p = p′ + ip′′ with p′ = k and p′′ = Im(p) < 0. Thus, in this special case, the integral above over the variable 𝜉1 is convergent due to the inequality Im(p) < 0. The next work consists of a thorough analysis of the integral over the counter D. Its integrand possesses poles of two types. There are two poles, 𝜂1 = 𝜑0 and 𝜂2 = −𝜑0

TRANSFORMATION OF TRIPLE INTEGRALS INTO ONE-DIMENSIONAL INTEGRALS

221

[related to the functions 𝑤−1 (𝜂 ∓ 𝜑0 )], and two other poles, 𝜂3 = 𝜎 and 𝜂4 = −𝜎, that are the zeros of the denominator p2 + k2 cos 𝜂 when ( ) p p cos 𝜎 = − 2 and 𝜎 = arccos − 2 . (7.37) k2 k2 It is clear that 𝜎 = 𝜋 − arccos

p2 k2

if

|p2 | ≤ k2 ,

(7.38)

where for the inverse cosine we take its principal values, 0 ≤ arccos(x) ≤ 𝜋. The condition |p2 | ≤ k2 is satisfied for points p on the real axis in the interval k cos(2𝛾0 ) ≤ p ≤ k. In this case, 0 ≤ 𝜎 ≤ 𝜋. To define 𝜎 for the values ||p2 || > k2 , one should use the Euler formula for the cosine function. Together with Equation (7.37), they lead to the equation p 1 i𝜎 (e + e−i𝜎 ) = − 2 . 2 k2

(7.39)

By using the replacement e−i𝜎 = t, it is reduced to the quadratic equation with the solution √ −p2 ± p22 − k22 . (7.40) 𝜎 = i ln k2 √ Now it is necessary to define the appropriate branches for the square root p22 − k22 and for the logarithm function. The square root has two branch points, p2 = ±k2 . To make this function single-valued, we introduce √ two branch cuts (−∞ ≤ p2 ≤ −k2 and k2 ≤ p2 ≤ ∞) and choose the branch where p22 − k22 ≥ 0 for the points on the upper (lower) side of the left (right) branch cut. To define the appropriate sign in front of the square root in Equation (7.40) and to define the appropriate branch of the logarithm, we use the conditions 𝜎 = 𝜋 for p2 = k2 and 𝜎 = 0 for p2 = −k2 . These conditions lead to the function √ −p2 + p22 − k22 𝜎 = i ln , (7.41) k2 where ln(−1) = −i𝜋 and ln(1) = 0. It follows from this equation that √ 𝜎 = 𝜋 + i ln

p2 − √

𝜎 = i ln

−p2 + k2

p22 − k22

k2 p22 − k22

with p2 ≥ k2

(7.42)

with p2 ≤ −k2 .

(7.43)

222

ELEMENTARY ACOUSTIC AND ELECTROMAGNETIC EDGE WAVES

Im(η)

F+

η4 = –σ

η3 = σ

D Re(η )

η2 = –φ0 η1 = φ0

–3π /2

–π

η3 = σ F–

η4 = –σ

D

– π /2

π /2

0

π

3π /2

Figure 7.5 Integration contours in the complex plane (𝜂).

Because the quantity p2 is real and the argument of the logarithm in Equations (7.42) and (7.43) is always positive, we take the regular arithmetic branch for this logarithm, where ln(0) = −∞, ln(1) = 0, and ln(∞) = +∞. Now we can trace the position of pole 𝜎 in the complex plane (𝜂) as the function of the variable p in Equations (7.35) and (7.36). When this variable changes from p = −∞ to p = k cos(2𝛾0 ), the pole 𝜂3 = 𝜎 runs in the complex plane (𝜂) along the imaginary axis Im(𝜂) from 𝜂 = +i∞ to 𝜂 = 0. When point p moves from p = k cos(2𝛾0 ) to p = k, the pole 𝜂3 = 𝜎 runs along the real axis Re(𝜂) from 𝜂 = 0 to 𝜂 = 𝜋. When point p moves from p = k to p = +∞, the pole 𝜂3 = 𝜎 runs down along the vertical line from 𝜂 = 𝜋 to 𝜂 = 𝜋 − i∞. The location of the pole 𝜂3 = 𝜎 in the complex plane (𝜂) is shown in Figure 7.5 by the thick solid line. The short-dashed line shows the location of the pole 𝜂4 = −𝜎. To calculate the integrals over the contour D, we connect its branches with the additional contours F+ and F− (Fig. 7.5), where Im(𝜂) = A and Im(𝜂) = −A, respectively. First, we place these contours at a large finite distance from the real axis (A ≫ 1) and apply the Cauchy residue theorem to the integrals over the closed contour C = D + F+ + F− . Then the results of this theorem are extended to the case when the contours F+ and F− are shifted to the infinite distance (A → ∞). This procedure is realized in the following for the integrals: Ju (C) =

𝑤−1 (𝜂 + 𝜑0 ) − 𝑤−1 (𝜂 − 𝜑0 ) sin 𝜂 d𝜂, ∫C p2 + k2 cos 𝜂

(7.44)

J𝑣 (C) =

𝑤−1 (𝜂 + 𝜑0 ) + 𝑤−1 (𝜂 − 𝜑0 ) d𝜂. ∫C p2 + k2 cos 𝜂

(7.45)

The basic details of this procedure are the same for both integrals. That is why we demonstrate these details only for the integral Ju and then establish the final result for J𝑣 . When all poles of the integrand of Ju are inside the region closed by the contour C = D + F+ + F− , the Cauchy theorem states that Ju (C) = 2𝜋i

4 ∑ m=1

Resm .

(7.46)

TRANSFORMATION OF TRIPLE INTEGRALS INTO ONE-DIMENSIONAL INTEGRALS

223

Here, the quantities Res1 and Res2 are the residues at the poles 𝜂1 = 𝜑0 and 𝜂2 = −𝜑0 : Res1 = Res2 =

𝛼 𝜀(𝜑0 ) sin 𝜑0 i𝜋 p2 + k2 cos 𝜑0

(7.47)

with { 𝜀(x) =

1 0

if if

0 ≤ x ≤ 𝜋, 𝜋 0) and is measured in the positive direction of the local polar axis ẑ = ̂t. The quantity ⃗ (1) (𝜗, 𝜑)]eik𝜙i (𝜁 ) ⃗ (1) (𝜗, 𝜑) + Z0 H0t (𝜁 )G ⃗(1) (𝜗, 𝜑) = [E0t (𝜁 )F

(7.137)

is the directivity pattern of the EEWs, and k𝜙i (𝜁 ) is the phase of the incident wave ⃗ (1) can be interpreted as the differential ⃗ (1) and G at the edge point 𝜁 . The vectors F diffraction coefficients.

240

ELEMENTARY ACOUSTIC AND ELECTROMAGNETIC EDGE WAVES

At this point it is also necessary to repeat the note from Section 7.2: The local cylindrical coordinates r, 𝜑, z and spherical coordinates R, 𝜗, 𝜑 are introduced at every edge point 𝜁 according to the right-hand rule (with respect to their unit vectors, r̂ × 𝜑̂ = ẑ , R̂ × 𝜗̂ = 𝜑). ̂ One should remember that we introduce these coordinates in such a way that the angle 𝜑 is measured from the illuminated face of the edge and the tangent ̂t to the edge is directed along the local polar axis ẑ (̂t = ẑ ). When both faces are illuminated, one can measure the angle 𝜑 from any face, but in this case one should choose the correct direction of the polar axis z and the tangent ̂t (̂z = ̂t = r̂ × 𝜑). ̂ This note is important for correct applications of the theory of electromagnetic EEWs. The quantities i

E0t (𝜁 )eik𝜙 (𝜁) = Etinc (𝜁 ),

i

H0t (𝜁 )eik𝜙 (𝜁 ) = Htinc (𝜁 )

(7.138)

in Equation (7.137) can be interpreted as the electric and magnetic components of an arbitrary incident wave, which are tangential to the edge at the diffraction point 𝜁 . In the high-frequency approximation, this wave in the vicinity of the diffraction point 𝜁 can be considered as a plane wave propagating in the direction k̂ i = ∇𝜙i ≡ grad 𝜙i and forming the angle 𝜋 − 𝛾0 with the local tangent ̂t ≡ ẑ loc (Fig. 7.3). The angle 𝛾0 is determined by the equation k̂ i ⋅ ̂t = − cos 𝛾0 . ⃗ (1) are specified by their spherical components: ⃗ (1) and G The vectors F F𝜗(1) (𝜗, 𝜑) = [U(𝜎1 , 𝜑0 ) + U(𝜎2 , 𝛼 − 𝜑0 )] sin 𝜗, (𝜗, 𝜑) = G(1) 𝜗

sin 𝜗 cos 𝛾0 sin2 𝛾0

F𝜑(1) (𝜗, 𝜑) = 0

(7.139)

[𝜀(𝜑0 ) − 𝜀(𝛼 − 𝜑0 )]

+ (sin 𝛾0 cos 𝜗 cos 𝜑 − cos 𝛾0 sin 𝜗 cos 𝜎1 )V(𝜎1 , 𝜑0 ) − [sin 𝛾0 cos 𝜗 cos(𝛼 − 𝜑) − cos 𝛾0 sin 𝜗 cos 𝜎2 ]V(𝜎2 , 𝛼 − 𝜑0 ), (7.140) G(1) 𝜑 (𝜗, 𝜑)

= −[V(𝜎1 , 𝜑0 ) sin 𝜑 + V(𝜎2 , 𝛼 − 𝜑0 ) sin(𝛼 − 𝜑)] sin 𝛾0 .

(7.141)

All functions and parameters in Equations (7.139) to (7.141) are the same as those introduced for acoustic waves in previous sections. The EEWs radiated by the total current ⃗j(t) = ⃗j(1) + ⃗j(0) are determined by ⃗ (t) = dE ⃗ (t)

dH

d𝜁 ⃗(t) eikR  (𝜗, 𝜑) , 2𝜋 R

d𝜁 ⃗ (t) eikR ⃗ (t) =  (𝜗, 𝜑) ,  (𝜗, 𝜑) = Y0 [R̂ × ⃗(t) (𝜗, 𝜑)] 2𝜋 R

(7.142)

with ⃗ (t) (𝜗, 𝜑), ⃗ (t) (𝜗, 𝜑) + Z0 H inc (𝜁 )G ⃗(t) (𝜗, 𝜑) = Einc (𝜁 )F t

t

(7.143)

241

ELECTROMAGNETIC ELEMENTARY EDGE WAVES

where F𝜗(t) (𝜗, 𝜑) = [Ut (𝜎1 , 𝜑0 ) + Ut (𝜎2 , 𝛼 − 𝜑0 )] sin 𝜗,

F𝜑(t) (𝜗, 𝜑) = 0,

(7.144)

G(t) (𝜗, 𝜑) = (sin 𝛾0 cos 𝜗 cos 𝜑 − cos 𝛾0 sin 𝜗 cos 𝜎1 )Vt (𝜎1 , 𝜑0 ) 𝜗 − [sin 𝛾0 cos 𝜗 cos(𝛼 − 𝜑) − cos 𝛾0 sin 𝜗 cos 𝜎2 ]Vt (𝜎2 , 𝛼 − 𝜑0 ), (7.145) G(t) 𝜑 (𝜗, 𝜑) = −[Vt (𝜎1 , 𝜑0 ) sin 𝜑 + Vt (𝜎2 , 𝛼 − 𝜑0 ) sin(𝛼 − 𝜑)] sin 𝛾0 .

(7.146)

The properties of functions U(𝜎, 𝜓), V(𝜎, 𝜓) and Ut (𝜎, 𝜓), Vt (𝜎, 𝜓) are described in Sections 7.5 and 7.6. The difference ⃗ (t) − dE ⃗ (1) , ⃗ (0) = dE dE

⃗ (0) = dH ⃗ (t) − dH ⃗ (1) dH

(7.147)

is the electromagnetic field of EEWs generated by the uniform current ⃗j(0) . 7.8.1

Electromagnetic EEWs on the Diffraction Cone Outside the Wedge

For the observation points outside the scattering wedge (0 ≤ 𝜑 ≤ 𝛼) which are located ⃗ (1) , in the directions of the diffraction cone (𝜗 = 𝜋 − 𝛾0 ), the components of vectors F (t) (1) (t) ⃗ ⃗ ⃗ F and G , G are determined as follows: F𝜗(1) (𝜋 − 𝛾0 , 𝜑) = −

1 (1) f (𝜑, 𝜑0 , 𝛼), sin 𝛾0

(7.148)

F𝜗(t) (𝜋 − 𝛾0 , 𝜑) = −

1 f (𝜑, 𝜑0 , 𝛼), sin 𝛾0

(7.149)

F𝜑(1) (𝜋 − 𝛾0 , 𝜑) = 0,

F𝜑(t) (𝜋 − 𝛾0 , 𝜑) = 0

(7.150)

and G(1) 𝜑 (𝜋 − 𝛾0 , 𝜑) =

1 (1) g (𝜑, 𝜑0 , 𝛼), sin 𝛾0

(7.151)

G(t) 𝜑 (𝜋 − 𝛾0 , 𝜑) =

1 g(𝜑, 𝜑0 , 𝛼), sin 𝛾0

(7.152)

G(1) (𝜋 − 𝛾0 , 𝜑) = [𝜀(𝜑0 ) − 𝜀(𝛼 − 𝜑0 )] cot 𝛾0 , 𝜗

(7.153)

G(t) (𝜋 − 𝛾0 , 𝜑) = 0. 𝜗

(7.154)

242

ELEMENTARY ACOUSTIC AND ELECTROMAGNETIC EDGE WAVES

According to these equations, one can represent the electromagnetic EEWs for the directions 𝜗 = 𝜋 − 𝛾0 as dE𝜗(0) = Z0 dH𝜑(0) ⎫ ⎪ d𝜁 1 dE𝜗(1) = Z0 dH𝜑(1) ⎬ = −Etinc (𝜁 ) 2𝜋 sin 𝛾0 ⎪ dE𝜗(t) = Z0 dH𝜑(t) ⎭

⎧f (0) (𝜑, 𝜑0 , 𝛼)⎫ ⎪ (1) ⎪ eikR ⎨f (𝜑, 𝜑0 , 𝛼)⎬ R , ⎪ f (𝜑, 𝜑 , 𝛼) ⎪ 0 ⎩ ⎭

(7.155)

dH𝜗(0) = −Y0 dE𝜑(0) ⎫ ⎪ d𝜁 1 dH𝜗(1) = −Y0 dE𝜑(1) ⎬ = −Htinc (𝜁 ) 2𝜋 sin 𝛾0 ⎪ dH𝜗(t) = −Y0 dE𝜑(t) ⎭

⎧g(0) (𝜑, 𝜑0 , 𝛼)⎫ ⎪ (1) ⎪ eikR ⎨g (𝜑, 𝜑0 , 𝛼)⎬ R , ⎪ g(𝜑, 𝜑 , 𝛼) ⎪ 0 ⎩ ⎭

(7.156)

where Y0 = 1∕Z0 and ẑ = ̂t is the unit vector tangential to the scattering edge at the diffraction point 𝜁 . The first two rows in Equation (7.155) are valid only in the absence of the polarization coupling in the PO field (see Section 4.3). In the general case, they have the following form: dE𝜗(0) = Z0 dH𝜑(0) =

d𝜁 1 {−Etinc (𝜁 )f (0) (𝜑, 𝜑0 , 𝛼) 2𝜋 sin 𝛾0

− Z0 Htinc (𝜁 )[𝜀(𝜑0 ) − 𝜀(𝛼 − 𝜑0 )] cos 𝛾0 } dE𝜗(1) = Z0 dH𝜑(1) =

eikR , R

(7.157)

d𝜁 1 {−Etinc (𝜁 )f (1) (𝜑, 𝜑0 , 𝛼) 2𝜋 sin 𝛾0

+ Z0 Htinc (𝜁 )[𝜀(𝜑0 ) − 𝜀(𝛼 − 𝜑0 )] cos 𝛾0 }

eikR , R

(7.158)

Notice also that the radial components of the far field are on the order 1∕R2 ; they are neglected here. Because of that, in general, E𝜗 = −

Et , sin 𝜗

H𝜗 = −

Ht sin 𝜗

dE𝜗 = −

dEt , sin 𝜗

dH𝜗 = −

dHt sin 𝜗

(7.159)

(7.160)

where ẑ = ̂t. Taking into account these equations and the condition 𝜗 = 𝜋 − 𝛾0 , one can rewrite the asymptotics (7.155) and (7.156) in the form dEt(0) ⎫ ⎪ d𝜁 dEt(1) ⎬ = Etinc (𝜁 ) 2𝜋 ⎪ dEt(t) ⎭

⎧ f (0) (𝜑, 𝜑0 , 𝛼)⎫ ⎪ eikR ⎪ (1) ⎨ f (𝜑, 𝜑0 , 𝛼)⎬ R , ⎪ f (𝜑, 𝜑 , 𝛼) ⎪ 0 ⎭ ⎩

(7.161)

243

ELECTROMAGNETIC ELEMENTARY EDGE WAVES

dHt(0) ⎫ ⎪ d𝜁 dHt(1) ⎬ = Htinc (𝜁 ) 2𝜋 ⎪ dHt(t) ⎭

⎧g(0) (𝜑, 𝜑0 , 𝛼)⎫ ⎪ eikR ⎪ (1) ⎨g (𝜑, 𝜑0 , 𝛼)⎬ R . ⎪ g(𝜑, 𝜑 , 𝛼) ⎪ 0 ⎭ ⎩

(7.162)

The first two rows in Equation (7.161) are valid only in the absence of polarization coupling. In the general case, according to Equations (7.157) and (7.158) they have the following form: dEt(0) =

d𝜁 inc eikR {Et (𝜁 )f (0) (𝜑, 𝜑0 , 𝛼) + Z0 Htinc (𝜁 )[𝜀(𝜑0 ) − 𝜀(𝛼 − 𝜑0 )] cos 𝛾0 } , 2𝜋 R (7.163)

dEt(1) =

d𝜁 inc eikR {Et (𝜁 )f (1) (𝜑, 𝜑0 , 𝛼) − Z0 Htinc (𝜁 )[𝜀(𝜑0 ) − 𝜀(𝛼 − 𝜑0 )] cos 𝛾0 } . 2𝜋 R (7.164)

Comparison of Equations (7.161) and (7.162) with Equations (7.120) and (7.121) allows one to establish the following relationships between the acoustic and electromagnetic EEWs for the directions belonging to the diffraction cone:

dEt = dus

if

Etinc (𝜁 ) = uinc (𝜁 ),

(7.165)

dHt = duh

if

Htinc (𝜁 ) = uinc (𝜁 ).

(7.166)

are valid only in the absence of the polarThe relationships dEt(0,1) = du(0,1) s ization coupling, when 𝛼 − 𝜋 < 𝜑0 < 𝜋 or 𝛾0 = 𝜋∕2.

Note also that Ufimtsev (1991) has investigated in detail the ray, caustic, and focal asymptotics of electromagnetic EEWs, as well as their multiple and slope diffraction. The results of this investigation are presented in Chapters 8 to 10. 7.8.2

Electromagnetic EEWs on the Diffraction Cone Inside the Wedge

For the observation points inside the scattering wedge (𝛼 ≤ 𝜑 ≤ 2𝜋), which are located in the directions of the diffraction cone (𝜗 = 𝜋 − 𝛾0 ), the components of the ⃗ (1) , G ⃗ (t) are determined by the expressions different from those ⃗ (t) and G ⃗ (1) , F vectors F valid outside the wedge: F𝜗(1) (𝜋 − 𝛾0 , 𝜑) =

1 (0) f (𝜑, 𝜑0 , 𝛼), sin 𝛾0

F𝜗(t) (𝜋 − 𝛾0 ) = 0,

(7.167)

244

ELEMENTARY ACOUSTIC AND ELECTROMAGNETIC EDGE WAVES

G(1) (𝜋 − 𝛾0 , 𝜑) = [𝜀(𝜑0 ) − 𝜀(𝛼 − 𝜑0 )] cot 𝛾0 , 𝜗 G(1) 𝜑 (𝜋 − 𝛾0 , 𝜑) = −

1 (0) g (𝜑, 𝜑0 , 𝛼), sin 𝛾0

G(t) (𝜋 − 𝛾0 , 𝜑) = 0, 𝜗 G(t) 𝜑 (𝜋 − 𝛾0 , 𝜑) = 0,

(7.168) (7.169)

where the functions f (0) (𝜑, 𝜑0 , 𝛼) and g(0) (𝜑, 𝜑0 , 𝛼) are defined in Equations (7.118) and (7.119). In the direction of the incident wave (𝜑 = 𝜋 + 𝜑0 ) inside the wedge, the functions f (0) (𝜑, 𝜑0 , 𝛼) and g(0) (𝜑, 𝜑0 , 𝛼) are determined by Equations (7.118a) and (7.119a). In terms of the components parallel to the edge, the field is written as ⎫ dEt(0) ⎪ ⎪ d𝜁 dEt(1) ⎬ = Etinc (𝜁 ) 2𝜋 ⎪ dEt(t) ⎪ ⎭

⎧ f (0) (𝜑, 𝜑0 , 𝛼) ⎫ ⎪ eikR ⎪ (0) ⎨−f (𝜑, 𝜑0 , 𝛼)⎬ R , ⎪ ⎪ 0 ⎭ ⎩

(7.170)

⎫ dHt(0) ⎪ ⎪ d𝜁 dHt(1) ⎬ = Htinc (𝜁 ) 2𝜋 ⎪ dHt(t) ⎪ ⎭

⎧ g(0) (𝜑, 𝜑0 , 𝛼) ⎫ ⎪ eikR ⎪ (0) ⎨−g (𝜑, 𝜑0 , 𝛼)⎬ R . ⎪ ⎪ 0 ⎭ ⎩

(7.171)

The first two rows in Equation (7.170) are valid in the absence of polarization coupling. In the general case they should be written as dEt(0) =

d𝜁 inc {E (𝜁 )f (0) (𝜑, 𝜑0 , 𝛼) 2𝜋 t + Z0 Htinc (𝜁 )[𝜀(𝜑0 ) − 𝜀(𝛼 − 𝜑0 )] cos 𝛾0 }

dEt(1) = −dEt(0) .

eikR R

(7.172) (7.173)

Note that Equation (7.172) represents the analytic continuation of Equation (7.163) into the wedge region (𝛼 < 𝜑 < 2𝜋). Expressions (7.170) to (7.173) have a clear physical interpretation. The PO fields generated by the uniform currents propagate inside the perfectly conducting wedge, where they are canceled totally by the field generated by the nonuniform currents, in complete agreement with the physics. Here it is pertinent to remind readers that the diffraction theories dealing with the fields radiated by the surface scattering sources/currents are based on the equivalence principle. According to this principle, the scattering object is actually replaced by the equivalent surface currents distributed in free space over a surface conformal to the object surface and radiating in all directions.

IMPROVED THEORY OF ELEMENTARY EDGE WAVES: REMOVAL OF THE GRAZING SINGULARITY

245

90 120

0

60

–10

150

30

–20

(1)

Gϑ

180

(1)

Gφ

0

(1)

Fϑ

210

330

300

240 270

Figure 7.9 Analog of Figure 7.6. Here 𝛼 = 315◦ , 𝛾0 = 45◦ , 𝜑0 = 45◦ , 𝜗 = 90◦ , and 0◦ ≤ 𝜑 ≤ 360◦ . (Produced by F. Hacivelioglu and L. Sevgi.)

7.8.3

Numerical Calculations of Electromagnetic Elementary Fringe Waves

For comparison purposes, illustrative figures are presented here for the same parameters as in Figures 7.6 to 7.8 for acoustic waves. The equations F𝜗(1) (𝜗, 𝜑) = −Fs(1) (𝜗, 𝜑)

sin 𝜗 , sin2 𝛾0

(1) G(1) 𝜑 (𝜗, 𝜑) = Fh (𝜗, 𝜑)

1 sin 𝜗

clarify the relationships between electromagnetic and acoustic waves. No acoustic analog exists for the electromagnetic component G(1) (𝜗, 𝜑) that describes the 𝜗 polarization coupling. The calculations were performed with the MATLAB codes presented in the Appendix to Section 7.8.3. The results are illustrated in Figures 7.9 |. to 7.11. They represent the quantities 10 log |F𝜗(1) | and 10 log |G(1) 𝜗,𝜑 7.9 IMPROVED THEORY OF ELEMENTARY EDGE WAVES: REMOVAL OF THE GRAZING SINGULARITY The central idea of PTD is the separation of surface scattering sources into uniform and nonuniform components in such a way that they would be the most

246

ELEMENTARY ACOUSTIC AND ELECTROMAGNETIC EDGE WAVES

90 120

0

60

–10

150

30

–20

(1)

Gϑ 0

180

(1)

Gφ

(1)

Fϑ

330

210

300

240 270

Figure 7.10 Analog of Figure 7.7. The field in the bisecting plane: 𝜑 = 𝛼∕2 and 𝜑 = 𝛼∕2 + 180◦ . Here 𝛼 = 315◦ , 𝛾0 = 45◦ and 𝜑0 = 45◦ . The polar angle 𝜃 is defined through the ordinary spherical coordinate 𝜗 as 𝜃 = 𝜗 for 𝜑 = 𝛼∕2 and 𝜃 = 2𝜋 − 𝜗 for 𝜑 = 𝛼∕2 + 180◦ . The points with 𝜑 = 𝛼∕2 + 180◦ and 180◦ ≤ 𝜃 ≤ 360◦ are inside the wedge. (Produced by F. Hacivelioglu and L. Sevgi.)

appropriate for calculation of the scattered field. For scattering objects with edges, the nonuniform component is defined as that part of the field that concentrates near edges. Equations (7.7) and (7.8) determine it as the difference between the total field on the tangential wedge and its geometrical optics part. The latter is considered as the uniform component. As shown in this book and in other publications, utilization of these components is really helpful for the investigation of many scattering problems. However, it is not the case for forward scattering in the directions grazing to the edge faces (Fig. 7.12), where the theory of elementary edge waves above predicts infinite values for the functions f (1) = f − f (0) and g(1) = g − g(0) . It turns out that for these directions, either function f or f (0) (either g or g(0) ) becomes singular. Which of them becomes singular depends on how the grazing direction is approached: 𝜑 = 0 and then 𝜑0 → 𝜋, or 𝜑0 = 𝜋 and then 𝜑 → 0. These singularities indicate that the definitions (1.31) and (7.7), (7.8) for uniform and nonuniform components are not adequate for the actual surface field in this case. The physical reason for the grazing singularity is that the geometrical optics used above in the definition of the uniform component j(0) is not valid for the grazing incidence. To clarify this situation one can consider the scattering sources induced on the illuminated face of a half-plane tangential to the wedge face 𝜑 = 0. According to

247

IMPROVED THEORY OF ELEMENTARY EDGE WAVES: REMOVAL OF THE GRAZING SINGULARITY

90 0

120

60

–10

150

30

–20

(1)

Gϑ 0

180

(1)

Gφ

(1)

Fϑ

330

210

300

240 270

Figure 7.11 Analog of Figure 7.8. The field on the diffraction cone, 𝜗 = 𝜋 − 𝛾0 . Here 𝛼 = 315◦ , 𝛾0 = 45◦ , 𝜑0 = 45◦ and 0◦ ≤ 𝜑 ≤ 315◦ . Here also G(1) = 1 and G(1) = 0. (Produced by F. Hacivelioglu and L. Sevgi.) 𝜗 𝜗

Equations (2.40), (2.41), (2,52), and (2.82), the total scattering sources/currents on strip 1 of the half-plane 𝜑 = 0 (Fig. 7.3) are determined by hp,tot

jh,s

(1)hp

= jh,s

+ j(0) . h,s

(7.174)

Here the quantities j(1)hp and j(0) represent the ordinary nonuniform and uniform components, (1)hp

jh

= −2u0 e−ik𝜁 cos 𝛾0 eik𝜉1 (cos

2 𝛾 −sin2 𝛾 cos 𝜑 ) 0 0 0

∞

2 e−𝜋∕4 eit dt, √ √ ∫ 2k𝜉1 sin 𝛾0 cos(𝜑0 ∕2) 𝜋

(7.175) φ0 = π

φ =0 φ =0

φ =α Figure 7.12 Grazing incidence on the wedge under the angle 𝜑0 = 𝜋 and grazing scattering in the direction 𝜑 = 0. No singularity exists in the reverse situation when 𝜑0 = 0 and 𝜑 = 𝜋.

248

ELEMENTARY ACOUSTIC AND ELECTROMAGNETIC EDGE WAVES

j(0) = 2u0 e−ik𝜁 cos 𝛾0 eik𝜉1 (cos h

2 𝛾 −sin2 𝛾 cos 𝜑 ) 0 0 0

(7.176)

and [ (1)hp js

−ik𝜁 cos 𝛾0 ik𝜉1 cos2 𝛾0

= 2u0 e

√

∞

×

2 𝛾 cos 𝜑 0 0

ik sin 𝛾0 sin 𝜑0 e−ik𝜉1 sin

e

∫√2k𝜉1 sin 𝛾0 cos(𝜑0 ∕2)

it2

e dt +

e−i𝜋∕4 √ 𝜋 ] 2

𝜑 e−i𝜋∕4 eik𝜉1 sin k sin 0 √ √ 2 2 𝜉1 𝜋

−ik𝜁 cos 𝛾0 ik sin 𝛾0 sin 𝜑0 eik𝜉1 (cos j(0) s = −2u0 e

2 𝛾 −sin2 𝛾 cos 𝜑 ) 0 0 0

𝛾0

.

,

(7.177) (7.178)

hp,tot

Note also that the quantities jh,s on strip 2 (of the half-plane 𝜑 = 𝛼) can be found from the equations above with the replacement of 𝜑0 by 𝛼 − 𝜑0 and 𝜉1 by 𝜉2 . Here we suppose that 𝛼 − 𝜋 ≤ 𝜑0 ≤ 𝜋. When 𝜑0 → 𝜋, Equations (7.170) to (7.173) reduce to (1)hp

= u0 e−ik𝜁 cos 𝛾0 eik𝜉1 ,

(1)hp js

−ik𝜁 cos 𝛾0

jh

= u0 e

√

j(0) = 2u0 e−ik𝜁 cos 𝛾0 eik𝜉1 , h

2k i(k𝜉1 −𝜋∕4) e , 𝜋𝜉1

j(0) s = 0.

(7.179)

(7.180)

(1)hp

transforms into the traveling Now it is seen that the nonuniform component jh wave with nonvanishing amplitude, and the uniform component j(0) contains the h absent reflected wave. Thus, the ordinary definitions of uniform and nonuniform components indeed are not satisfactory in the case of gazing incidence. To remove the grazing singularity we next introduce new definitions for components j(0) and j(1) , which are more appropriate when the incident wave illuminates both faces of the edge and propagates in the direction close to the face orientation. 7.9.1

Acoustic EEWs

This theory is based on a paper by Ufimtsev (2006a).

The appropriate candidate for a new modified uniform component j(0) mod is the component (7.174) induced by the incident wave (7.2) on the illuminated side of the tangential half-plane, j(0) mod ≡ jhp,tot = j(1)hp + j(0) .

(7.181)

IMPROVED THEORY OF ELEMENTARY EDGE WAVES: REMOVAL OF THE GRAZING SINGULARITY

249

The modified nonuniform component is defined as j(1) mod = jwedge,tot − j(0) mod ,

(7.182)

jwedge,tot = j(1)wedge + j(0)

(7.183)

where

is the total surface field on the wedge. The right-hand-side terms in Equations (7.181) and (7.183) represent the ordinary nonuniform and uniform components. The geometrical optics components j(0) in Equations (7.181) and (7.183) are identical and cancel each other in Equation (7.182), which transforms to j(1) mod = j(1)wedge − j(1)hp .

(7.184)

This current structure entails the corresponding structure of the field generated by the modified nonuniform current, du(1) mod = du(1)wedge − du(1)hp .

(7.185)

Here and below we assume that 𝛼 − 𝜋 ≤ 𝜑0 ≤ 𝜋. EEWs Generated by the Modified Nonuniform Component j(1) 𝐦𝐨𝐝 The field (7.185) generated by the modified nonuniform component j(1) mod is easily found by taking the following observations into account:

r The wedge-related components j(1)wedge are described by Equations (7.12) to h,s (1)hp

(7.15). The half-plane-related components jh,s are described by the same equations, where one should set 2𝜋 for the wedge angle 𝛼, that is, replace 1 𝛼𝑤(𝜂 + 𝜓)

by

1 2𝜋𝑤hp (𝜂 + 𝜓)

(7.186)

with 𝑤hp (x) = 1 − eix∕2 .

(7.187)

r The fields du(1)wedge generated by the components j(1)wedge are described by h,s h,s (1)hp

Equations (7.89) to (7.94). The fields duh,s generated by the components (1)hp jh,s are described by the same equations, where one should implement the replacement (7.186).

r These equations for the fields du(1)wedge and du(1)hp contain the same terms h,s h,s

generated by the geometrical optics U0 and V0 , which describe the field du(0) h,s components j(0) and which cancel each other in the difference (7.185).

250

ELEMENTARY ACOUSTIC AND ELECTROMAGNETIC EDGE WAVES

Hence, the field (7.185) generated by the modified nonuniform component j(1) mod can be represented in the form d𝜁 (1) eikR h (𝜗, 𝜑) , 2𝜋 R d𝜁 eikR = uinc (𝜁 ) s(1) (𝜗, 𝜑) . 2𝜋 R

mod = uinc (𝜁 ) du(1) h

(7.188)

mod du(1) s

(7.189)

Here, hp

h(1) (𝜗, 𝜑) = {[−Vt (𝜎1 , 𝜑0 ) + Vt (𝜎1 , 𝜑0 )] sin 𝜑 hp

+ [−Vt (𝜎2 , 𝛼 − 𝜑0 ) + Vt (𝜎2 , 𝛼 − 𝜑0 )] sin(𝛼 − 𝜑)} sin 𝛾0 sin 𝜗, (7.190) hp

hp

s(1) = [−Ut (𝜎1 , 𝜑0 ) + Ut (𝜎1 , 𝜑0 ) − Ut (𝜎2 , 𝛼 − 𝜑0 ) + Ut (𝜎2 , 𝛼 − 𝜑0 )] sin2 𝛾0 . (7.191) Here and below we assume that both faces of the wedge are illuminated (𝛼 − 𝜋 ≤ hp 𝜑0 ≤ 𝜋). The functions Ut and Vt are defined in Equations (7.85) and (7.87), but Ut hp and Vt are the similar functions associated with the tangential half-plane: ( 𝜎+𝜓 𝜎−𝜓) 1 , cot + cot 4 4 4 sin2 𝛾0 sin 𝜎 ( 𝜎+𝜓 𝜎−𝜓) 1 hp Ut (𝜎, 𝜓) = cot − cot . 4 4 4 sin2 𝛾0 hp

Vt (𝜎, 𝜓) =

(7.192) (7.193)

For the directions of the diffraction cone (𝜗 = 𝜋 − 𝛾0 ), the functions (7.190) and (7.191) can be represented in the form h(1) (𝜋 − 𝛾0 , 𝜑) = g(𝜑, 𝜑0 , 𝛼) − ghp (𝜑, 𝜑0 ),

(7.194)

s(1) (𝜋 − 𝛾0 , 𝜑) = f (𝜑, 𝜑0 , 𝛼) − f hp (𝜑, 𝜑0 ),

(7.195)

where f and g are the Sommerfeld functions (2.62) and (2.64), and ( ) 𝜋 − 𝜑 − 𝜑0 𝜋 − 𝜑 + 𝜑0 cot + cot 4 4 ( ) 𝜋 + 𝜑 − 𝜑0 𝜋 + 𝜑 + 𝜑0 − 2𝛼 1 − cot + cot , 4 4 4

1 g (𝜑, 𝜑0 ) = − 4 hp

(7.196)

IMPROVED THEORY OF ELEMENTARY EDGE WAVES: REMOVAL OF THE GRAZING SINGULARITY

f hp (𝜑, 𝜑0 ) =

1 4 −

(

𝜋 − 𝜑 − 𝜑0 𝜋 − 𝜑 + 𝜑0 − cot 4 4

cot 1 4

(

)

𝜋 + 𝜑 − 𝜑0 𝜋 + 𝜑 + 𝜑0 − 2𝛼 − cot 4 4

cot

251

) (7.197)

are the functions associated with the field scattered by the half-planes tangential to the faces 𝜑 = 0 and 𝜑 = 𝛼. Note also that for analytic analysis and numerical calculation, the cotangent forms of functions f and g are more convenient than the Sommerfeld expressions (2.62) and (2.64). These forms are identical to Equations (2.128) and (2.129) and determined as (

𝜋 − 𝜑 − 𝜑0 𝜋 − 𝜑 + 𝜑0 + cot 2n 2n ) 𝜋 + 𝜑 + 𝜑0 𝜋 + 𝜑 − 𝜑0 + cot + cot , 2n 2n

g(𝜑, 𝜑0 , 𝛼) = −

f (𝜑, 𝜑0 , 𝛼) =

1 2n

cot

(

𝜋 − 𝜑 − 𝜑0 𝜋 − 𝜑 + 𝜑0 − cot 2n 2n ) 𝜋 + 𝜑 + 𝜑0 𝜋 + 𝜑 − 𝜑0 + cot − cot 2n 2n 1 2n

(7.198)

cot

(7.199)

with n = 𝛼∕𝜋. The functions f , g and f hp , ghp are singular in the directions of the incident and reflected plane waves (𝜑 = 𝜋 + 𝜑0 , 𝜑 = 𝜋 − 𝜑0 , 𝜑 = 2𝛼 − 𝜋 − 𝜑0 ), but their difference is finite. For example, 𝜑 𝜑 1 1 cot 0 + cot 0 2n n 4 2 1 𝛼−𝜋 1 + cot − cot 2n n 4 𝜑 𝜑 1 1 s(1) (𝜋 − 𝛾0 , 𝜋 − 𝜑0 ) = − cot 0 + cot 0 2n n 4 2 1 𝛼−𝜋 1 − cot + cot 2n n 4

h(1) (𝜋 − 𝛾0 , 𝜋 − 𝜑0 ) = −

𝜋 − 𝜑0 1 𝜋 − 𝜑0 1 cot + cot 2n n 4 2 𝛼−𝜋 , (7.200) 2 𝜋 − 𝜑0 1 𝜋 − 𝜑0 1 − cot + cot 2n n 4 2 𝛼−𝜋 , (7.201) 2 −

𝜑 − (𝛼 − 𝜋) 1 𝜑 − (𝛼 − 𝜋) 1 cot 0 + cot 0 2n n 4 2 1 𝛼−𝜋 1 𝛼−𝜋 + cot − cot 2n n 4 2 𝛼 − 𝜑 𝛼 − 𝜑0 1 1 0 − cot + cot , (7.202) 2n n 4 2

h(1) (𝜋 − 𝛾0 , 2𝛼 − 𝜋 − 𝜑0 ) = −

252

ELEMENTARY ACOUSTIC AND ELECTROMAGNETIC EDGE WAVES

and 𝜑 − (𝛼 − 𝜋) 1 𝜑 − (𝛼 − 𝜋) 1 cot 0 + cot 0 2n n 4 2 1 𝛼−𝜋 1 𝛼−𝜋 − cot + cot 2n n 4 2 𝛼 − 𝜑0 1 𝛼 − 𝜑0 1 − cot + cot . (7.203) 2n n 4 2

s(1) (𝜋 − 𝛾0 , 2𝛼 − 𝜋 − 𝜑0 ) = −

(1) These equations show that the functions h,s are free of the grazing singularity. Indeed, for the grazing directions of the incident wave (𝜑0 = 𝜋, 𝜑0 = 𝛼 − 𝜋), these functions have the finite values

1 𝜋 1 𝛼 h(1) (𝜋 − 𝛾0 , 0) = h(1) (𝜋 − 𝛾0 , 𝛼) = − cot + tan , n n 4 2 1 𝛼 s(1) (𝜋 − 𝛾0 , 0) = s(1) (𝜋 − 𝛾0 , 𝛼) = − tan . 4 2

(7.204) (7.205)

(1) It also follows from these equations that the functions h,s are equal to zero when 𝛼 = 2𝜋 and the wedge transforms into the half-plane. This result is the obvious consequence of the general expressions (7.190) and (7.191) and the identities Vt = hp hp Vt and Ut = Ut , which are valid in the case 𝛼 = 2𝜋.

Field Generated by the Modified Uniform Component j(0) 𝐦𝐨𝐝 ≡ jhp,tot To find the total field scattered by finite objects, one should also calculate the contribution hp,tot mod mod generated by the modified uniform component j(0) ≡ jh,s distributed du(0) h,s h,s over the finite elementary strips (0 ≤ 𝜉1,2 ≤ l). In the far zone (R ≫ kl2 ) it is determined by the integrals mod = du(0) h,1

l d𝜁 eikR j(0) mod (𝜉1 , 𝜑0 )e−ik𝜉1 cos 𝛽1 d𝜉1 , ik sin 𝛾0 sin 𝜗 sin 𝜑 4𝜋 R ∫0 h,1

mod du(0) =− s,1

l d𝜁 eikR j(0) mod (𝜉1 , 𝜑0 )e−ik𝜉1 cos 𝛽1 d𝜉1 . sin 𝛾0 4𝜋 R ∫0 s,1

(7.206) (7.207)

mod are determined by Equations (7.181) together with EquaThe quantities j(0) h,s tions (7.174) to (7.178). Replacing 𝜉1 , 𝛽1 , 𝜑, and 𝜑0 in Equations (7.206) and (7.207) by 𝜉2 , 𝛽2 , 𝛼 − 𝜑, and 𝛼 − 𝜑0 , one obtains the equations associated with the field from strip 2 (0 ≤ 𝜉2 ≤ l). These integrals are calculated in closed form. We show only those results that relate to the grazing incidence (𝜑0 = 𝜋), mod du(0) = u0 e−ik𝜁 cos 𝛾0 h,1

d𝜁 eikR sin 𝛾0 sin 𝜗 sin 𝜑 ikl(1−cos 𝛽1 ) [e − 1], 4𝜋 R 1 − cos 𝛽1

(7.208)

253

IMPROVED THEORY OF ELEMENTARY EDGE WAVES: REMOVAL OF THE GRAZING SINGULARITY

mod du(0) s,1

d𝜁 eikR = u0 e−ik𝜁 cos 𝛾0 sin 𝛾0 2𝜋 R

√

2 ei⋅3𝜋∕4 √ 1 − cos 𝛽1 𝜋 ∫0

√ kl(1−cos 𝛽1 )

2

eit dt. (7.209)

For the grazing scattering direction (𝛽1 = 0, 𝜑 = 0), it follows from these equations that mod du(0) = 0, h,1 mod du(0) s,1

(7.210) −ik𝜁 cos 𝛾0

= u0 e

d𝜁 sin 𝛾0 2𝜋

√

2kl i⋅3𝜋∕4 eikR e 𝜋 R

(7.211)

mod where R ≫ kl2 . As expected, the field du(0) is also free from the grazing singuh,s larity.

7.9.2 Electromagnetic EEWs Generated by the Modified Nonuniform Current

This theory is based on a paper by Ufimtsev (2006b).

The situation with the grazing singularity for electromagnetic waves is similar to that for acoustic waves. Suppose that the incident wave illuminates both faces of the wedge (𝛼 − 𝜋 ≤ 𝜑0 ≤ 𝜋). As in the case of acoustic waves, it turns out that the original definitions of quantities ⃗j(0) and ⃗j(1) are not adequate for actual surface currents induced under the grazing incidence (𝜑0 = 𝜋 or 𝜑0 = 𝛼 − 𝜋). For example, the component j(1) x normal to the edge does not vanish away from the edge; instead, it transforms into a traveling wave with constant amplitude. Also, the component j(0) x includes the absent reflected wave. Therefore, to avoid the grazing singularity, one should modify the original definitions of ⃗j(0) and ⃗j(1) . With this purpose we introduce a new modified uniform component ⃗j(0) mod ≡ ⃗jhp,tot , identical to the total electric surface current induced on the illuminated side of the tangential perfectly conducting half-plane. Then we refer to Equations (7.181) to (7.184), which are also applicable for electromagnetic waves and define the modified nonuniform current as ⃗j(1) mod = ⃗j(1)w − ⃗j(1)hp .

(7.212)

254

ELEMENTARY ACOUSTIC AND ELECTROMAGNETIC EDGE WAVES

Here the right-hand-side terms are the ordinary nonuniform currents induced by the incident wave (7.128) on the wedge and tangential half-plane. The field generated by the current (7.212) is represented in the form ⃗ (1)w − dE ⃗ (1)hp ⃗ (1) mod = dE dE

(7.213)

and is calculated using a procedure similar to that described in Section 7.9.1 for acoustic waves:

r The ordinary nonuniform currents ⃗j(1)w on strips 1 and 2 of the wedge faces are determined by Equations (7.130) and (7.131). The ordinary nonuniform currents ⃗j(1)hp on strips 1 and 2 of the tangential half-planes are determined by the same equations where one should implement the replacement (7.186). r The field dE⃗ (1)w radiated by the currents ⃗j(1)w is described by Equations (7.136) ⃗ (1)hp radiated by the currents ⃗j(1)hp is also described by to (7.141). The field dE the same equations, where one should set 𝛼 = 2𝜋 in the functions Ut (𝜎, 𝜓) and Vt (𝜎, 𝜓), that is, to replace hp

Ut (𝜎, 𝜓) by Ut (𝜎, 𝜓), hp

Vt (𝜎, 𝜓) by

hp

Vt (𝜎, 𝜓)

(7.214)

hp

with functions Ut (𝜎, 𝜓) and Vt (𝜎, 𝜓) defined in Equations (7.192) and (7.193). Thus, the modified nonuniform current ⃗j(1) mod radiates the field ⃗ (1) mod = dE

d𝜁 ⃗(1) mod eikR (𝜗, 𝜑)  , 2𝜋 R

⃗ (1) mod = dH

⃗ (1) mod ∇R × dE Z0

(7.215)

where i ⃗(1) mod (𝜗, 𝜑) = [E0t (𝜁 )⃗ (1) (𝜗, 𝜑) + Z0 H0t (𝜁 )⃗(1) (𝜗, 𝜑)]eik𝜙 (𝜁 ) ,

(7.216)

hp

𝜗(1) (𝜗, 𝜑) = [Ut (𝜎1 , 𝜑0 ) − Ut (𝜎1 , 𝜑0 )] sin 𝜗 hp

+ [Ut (𝜎2 , 𝛼 − 𝜑0 ) − Ut (𝜎2 , 𝛼 − 𝜑0 )] sin 𝜗, 𝜑(1) (𝜗, 𝜑) = 0,

(7.217) (7.218)

(1) (𝜗, 𝜑) = (sin 𝛾0 cos 𝜗 cos 𝜑 − cos 𝛾0 sin 𝜗 cos 𝜎1 ) 𝜗 hp

× [Vt (𝜎1 , 𝜑0 ) − Vt (𝜎1 , 𝜑0 )] − [sin 𝛾0 cos 𝜗 cos(𝛼 − 𝜑) − cos 𝛾0 sin 𝜗 cos 𝜎2 ] hp

× [Vt (𝜎2 , 𝛼 − 𝜑0 ) − Vt (𝜎2 , 𝛼 − 𝜑0 )],

(7.219)

IMPROVED THEORY OF ELEMENTARY EDGE WAVES: REMOVAL OF THE GRAZING SINGULARITY

255

hp

(1) 𝜑 (𝜗, 𝜑) = −[Vt (𝜎1 , 𝜑0 ) − Vt (𝜎1 , 𝜑0 )] sin 𝜑 sin 𝛾0 hp

− [Vt (𝜎2 , 𝛼 − 𝜑0 ) − Vt (𝜎2 , 𝛼 − 𝜑0 )] sin(𝛼 − 𝜑) sin 𝛾0 .

(7.220)

It is supposed here that 0 < 𝜑0 ≤ 𝜋. In the directions 𝜗 = 𝜋 − 𝛾0 associated with the diffraction cone, these expressions are simplified as 𝜗(1) (𝜋 − 𝛾0 , 𝜑) = −

1 [f (𝜑, 𝜑0 , 𝛼) − f hp (𝜑, 𝜑0 )], sin 𝛾0

𝜑(1) (𝜗, 𝜑) = 0, (1) 𝜑 (𝜋 − 𝛾0 , 𝜑) =

(7.221) (7.222)

1 [g(𝜑, 𝜑0 , 𝛼) − ghp (𝜑, 𝜑0 )], sin 𝛾0

(𝜋 − 𝛾0 , 𝜑) = 0, (1) 𝜗

(7.223) (7.224)

with the functions f , g and f hp , ghp as defined in Section 7.9.1. Comparison with Equations (7.194) and (7.195) reveals the following relationships between the electromagnetic and acoustic EEWs generated by the modified nonuniform scattering sources/currents: 𝜗(1) (𝜋 − 𝛾0 , 𝜑) = − (1) 𝜑 (𝜋 − 𝛾0 , 𝜑) =

1  (1) (𝜋 − 𝛾0 , 𝜑), sin 𝛾0 s

1  (1) (𝜋 − 𝛾0 , 𝜑). sin 𝛾0 h

(7.225)

(7.226)

According to Section 7.9.1, these functions are free of the grazing singularities as well as from the singularities in the directions of the incident and reflected rays. To find the total field scattered by finite objects, we should also calculate the ⃗ (0) mod generated by the modified uniform currents ⃗j(0) mod ≡ ⃗jhp contribution dE distributed over the finite elementary strips (0 ≤ 𝜉1,2 ≤ l1,2 ). According to Chapter 2, these currents are expressed in terms of the Fresnel function and its derivative. In ⃗ (0) mod is determined by the integrals, which the far zone (R ≫ kl2 ), the field dE are calculated in closed form. These calculations are presented in the Appendix to Section 7.9.2. The asymptotic theory developed in Sections 7.9.1 and 7.9.2 is well suited for the calculation of bistatic scattering in the case, when both planar faces of the edge are illuminated by the incident wave (𝛼 − 𝜋 ≤ 𝜑0 ≤ 𝜋). For other incidence directions 𝜑0 , one can utilize the original theory presented in Sections 7.1 to 7.8. Here it is pertinent to mention the alternative approach (Michaeli, 1987; Breinbjerg, 1992; Johanson, 1996) for elimination of the grazing singularity. The

256

ELEMENTARY ACOUSTIC AND ELECTROMAGNETIC EDGE WAVES

uniform and nonuniform components of the surface current are defined there according to the original PTD, and the grazing singularity is eliminated by truncation of elementary strips (0 ≤ x1,2 ≤ l). Compared to this approach, a distinctive feature of the present theory is as follows: It introduces a new nonuniform scattering source ⃗j(1) mod that generates an elementary edge wave regular in all scattering directions. In other words, it allows extraction of the fringe component from the total field in a pure explicit form.

7.10 SOME REFERENCES RELATED TO ELEMENTARY EDGE WAVES The investigation of EEWs has a long history. In Kirchhoff’s approach, the EEWs were as first discovered by Maggi (Maggi, 1888; Baker and Copson, 1939). The same result was rediscovered by Rubinowicz (1917). A similar approach to electromagnetic EEWs was developed by Kottler (1923b). Attempts to define EEWs more strictly [on the basis of the Sommerfeld (1896, 1935) exact solution of the wedge diffraction problem] were first undertaken by Bateman (1955) and Rubinowicz (1965). However, their expressions for EEWs satisfy the Dirichlet or Neuman boundary conditions everywhere at the faces of the canonical tangent wedge. For this reason, these EEWs predict incorrect values for the diffracted field at those parts of the virtual tangent wedge that are extended outside the real scattering object (as shown in Fig. 7.13 by dotted lines). In fact, according to this definition of EEWs, the infinite plane areas of free space (outside the scattering object) formally become perfectly reflecting. The same drawback exists in another theory of EEWs, suggested by Tiberio et al. (1994, 1995, 2004). In PTD a similar shortcoming occurs only at the extensions of infinitely narrow elementary strips (Fig. 7.3). As PTD is a source-based theory, this shortcoming can be removed completely by truncation of elementary strips (Johansen, 1996). The directivity patterns of electromagnetic EEWs can be interpreted as equivalent edge currents (EECs). The EECs introduced by Knott and Senior (1973) and Knott (1985) are based on GTD and are valid only for the directions of diffracted rays. The EECs based on PTD are applicable for arbitrary scattering directions (Michaeli,

∞

∞

∞

∞

∞

Figure 7.13 Perfectly reflecting solid prism of finite size (solid and dashed lines) and infinite faces of the virtual tangent wedge (dotted lines).

PROBLEMS

257

1986, 1987; Breinbjerg, 1992; Johansen, 1996). The paper by Johansen (1996)] also contains additional references related to the EEC concept. Another interpretation of the directivity pattern of EEWs is the incremental length diffraction coefficient (ILDC). The term was introduced by Mitzner (1974), who determined the ILDCs on the basis of PTD. The ILDC concept was developed further in the work of Tiberio et al. (2004). Here it is pertinent to point out that all of these coefficients (EEC and ILDC), as well as the directivity pattern of EEWs, can be interpreted as the differential diffraction coefficients in contrast to the integral diffraction coefficients introduced by Keller. Survey of these techniques one can find in (Kravtsov and Zhu, 2010). Note also the asymptotic theory for plane screens (Wolf, 1967), which is similar to PTD, and the method of matched asymptotic expansions (Tran Van Nhieu, 1995, 1996).

PROBLEMS 7.1

Start with Equations (7.3) and (7.4) and derive Equations (7.12) and (7.13) for (1) the scattering surface sources j(1) s and jh on strip 1. Solution In view of Equations (7.7) and (7.8), the geometrical optics components u(0) are excluded from expressions for the nonuniform components s,h j(1) . Derivatives of the functions 𝑣(k1 r, 𝜑 ± 𝜑0 ) are calculated according to s,h Equation (7.10). Subsequent substitutions (7.11) lead to Equations (7.12) and (7.13).

7.2

Start with Equations (7.13) and (7.20), and verify the field expression (7.26) related to the field generated by hard elementary strip 1. Solution The derivative in Equation (7.20) is calculated according to Equation (7.22). Then we substitute the function (7.13) into Equation (7.20) and obtain the integral form (7.26).

7.3

Apply the Cauchy residue theorem to the integral (7.36) and verify its transformation into the form (7.65). Solution The problem is reduced to calculation of the integral (7.45) over the contour C shown in Figure 7.5. Due to the Cauchy theorem, it equals the sum of residues. The residues are determined by Equations (7.47) to (7.51). The contour C consists of branches D, F+ , and F− . Due to the properties (7.58) and p2 + k2 cos 𝜂 → ∞ with 𝜂 → ±i∞, the integral over branches F± tends to zero with 𝜂 → ±i∞. Hence, the integral (7.45) is exactly equal to that in Equation (7.36) over the contour D and allows transforming this equation into the form (7.65).

7.4

(1) of EEWs in Equations (7.89) and (7.90) Verify that the directivity patterns Fs,h are always real functions, although their arguments 𝜎1,2 can be imaginary quantities.

258

ELEMENTARY ACOUSTIC AND ELECTROMAGNETIC EDGE WAVES

(1) Solution It is clear that for real 𝜎 the functions Fs,h are real. To verify this property for imaginary 𝜎 = iIm(𝜎), it is sufficient to consider the cotangent terms in Equations (7.85) and (7.87), which take the forms

cot(x + iy) − cot(−x + iy) = −

sin(2x) sin(x + iy) sin(−x + iy)

= −2

sin(2x) , cos(2x) − cos(i2y)

sin(i2y) 1 [cot(x + iy) + cot(−x + iy)] = sin(iz) sin(x + iy) sin(−x + iy) =

sin(i2y) 2 , sin(iz) cos(2x) − cos(i2y)

where x = 𝜋𝜑0 ∕2𝛼 and y = (𝜋∕2𝛼) Im(𝜎), z = Im(𝜎). Observing that sin(i2y) = i sinh(2y), sin(iz) = i sinh(z) and cos(i2x) = cosh(2x), one can see that these (1) expressions are real. Hence, the functions Fs,h are also real. 7.5 The functions Ut (𝜎1 , 𝜑0 ) and Vt (𝜎1 , 𝜑0 ), as well as U0 (𝛽1 , 𝜑0 ) and V0 (𝛽1 , 𝜑0 ), are singular at the point 𝜎1 = 𝜑0 . Show that their differences, the functions U = Ut − U0 and V = Vt − V0 , remain finite there. Prove Equations (7.106) and (7.107). Solution This property of functions U and V is the direct consequence of Equations (7.85), (7.87) and (7.104), (7.105). For example, the difference U = Ut − U0 contains the singular terms −

𝜋(𝜎1 − 𝜑0 ) 1 𝜎 − 𝜑0 𝜋 cot + cot 1 , 2𝛼 2𝛼 2 2

which cancel each other completely when 𝜎1 → 𝜑0 . 7.6 Show that for the scattering directions 𝜗 = 𝜋 − 𝛾0 , 0 ≤ 𝜑 ≤ 𝛼 (which belong to the diffraction cone, outside the wedge), the functions Fs(1) and Fh(1) transform into the functions f (1) and g(1) , respectively. Prove Equations (7.115). Solution For simplicity, consider the region 0 < 𝜑 < 𝛼 − 𝜋, where according to Equations (7.113) and (7.114) the quantities 𝜎1,2 are determined as 𝜎1 = 𝜋 − 𝜑 and 𝜎2 = 𝛼 − 𝜋 − 𝜑. Utilize the cotangent forms (7.104) and (7.105) for the functions U0 and V0 . Simple manipulations lead to expressions (7.115), where the functions f (1) and g(1) are presented in cotangent form. 7.7 Derive Equations (7.130) and (7.131) for the nonuniform current ⃗j(1) induced by the incident wave (7.128) on elementary strip 1 (Fig. 7.3). Solution The current dif and j(1) z = −Hx , where

dif ⃗j(1) on strip 1 contains two components, j(1) x = Hz dif ⃗ H is the diffracted part of the field. According to

PROBLEMS

259

Equations (7.3) and (7.4) adopted for electromagnetic waves, Hzdif = H0z e−ikz cos 𝛾0 [𝑣(k1 r, 𝜑 − 𝜑0 ) + 𝑣(k1 r, 𝜑 + 𝜑0 )], Ezdif = E0z e−ikz cos 𝛾0 [𝑣(k1 r, 𝜑 − 𝜑0 ) + 𝑣(k1 r, 𝜑 + 𝜑0 )], where the function 𝑣(k1 r, 𝜑 ± 𝜑0 ) is defined by Equations (7.5) and (7.6) and k1 = k sin 𝛾0 . All other cylindrical components of the field are expressed through the z-components. To find them, substitute Ezdif and Hzdif into the Maxwell equations ⃗, ⃗ = −ikY0 E ∇×H

⃗ = ikZ0 H ⃗ ∇×E

written in cylindrical coordinates. Keep in mind that only the factor exp(−ikz cos 𝛾0 ) depends on the variable z. Solve the Maxwell equations for dif and H dif . The results were shown by Ufimtsev [1962, Equacomponents Er,𝜑 r,𝜑 tion (5.04)], where 𝛾 = 𝜋 − 𝛾0 . In particular, ( Hrdif

1 = ik sin2 𝛾0

𝜕Hz 1 𝜕Ez + cos 𝛾0 Y0 r 𝜕𝜑 𝜕r dif

dif

) .

On the face 𝜑 = 0, r = x and Hrdif ≡ Hxdif . Differentiation with respect to variable r is straightforward: ik 𝜕𝑣(k1 r, 𝜓) e−ik1 r cos 𝜂 𝑤−1 (𝜂 + 𝜓) cos 𝜂 d𝜂. =− 1 𝜕r 2𝛼 ∫D Differentiation with respect to variable 𝜑 is performed according to Equation (7.10): 𝜕𝑣(k1 r, 𝜓) ik r e−ik1 r cos 𝜂 𝑤−1 (𝜂 + 𝜓) sin 𝜂 d𝜂. =− 1 𝜕𝜓 2𝛼 ∫D Hence, dif

𝜕Hz 𝜕r

= −H0z e−ikz cos 𝛾0 ×

∫D

i 2𝛼

e−ik1 r cos 𝜂 [𝑤−1 (𝜂 + 𝜑 − 𝜑0 ) + 𝑤−1 (𝜂 + 𝜑 + 𝜑0 )] cos 𝜂 d𝜂,

dif

𝜕Ez ik r = −E0z e−ikz cos 𝛾0 1 𝜕𝜑 2𝛼 ×

∫D

e−ik1 r cos 𝜂 [𝑤−1 (𝜂 + 𝜑 − 𝜑0 ) − 𝑤−1 (𝜂 + 𝜑 + 𝜑0 )] sin 𝜂 d𝜂.

260

ELEMENTARY ACOUSTIC AND ELECTROMAGNETIC EDGE WAVES

Notice that according to Figure 7.3, the coordinates of the point on strip 1 are equal to z = 𝜁 − 𝜉1 cos 𝛾0 and r = x = 𝜉1 sin 𝛾0 . Utilizing the expressions above and setting 𝜑 = 0, one obtains the currents on strip 1: −ik(𝜁−𝜉1 cos 𝛾0 ) cos 𝛾0 j(1) x = H0z e

× j(1) z =

∫D

2 𝛾 cos 𝜂 0

e−ik𝜉1 sin

1 2𝛼

[𝑤−1 (𝜂 + 𝜑0 ) + 𝑤−1 (𝜂 − 𝜑0 )] d𝜂,

1 e−ik(𝜁−𝜉1 cos 𝛾0 ) cos 𝛾0 2𝛼 sin 𝛾0 ×

∫D

2 𝛾 cos 𝜂 0

e−ik𝜉1 sin

{H0z cos 𝛾0 [𝑤−1 (𝜂 + 𝜑0 ) + 𝑤−1 (𝜂 − 𝜑0 )] cos 𝜂

− Y0 E0z [𝑤−1 (𝜂 + 𝜑0 ) − 𝑤−1 (𝜂 − 𝜑0 )] sin 𝜂]} d𝜂. 7.8 The asymptotics of acoustic EEWs free of the grazing singularity are developed in Section 7.9.1. Show that for the directions of the diffraction cone (𝜗 = (1) take the forms (7.194) and (7.195). 𝜋 − 𝛾0 ), the functions h,s (1) Solution According to Equations (7.190) and (7.191), the functions h,s hp hp are expressed through the differences Ut − Ut and Vt − Vt . For simplicity, consider the region where according to Equations (7.113) and (7.114), the quantities 𝜎1,2 are determined as 𝜎1 = 𝜋 − 𝜑 and 𝜎2 = 𝛼 − 𝜋 − 𝜑. Utilize the cotangent forms (7.85) and (7.87) for the functions Ut and Vt and cotangent hp hp forms (7.192) and (7.193) for the functions Ut and Vt . Simple manipulations lead to expressions (7.194) and (7.195).

8 Ray and Caustic Asymptotics for Edge Diffracted Waves 8.1

RAY ASYMPTOTICS

The following relationships exist between the acoustic and electromagnetic diffracted rays: us = Et

if

uinc (𝜁 ) = Etinc (𝜁 ),

uh = Ht

if

uinc (𝜁 ) = Htinc (𝜁 ),

where ̂t is tangent to the edge at diffraction point 𝜁 . Here the relationships u(0,1) = Et(0,1) are valid only in the absence of polarization coupling in the PO s field.

8.1.1

Acoustic Waves

The theory of EEWs is employed here for the calculation of scattering at a smoothly curved edge L with a slowly changing angle 𝛼(𝜁 ) between its faces (Fig. 8.1). In close proximity to point 𝜁 on the edge, an arbitrary incident field i

uinc (𝜁 ) = u0 (𝜁 )eik𝜙 (𝜁 )

(8.1)

Fundamentals of the Physical Theory of Diffraction, Second Edition. Pyotr Ya. Ufimtsev. © 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

261

262

RAY AND CAUSTIC ASYMPTOTICS FOR EDGE DIFFRACTED WAVES

P

Q R ˆ m

γ0 tˆ L

ξ

kˆ i

Figure 8.1 Element of a scattering edge L with a curvilinear coordinate 𝜁 along the edge; ̂t is the unit vector tangential to the edge at point 𝜁 .

can be considered locally as a plane wave propagating in the direction k̂ i = ∇′ 𝜙i = grad′ 𝜙i .

(8.2)

Therefore, replacing the quantity uinc (𝜁 ) in Equations (7.89) and (7.90) and in (7.96) and (7.97) by Equation (8.1), we obtain asymptotic expressions for EEWs generated by an arbitrary incident wave. The resulting diffracted wave arising at the edge L is a linear superposition of EEWs and created by the nonuniform/fringe sources j(1) s,h (7.89) and (7.90), u(1) = s,h

1 eikΦ(𝜁 ) (1) ̂ u0 (𝜁 )Fs,h (𝜁 , m) d𝜁 , 2𝜋 ∫L R(𝜁 )

(8.3)

= j(0) + j(1) is a linear and the edge wave generated by the total scattering sources j(t) s,h s,h s,h superposition of EEWs (7.96) and (7.97), u(t) = s,h

1 eikΦ(𝜁 ) (t) ̂ u0 (𝜁 )Fs,h (𝜁 , m) d𝜁 . ∫ 2𝜋 L R(𝜁 )

(8.4)

̂ = ∇R, m

(8.5)

Here Φ = 𝜙i + R

and R is the distance between the edge point 𝜁 and the observation point P(x, y, z). Notice that the differential operator ∇′ in Equation (8.2) acts on coordinates of the edge point 𝜁 , but the operator ∇ in Equation (8.5) acts on coordinates x, y, z of the ̂ observation point P. Hence, ∇′ R = −∇R = −m. A high-frequency approximation (with k ≫ 1) of the scattered field can be obtained using the stationary-phase technique (Copson, 1965; Murray, 1984), whose details

RAY ASYMPTOTICS

263

were considered in Section 6.1.2. The stationary point 𝜁st is determined by the equation dΦ ̂ ⋅ ̂t = 0. = ∇′ Φ ⋅ ̂t = ∇′ (𝜙i + R) ⋅ ̂t = (k̂ i − m) d𝜁

(8.6)

̂ directed from the stationary point 𝜁st to the observation Denote by k̂ s the unit vector m point. Then Equation (8.6) can be rewritten as k̂ s ⋅ ̂t = k̂ i ⋅ ̂t = − cos 𝛾0 .

(8.7)

Thus, the scattering directions k̂ s form a cone with its axis along the tangent ̂t to the edge at the stationary point. Such a cone is shown in Figure 4.4. The function Φ describes the distance between points Q and P along the straight lines Q𝜁 and 𝜁 P (Fig. 8.1). Hence, Equation (8.6) indicates that this distance is extremal (minimal or maximal) when the point 𝜁 is stationary. In other words, the location of the stationary point 𝜁st on the edge L satisfies the Fermat principle. In accordance with the stationary-phase technique, the first term of the asymptotic expansion for the field (8.3) equals = u(1) s,h

) eikR ′′ 2 1 inc (1) ( 𝜁st , k̂ s eik[Φ (𝜁st )∕2](𝜁−𝜁st ) d𝜁 , u (𝜁st )Fs,h 2𝜋 R ∫−∞ ∞

(8.8)

where Φ′′ (𝜁st ) = d2 Φ(𝜁st )∕d𝜁 2 and R is the distance between the stationary point 𝜁st and the observation point P. Due to the equality ∞

∫−∞

2

e±ix dx =

√ ±i𝜋∕4 , 𝜋e

(8.9)

Equation (8.8) can be written as i𝜋∕4

ikR

e e (1) u(1) = uinc (𝜁st )Fs,h (𝜁st , k̂ s ) √ , s,h ′′ 2𝜋kΦ (𝜁st ) R

(8.10)

where √

Φ′′ (𝜁st ) =

√

|Φ′′ (𝜁st )|ei(𝜋∕2)

if

Φ′′ (𝜁st ) < 0.

(8.11)

In terms of the local spherical coordinates R, 𝜗, 𝜑 (introduced in Fig. 7.3), the unit vectors k̂ s have the directions 𝜗 = 𝜋 − 𝛾0 and 0 ≤ 𝜑 ≤ 2𝜋. For these directions, the (1) (𝜁st , k̂ s ) are determined by Equations (7.115) and (7.116). Hence, functions Fs,h

264

RAY AND CAUSTIC ASYMPTOTICS FOR EDGE DIFFRACTED WAVES

eikR , 2𝜋kΦ′′ (𝜁st ) R

(8.12)

ei𝜋∕4 eikR inc (1) u(1) = u (𝜁 )g (𝜑, 𝜑 , 𝛼) √ st 0 h 2𝜋kΦ′′ (𝜁st ) R

(8.13)

inc (1) u(1) s = u (𝜁st )f (𝜑, 𝜑0 , 𝛼) √

ei𝜋∕4

in the directions 0 ≤ 𝜑 ≤ 𝛼, and eikR , 2𝜋kΦ′′ (𝜁st ) R

(8.14)

ei𝜋∕4 eikR inc (0) = −u (𝜁 )g (𝜑, 𝜑 , 𝛼) u(1) √ st 0 h 2𝜋kΦ′′ (𝜁st ) R

(8.15)

inc (0) u(1) s = −u (𝜁st )f (𝜑, 𝜑0 , 𝛼) √

ei𝜋∕4

in the directions 𝛼 < 𝜑 < 2𝜋 related to the region inside the tangential wedge. radiated by the total sources jtot = j(1) + j(0) The total diffracted field utot s,h s,h s,h s,h is described by Equations (8.12) and (8.13), where we should replace f (1) (𝜑, 𝜑0 , 𝛼) and g(1) (𝜑, 𝜑0 , 𝛼) by the functions f (𝜑, 𝜑0 , 𝛼) and g(𝜑, 𝜑0 , 𝛼). In the region 𝛼 < 𝜑 < 2𝜋 (inside the tangential wedge), the total diffracted field u(1) + u(0) equals zero, since u(1) = −u(0) , in accordance with Equations (8.14) and (8.15). The asymptotics above for edge-diffracted waves can be presented in another form that reveals their ray structure. To do this, we utilize the following differential operations: Φ′ =

dΦ ̂ ′ inc = t ⋅ ∇ (𝜙 + R) = ̂t ⋅ k̂ i + ̂t ⋅ ∇′ R = − cos 𝛾0 + ̂t ⋅ ∇′ R, d𝜁

(8.16)

Φ′′ =

d𝛾 dΦ′ d̂t d∇′ R ̂ = sin 𝛾0 0 + ⋅ t + ∇′ R ⋅ , d𝜁 d𝜁 d𝜁 d𝜁

(8.17)

d∇′ R ̂ 1 ⋅ t = [1 − (̂t ⋅ ∇′ R)2 ], d𝜁 R d̂t 𝑣̂ = . d𝜁 a

(8.18)

(8.19)

Here 𝑣̂ is the unit vector of the principal normal to the edge L, and a is the radius of curvature of the edge.

265

RAY ASYMPTOTICS

At the stationary point, ∇′ R = −k̂ s and ̂t ⋅ ∇′ R = −̂t ⋅ k̂ s = cos 𝛾0 . Therefore, 2 d∇′ R ̂ sin 𝛾0 ⋅t = , d𝜁 R

∇′ R ⋅

(8.20)

d̂t k̂ s ⋅ 𝑣̂ =− . d𝜁 a

(8.21)

In view of the relationships (8.16) to (8.21), 1 Φ (𝜁st ) = R

(

R 1+ 𝜌

′′

) sin2 𝛾0 ,

(8.22)

where 1 1 = 𝜌 sin 𝛾0

(

d𝛾0 k̂ s ⋅ 𝑣̂ − d𝜁 a sin 𝛾0

) .

(8.23)

The quantity 𝜌 is a caustic parameter; it determines the distance (R = −𝜌) along the ray from the edge to the caustic. Now the edge-diffracted field can be written in the ray form = uinc (𝜁st ) ⋅ (DF) ⋅ (DC) ⋅ eikR , u(1) s,h

(8.24)

1 DF = √ R|1 + R∕𝜌|

(8.25)

where

is the ray’s divergence factor, and e±i(𝜋∕4) DC = √ sin 𝛾0 2𝜋k

{

f (1) (𝜑, 𝜑0 , 𝛼) g(1) (𝜑, 𝜑0 , 𝛼)

} (8.26)

can be interpreted as the integral diffraction coefficient. The quantities R and R + 𝜌 are two principal radii of curvature of the diffracted phase front. The upper sign in e±i𝜋∕4 is taken if Φ′′ (𝜁st ) > 0 and the lower one if Φ′′ (𝜁st ) < 0. The last multiplier in Equation (8.24), eikR , is the phase factor. The divergence factor shows how the edge waves, being cylindrical-like waves in the vicinity of the edge (R ≪ |𝜌|), eikR (DF)eikR ≈ √ , R

(8.27)

266

RAY AND CAUSTIC ASYMPTOTICS FOR EDGE DIFFRACTED WAVES

transform into spherical waves, (DF)eikR ≈

√ eikR |𝜌| , R

(8.28)

at a large distance from the edge (R ≫ |𝜌|). The total edge-diffracted fields can also be represented in the ray form

ei(𝜋∕4) 1 u(t) = uinc (𝜁st ) √ √ s,h R(1 + R∕𝜌) sin 𝛾0 2𝜋k

{

f (𝜑, 𝜑0 , 𝛼) g(𝜑, 𝜑0 , 𝛼)

} eikR .

(8.29)

We remind readers that all variable parameters and coordinates in Equation (8.29) relate to the stationary point 𝜁st . Notice that the PTD ray asymptotics (8.12) and (8.13) with the second derivative Φ′′ (𝜁 ) are much easier to calculate than the GTD form (8.29), which involves complicated calculations of the caustic parameter 𝜌(𝜁 ). 8.1.2

Electromagnetic Waves

According to Equations (7.136) and (7.142), the EEWs diverging from a scattering edge L create the combined wave eikR(𝜁 ) ⃗ (1,t) = 1 E ⃗(1,t) (𝜁 ) d𝜁 , 2𝜋 ∫L R(𝜁 ) [ ] ikR(𝜁 ) e ⃗ (1,t) = 1 ∇R(𝜁 ) × ⃗(1,t) (𝜁 ) d𝜁 H 2𝜋Z0 ∫L R(𝜁 )

(8.30) (8.31)

with ⃗ (1,t) (𝜁 )]eik𝜙i (𝜁 ) . ⃗ (1,t) (𝜁 ) + Z0 H0t (𝜁 )G ⃗(1,t) (𝜁 ) = [E0t (𝜁 )F

(8.32)

Here the integrands contain the fast oscillating factor exp[ikΦ(𝜁 )], where Φ(𝜁 ) = R(𝜁 ) + 𝜙i (𝜁 ). Therefore, application of the stationary-phase technique to these integrals results in the following ray asymptotics: E𝜗(1) = Z0 H𝜑(1) =

1 { inc −Et (𝜁st )f (1) (𝜑, 𝜑0 , 𝛼) sin 𝛾0

} eikR ei𝜋∕4 , + Z0 Htinc [𝜀(𝜑0 ) − 𝜀(𝛼 − 𝜑0 )] cos 𝛾0 √ 2𝜋kΦ′′ (𝜁st ) R E𝜗(t) = Z0 H𝜑(t) = −

Etinc (𝜁st ) sin 𝛾0

ei𝜋∕4 eikR f (𝜑, 𝜑0 , 𝛼) √ 2𝜋kΦ′′ (𝜁st ) R

(8.33)

(8.34)

RAY ASYMPTOTICS

267

and E𝜑(1) = −Z0 H𝜗(1)

}

E𝜑(t) = −Z0 H𝜗(t)

] [ Z0 Htinc (𝜁st ) g(1) (𝜑, 𝜑0 , 𝛼) ei𝜋∕4 eikR = . √ sin 𝛾0 g(𝜑, 𝜑0 , 𝛼) 2𝜋kΦ′′ (𝜁st ) R

(8.35)

Here we utilized Equations (7.148) to (7.154) related to the diffraction cone where 𝜗 = 𝜋 − 𝛾0 . Notice that the asymptotic (8.33) is the obvious consequence of Equation (7.158). Recall also that the terms with functions 𝜀(𝜑0 ) and 𝜀(𝛼 − 𝜑0 ) in Equation (8.33) describe the polarization coupling. The function 𝜀(x) is defined by Equation (7.48). The asymptotics above can also be written in the form of Equation (8.29). For example, E𝜑(1) = −Z0 H𝜗(1)

}

E𝜑(t) = −Z0 H𝜗(t)

] [ Z0 Htinc (𝜁st ) g(1) (𝜑, 𝜑0 , 𝛼) ei𝜋∕4 eikR . (8.36) = √ √ 2 g(𝜑, 𝜑0 , 𝛼) sin 𝛾0 2𝜋k R(1 + R∕𝜌)

Taking Equations (7.159) and (7.160) into account, one can represent these asymptotics in terms of the field components tangential to the scattering edge: } { Et(1) = Etinc (𝜁st )f (1) (𝜑, 𝜑0 , 𝛼) − Z0 Htinc [𝜀(𝜑0 ) − 𝜀(𝛼 − 𝜑0 )] cos 𝛾0 ei𝜋∕4 eikR ×√ , 2𝜋kΦ′′ (𝜁st ) R

(8.37)

ei𝜋∕4 eikR Et(t) = Etinc (𝜁st )f (𝜑, 𝜑0 , 𝛼) √ 2𝜋kΦ′′ (𝜁st ) R

(8.38)

and Ht(1) Ht(t)

}

[ =

Htinc (𝜁st )

g(1) (𝜑, 𝜑0 , 𝛼) g(𝜑, 𝜑0 , 𝛼)

]

ei𝜋∕4 eikR , √ 2𝜋kΦ′′ (𝜁st ) R

(8.39)

where Φ′′ (𝜁 ) = d2 Φ(𝜁 )∕d𝜁 2 .

8.1.3

Comments on Ray Asymptotics

r Comparison of Equations (8.12), (8.13) and (8.37) to (8.39) reveals the following relationships between acoustic and electromagnetic diffracted rays:

268

RAY AND CAUSTIC ASYMPTOTICS FOR EDGE DIFFRACTED WAVES

us = Et

if

uinc (𝜁 ) = Etinc (𝜁 ),

(8.40)

uh = Ht

if

uinc (𝜁 ) = Htinc (𝜁 ).

(8.41)

= Et(0,1) are valid only in the absence of the polarizaThe equalities u(0,1) s tion coupling, when 𝛼 − 𝜋 < 𝜑0 < 𝜋 or 𝛾0 = 𝜋∕2.

Here 𝜁 is the diffraction point on a scattering edge. These relationships [together with Equations (7.159) and (7.160)] allow us to determine completely the field of electromagnetic rays diffracted at a perfectly conducting object if we know the acoustic rays diffracted at soft and rigid objects of the same shape and size. Notice also that these relationships were established earlier by Ufimtsev (1995). r The ray asymptotics (8.29) and (8.36) (for the field generated by the total scattering sources j(t) = j(0) + j(1) ) were postulated in the geometrical theory of diffraction (GTD) (Keller, 1962). Now it is seen that GTD can be interpreted as the ray asymptotic form of PTD for total diffracted field. Notice that the ray asymptotics of the Equation (8.29) type (but in the Kirchhoff approximation) were obtained first by Rubinowicz (1924). Ufimtsev (1995) has shown that the ray asymptotics above can easily be obtained by direct extension of the Rubinowicz theory. r In contrast to PTD, GTD is not applicable in regions where the field does not have a ray structure and where the actual diffraction phenomena happen (GO boundaries, foci, caustics). Several ray-based techniques have been developed to overcome the deficiencies of GTD (Ahluwalia et al., 1968; Kouyoumjian and Pathak, 1974; Lee and Deshamps, 1976; James, 1980; Borovikov and Kinber, 1994). Among them the most developed for practical applications is the uniform theory of diffraction (Kouyoumjian and Pathak, 1974; McNamara et al., 1990). r The ray asymptotics (8.29) for acoustic fields u(t) , as well as the ray asymptotics s,h for electromagnetic fields Et(t) and Ht(t) , are invariant with respect to the permutations 𝜗 ↔ 𝛾0 and 𝜑 ↔ 𝜑0 , and therefore they satisfy the reciprocity principle. We note that these expressions are valid only in the directions of the diffraction , Et(t) , and Ht(t) are cone (𝜗 = 𝜋 − 𝛾0 ). Away from this cone, the total fields utot s,h asymptotically (with k → ∞) equal to zero, due to the absence of the stationary point on the edge. In this region, the individual elementary edge waves cancel each other asymptotically.

r The ray asymptotics for the total fields utot , E(t) , and H (t) satisfy the boundary t t s,h

conditions on the planar faces of the edge. A situation with these conditions is illustrated in Figure 8.2. r The ray asymptotics are not valid at caustics (R = 0 and R = −𝜌), where they predict an infinitely large field intensity. The caustic R = 0 is located at the edge

CAUSTIC ASYMPTOTICS

A

269

B A

B

Figure 8.2 Rectangular facet of the scattering edge. The edge-diffracted rays (solid arrows) exist only in region A and satisfy the boundary conditions there. Individual elementary edge rays (dotted arrows) in region B do not satisfy the boundary conditions, but they cancel each other there asymptotically.

itself. The caustic R = −𝜌 can be real or imaginary. A real caustic occurs outside the scattering object in the positive direction of the vector k̂ s . An imaginary caustic is located in the direction contrary to k̂ s . In particular, imaginary caustics may be inside the scattering body. The value Φ′′ (𝜁st ) > 0 relates to the case when the edge-diffracted ray has not yet reached a caustic, and the value Φ′′ (𝜁st ) < 0 corresponds to the ray that has already passed a caustic and acquired the additional phase shift equal to −𝜋∕2, according to Equation (8.11). r The theory of EEWs developed in Chapter 7 allows one to calculate the edgediffracted field in the vicinity of any caustics away from the scattering edge. An example of such a calculation is given in the following section.

8.2

CAUSTIC ASYMPTOTICS

Caustic asymptotics are presented here for both acoustic and electromagnetic waves. These asymptotics have the same structure and differ only in the coefficients.

8.2.1

Acoustic waves

Suppose that the edge-diffracted rays form a smooth caustic C (Fig. 8.3). It is the envelope of diffracted rays, where a high-intensity field is concentrated. According to Section 8.1, the diffracted field away from the caustic (and in front of the caustic) is the sum of two rays coming from stationary points 𝜁1 and 𝜁2 on the scattering edge L: ei𝜋∕4 eikΦ(𝜁1 ) F(𝜁1 ) u = u0 (𝜁1 ) √ R1 2𝜋kΦ′′ (𝜁1 ) e−i𝜋∕4 eikΦ(𝜁2 ) F(𝜁2 ) + u0 (𝜁2 ) √ , R2 2𝜋k|Φ′′ (𝜁2 )|

(8.42)

270

RAY AND CAUSTIC ASYMPTOTICS FOR EDGE DIFFRACTED WAVES

ζ2 R2

C P

L R1

ζ1 Figure 8.3 Edge-diffracted rays at point P in the vicinity of caustic C.

where Φ(𝜁 ) = 𝜙i (𝜁 ) + R(𝜁 ) and 𝜙i (𝜁 ) is the phase of the incident wave at the point 𝜁 on the scattering edge. Depending on the type of the function F(𝜁1,2 ), Equation generated by the nonuniform/fringe sources j(1) (8.42) represents either the field u(1) s,h s,h or the field utot generated by the total scattering sources j(t) = j(1) + j(0) . Specifically, s,h s,h s,h s,h the functions Fs(1) (𝜁1,2 ) = f (1) (𝜑1,2 , 𝜑01,02 , 𝛼1,2 ),

Fh(1) (𝜁1,2 ) = g(1) (𝜑1,2 , 𝜑01,02 , 𝛼1,2 ) (8.43)

relate to the field u(1) , and the functions s,h Fs(t) (𝜁1,2 ) = f (𝜑1,2 , 𝜑01,02 , 𝛼1,2 ),

Fh(t) (𝜁1,2 ) = g(𝜑1,2 , 𝜑01,02 , 𝛼1,2 )

(8.44)

. We note that functions f , g, f (1) , and g(1) are defined in correspond to the field u(t) s,h Equations (2.62), (2.64), (4.24) to (4.28), and (3.59) to (3.61). The function Φ′′ (𝜁1,2 ) can be represented in the form of Equation (8.22). The first term in Equation (8.42) describes the ray that did not yet reach the caustic, and the second term relates to the ray that has already touched the caustic. For the observation points behind the caustic, no stationary points exist on the edge, and therefore no diffracted rays come here. This is a shadow region for diffracted rays. Below we develop the asymptotic approximation for the field in the illuminated region, in front of the caustic, including the points on the caustic itself. We proceed with the general integral expression for the edge-diffracted field of the Equations (8.3), (8.4) type, u=

eikΦ 1 ̂ u0 (𝜁 )F(𝜁 , m) d𝜁 , 2𝜋 ∫L R

̂ = grad R. m

(8.45)

In front of the caustic, the integrand in Equation (8.45) has two stationary points, 𝜁1 and 𝜁2 . As the observation point P approaches the caustic, the stationary points move toward each other and merge when P reaches the caustic. In this case, Φ′′ (𝜁1,2 ) → 0 and the ray asymptotics (8.42) become invalid. One can define a smooth caustic as a surface where both Φ′ (𝜁 ) = 0 and Φ′′ (𝜁 ) = 0.

CAUSTIC ASYMPTOTICS

271

A main contribution to the integral (8.45) is provided by the stationary points. Real edges usually have the ends. To extract the caustic effect in pure form, we extend the integration limits in (8.45) to infinity (−∞ ≤ 𝜁 ≤ ∞) and in this way remove the ends’ contributions to the field, which are usually lower in magnitude than those of Equation (8.42). Uniform asymptotics valid in the entire illuminated region, including the caustic, can be found using the stationary-phase method, extended for the case of two merging stationary points (Chester et al., 1957). In the following we provide the basic details of this technique and derive the caustic asymptotics.

r As the first and second derivatives of the phase function Φ(𝜁 ) equal zero at the caustic, it is expedient to represent this function as a cubic polynomial Φ(𝜁 ) = 13 𝜏 3 − 𝜇𝜏 + 𝜓

(8.46)

under the condition 𝜏 2 = 𝜇 = 0 for the observation point on the caustic. r The function 𝜏(𝜁 ) shapes a three-sheeted Riemann surface. The regular branch of this function is selected by setting 𝜏(𝜁1 ) = 𝜏1 = 𝜇1∕2 ,

𝜏(𝜁2 ) = −𝜇1∕2 .

(8.47)

The quantities 𝜏(𝜁1,2 ) are the stationary points of the function Φ[𝜁 (𝜏)]. The parameters 𝜇 and 𝜓 are found from Equation (8.46), setting 𝜁 = 𝜁1 and 𝜁 = 𝜁2 , 𝜇3∕2 = 34 [Φ(𝜁2 ) − Φ(𝜁1 )],

𝜓 = 12 [Φ(𝜁1 ) + Φ(𝜁2 )].

(8.48)

r According to Equation (8.42) and in agreement with Figure 8.3, the function Φ(𝜁 ) possesses the following properties: b Φ′′ (𝜁 ) ≥ 0 and Φ′′ (𝜁 ) ≤ 0. This means that Φ(𝜁 ) ≥ Φ(𝜁 ) and 𝜇 ≥ 0, 1 2 2 1 𝜇1∕2 ≥ 0, and 𝜇3∕2 ≥ 0. b Φ′ (𝜁 ) > 0 for 𝜁 < 𝜁 < 𝜁 and Φ′ (𝜁 ) < 0 for 𝜁 < 𝜁 and 𝜁 > 𝜁 . 1 2 1 2 b Φ′′′ (𝜁 ) < 0 in the vicinity of the merging point 𝜁 = lim 𝜁 0 P→C 1,2 . r It follows from Equation (8.46) that 𝜏2 − 𝜇 d𝜁 = ′ d𝜏 Φ (𝜁 )

if

Φ′ (𝜁 ) ≠ 0.

(8.49)

Due to the properties of function Φ(𝜁 ) and in view of the relationships (8.47), this derivative is negative everywhere (d𝜁 ∕d𝜏 ≤ 0). This observation is helpful in choosing the correct sign of d𝜁 ∕d𝜏 at the stationary points 𝜁1,2 where

272

RAY AND CAUSTIC ASYMPTOTICS FOR EDGE DIFFRACTED WAVES

Φ′ (𝜁1,2 ) = 0 and Φ′′ (𝜁0 ) = 0. The corresponding expressions for d𝜁 ∕d𝜏 at these points are found by the subsequent differentiation of Equation (8.46): √ 2𝜇1∕2 d𝜁 =− d𝜏 |Φ′′ (𝜁1,2 )| [ ]3∕2 d𝜁 2 =− d𝜏 |Φ′′′ (𝜁0 )|

if

Φ′ (𝜁1,2 ) = 0,

(8.50)

if

Φ′ (𝜁0 ) = Φ′′ (𝜁0 ) = 0,

(8.51)

√ where |x| > 0 and (|x|)3∕2 > 0. r After transition to the new variable 𝜏 = 𝜏(𝜁 ) in the integral (8.45), we again avoid contributions from the endpoints by extending the integration limits to infinity. Consistent with Equation (8.46), we choose the integration limit 𝜏 = −∞(𝜏 = ∞) when 𝜁 = ∞ (𝜁 = −∞). In addition, we take into account that d𝜁 ∕d𝜏 = − |d𝜁 ∕d𝜏|. Finally, the integral (8.45) can be represented as ∞

u=

3 1 ik𝜓 G(𝜏)eik(𝜏 ∕3−𝜇𝜏) d𝜏, e ∫−∞ 2𝜋

(8.52)

where G(𝜏) =

| d𝜁 | u0 (𝜁 ) ̂ || || . F(𝜁 , m) R(𝜁 ) | d𝜏 |

(8.53)

r Then the function G(𝜏) is expanded into the series G(𝜏) =

∞ ∑

pn (𝜏 − 𝜇)n +

n=0

∞ ∑

qn 𝜏(𝜏 − 𝜇)n .

(8.54)

n=0

By integration of this series in Equation (8.52), one can obtain an asymptotic expansion valid for k → ∞. We retain here only two leading terms in Equation (8.54): G(𝜏) = p + q𝜏,

(8.55)

p = p0 and q = q0 . Setting 𝜏 = 𝜏1 and 𝜏 = 𝜏2 here, one finds that p=

1 [G(𝜇1∕2 ) + G(−𝜇1∕2 )], 2

q=

1 [G(𝜇 1∕2 ) − G(−𝜇1∕2 )]. (8.56) 2𝜇1∕2

r The further substitution of Equation (8.54) into (8.52) leads to the asymptotic expression

CAUSTIC ASYMPTOTICS

u = k−1∕3 eik𝜓 [pAi(−k2∕3 𝜇) − ik−1∕3 qAi′ (−k2∕3 𝜇)]

with

273

k → ∞, (8.57)

where ∞

Ai(t) =

3 1 1 ei(x ∕3+tx) dx = 2𝜋 ∫−∞ 𝜋 ∫0

(

∞

cos

) x3 + tx dx 3

(8.58)

is the Airy function (Abramowitz and Stegun, 1972) and d 1 Ai (t) = Ai(t) = − dt 𝜋 ∫0 ′

(

∞

x sin

) x3 + tx dx. 3

(8.59)

Note also that Ai′ (−t) =

d d Ai(−t) = − Ai(−t). d(−t) dt

(8.60)

The asymptotic approximation (8.57) is uniformly valid in the entire illuminated region, including the caustic. In particular, on the caustic itself, u = k−1∕3 p(𝜁0 ) Ai(0)eikΦ(𝜁0 ) + O(k−2∕3 ),

(8.61)

where according to Abramowitz and Stegun [1972, Equation (10.4.4)] Ai(0) =

3−2∕3 ≈ 0.35502. Γ(2∕3)

(8.62)

, we obtain the explicit To specify the final asymptotics (8.57) for the fields u(1) s,h expressions for coefficients p and q: }

[ ] u0 (𝜁1 ) f (1) (𝜑1 , 𝜑01 , 𝛼1) || d𝜁 (𝜏1 ) || R(𝜁1 ) g(1) (𝜑1 , 𝜑01 , 𝛼1 ) || d𝜏 || p(1) h } [ ] u (𝜁 ) f (1) (𝜑2 , 𝜑02 , 𝛼2 ) || d𝜁 (𝜏2 ) || , + 0 2 R(𝜁2 ) g(1) (𝜑2 , 𝜑02 , 𝛼2 ) || d𝜏 || } ] [ { u0 (𝜁1 ) f (1) (𝜑1 , 𝜑01 , 𝛼1) || d𝜁 (𝜏1 ) || q(1) 1 s = R(𝜁1 ) g(1) (𝜑1 , 𝜑01 , 𝛼1 ) || d𝜏 || q(1) 2𝜇1∕2 h } [ ] u (𝜁 ) f (1) (𝜑2 , 𝜑02 , 𝛼2 ) || d𝜁 (𝜏2 ) || , − 0 2 R(𝜁2 ) g(1) (𝜑2 , 𝜑02 , 𝛼2 ) || d𝜏 || p(1) s

1 = 2

{

(8.63)

(8.64)

274

RAY AND CAUSTIC ASYMPTOTICS FOR EDGE DIFFRACTED WAVES

where the quantity 𝜇 is defined in Equation (8.48). After replacement of functions f (1) and g(1) by f and g (or by f (0) and g(0) ), the expressions (8.63) and (8.64) determine the coefficients p(t) and q(t) (or p(0) and q(0) ) related to the fields u(t) (or to u(0) ). s,h s,h s,h s,h s,h s,h For large real arguments (t ≫ 1), the Airy function and its derivative are determined by the asymptotic expressions (Abramowitz and Stegun, 1972) (

) 2 3∕2 𝜋 t + , 3 4 ) ( 2 𝜋 . Ai′ (−t) ∼ −𝜋 −1∕2 t1∕4 cos t3∕2 + 3 4 Ai(−t) ∼ 𝜋 −1∕2 t−1∕4 sin

(8.65) (8.66)

As shown in Problem 8.5, the general asymptotics (8.57) transform into the ray asymptotics (8.42) in the region far from the caustic (k2∕3 𝜇 ≫ 1). 8.2.2

Electromagnetic Waves

For electromagnetic waves the caustic asymptotics are derived in the same way (Ufimtsev, 1991), and they can be written as [

⃗ (1) E ⃗ (1) H

]

{[ = k−1∕3 eik𝜓

p⃗ (1) e p⃗ (1) h

]

[ Ai(−k2∕3 𝜇) − ik−1∕3

q⃗ (1) e q⃗ (1) h

}

]

Ai′ (−k2∕3 𝜇)

.

(8.67)

Here [ ] 1 ⃗(1) 1 || d𝜁 (𝜏1 ) || ⃗(1) 1 || d𝜁 (𝜏2 ) || (𝜁 )  (𝜁1 ) +  , 2 2 R(𝜁1 ) || d𝜏 || R(𝜁2 ) || d𝜏 || ] [ 1 1 || d𝜁 (𝜏1 ) || ⃗(1) 1 || d𝜁 (𝜏2 ) || (1) ⃗ = −  (𝜁2 ) ,  (𝜁1 ) R(𝜁1 ) || d𝜏 || R(𝜁2 ) || d𝜏 || 2𝜇1∕2

p⃗ (1) e =

(8.68)

q⃗ (1) e

(8.69)

and according to Equations (7.137), (7.156), and (7.158), 𝜗(1) (𝜁m ) = m

1 {−Etinc (𝜁m )f (1) (𝜑m , 𝜑0m , 𝛼m ) m sin 𝛾0m + Z0 Htinc (𝜁m )[𝜀(𝜑0m ) − 𝜀(𝛼 − 𝜑0m )] cos 𝛾0m }, m

𝜑(1) (𝜁m ) = Z0 Htinc m

m

1 g(1) (𝜑m , 𝜑0m , 𝛼m ). sin 𝛾0m

(8.70) (8.71)

The subscript m = 1, 2 indicates that a quantity with this subscript relates to the stationary point 𝜁1 or 𝜁2 at the edge.

RELATIONSHIPS BETWEEN PTD AND GTD

275

Formulas such as Equations (8.68) and (8.69) define the vectors p⃗ (1) and q⃗ (1) . It is h h ⃗ (1) = [∇R × ⃗(1) ]∕Z0 . only necessary to replace the vector ⃗(1) by  After replacement of functions f (1) and g(1) by f and g (or by f (0) and g(0) ), the , q⃗ (t) (or p⃗ (0) , q⃗ (0) ) related expressions (8.68) and (8.69) determine the coefficients p⃗ (t) e,h e,h e,h e,h ⃗ (t) (or to E ⃗ (0) , H ⃗ (0) ). ⃗ (t) , H to the fields E ⃗ (1) , H ⃗ (1) (with functions f (1) and g(1) ) ,E The asymptotics derived for the field u(1) s,h , have an important advantage compared to the asymptotics for the total field u(t) s,h ⃗ (t) . They remain finite at the boundaries of ordinary incident and reflected rays ⃗ (t) , H E (𝜑 = 𝜋 ± 𝜑0 and 𝜑 = 2𝛼 − 𝜋 − 𝜑0 ), where the functions f and g become singular. It follows from Equations (8.57) and (8.67) that the structure of the caustic field is the same for acoustic and electromagnetic waves. The difference is only in the coefficients p and q, which are scalar quantities for acoustic waves and vectors for electromagnetic waves. Note also that asymptotics (8.57) and (8.67) are valid for the field calculation only in the illuminated region, in front of the caustic. When the observation point moves across the caustic into the shadow region, the diffracted field changes continuously and attenuates exponentially, because the elementary edge waves cancel each other there asymptotically. We do not consider this topic here. Details regarding the wave field in the vicinity of arbitrary caustics are available in a review paper by Kravtsov and Orlov (1983).

8.3

RELATIONSHIPS BETWEEN PTD AND GTD

Here we continue to discuss the relationships between PTD and GTD. PTD is a source-based technique and GTD belongs to ray-type theory. One may think that they are completely different and independent. However, a tight connection exists between them. First, both of them are founded on the localization principle: that in the high-frequency limit (k = 2𝜋f ∕c → ∞), the diffraction field in the vicinity of a scattering object depends on its local properties. But this principle is utilized in PTD and GTD in different ways; in PTD it is applied to the field (currents) on the object surface, and in GTD, to the diffracted rays (i.e., only to the ray part of the field radiated by these currents). Thus, even on this foundation level it is understood that GTD can be interpreted as the asymptotic ray form of PTD. That is why it is not surprising that GTD edge-diffracted rays are derived from the PTD field integrals by asymptotic evaluation, as shown in Sections 8.1.1 and 8.1.2 (and in Ufimtsev, 1977). It is pertinent to remind readers again of the central idea of PTD. It consists of the separation of the surface sources/currents into uniform and nonuniform components. The nonuniform component is the diffracted part of the current caused by any deviation of the object surface from a homogeneous infinite plane. A creeping wave on a smooth convex object is a typical example of a nonuniform component. Thus, the GTD surface-diffracted rays represent the waves radiated by this PTD current component.

276

RAY AND CAUSTIC ASYMPTOTICS FOR EDGE DIFFRACTED WAVES

It is worth noting that basic items of GTD such as edge-diffracted rays, diffraction cones, and surface-diffracted rays were introduced initially in source-based theory. In particular, Rubinowicz (1924) established the ray structure of the edgediffracted field and found that these rays satisfy the Fermat principle and are distributed over the cone surface. He developed this on the basis of the Kirchhoff (source-based) approximation by asymptotic evaluation of the diffraction integral in the general case of arbitrary curved edges. To compare his results with GTD, it is convenient to represent them in the GTD format (Ufimtsev, 1983, 1995). Then it becomes clear that GTD follows from the Rubinowicz theory with obvious replacement of the Kirchhoff diffraction coefficients by the Sommerfeld diffraction coefficients. A similar situation happened with the origination of surface-diffracted rays. They were introduced by integration of the creeping currents (Franz and Depperman, 1952; Depperman and Franz, 1954). What these authors did can be interpreted as the typical PTD procedure since the creeping waves are the nonuniform currents. These comments stress the role of source-based theories in origination of the concepts of diffracted rays. One should also notice the important role of the ray concepts in source-based techniques. In particular, the ray concept is useful in calculating the PTD multiple diffraction, as shown in Chapters 9 to 12, and beneficial in clarifying the physical structure of the diffracted field. In modern high-frequency diffraction theory the concepts of the scattering sources and diffracted rays supplement each other successfully.

PROBLEMS 8.1 Calculate the edge wave scattered by an infinite straight edge of a wedge (Fig. 4.3). Find the wave generated by the nonuniform/fringe scattering sources. The incident acoustic wave is given by uinc = u0 e−ikz cos 𝛾0 e−ikr sin 𝛾0 cos(𝜑−𝜑0 ) . Notice that the angle 𝛾0 is defined here as 𝛾0 = 𝜋 − 𝛾, where the angle 𝛾 is shown in Figure 4.3. Consider scattering by soft and hard wedges. Integrate the EEWs over the entire edge. Utilize the stationary-phase method and obtain the asymptotics (4.61) and (4.62). Solution Start with Equation (8.3). Specify the integration contour L as −∞ ≤ 𝜁 ≤ ∞. Follow Equations (8.8) to (8.12), (8.13). Determine the stationary point 𝜁st and the function Φ′′ (𝜁st ). Here Φ(𝜁 ) = −𝜁 cos 𝛾0 + R. Figure P8.1 illustrates the problem. √ For simplicity we consider the field in the (x, y)-plane where z = 0 and R = x2 + y2 + 𝜁 2 . The stationary point 𝜁st is found from the Equation Φ′ (𝜁 ) = − cos 𝛾0 + R′ (𝜁 ) = − cos 𝛾0 + 𝜁 ∕R = 0.

PROBLEMS

277

Z r

z=0

ϑ

R

ξ Figure P8.1 Geometry of the problem.

According to Figure P8.1, 𝜁 = −R cos 𝜗. Hence, at the stationary point, cos 𝛾0 + cos 𝜗st = 0, 𝜗st = 𝜋 − 𝛾0 , 𝜁st = R cos 𝛾0 , and Φ′′ (𝜁st ) = sin2 𝛾0 ∕R. Also, 2 2 eik(R−𝜁st cos 𝛾0 ) = eik(R−R cos 𝛾0 ) = eikR sin 𝛾0 = eik1 r and 1 1 1 =√ =√ , √ ′′ 2𝜋k1 r R 2𝜋kΦ (𝜁st ) 2𝜋kR sin2 𝛾0 where k1 = k sin 𝛾0 and r = R sin 𝛾0 . Finally, we obtain the expressions ei(k1 r+𝜋∕4) (1) ei(k1 r+𝜋∕4) (1) u(1) , uh = u0 g(1) (𝜑, 𝜑0 , 𝛼) √ s = u0 f (𝜑, 𝜑0 , 𝛼) √ 2𝜋k1 r 2𝜋k1 r which coincide with Equations (4.61), and (4.62) in the plane z = 0. 8.2

Calculate the electromagnetic wave scattered by an infinite straight edge of a perfectly conducting wedge (Fig. 4.3). Find the wave generated by the nonuniform/fringe currents. The incident electromagnetic wave is given by Ezinc = E0z e−ikz cos 𝛾0 e−ikr sin 𝛾0 cos(𝜑−𝜑0 ) , Hzinc = H0z e−ikz cos 𝛾0 e−ikr sin 𝛾0 cos(𝜑−𝜑0 ) . Note that the angle 𝛾0 is defined here as 𝛾0 = 𝜋 − 𝛾, where the angle 𝛾 is shown in Figure 4.3. Prove Equations (4.69) and (4.70). Solution In general form, the solution is presented by Equations (8.37) and (8.39). One need only specify the stationary point 𝜁st and the function Φ′′ (𝜁√ st ). For simplicity we consider the field in the (x, y)-plane where z = 0 and R = x2 + y2 + 𝜁 2 . The quantities 𝜁st and Φ′′ (𝜁st ) have already been found, in

278

RAY AND CAUSTIC ASYMPTOTICS FOR EDGE DIFFRACTED WAVES

Problem 8.1. Utilizing those results, one obtains the asymptotics } ei(k1 r+𝜋∕4) { Ez(1) = E0z f (1) (𝜑, 𝜑0 , 𝛼) − Z0 H0z [𝜀(𝜑0 ) − 𝜀(𝛼 − 𝜑0 )] cos 𝛾0 √ , 2𝜋k1 r ei(k1 r+𝜋∕4) Hz(1) = H0z g(1) (𝜑, 𝜑0 , 𝛼) √ . 2𝜋k1 r They coincide with Equations (4.69) and (4.70) in the plane z = 0 if one takes into account that 𝛾0 = 𝜋 − 𝛾. 8.3 Use Equations (8.3) and (8.4) and calculate the edge waves u(1) and u(t) in s,h s,h 2 the far zone (R ≫ ka ) scattered by a circular disk (Fig. P8.2). The incident acoustic wave is given by uinc = u0 eikz . Integrate the EEWs over the entire edge (d𝜁 = a d𝜓, 0 ≤ 𝜓 ≤ 2𝜋). Carefully read the box at the beginning of Section 7.2 regarding the local coordinates associated with edge points 𝜁 = a𝜓. Remember that the local angle 𝜑(𝜓)is measured from the illuminated face of the disk. That is why on Figure P8.2 the tangent ̂t = 𝜓̂ is directed counterclockwise and the polar angle 𝜓 is measured counterclockwise. . Compare with Equations (a) Find the focal asymptotics for the fields u(1) s,h (6.81) and (6.82). (b) Use the stationary-phase method and obtain the ray asymptotics for the field u(t) in the half-plane x = 0, y ≥ 0. Compare with Equations (6.84) and s,h (6.85).

y

ρˆ

1

ψ x

tˆ

2 Figure P8.2 Illuminated face of a disk of radius a and related coordinates.

Solution (a) For the observation points on the focal line (𝜗 = 0 or 𝜗 = 𝜋) every edge point 𝜍 is stationary. Therefore, according to Equations (7.115), the functions Fs(1)

279

PROBLEMS

and Fh(1) are reduced to the functions f (1) and g(1) , respectively. Thus, in the far-zone approximation, u(1) s u(1) h

} 1 = u0 2𝜋

(

f (1) g(1)

)

eikR R ∫0

(

2𝜋

a d𝜓 = u0 a

f (1)

)

g(1)

eikR . R

For the focal line the functions f (1) and g(1) are determined according to Equations (6.79) and (6.80). Taking them into account, one obtains Equations (6.81) and (6.82). (b) Away from the focal line the integral (8.4) is evaluated utilizing the stationary-phase technique. For simplicity, consider the scattered field in the far zone (R ≫ ka2 ) for the observation points in the half-plane x = 0, y ≥ 0 (Fig. P8.2). √ In Equation (8.4) √ we denote R(𝜁 ) as r(𝜓) = 𝜉 2 + (y − 𝜂)2 + z2 and retain the letter R for y2 + z2 , where y = R sin 𝜗, z = R cos 𝜗, 𝜉 = a sin 𝜓, and 𝜂 = a cos 𝜓. In the far-zone approximation, 2𝜋

a eikR eikr eikR −iika sin 𝜗 cos 𝜓 and u(t) = u0 F (t) (𝜓)e−ika sin 𝜗 cos 𝜓 d𝜓. ≈ e s,h r R 2𝜋 R ∫0 s,h Here the phase function equals Φ(𝜓) = −a sin 𝜗 cos 𝜓. It contains two stationary points, 𝜓1 = 0 and 𝜓2 = 𝜋, shown in Figure P8.2 by points 1 and 2. The sec= Φ′′ (𝜋∕2) = a sin 𝜗 and Φ′′ = ond derivatives of the phase function equal Φ′′ 1 2 √ √ √ √ ′′ ′′ ′′ Φ (−𝜋∕2) = −a sin 𝜗 with Φ1 = a sin 𝜗 and Φ2 = a sin 𝜗 exp(i𝜋∕2). According to the stationary-phase technique, 2𝜋

∫0

(t) (t) Fs,h (𝜓)eikΦ(𝜓) d𝜓 ∼ Fs,h (𝜓1 )eikΦ(𝜓1 )

∞

∫−∞

(t) (𝜓2 )eikΦ(𝜓2 ) + Fs,h

′′ (𝜓 )(𝜓 −𝜓)2 1 1

e(ik∕2)Φ ∞

∫−∞

d𝜓

′′ (𝜓 )(𝜓 −𝜓)2 2 2

e(ik∕2)Φ

d𝜓,

or in view of Equation (8.9), 2𝜋

∫0

√ (t) (t) Fs,h (𝜓)eikΦ(𝜓) d𝜓 ∼ Fs,h (𝜓1 )eikΦ(𝜓1 )

2𝜋 ei𝜋∕4 kΦ′′ (𝜓1 )

√ (t) + Fs,h (𝜓2 )eikΦ(𝜓2 )

2𝜋 e−i𝜋∕4 . | k |Φ′′ (𝜓2 )||

280

RAY AND CAUSTIC ASYMPTOTICS FOR EDGE DIFFRACTED WAVES

(t) Here Fs(t) (𝜓1,2 ) = f (1, 2), F√ (𝜓1,2 ) = g(1, 2), Φ(𝜓1 ) = −a sin 𝜗, Φ(𝜓2 ) = h √ ′′ a sin 𝜗, and k|Φ (𝜓1,2 | = ka sin 𝜗. These asymptotics lead to Equations (6.84) and (6.85) for the fields u(t) . s,h

8.4 Use Equations (8.30) and (8.31) and calculate the far fields Ex(1) and Ex(t) scattered by a circular perfectly conducting disk (Fig. P8.2). The incident electromagnetic wave is given by Exinc = E0x eikz ,

H0y = Y0 E0x eikz .

Integrate the EEWs over the entire edge. Carefully read the box at the beginning of Section 7.2 regarding the local coordinates associated with edge points 𝜁 = a𝜓. (a) Find the focal asymptotics for the field Ex(1) generated by the nonuniform/fringe current. (b) Use the stationary-phase method to obtain the ray asymptotics for the field Ex(t) generated by the total current in the half-plane x = 0, y ≥ 0. Compare with Equation (6.84). Solution (a) First, it is necessary to establish the relationships between the local coordinates R(𝜁 ) ≡ r(𝜓), 𝜗(𝜓), 𝜑(𝜓), and ̂t(𝜓) = 𝜓̂ associated with the edge point 𝜁 (𝜓) and the basic coordinates x, y, z and R, 𝜗, 𝜑. Keep in mind that the angle 𝜑(𝜓) is measured from the illuminated face of the disk. That is why on Figure P8.2 the tangent ̂t = 𝜓̂ is directed counterclockwise and the polar angle 𝜓 is measured counterclockwise. According to this figure, ̂t(𝜓) = x̂ cos 𝜓 − ŷ sin 𝜓 and 𝜑(𝜓) ̂ = 𝜌(𝜓) ̂ = ̂ = x̂ sin 𝜓 + ŷ cos 𝜓 for the focal line 𝜗 = 𝜋 where 𝜑(𝜓) = 𝜋∕2, but 𝜑(𝜓) −𝜌(𝜓) ̂ = −̂x sin 𝜓 − ŷ cos 𝜓 for the focal line 𝜗 = 0, where 𝜑(𝜓) = 3𝜋∕2. ̂ In addition, for the focal points, 𝜗(𝜓) = 𝜋∕2 and 𝜗(𝜓) = −̂t(𝜓). Note also that in this problem the direction of the incident wave is normal to the edge, and therefore the angle 𝛾0 equals 𝜋∕2. We calculate the field Ex(1) and utilize Equations (8.30) and (8.32). They ⃗ (1) , whose components are defined in Section ⃗ (1) and G contain the vectors F 7.8. For the stationary points they are determined by Equations (7.148) and (7.151). For the focal line, every edge point 𝜁 = a𝜓 is stationary and, therefore, (1) = −f (1) F𝜗(𝜓)

(1) F𝜗(𝜓) = −f (1)

(

(

) 𝜋 𝜋 , , 2𝜋 , 2 2

= g(1) G(1) 𝜑(𝜓)

) 3𝜋 𝜋 , , 2𝜋 , 2 2

= g(1) G(1) 𝜑(𝜓)

( (

𝜋 𝜋 , , 2𝜋 2 2

)

3𝜋 𝜋 , , 2𝜋 2 2

for ) for

𝜗 = 𝜋, 𝜗 = 0.

PROBLEMS

281

To find Ex(1) one needs to calculate the scalar/dot product ̂ ⋅ x̂ ] + Z0 H0t (𝜓)G(1) [𝜑(𝜓) ̂ ⋅ x̂ ]. ⃗(1) (𝜓) ⋅ x̂ = E0t (𝜓)F𝜗(𝜓) [𝜗(𝜓) 𝜑(𝜓) ̂ ⋅ x̂ = −̂t ⋅ x̂ = − cos 𝜓, H0t (𝜓) = −Y0 E0x sin 𝜓, Here E0t = E0x cos 𝜓, 𝜗(𝜓) 𝜑(𝜓) ̂ ⋅ x̂ = sin 𝜓 for 𝜗 = 𝜋, and 𝜑(𝜓) ̂ ⋅ x̂ = − sin 𝜓 for 𝜗 = 0. Thus, ⃗(1) (𝜓) ⋅ x̂ = E0x [f (1) cos2 (𝜓) − g(1) sin2 𝜓] for the direction 𝜗 = 𝜋, ⃗(1) (𝜓) ⋅ x̂ = E0x [f (1) cos2 (𝜓) + g(1) sin2 𝜓] for the direction 𝜗 = 0. The integrals from sin2 𝜓 and cos2 𝜓 over 0 ≤ 𝜓 ≤ 2𝜋 are equal to 12 . Taking these relationships into account, we obtain Ex(1) = E0x

( ( [ ) )] ikR 3𝜋 𝜋 a (1) 3𝜋 𝜋 e f , , 2𝜋 + g(1) , , 2𝜋 2 2 2 2 2 R

in the direction 𝜗 = 0, Ex(1) = E0x

( ( [ ) )] ikR 𝜋 𝜋 a (1) 𝜋 𝜋 e f , , 2𝜋 − g(1) , , 2𝜋 2 2 2 2 2 R

in the direction 𝜗 = 𝜋. These expressions agree with Equations (2.3.18) and (2.3.19) in Ufimtsev’s book (2009). According to Equations (4.29) and (4.31), ( ) ) 𝜋 𝜋 𝜋 𝜋 1 , , 2𝜋 = g(1) , , 2𝜋 = − in the direction 𝜗 = 𝜋, 2 2 2 2 2 ( ( ) ) 3𝜋 𝜋 3𝜋 𝜋 1 f (1) , , 2𝜋 = −g(1) , , 2𝜋 = − in the direction 𝜗 = 0. 2 2 2 2 2 f (1)

(

Thus, Ex(1) = 0; that is, the electromagnetic field generated by the nonuniform/fringe currents equals zero on the focal line, in contrast to the nonzero acoustic focal field found in Problem 8.3. (b) For simplicity we consider the field in the half-plane x = 0, y ≥ 0. For observation points in this half-plane, the phase function in Equation (8.30) contains two stationary points, 𝜓1 = 0 and 𝜓2 = 𝜋 (see Fig. P8.2). Denote them as √ points 1 and 2. The phase function in Equation (8.30) equals Φ(𝜓) = 𝜉 2 + (y − 𝜂)2 + z2 , where 𝜉 = a sin 𝜓 and 𝜂 = a cos 𝜓. In the

282

RAY AND CAUSTIC ASYMPTOTICS FOR EDGE DIFFRACTED WAVES

far-zone approximation, Φ(𝜓) = R − a sin 𝜗 cos 𝜓, Φ′′ (𝜓) = d2 Φ(𝜓)∕ d𝜓 2 = a sin 𝜗 cos 𝜓 and Φ′′ (𝜓1 ) = a sin 𝜗, Φ′′ (𝜓2 ) = −a sin 𝜗. The field contribution of each stationary point is determined by the general Equation (8.38), where one should set Et(t) = Ex(t) , Et(t)

=

−Ex(t) ,

Etinc = Exinc

for the contribution of point 1,

Etinc = −Exinc

for the contribution of point 2, √ √ f = f (1), R1 = R − a sin 𝜗, Φ′′ (𝜓1 ) = a sin 𝜗 for point 1, √ √ f = f (2), R2 = R + a sin 𝜗, Φ′′ (𝜓2 ) = a sin 𝜗 exp(i𝜋∕2) for point 2.

Note also that in Equation (8.38), Φ′′ (𝜁 ) = d2 Φ[𝜁 (𝜓)]∕d𝜁 2 = (1∕a2 ) d2 Φ∕d𝜓 2 . Summation of the contributions from stationary points 1 and 2 leads to the equation Ex(t) = E0x √

a 2𝜋ka sin 𝜗

[f (1)e−i(ka sin 𝜗−𝜋∕4) + f (2)ei(ka sin 𝜗−𝜋∕4) ]

eikR , R

which is consistent with Equation (6.84) for acoustic waves. 8.5 Show that the caustic asymptotics (8.57) for acoustic waves transform into the ray asymptotics (8.42) when k2∕3 𝜇 ≫ 1. Solution Represent Equation (8.57) as the sum u = u1 + u2 . Utilize the Euler formulas for the sine and cosine functions in the asymptotics (8.65) and (8.66). These asymptotics obtain the factors (eix − e−ix ) and (eix + e−ix ), respectively, where x = 12 k[Φ(𝜁2 ) − Φ(𝜁1 )] + 𝜋∕4.

Being multiplied by exp(ik𝜓) = exp[i 12 k[Φ(𝜁2 ) + Φ(𝜁1 )], these factors transform to [eikΦ(𝜁2 )+i(𝜋∕4) − eikΦ(𝜁1 )−i(𝜋∕4) ] and [eikΦ(𝜁2 )+i(𝜋∕4) + eikΦ(𝜁1 )−i(𝜋∕4) ], respectively. Represent coefficients p and q as 𝜇1∕4 p = √ [a(𝜁1 )F(𝜁1 ) + a(𝜁2 )F(𝜁2 )], 2

𝜇 −1∕4 q = √ [a(𝜁1 )F(𝜁1 ) − a(𝜁2 )F(𝜁2 )], 2

√ where a(𝜁 ) = [u0 (𝜁 )∕R(𝜁 )][1∕ |Φ′′ (𝜁 )|]. Then the terms u1 and u2 can be written as follows: i [eikΦ(𝜁2 )+i𝜋∕4 − eikΦ(𝜁1 )−i𝜋∕4 ][a(𝜁1 )F(𝜁1 ) + a(𝜁2 )F(𝜁2 )], u1 = − √ 2 2𝜋k i [eikΦ(𝜁2 )+i𝜋∕4 + eikΦ(𝜁1 )−i𝜋∕4 ][a(𝜁1 )F(𝜁1 ) − a(𝜁2 )F(𝜁2 )] u2 = √ 2 2𝜋k

PROBLEMS

or 1 [eikΦ(𝜁2 )−i𝜋∕4 + eikΦ(𝜁1 )+i𝜋∕4 ][a(𝜁1 )F(𝜁1 ) + a(𝜁2 )F(𝜁2 )], u1 = √ 2 2𝜋k 1 [−eikΦ(𝜁2 )−i𝜋∕4) + eikΦ(𝜁1 )+i𝜋∕4 ][a(𝜁1 )F(𝜁1 ) − a(𝜁2 )F(𝜁2 )]. u2 = √ 2 2𝜋k The summation of these terms gives exactly the ray asymptotics (8.42).

283

9 Multiple Diffraction of Edge Waves: Grazing Incidence and Slope Diffraction 9.1

STATEMENT OF THE PROBLEM AND RELATED REFERENCES

Clearly, the theory developed in Chapter 8 can be utilized in the investigation of multiple diffraction at edges that are spaced apart. Only two special cases require an individual investigation. The first case is a grazing incidence of edge waves on acoustically hard planar plates. In the asymptotic theory, the incident wave is approximated by an equivalent plane wave. However, a plane wave does not undergo diffraction at an infinitely thin plate under the grazing incidence, for the following reason. When this wave propagates in the direction parallel to the plate, its wave and amplitude fronts are perpendicular to the plate. As this incident field is constant in the direction normal to the plate, it automatically satisfies the boundary condition du∕dn = 0 on the plate. Such a wave does not “see” the plate and propagates as if a free space is on its path. Because of this, the foregoing theory predicts a zero-diffracted field in this case. However, in the process of multiple diffraction, every diffracted wave is not plane. If its normal derivative du∕dn is not a zero on the plate, it undergoes diffraction. Such a grazing diffraction is studied in Section 9.2. The second case that also needs special treatment occurs when the scattering edge is located in the zero of the incident wave. This is the case for slope diffraction. One distinguishes a slope diffraction of different orders, depending on the zero orders.

Fundamentals of the Physical Theory of Diffraction, Second Edition. Pyotr Ya. Ufimtsev. © 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

285

286

MULTIPLE DIFFRACTION OF EDGE WAVES: GRAZING INCIDENCE AND SLOPE DIFFRACTION

Here we consider the most important one to be the slope diffraction of the first order, when the first derivative of the incident wave is not equal to zero. Such a situation occurs, for example, in reflector antennas, when one tries to decrease side lobes, and in the process of multiple diffraction between several scatterers or between different parts of the same scatterer. Many authors have studied the phenomenon of slope diffraction. Ufimtsev (1958b,c, 1962b) suggested uniform asymptotics for the secondary edge waves arising due to the slope diffraction on plane screens (strip, disk). Karp and Keller (1961) derived nonuniform ray asymptotics for the same edge waves. Mentzer et al. (1975) published uniform asymptotics similar to those found by Ufimtsev (1958b,c). The spectral theory of diffraction (Rahmat-Samii and Mitra, 1978) also enables one to investigate the slope diffraction. In the particular case of the half-plane diffraction problem, Boersma and Rahmat-Samii (1980) analyzed this phenomenon in the framework of ray-based theories [uniform asymptotic theory (UAT) and uniform theory of diffraction (UTD)]. Pathak (1988) constructed the general UTD for slope diffraction at the wedge. The general PTD for the slope diffraction of electromagnetic waves (based on the concept of elementary edge waves) was elaborated in papers by Ufimtsev (1991) and Ufimtsev and Rahmat-Samii (1995). A similar theory for both the grazing diffraction and the slope diffraction of acoustic and electromagnetic waves was developed earlier by Ufimtsev (1989, 1991). The theory presented below is based on papers by Ufimtsev (1989, 1991) and Ufimtsev and Rahmat-Samii (1995).

9.2 GRAZING DIFFRACTION The following relationship exists between the acoustic and electromagnetic diffracted rays arising due to the grazing diffraction at the plate S1 (Fig. 9.1): uh = Ht

if

inc 𝜕uinc (𝜁 ) 𝜕Ht (𝜁 ) = 𝜕n 𝜕n

at the scattering edge L1 .

Here ̂t is the tangent to the edge L1 and n̂ is the normal to the plate S1 at the diffraction point 𝜁 .

9.2.1

Acoustic Waves

Figure 9.1 shows a configuration appropriate for studying both the grazing diffraction and the slope diffraction. There are two acoustically hard scattering objects with edges L1 and L2 . One of them is a planar plate S1 . The boundary conditions du∕dn = 0 are imposed on the surfaces S1 and S2 . Edge L2 is located in the plane containing plate S1 . No more than one edge-diffracted ray comes to every point on edge L1 (L2 ) from edge L2 (L1 ). The wave initially diffracted at edge L2 propagates to edge L1 and undergoes the grazing diffraction at plate S1 . This problem is investigated in the present section. The wave field diffracted at edge L1 equals zero in the direction to edge L2 , where it undergoes slope diffraction. This problem is considered in the next section.

287

GRAZING DIFFRACTION

r2

φ2

S2

r1

φ02 L2

R21

φ1

S1

L1

Figure 9.1 Problem of multiple diffraction. Edge L2 is in the plane containing plate S1 with edge L1 . Edge L1 is perpendicular to the figure plane at the intersection point. Edge L2 intersects the figure plane at an oblique angle. Line R21 shows the diffracted ray coming from L2 to L1 . [Reprinted from Ufimtsev (1989) with the permission of the Journal of the Acoustical Society of America.]

Suppose that the wave ikR2 uinc 2 = 𝑣2 (R2 , 𝜑2 , 𝜑02 )e

(9.1)

propagates from edge L2 and undergoes grazing diffraction at plate S1 . This is a wave with a ray structure of the type of Equation (8.29). Consider the two first terms of its Taylor expansion in the vicinity of the grazing direction 𝜑2 = 𝜑02 : ikR2 + uinc 2 = 𝑣2 (R2 , 𝜑02 , 𝜑02 )e

𝜕𝑣2 (R2 , 𝜑02 , 𝜑02 ) (𝜑2 − 𝜑02 )eikR2 + ⋯ . 𝜕𝜑2

(9.2)

Here the first term represents the wave that does not undergo diffraction at plate S1 because its normal derivative on the plate equals zero [𝜕𝑣2 (R2 , 𝜑02 , 𝜑02 )∕𝜕𝜑2 = 0]. Therefore, that part of the incident wave that experiences diffraction at the plate can be approximated by the wave 𝜕𝑣2 (R2 , 𝜑02, 𝜑02 ) 𝜕𝜑2

(𝜑2 − 𝜑02 )eikR2 ,

(9.3)

which has a zero field in the grazing direction. Thus, we see that the grazing diffraction of the wave (9.1) actually itself represents a particular case of the slope diffraction. We approximate the wave (9.3) by the equivalent canonical wave

eq

u2 = u02

𝜕 −ikz1 cos 𝛾01 −ikr1 sin 𝛾01 cos(𝜑1 −𝜑01 ) | e e | |𝜑01 =𝜋 𝜕𝜑01

= u02 ikr1 sin 𝛾01 sin 𝜑1

(9.4)

e−ikz1 cos 𝛾01 eikr1 sin 𝛾01 cos 𝜑1 ,

obtained by differentiation of the plane wave. The quantities r1 , 𝜑1 , and z1 are local polar coordinates with the origin at the diffraction point on edge L1 (Fig. 9.1), and the angle 𝛾01 is shown in Figure 9.2, where k̂ 1i = ∇R21 .

288

MULTIPLE DIFFRACTION OF EDGE WAVES: GRAZING INCIDENCE AND SLOPE DIFFRACTION

z2 tˆ2 r21

02

L1

R21

L2

01

tˆ1

kˆi1

z1

Figure 9.2 An edge wave arising at edge L2 propagates in the direction k̂ 1i = ∇R21 and undergoes the next diffraction at edge L1 . The unit vectors ̂t1 and ̂t2 are tangents to edges L1 and L2 . [Reprinted from Ufimtsev (1989) with the permission of the Journal of the Acoustical Society of America.]

The amplitude u02 of the equivalent canonical wave is defined by equating the normal derivatives of the real and equivalent incident waves at the diffraction point z1 = r1 = 0: eq 𝜕𝑣2 (R2 , 𝜑2 , 𝜑02 ) ikR || 1 1 𝜕u2 || e 2| = . | R21 sin 𝛾02 𝜕𝜑2 |R2 =R21 , 𝜑2 =𝜑02 r1 𝜕𝜑1 ||z =r =0, 𝜑 =𝜋 1 1 1

(9.5)

According to this equation, u02 = −

𝜕𝑣2 (R21 , 𝜑2 , 𝜑02 ) ikR 1 e 21 ikR21 sin 𝛾01 sin 𝛾02 𝜕𝜑2

with 𝜑2 = 𝜑02

(9.6)

or u02 = 𝑤02 eikR21

(9.7)

with

𝑤02 = −

𝜕𝑣2 (R21 , 𝜑2 , 𝜑02 ) || 1 . | ikR21 sin 𝛾01 sin 𝛾02 𝜕𝜑2 |𝜑2 =𝜑02

(9.8)

Now we note that the Helmholtz equation governing the wave field and its diffraction can be differentiated with respect to the parameter 𝜑01 without changing the type of these equations. This means that the derivative (with respect to a free parameter)

GRAZING DIFFRACTION

289

of any solution of the Helmholtz equations is also a solution of the same equation. The boundary conditions also admit the differentiation with respect to 𝜑01 . In Chapter 7 we found the edge-diffracted field generated by the incident plane wave (7.2). The incident wave (9.4) is the derivative of the wave (7.2). Therefore, the diffracted field generated by the wave (9.4) can be found by the differentiation of the edge-diffracted fields (7.97) with respect to 𝜑01 if we also replace uinc (𝜁 ) there by u02 = 𝑤02 exp(ikR21 ). Before completing this procedure, however, we make another observation. The incident wave propagating in the direction grazing to the plate creates the = 2uinc on both sides of the plate. According to identical scattering sources j(0) h (1.10), the field generated by the sources induced on one side of the plate is canceled completely by the field generated by the identical sources induced on the opposite side of the same plate. Therefore, in this particular case of the grazing incidence, the equals zero and u(1) ≡ u(t) . field u(0) h h h In view of these observations, the elementary edge wave generated at edge L1 is found by the differentiation of Equation (7.97) with the simultaneous replacement uinc (𝜁 ) by 𝑤02 (𝜁 ) exp(ikR21 ): = du(t) h

d𝜁 eik[R1 (𝜁 )+R21 (𝜁 )] 𝜕 | ̂ | Fh(t) (𝜁 , m) 𝑤02 (𝜁 ) . |𝜑01 =𝜋 2𝜋 𝜕𝜑01 R1 (𝜁 )

(9.9)

The diffracted wave diverging from the entire edge L1 is determined respectively by the integral

u(t) = h

eik[R1 (𝜁 )+R21 (𝜁 )] 1 𝜕 | ̂ | 𝑤02 (𝜁 ) Fh(t) (𝜁 , m) d𝜁 . |𝜑01 =𝜋 2𝜋 ∫L1 𝜕𝜑01 R1 (𝜁 )

(9.10)

The ray asymptotics of this field is found using the stationary-phase technique described in Section 8.1. We omit all intermediate details and bring the final expression

ei𝜋∕4 1 u(t) = 𝑤 (𝜁 ) √ √ 02 st h R1 (1 + R1 ∕𝜌1 ) sin 𝛾01 2𝜋k ×

𝜕g(𝜑1 , 𝜑01 , 𝛼1 ) ik(R +R ) | e 1 21 | , |𝜑01 =𝜋 𝜕𝜑01

(9.11)

√ √ where 𝛼1 = 2𝜋, R1 (1 + R1 ∕𝜌1 ) > 0 if (1 + R1 ∕𝜌1 ) > 0, and R1 (1 + R1 ∕𝜌1 ) = √ | | i | R1 (1 + R1 ∕𝜌1 )| if (1 + R1 ∕𝜌1 ) < 0. Here all variable parameters and coordinates | |

290

MULTIPLE DIFFRACTION OF EDGE WAVES: GRAZING INCIDENCE AND SLOPE DIFFRACTION

are the functions of the stationary point 𝜁st : for example, 𝛾01 (𝜁st ) and 𝜑1 (𝜁st ). The caustic parameter 𝜌1 (𝜁st ) is determined according to Equation (8.23) as ( 1 1 = 𝜌1 sin 𝛾01

k̂ s ⋅ 𝑣̂ 1 d𝛾01 − 1 d𝜁 a1 sin 𝛾01

) ,

(9.12)

where the unit vector 𝑣̂ 1 = a1

d̂t1 d𝜁

(9.13)

is the principal normal to edge L1 and a1 is the radius of curvature of this edge at the stationary point 𝜁st . The unit vector k̂ 1s shows the directions of the diffracted rays (9.11) that form the diffraction cone. In accordance with Equation (8.7), this vector is defined by the equation k̂ s ⋅ ̂t1 = k̂ 1i ⋅ ̂t1 = − cos 𝛾01 .

(9.14)

Asymptotic expression (9.11) can also be written in the form of Equation (8.13): ≈ 𝑤02 (𝜁st ) u(t) h

𝜕g(𝜑1 , 𝜑01 , 𝛼1 ) ei𝜋∕4 eik(R1 +R21 ) , √ 𝜕𝜑01 R1 2𝜋kΦ′′ (𝜁st )

(9.15)

where Φ(𝜁st ) = R21 (𝜁st ) + R1 (𝜁st ). Note that the quantity Φ′′ (𝜁 ) = d2 Φ(𝜁 )∕d𝜁 2 is easier to calculate than the caustic parameter 𝜌1 (𝜁 ) in Equation (9.11). The approximations (9.11) and (9.15) are nonuniform asymptotics. They are singular in the directions 𝜑1 = 0, 2𝜋 because 𝜕 1 cos(𝜑1 ∕2) g(𝜑1 , 𝜑01 , 2𝜋)||𝜑 =𝜋 = − →∞ 01 𝜕𝜑01 2 sin2 (𝜑1 ∕2)

with 𝜑1 → 0, 2𝜋. (9.16)

This singularity is a consequence of the singularity of function g(𝜑1 , 𝜑01 , 𝛼1 ) in the directions 𝜑1 = 𝜋 ± 𝜑0 . It can be treated as shown in Section 7.9. It is worth noting that in view of Equation (9.8), the wave (9.11), (9.15) arising due to the grazing/slope diffraction is less in magnitude by a small factor of 1∕kR21 compared to the wave (8.29) generated by ordinary edge diffraction. 9.2.2

Electromagnetic Waves

A similar problem for the grazing diffraction of electromagnetic waves was investigated in a paper by Ufimtsev (1991). Its solution is found in the same way as for acoustic waves. Here it is supposed that the edge wave traveling from edge L2 to edge

GRAZING DIFFRACTION

291

⃗ tot ⊥S1 ), and its magnetic vector is parallel L1 is polarized perpendicular to plate S1 (E 2 to this plate, tot = 𝑣2 (R2 , 𝜑2 , 𝜑02 )eikR2 . H2z 1

(9.17)

In the vicinity of edge L1 , this wave is approximated by the equivalent wave (9.4). The quantity u02 in Equation (9.4) is defined by Equation (9.6), and the equivalent wave is written as eq

H2z = 𝑤02 eikR21 1

(9.18)

with the quantity 𝑤02 defined by Equation (9.8). The elementary edge wave arising at edge L1 is calculated by differentiation of the EEW given in Equations (7.142) and (7.143), assuming that Etinc (𝜁 ) = 0 and Htinc (𝜁 ) = 𝑤02 exp(ikR21 ). Notice that this EEW is radiated by the total surface current ⃗jtot = ⃗j(0) + ⃗j(1) induced by the incident wave on plate S1 in the vicinity of its edge L1 . By the integration of EEWs over edge L1 , one finds the wave diffracted at edge L1 :

⃗ (t) E 21 ⃗ (t) H 21

[ ] ⃗ ) || 𝜕 G(𝜁 eikR1 1 ikR21 = 𝑤02 (𝜁 )e d𝜁1 , Z0 | 2𝜋 ∫L1 𝜕𝜑01 ||𝜑 =𝜋 R1 01 [ ] | ⃗ 𝜕 G(𝜁 ) eikR1 1 | = 𝑤02 (𝜁 )eikR21 ∇R1 × d𝜁1 . | 2𝜋 ∫L1 𝜕𝜑01 ||𝜑 =𝜋 R1 01

(9.19)

(9.20)

Here are its ray asymptotics derived using the stationary-phase technique: ⃗ (t) = ê 𝜑 Z0 E 21 1

𝑤02 (𝜁st ) 𝜕g(𝜑1 , 𝜋, 𝛼1 ) eik(R1 +R21 )+i𝜋∕4 , √ 𝜕𝜑01 sin2 𝛾01 2𝜋kR1 (1 + R1 ∕𝜌1 )

⃗ (t) ]∕Z0 ⃗ (t) = [∇R1 × E H 21 21

(9.21) (9.22)

where ê 𝜑1 is the unit vector associated with the polar angle 𝜑1 (Fig. 9.1). Therefore, in the first asymptotic approximation,

E𝜑(t) = −Z0 H𝜗1 = Z0 1

𝑤02 (𝜁st ) 𝜕g(𝜑1 , 𝜋, 𝛼1 ) eik(R1 +R21 )+i𝜋∕4 . √ 𝜕𝜑01 sin2 𝛾01 2𝜋kR1 (1 + R1 ∕𝜌1 )

(9.23)

292

MULTIPLE DIFFRACTION OF EDGE WAVES: GRAZING INCIDENCE AND SLOPE DIFFRACTION

The caustic parameter 𝜌1 is defined by Equation (9.12). According to Equation (7.159), H𝜗(t) = − 1

Ht(t) 1

(9.24)

sin 𝛾01

and hence,

Ht(t) = 𝑤02 (𝜁st ) 1

1 𝜕g(𝜑1 , 𝜋, 𝛼1 ) eik(R1 +R21 )+i𝜋∕4 . √ sin 𝛾01 𝜕𝜑01 2𝜋kR1 (1 + R1 ∕𝜌1 )

(9.25)

This is exactly the same ray asymptotic (9.11) as that found above for the acoustic waves. Thus, the relationship

Ht = uh

if

𝜕Htinc 𝜕n

=

𝜕uinc 𝜕n

at scattering edge L1

(9.26)

exists between the acoustic and electromagnetic waves arising due to the grazing diffraction.

9.3 SLOPE DIFFRACTION IN CONFIGURATION OF FIGURE 9.1 The following relationship exists between acoustic and electromagnetic waves arising due to the slope diffraction at edge L2 (Fig. 9.1): uh = Ht

if

𝜕Htinc 𝜕uinc = 𝜕n 𝜕n

at the diffraction point on edge L2 .

Here ̂t ≡ ̂t2 is the tangent to the edge L2 and n̂ is the normal to plate S1 (Fig. 9.1).

9.3.1

Acoustic Waves

Suppose that an external wave uext = u0 eik𝜙0

(9.27)

SLOPE DIFFRACTION IN CONFIGURATION OF FIGURE 9.1

293

generates the diffracted wave at edge L1 [of the type of Equation (8.29)]: ikR1 uinc 1h = 𝑣1 (R1 , 𝜑1 , 𝜑01 )e

= u0

ei𝜋∕4 1 g(𝜑1 , 𝜑01 , 𝛼1 )eikR1 √ √ sin 𝛾0 2𝜋k R1 (1 + R1 ∕𝜌1 )

(9.28)

with 𝛼1 = 2𝜋, 𝛾0 = 𝜋 − 𝛾01 , cos 𝛾01 = ̂t1 × ∇𝜙0 , and the angle 𝛾01 is shown in Figure 9.2. This wave hits the edge L2 , where it undergoes diffraction. That is why . The function g(𝜑1 , 𝜑01 , 2𝜋) is defined by Equation (2.64) with we denote it as uinc 1h parameter n = 𝛼∕𝜋 = 2 and equal to zero in the direction 𝜑1 = 𝜋 to edge L2 . To calculate the slope diffraction of the wave (9.28) at edge L2 , we approximate it by the equivalent wave eq

u1 = u01

𝜕 ikz2 cos 𝛾02 −ikr2 sin 𝛾02 cos(𝜑2 −𝜑02 ) e e 𝜕𝜑02

= −u01 ikr2 sin 𝛾02 sin(𝜑2 − 𝜑02 )eikz2 cos 𝛾02 e−ikr2 sin 𝛾02 cos(𝜑2 −𝜑02 )

. (9.29)

The angle 𝛾02 is shown in Figure 9.2. The amplitude of this wave is determined by the requirement that eq

𝜕u1h | 1 1 𝜕u1 | |z2 =r2 =0, 𝜑2 =𝜑02 = | r2 𝜕𝜑2 | R1 sin 𝛾01 𝜕𝜑1 |R1 =R21 , 𝜑1 =𝜋 inc

(9.30)

and equals u01 = −

𝜕𝑣1 (R1 , 𝜑1 , 𝜑01 ) ikR | 1 e 21 | |R1 =R21 , 𝜑1 =𝜋 ikR21 sin 𝛾01 sin 𝛾02 𝜕𝜑1

(9.31)

or u01 = 𝑤01 eikR21 ,

(9.32)

where 𝑤01 = −

𝜕𝑣1 (R21 , 𝜑1 , 𝜑01 ) || 1 . | ikR21 sin 𝛾01 sin 𝛾02 𝜕𝜑1 |𝜑1 =𝜋

(9.33)

According to the idea demonstrated in Section 9.2, the elementary edge waves generated at edge L2 due to the slope diffraction are found by the differentiation of Equations (7.90) and (7.97) with respect to 𝜑0 and with the simultaneous replacement of uinc (𝜁 ) by (9.32) and 𝜑0 by 𝜑02 : du(1) = h

d𝜁 eik[R2 (𝜁 )+R21 (𝜁 )] 𝜕 Fh(1) (𝜁 ) 𝑤01 (𝜁 ) , 2𝜋 𝜕𝜑02 R2 (𝜁 )

(9.34)

294

MULTIPLE DIFFRACTION OF EDGE WAVES: GRAZING INCIDENCE AND SLOPE DIFFRACTION

du(t) = h

d𝜁 eik[R2 (𝜁 )+R21 (𝜁 )] 𝜕 Fh(t) (𝜁 ) 𝑤01 (𝜁 ) . 2𝜋 𝜕𝜑02 R2 (𝜁 )

(9.35)

and du(t) are the EEWs generated by the nonuniform We recall that the quantities du(1) h h ) and total (j(t) = j(0) + j(1) ) scattering sources, respectively. (j(1) h h h h The diffracted waves diverging from the whole edge are determined, respectively, by the integrals: = u(1) h

eik[R2 (𝜁 )+R21 (𝜁 )] 1 𝜕 𝑤01 (𝜁 ) Fh(1) (𝜁 ) d𝜁 , 2𝜋 ∫L2 𝜕𝜑02 R2 (𝜁 )

(9.36)

= u(t) h

eik[R2 (𝜁 )+R21 (𝜁 )] 1 𝜕 𝑤01 (𝜁 ) Fh(t) (𝜁 ) d𝜁 . 2𝜋 ∫L2 𝜕𝜑02 R2 (𝜁 )

(9.37)

The ray asymptotics of these waves are found using the stationary-phase technique:

u(1) = 𝑤01 (𝜁st ) h

𝜕g(1) (𝜑2 , 𝜑02 , 𝛼2 ) eik(R2 +R21 ) ei𝜋∕4 , √ √ 𝜕𝜑02 R2 (1 + R2 ∕𝜌2 ) sin 𝛾02 2𝜋k

(9.38)

= 𝑤01 (𝜁st ) u(t) h

𝜕g(𝜑2 , 𝜑02 , 𝛼2 ) eik(R2 +R21 ) ei𝜋∕4 . √ √ 𝜕𝜑02 R2 (1 + R2 ∕𝜌2 ) sin 𝛾02 2𝜋k

(9.39)

Here the caustic parameter 𝜌2 is defined according to Equation (8.23) as ( 1 1 = 𝜌2 sin 𝛾02

k̂ s ⋅ 𝑣̂ 2 d𝛾 − 02 − 2 d𝜁 a2 sin 𝛾02

) (9.40)

where 𝑣̂ 2 = a2 d̂t2 ∕d𝜁 is the principal normal to edge L2 with radius a2 at the stationary point 𝜁st . The diffracted rays (9.38) and (9.39) propagate in the direction of the vector k̂ 2s determined by the equation k̂ 2s ⋅ ̂t2 = k̂ 2i ⋅ ̂t2 = cos 𝛾02 . The functions g(𝜑2 , 𝜑02 , 𝛼2 ) and g(1) (𝜑2 , 𝜑02 , 𝛼2 ) are defined according to Equations (2.64) and (4.23). Notice also that the angle 𝛾0 in Equation (8.23) relates to Figure 8.1 and according to Figure 9.2 equals 𝜋 − 𝛾02 . That is why the first term in Equation (9.40) acquires the minus sign. As well as in the case of grazing diffraction considered in Section 9.2, the waves (9.38) and (9.39) arising due to the slope diffraction are also lower in magnitude by a factor of 1∕kR21 compared to the waves (8.13) and (8.29) generated by ordinary edge diffraction. This result is not surprising, because the intensity of the incident field hitting the edge in the case of the slope diffraction is significantly less.

SLOPE DIFFRACTION IN CONFIGURATION OF FIGURE 9.1

9.3.2

295

Electromagnetic Waves

Here the theory above is extended for electromagnetic waves (Ufimtsev, 1991). The edge wave traveling from edge L1 to edge L2 can be considered as the sum of two waves with orthogonal polarization. One of them contains the component Ht(t) = 𝑣1 (R1 , 𝜑1 , 𝜑01 )eikR1 1

(9.41)

with the function 𝑣1 shown in Equation (9.28) and equal to zero in direction to edge L2 . It is clear that the diffraction of this wave at edge L2 can be investigated in the same way as the diffraction of the acoustic wave (9.27). One approximates the incident wave by the equivalent wave eq

H1t = 𝑤01 eikR21 2

𝜕 ikz2 cos 𝛾02 −ikr2 sin 𝛾02 cos(𝜑2 −cos 𝜑02 ) e e 𝜕𝜑02

(9.42)

with 𝑤01 defined in Equation (9.33). The EEWs are found by the differentiation of Equations (7.142) and (7.143) (with respect to the angle 𝜑0 = 𝜑02 ), where we should set E0t = 0 and Htinc (𝜁 ) = 𝑤01 exp(ikR21 ). Then, for the total edge wave arising at the wedge L2 , one obtains ⃗ (t) (𝜁 ) eik[R2 (𝜁 )+R21 (𝜁 )] Z0 𝜕G 𝑤01 (𝜁 ) d𝜁 , 2𝜋 ∫L2 𝜕𝜑02 R2 (𝜁 ) [ ] ⃗ (t) (𝜁 ) eik[R2 (𝜁 )+R21 (𝜁 )] 𝜕G 1 = 𝑤 (𝜁 ) ∇R2 × d𝜁 . 2𝜋 ∫L2 01 𝜕𝜑02 R2 (𝜁 )

⃗ (t) = E

(9.43)

⃗ (t) H

(9.44)

Asymptotic evaluation of these integrals leads to ray asymptotics:

𝜕g(𝜑2 , 𝜑02 , 𝛼2 ) eik(R2 +R21 ) ei𝜋∕4 , √ √ 𝜕𝜑02 R2 (1 + R2 ∕𝜌2 ) sin2 𝛾02 2𝜋k

(9.45)

𝜕g(𝜑2 , 𝜑02 , 𝛼2 ) eik(R2 +R21 ) ei𝜋∕4 . √ √ 𝜕𝜑02 R2 (1 + R2 ∕𝜌2 ) sin2 𝛾02 2𝜋k

(9.46)

E𝜑(t) = Z0 𝑤01 (𝜁st ) 2

H𝜗(t) = −𝑤01 (𝜁st ) 2

As H (t) = −Ht(t) ∕ sin 𝛾02 , one obtains the expression 𝜗2

2

Ht(t) = 𝑤01 (𝜁st ) 2

𝜕g(𝜑2 , 𝜑02 , 𝛼2 ) eik(R2 +R21 ) ei𝜋∕4 , √ √ 𝜕𝜑02 R2 (1 + R2 ∕𝜌2 ) sin 𝛾02 2𝜋k

(9.47)

296

MULTIPLE DIFFRACTION OF EDGE WAVES: GRAZING INCIDENCE AND SLOPE DIFFRACTION

which agrees completely with Equation (9.39) and allows formulation of the relationship

uh = Ht

if

𝜕 inc 𝜕 inc uh = H 𝜕n 𝜕n t

on the scattering edge at 𝜁 = 𝜁st

(9.48)

between the acoustic and electromagnetic rays arising due to the slope diffraction.

9.4 SLOPE DIFFRACTION: GENERAL CASE The following relationships exist between acoustic and electromagnetic diffracted rays arising due to slope diffraction: us = Et

if

uh = Ht

if

𝜕 inc 𝜕 inc u (𝜁st ) = E (𝜁st ), 𝜕n 𝜕n t 𝜕 inc 𝜕 inc u (𝜁st ) = H (𝜁st ), 𝜕n 𝜕n t

where 𝜁st is the diffraction point on the scattering edge and ̂t is the tangent to the edge.

9.4.1

Acoustic Waves

Suppose that the wave uinc = 𝑣0 eikRQ

(9.49)

[with a ray structure of the type of Equation (8.29)] undergoes diffraction at the scattering object with edge L. The object can be acoustically soft or hard (with boundary conditions u = 0 or du∕dn = 0). The geometry of the problem is illustrated in Figures 9.3 and 9.4. Point Q there belongs to the caustic of the incident wave. It is assumed that uinc = 0 and 𝜕uinc ∕𝜕n ≠ 0 at point 𝜁 on scattering edge L. The diffracted wave is calculated in the same manner as in Sections 9.2 and 9.3. The incident wave (9.49) is approximated by the equivalent wave 𝜕 −ikz cos 𝛾0 −ikr sin 𝛾0 cos(𝜑−𝜑0 ) e e 𝜕𝜑0 ) ( = −u0 ikr sin 𝛾0 sin 𝜑 − 𝜑0 e−ikz cos 𝛾0 e−ikr sin 𝛾0 cos(𝜑−𝜑0 ) .

ueq = u0

(9.50)

The local polar coordinates r, 𝜑, z are shown in Figure 9.4. The angle 𝜑 is measured from the illuminated face of the edge (0 ≤ 𝜑 ≤ 𝛼, 0 ≤ 𝜑0 < 𝜋).

SLOPE DIFFRACTION: GENERAL CASE

297

z Q →

t

ζ

ϑQ

→

n

RQ

φq

rq

ζ

L

q

(P)

Figure 9.3 Plane P contains tangent ̂t to edge L, the incident ray Q𝜁, and point q (which is the projection of point Q on the perpendicular rq to the tangent ̂t). The vector n̂ is the unit normal to plane P. [Reprinted from (Ufimtsev and Rahmat-Samii, 1995) with the permission of Annales des Telecommunications.]

z

P

ϑ

ζ

R

→

t

Q

γ0

→

x

q

φ0

τ

ζ

L

T

W

Figure 9.4 Here W is the tangential wedge to the scattering edge L, the plane T is the face of wedge W, and the vector 𝜏⃗ is perpendicular to the tangent ⃗t and belongs to plane T. The angle 𝛾0 indicates the direction of the incident ray. The point P(R, 𝜗, 𝜑) is the observation point. [Reprinted from Ufimtsev and Rahmat-Samii (1995) with the permission of Annales des Telecommunications.]

298

MULTIPLE DIFFRACTION OF EDGE WAVES: GRAZING INCIDENCE AND SLOPE DIFFRACTION

The amplitude u0 of the equivalent wave is determined by the equation 𝜕ueq 𝜕uinc 1 𝜕uinc || 1 𝜕ueq || 1 𝜕uinc || = = (9.51) =− = | | | 𝜕n r 𝜕𝜑 |z=r=0, 𝜑=𝜑0 𝜕n RQ 𝜕𝜗Q ||𝜗 =0 rq 𝜕𝜑q ||𝜑 =0 Q q and equals u0 =

𝜕𝑣0 || eikRQ . | ikRQ sin 𝛾0 𝜕𝜗Q ||𝜗 =0 Q

(9.52)

It can be written in the form u0 = 𝑤0 eikRQ ,

(9.53)

where 𝑤0 =

𝜕𝑣0 || 1 . | ikRQ sin 𝛾0 𝜕𝜗Q ||𝜗 =0 Q

(9.54)

According to the idea introduced in Section 9.2, the waves diffracted at edge L are found by the differentiation of Equations (8.3) and (8.4) with the simultaneous replacement of uinc (𝜁 ) = u0 (𝜁 ) exp[ik𝜙i (𝜁 )] by the quantity (9.53). As a result, the ) are deteredge waves generated by the nonuniform/fringe scattering sources (j(1) s,h mined as 𝜕F (1) (𝜁 ) eik[R(𝜁)+RQ (𝜁 )] 1 𝑤0 (𝜁 ) s d𝜁 , 2𝜋 ∫L 𝜕𝜑0 R

(9.55)

𝜕Fh(1) (𝜁 ) eik[R(𝜁)+RQ (𝜁 )] 1 = 𝑤 (𝜁 ) d𝜁 , 2𝜋 ∫L 0 𝜕𝜑0 R

(9.56)

u(1) s = u(1) h

and the edge waves radiated by the total scattering sources (j(t) = j(1) + j(0) ) are s,h s,h s,h described by u(t) s =

𝜕F (t) (𝜁 ) eik[R(𝜁 )+RQ (𝜁 )] 1 𝑤0 (𝜁 ) s d𝜁 , 2𝜋 ∫L 𝜕𝜑0 R

(9.57)

= u(t) h

𝜕F (t) (𝜁 ) eik[R(𝜁 )+RQ (𝜁 )] 1 𝑤0 (𝜁 ) h d𝜁 . 2𝜋 ∫L 𝜕𝜑0 R

(9.58)

Their ray asymptotics can be derived using the stationary-phase technique demonstrated in Section 8.1. However, we can obtain them much faster by differentiation

SLOPE DIFFRACTION: GENERAL CASE

299

of the ray asymptotics (8.12) and (8.13) with the simultaneous replacement ui (𝜁st ) by (9.53): ⎡ 𝜕f (1) (𝜑, 𝜑0 , 𝛼) ⎤ ⎢ ⎥ ⎢ (1) 𝜕𝜑0 ⎥ ik(R+RQ ) ei𝜋∕4 1 = 𝑤0 (𝜁st ) √ , √ ⎢ 𝜕g (𝜑, 𝜑0 , 𝛼) ⎥ e R(1 + R∕𝜌) sin 𝛾 2𝜋k ⎢ ⎥ 0 ⎦ 𝜕𝜑0 ⎢ ⎥ ⎣ ⎦

(1) ⎡ us ⎤ ⎢ u(1) ⎥ ⎢ h ⎥

⎣

(t)

⎡ us ⎤ ei𝜋∕4 1 ⎢ u(t) ⎥ = 𝑤 (𝜁 ) √ √ 0 st ⎢ h ⎥ R(1 + R∕𝜌) sin 𝛾0 2𝜋k ⎣ ⎦

⎡ 𝜕f (𝜑, 𝜑0 , 𝛼) ⎤ ⎥ ⎢ 𝜕𝜑0 ⎢ 𝜕g(𝜑, 𝜑0 , 𝛼) ⎥ eik(R+RQ ) , ⎥ ⎢ 𝜕𝜑0 ⎥ ⎢ ⎦ ⎣

(9.59)

(9.60)

where the stationary point 𝜁st is calculated according to Equation (8.7) and the caustic parameter 𝜌 = 𝜌(𝜁st ) is defined in (8.23). In view of Equation (9.54), the fields arising due to the slope diffraction are lower in magnitude by a small factor 1∕kRQ than the ordinary diffracted fields (8.12) and (8.13). Notice that the integral representations (9.55), (9.56), (9.57), and (9.58) allow calculation of the slope-diffracted field in the vicinity of caustics and foci.

9.4.2

Electromagnetic Waves

Let us define an incident electromagnetic wave as ⃗ i eikRQ , ⃗ inc = E E 0

⃗ inc = H ⃗ i eikRQ , H 0

(9.61)

where ⃗ i × ∇RQ ]. ⃗ i = Z0 [H E 0 0

(9.62)

⃗ i and H ⃗ 0 have Suppose that in the direction to the scattering edge L, the quantities E 0 zeros but their first normal derivatives are not equal to zero: ⃗ 0 = 0, E ⃗i 𝜕E 0 𝜕n

=

⃗i = 0 H 0

⃗i 1 𝜕 E0 ≠ 0, RQ 𝜕𝜗Q

on ⃗i 𝜕H 0 𝜕n

L =

(9.63) ⃗i 1 𝜕 H0 ≠0 RQ 𝜕𝜗Q

on

L.

(9.64)

300

MULTIPLE DIFFRACTION OF EDGE WAVES: GRAZING INCIDENCE AND SLOPE DIFFRACTION

As in the preceding section, one approximates the actual incident wave (in the vicinity of edge L) by the equivalent wave with components Et = −ikrE0t sin 𝛾0 sin(𝜑 − 𝜑0 )e−ikz cos 𝛾0 e−ikr sin 𝛾0 cos(𝜑−𝜑0 ) ,

(9.65)

Ht = −ikrH0t sin 𝛾0 sin(𝜑 − 𝜑0 )e−ikz cos 𝛾0 e−ikr sin 𝛾0 cos(𝜑−𝜑0 ) ,

(9.66)

eq

eq

eq

eq

which are the derivatives of the ordinary plane wave with respect to the incidence angle 𝜑0 . The amplitudes of the equivalent wave are found with the requirement that the normal derivatives of this wave (on the scattering edge) are equal to those of the actual incident wave:

=−

eq i | 𝜕E0t 𝜕Et || | eq eikRQ | = ik sin 𝛾0 E0t = , | | r𝜕𝜑 ||z=r=0, 𝜑=𝜑 RQ 𝜕𝜗Q |𝜗Q =0 0

(9.67)

=−

eq i | 𝜕H0t 𝜕Ht || | eq eRQ | = ik sin 𝛾0 H0t = . | | r𝜕𝜑 ||z=r=0, 𝜑=𝜑 RQ 𝜕𝜗Q | 𝜗 =0 0 Q

(9.68)

eq

𝜕Et

𝜕n eq

𝜕Ht

𝜕n Hence,

eq

E0t = eq

H0t =

i | 𝜕E0t 1 | eikRQ , | ikRQ sin 𝛾0 𝜕𝜗Q ||𝜗 =0 Q

(9.69)

𝜕H i || 1 eikRQ 0t | ikRQ sin 𝛾0 𝜕𝜗Q ||𝜗

(9.70)

. Q =0

As in the preceding section, the elementary edge waves diffracted at edge L are found by the differentiation of Equations (7.142) and (7.143) with respect to the angle 𝜑0 and with the simultaneous replacement of E0t (𝜁 ) and H0t (𝜁 ) by the quantities (9.69) and (9.70): ⃗ (t) = dE

d𝜁 ⃗(t) eikR(𝜁 )  (𝜁 )eikRQ (𝜁 ) , 2𝜋 R(𝜁 )

(9.71)

⃗ (t) = dH

⃗ (t) ] [∇R × dE . Z0

(9.72)

Here 𝜕 ⃗ (t) 𝜕 ⃗ (t) eq eq G (𝜁 , 𝜗, 𝜑). F (𝜁 , 𝜗, 𝜑) + Z0 H0t (𝜁 ) ⃗(t) (𝜁 ) = E0t (𝜁 ) 𝜕𝜑0 𝜕𝜑0

(9.73)

301

SLOPE DIFFRACTION: GENERAL CASE

The total edge wave created by all EEWs is determined by the integrals eik[R(𝜁 )+RQ (𝜁 )] ⃗ (t) = 1 ⃗(t) (𝜁 ) d𝜁 , E 2𝜋 ∫L R(𝜁 ) ⃗ (t) = H

eik[R(𝜁 )+RQ (𝜁 )] 1 [∇R × ⃗(t) (𝜁 )] d𝜁 . 2𝜋Z0 ∫L R(𝜁 )

(9.74) (9.75)

The ray asymptotics of this wave are found by the stationary-phase technique:

eq

E𝜑(t) = −Z0 H𝜗(t) = Z0 H0t eq

E𝜗(t) = Z0 H𝜑(t) = −E0t

𝜕g(𝜑, 𝜑0 , 𝛼) eik(R+RQ ) ei𝜋∕4 , √ √ 𝜕𝜑0 R(1 + R∕𝜌) sin2 𝛾0 2𝜋k

𝜕f (𝜑, 𝜑0 , 𝛼) eik(R+RQ ) ei𝜋∕4 , √ √ 𝜕𝜑0 R(1 + R∕𝜌) sin2 𝛾0 2𝜋k

(9.76)

(9.77)

where for all functions of the variable 𝜁 , one should take their values at the stationary point 𝜁st . In view of Equation (7.159), these expressions can be written in terms of the components parallel to the tangent ̂t(𝜁st ), eq

Et(t) = E0t

𝜕f (𝜑, 𝜑0 , 𝛼) eik(R+RQ ) ei𝜋∕4 , √ √ 𝜕𝜑0 R(1 + R∕𝜌) sin 𝛾0 2𝜋k

(9.78)

𝜕g(𝜑, 𝜑0 , 𝛼) eik(R+RQ ) ei𝜋∕4 . √ √ 𝜕𝜑0 R(1 + R∕𝜌) sin 𝛾0 2𝜋k

(9.79)

eq

Ht(t) = H0t

Their comparison with Equation (9.60) leads to the equivalence relationships between the acoustic and electromagnetic diffracted rays arising due to the slope diffraction:

us = Et

if

uh = Ht

if

𝜕 inc 𝜕 inc u (𝜁st ) = E (𝜁st ), 𝜕n 𝜕n t 𝜕 inc 𝜕 inc u (𝜁st ) = H (𝜁st ), 𝜕n 𝜕n t

where 𝜁st is the diffraction point on the scattering edge.

(9.80) (9.81)

302

MULTIPLE DIFFRACTION OF EDGE WAVES: GRAZING INCIDENCE AND SLOPE DIFFRACTION

PROBLEMS 9.1 The function u(x, y, 𝜑0 ) = eik(x cos 𝜑0 −y sin 𝜑0 ) − eik(x cos 𝜑0 +y sin 𝜑0 ) satisfies the wave equation 𝜕2u 𝜕2u + + k2 u = 0 𝜕x2 𝜕y2 in the region −∞ ≤ x ≤ ∞ and y ≥ 0 and the boundary condition u = 0 on the plane y = 0. Prove that the derivative 𝜕u∕𝜕𝜑0 also satisfies the wave equation and the same boundary condition. Solution u(x, y, 𝜑0 ) = −2i sin(ky sin 𝜑0 )eikx cos 𝜑0 = −2i𝑣(x, y, 𝜑0 ), 𝑣(x, y, 𝜑0 ) = sin(ky sin 𝜑0 )eikx cos 𝜑0 , 𝑤(x, y, 𝜑0 ) =

𝜕𝑣(x, y, 𝜑0 ) 𝜕𝜑0

= k[y cos 𝜑0 cos(ky sin 𝜑0 ) − ix sin 𝜑0 sin(ky sin 𝜑0 )]eikx cos 𝜑0 . It is obvious that this function satisfies the boundary condition 𝑤 = 0 on the plane y = 0. Calculation of the derivatives leads to the equation 𝜕2𝑤 𝜕2𝑤 + 2 = k3 [−y cos 𝜑0 cos(ky sin 𝜑0 ) 𝜕x2 𝜕y +ix sin 𝜑0 sin(ky sin 𝜑0 )]eikx cos 𝜑0 = −k2 𝑤. Hence, 𝜕2𝑤 𝜕2𝑤 + 2 + k2 𝑤 = 0. 𝜕x2 𝜕y 9.2 The incident wave (9.49) is approximated by the equivalent wave (9.50) constructed by differentiation of the plane wave with respect to the incidence angle 𝜑0 . Prove that the diffracted wave u(t) s in Equation (9.60) satisfies the boundary = 0 on the wedge faces 𝜑 = 0 and 𝜑 = 𝛼. condition u(t) s Solution

As shown in Section 8.1.1, the incident plane wave uinc = e−ikz cos 𝛾0 e−ik1 r cos(𝜑−𝜑0 )

PROBLEMS

303

creates the diffracted wave u(t) s of Equation (8.29). This diffracted wave is determined by the function [ 𝜑 − 𝜑0 )−1 sin(𝜋∕n) ( 𝜋 cos − cos f (𝜑, 𝜑0 , 𝛼) = n n n ( )−1 ] 𝜑 + 𝜑0 𝜋 𝛼 − cos − cos with n = n n 𝜋 and satisfies the boundary condition u(t) s = 0 as f (0, 𝜑0 , 𝛼) = f (𝛼, 𝜑0 , 𝛼) = 0. We need to prove that the incident wave obtained by differentiation of the plane wave with respect to the angle 𝜑0 generates the diffracted field (9.60), which also satisfies the same boundary condition. That is, one should prove that 𝜕f (𝜑, 𝜑0 , 𝛼)∕𝜕𝜑0 = 0 on the faces 𝜑 = 0 and 𝜑 = 𝛼. One can verify that this derivative, [

sin[(𝜑 − 𝜑0 )∕n] ( )2 cos(𝜋∕n) − cos[(𝜑 − 𝜑0 )∕n] ] sin[(𝜑 + 𝜑0 )∕n] +( )2 , cos(𝜋∕n) − cos[(𝜑 + 𝜑0 )∕n]

sin(𝜋∕n) 𝜕f = 𝜕𝜑0 n2

indeed equals zero when 𝜑 = 0 and 𝜑 = 𝛼. 9.3

Use the asymptotic expression (9.10) for the grazing diffraction of acoustic waves, employ the stationary-phase technique, and confirm the ray approximation (9.11). Solution The geometry of the problem is shown in Figure 9.2. The phase function in Equation (9.10) equals Φ(𝜁 ) = R21 (𝜁 ) + R1 (𝜁 ). Here we assume that 𝜁 ≡ 𝜁1 and ̂t ≡ ̂t1 . The stationary point 𝜁st is determined by the equation 𝜕Φ = ∇′ Φ ⋅ ̂t = (∇′ R21 + ∇′ R1 ) ⋅ ̂t = 0. 𝜕𝜁 Here ∇′ R21 = k̂ i and ∇′ R1 = −∇R1 = −k̂ s , where the operator ∇′ (∇) acts on coordinates of the integration (observation) points. Thus, at the stationary point, k̂ i ⋅ ̂t = k̂ s ⋅ ̂t = − cos 𝛾01 , and the unit vector k̂ s of the scattered field is directed along the cone of diffracted rays. For these directions, Fh = g(𝜑1 , 𝜑01 , 𝛼1 ). Details of the related asymptotic calculations are demonstrated by Equations (8.8) and (8.9). They allow us to transform the integral (9.10) to the asymptotic form of Equation (8.10). Then, utilizing Equation (8.22) for the function Φ′′ (𝜁st ) we obtain Equation (9.11).

304

MULTIPLE DIFFRACTION OF EDGE WAVES: GRAZING INCIDENCE AND SLOPE DIFFRACTION

9.4 Use the asymptotic expressions (9.19) for the grazing diffraction of electromagnetic waves, employ the stationary-phase technique, and confirm the ray approximations (9.23). Solution Follow the instructions in the solution of Problem 9.3. Set G(t) 𝜑01 = (1∕ sin 𝛾01 )g(𝜑1 , 𝜑01 , 𝛼1 ) at the stationary point and apply Equation (8.22).

10 Diffraction Interaction of Neighboring Edges on a Ruled Surface The following relationships exist between acoustic and electromagnetic diffracted waves: 𝜕Etinc (𝜁 ) 𝜕uinc s (𝜁 ) inc us = Et if = . uh = Ht if uinc h (𝜁 ) = Ht (𝜁 ), 𝜕n 𝜕n Here ̂t is tangent to the edge and n̂ is the normal to the face at the diffraction point 𝜁 . These relationships, together with Equation (7.159), allow one to determine all components of the electromagnetic diffracted wave.

Consider a diffraction interaction of two edges with a common face. If this face is bent, the edge wave already undergoes diffraction on its way along the face to another edge. This problem is not amenable to theoretical treatment in a general case. However, the disturbing effect of the face can be neglected in a particular case illustrated in Figure 10.1. The common face S of edges L1 and L is a ruled surface whose generatrices coincide with the edge-diffracted rays arising at edge L1 and propagating to edge L. It is assumed that a plane tangential to the face does not change its orientation along the generatrix. Notice that a planar facet can be considered as a limiting case of a ruled surface. Therefore, the theory developed here is applicable in this case as well. The present section is based on papers by Ufimtsev (1989, 1991). Fundamentals of the Physical Theory of Diffraction, Second Edition. Pyotr Ya. Ufimtsev. © 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

305

306

DIFFRACTION INTERACTION OF NEIGHBORING EDGES ON A RULED SURFACE

R1 nˆ

φ1

L1

–

01

γ

z1

τ1

nˆ

φ

τ R10

ζ

kˆ i –γ0

ζ1

r

z

π

π

r1

t

R

t1

L

S

Figure 10.1 Element of a ruled surface S with two edges (L1 and L). The unit vectors ̂t1 and ̂t are tangential to the edges. The unit vectors 𝜏̂1 and 𝜏̂ are tangential to the surface S and perpendicular to ̂t1 and ̂t, respectively. The quantities r1 , 𝜑1 , z1 and r, 𝜑, z are local polar coordinates. The unit vector n̂ is normal to the plane tangential to S and containing the generatrix R10 as well as the tangents ̂t1 ,̂t and 𝜏̂1 ,𝜏. ̂ [Reprinted from Ufimtsev (1989) with the permission of the Journal of the Acoustical Society of America.]

10.1

DIFFRACTION AT AN ACOUSTICALLY HARD SURFACE

Suppose that the edge wave propagating from edge L1 has a ray structure and can be represented in the form of Equation (8.29) as u1h (R1 , 𝜑1 ) = u01 g(𝜑1 , 𝜑01 , 𝛼1 )

eikR1 +i𝜋∕4 , √ sin 𝛾01 2𝜋kR1 (1 + R1 ∕𝜌1 )

(10.1)

where the function g(𝜑1 , 𝜑01 , 𝛼1 ) is defined in Equation (2.64). In the vicinity of edge L, this wave can be approximated by the two merging plane waves, lim 1 u1h (R10 , 0)e−ikz cos 𝛾0 [e−ikr sin 𝛾0 cos(𝜑−𝜑0 ) 𝜑0 →0 2

+ e−ikr sin 𝛾0 cos(𝜑+𝜑0 )].

(10.2)

The first term here plays the role of the incident wave in the canonical wedge diffraction problem utilized in Chapter 8 to derive the asymptotic approximation (8.4). Therefore, replacing the quantity u0 (𝜁 ) exp(ik𝜙i ) in Equation (8.4) by 12 u1h (R10 , 0), we obtain the asymptotic expression = u(t) h

eikR(𝜁 ) 1 ̂ u1h (𝜁 )Fh(t) (𝜁 , m) d𝜁 4𝜋 ∫L R(𝜁 )

with u1h (𝜁 ) = u1h (R10 , 0)

(10.3)

= j(1) + j(0) . One should for the edge wave generated by the total scattering source j(t) h h h

= u1h (R1 , 0). note that in this particular case, j(0) h

DIFFRACTION AT AN ACOUSTICALLY HARD SURFACE

307

The ray asymptotics of this wave can be found using the stationary-phase technique. However, it can be obtained directly from Equation (8.29) if we replace uinc (𝜁st ) there by 12 u1h (𝜁st ) and set 𝜑0 = 0: = 12 u1h (𝜁st )g(𝜑, 0, 𝛼) u(t) h

eikR+i𝜋∕4 √ , sin 𝛾0 2𝜋kR(1+R∕𝜌)

(10.4)

where u1h (𝜁st ) = u1h (R10 , 0) = u01 g(0, 𝜑01 , 𝛼1 )

eikR10 +i𝜋∕4 . √ sin 𝛾01 2𝜋kR10 (1 + R10 ∕𝜌1 )

(10.5)

The symbols 𝛼1 and 𝛼 denote the external angles of edges L1 and L (𝜋 ≤ 𝛼1 ≤ 2𝜋, 𝜋 ≤ 𝛼 ≤ 2𝜋), and the caustic parameters 𝜌1 and 𝜌 are defined according to Equation (8.23). The ray asymptotic (10.4) is not applicable near the shadow boundary 𝜑 = 𝜋, where it becomes singular. Instead, one can suggest the following heuristic approximation of Equation (10.3) valid in all directions along the diffraction cone (𝜗 = 𝜋 − 𝛾0 , 0 ≤ 𝜑 ≤ 𝛼), including the shadow boundary:

u(t) = u1h (R10 , 0)Dh (𝜒, 𝜑, 𝛾0 ) h

eikR+i𝜋∕4 . √ sin 𝛾0 2𝜋kR(1 + R∕𝜌)

(10.6)

Here the diffraction coefficient is defined as √

2s cos(𝜑∕2) √ 𝜑 2 eit dt Dh (𝜒, 𝜑, 𝛾0 ) = i 2s g(𝜑, 0, 𝛼) cos e−is(1+cos 𝜑) ∫∞ cos (𝜑∕2) 2

(10.7)

where s = k𝜒 sin 𝛾0 , g(𝜑, 0, 𝛼) =

𝜒=

R10 R sin 𝛾0 , R10 + R

(2∕N) sin (𝜋∕N) , cos (𝜋∕N) − cos (𝜑∕N)

N=

(10.8) 𝛼 . 𝜋

√ It is assumed that 𝜒 ≫ 1 in Equation (10.6). Taking into account the asymptotic approximation √ 2s cos (𝜑∕2)

∫∞ cos (𝜑∕2)

2

eit dt ∼

√

1

2i 2s cos (𝜑∕2)

eis(1+cos 𝜑)

(10.9)

308

DIFFRACTION INTERACTION OF NEIGHBORING EDGES ON A RULED SURFACE

valid under the condition

√ 2s cos (𝜑∕2) ≫ 1, one can see that Dh (𝜒, 𝜑, 𝛾0 ) ∼ 12 g(𝜑, 0, 𝛼)

(10.10)

and Equation (10.6) transforms into the ray asymptotic (10.4). Next let us examine the asymptotic (10.6) at the shadow boundary of the incident wave 𝜑 = 𝜋 ∓ 0. Pertinent here is the following comment. To ensure the continuity of the total field, the wave (10.6) diffracted at edge L must have the same ray structure as the incident wave (10.1). This requirement entails the relationships R1 = R10 + R

and

𝜌 = 𝜌1 + R10

when

𝜑 = 𝜋.

(10.11)

Taking them into account, one can show that the diffracted wave (10.6) is discontinuous at the shadow boundary and equals = ∓ 12 u1h (R10 + R, 0) u(t) h

with 𝜑 = 𝜋 ∓ 0.

(10.12)

This equation is derived in Problem 10.1. It is clear that the sum of the diffracted and incident fields (i.e., u(t) + u1h is continuous there. h One can show that the normal derivative of this sum is also continuous at the shadow boundary. Notice that the derivative of the incident wave (10.1) equals zero there, 𝜕u1h (R1 , 0) 𝜕u1h (R1 , 𝜑1 ) = 0, = 𝜕n R1 sin 𝛾01 𝜕𝜑1

(10.13)

due to the boundary condition. Indeed, one can verify that 𝜕g(0, 𝜑01 , 𝛼1 )∕𝜕𝜑1 = 0 in the direction 𝜑1 = 0. It is also assumed that the incident field u1h equals zero in the entire shadow region 𝜋 ≤ 𝜑 ≤ 2𝜋. Therefore, its normal derivative at the shadow boundary is continuous. As shown in Problem 10.2, the normal derivative of the diffracted field (10.6) is also continuous there and equals 𝜕u(t) h

𝜕n

=−

| | | R sin 𝛾0 𝜕𝜑 || |𝜑=𝜋 𝜕u(t) h

1 = u01 g(0, 𝜑01 , 𝛼1 ) 2𝜋 sin 𝛾01 (R10 + R)

√

R10 eik(R10 +R) . √ R 1 + (R + R)∕𝜌 10

(10.14)

1

Thus, Equation (10.6) really provides the uniform approximation for the field everywhere along the diffraction cone. It is simple and convenient for estimation.

DIFFRACTION AT AN ACOUSTICALLY SOFT SURFACE

10.2

309

DIFFRACTION AT AN ACOUSTICALLY SOFT SURFACE

The geometry of the problem is shown in Figure 10.1. Suppose that the edge-diffracted wave u1s (R1 , 𝜑1 ) = u01 f (𝜑1 , 𝜑01 , 𝛼1 )

eikR1 +i𝜋∕4 √ sin 𝛾01 2𝜋kR1 (1 + R1 ∕𝜌1 )

(10.15)

propagates from edge L1 along face S. Due to Equation (2.62), the function f (𝜑1 , 𝜑01 , 𝛼1 ) equals zero in the direction 𝜑1 = 0 to edge L. Thus, the diffraction of this wave at edge L is a particular case of the slope diffraction, and it can be treated, as shown below, with a little modification of the technique developed in Chapter 9. First, we note that an appropriate wave equivalent to the incident wave (10.15) in the vicinity of edge L can be constructed from a combination of two plane waves, e−ikz cos 𝛾0 (e−ikr sin 𝛾0 cos(𝜑−𝜑0 ) − e−ikr sin 𝛾0 cos(𝜑+𝜑0 ) ),

(10.16)

running along the face to the edge. Namely, us = ueq e−ikz cos 𝛾0 eq

0

𝜕 | ( e−ikr sin 𝛾0 cos(𝜑−𝜑0 ) − e−ikr sin 𝛾0 cos(𝜑+𝜑0 ) )| |𝜑0 =0 𝜕𝜑0

= − ueq 2ikr sin 𝛾0 sin 𝜑 e−ikz cos 𝛾0 e−ikr sin 𝛾0 cos 𝜑 . 0

(10.17)

Here the term ueq e−ikz cos 𝛾0 0

𝜕 −ikr sin 𝛾0 cos(𝜑−𝜑0 ) | e | |𝜑0 =0,r=0 𝜕𝜑0

(10.18)

plays the role of the incident wave. Following the idea introduced in Chapter 9, we eq replace u0 (𝜁 )eik𝜙i in the integral (8.4) by the operator u0 𝜕∕𝜕𝜑0 and find the wave diffracted at edge L: u(t) s =

| eikR(𝜁 ) 1 𝜕 (t) eq ̂ || u0 (𝜁 ) Fs (𝜁 , m) d𝜁 . 2𝜋 ∫L 𝜕𝜑0 |𝜑0 =0 R(𝜁 )

(10.19)

eq

The amplitude factor u0 of the equivalent wave is determined by the requirement that its normal derivative at the diffraction point 𝜁 on the edge L equals the normal derivative of the incident wave (10.15): 𝜕u1s (R10 , 0) 𝜕u1s (R1 , 𝜑1 ) || 1 = | 𝜕n R10 sin 𝛾01 𝜕𝜑1 |R1 =R10 ,𝜑1 =0 =

eq 1 𝜕us || eq = −u0 2ik sin 𝛾0 . | r 𝜕𝜑 ||z=r=0,𝜑=0

(10.20)

310

DIFFRACTION INTERACTION OF NEIGHBORING EDGES ON A RULED SURFACE

Notice that normal derivatives are calculated here in local polar coordinates r1 , 𝜑1 and r, 𝜑, where r1 = R1 sin 𝛾01 and r = R sin 𝛾0 , respectively. It follows from Equation (10.20) that eq

u0 = −

𝜕u1s (R1 , 𝜑1 ) || 1 | | 2ikR10 sin 𝛾01 sin 𝛾0 𝜕𝜑1 |R

(10.21)

1 =R10 ,𝜑1 =0

or eq

u0 = u01

𝜕f (0, 𝜑01 , 𝛼1 ) eikR10 +i𝜋∕4 . (10.22) √ 𝜕𝜑1 2 sin2 𝛾01 sin 𝛾0 (kR10 )3∕2 2𝜋(1 + R10 ∕𝜌1 ) i

The ray asymptotics of the diffracted field (10.19) can be found using the stationary-phase technique or directly by the differentiation of Equation (8.29) and eq replacement uinc (𝜁 ) by u0 : eq

u(t) s = u0 (𝜁st )

𝜕f (𝜑, 0, 𝛼) eikR+i𝜋∕4 . √ 𝜕𝜑0 sin 𝛾0 2𝜋kR(1 + R∕𝜌)

(10.23)

This function is singular in the direction of the shadow boundary (𝜑 = 𝜋). Instead of it, one can suggest the following heuristic approximation of Equation (10.19) valid in all directions along the diffraction cone (𝜗 = 𝜋 − 𝛾0 , 0 ≤ 𝜑 ≤ 𝛼):

u(t) s =

𝜕u1s (𝜁st ) eikR+i𝜋∕4 , Ds (𝜒, 𝜑, 𝛾0 ) √ 𝜕n sin 𝛾0 2𝜋kR(1 + R∕𝜌)

(10.24)

where the diffraction coefficient equals Ds (𝜒, 𝜑, 𝛾0 ) = 2𝜒

𝜕f (𝜑, 0, 𝛼) 𝜑) 𝜑 (√ cos2 𝔽 2k𝜒 sin 𝛾0 cos 𝜕𝜑0 2 2

(10.25)

and sgn(x)∞

2

𝔽 (x) = 1 + i2xe−ix ⋅

∫x

2

eit dt.

(10.26)

For large arguments |x| ≫ 1, 𝔽 (x) ≈

i . 2x2

(10.27)

DIFFRACTION AT AN ACOUSTICALLY SOFT SURFACE

√

Therefore, far away from the shadow boundary 𝜑 = 𝜋, where |cos (𝜑∕2)| ≫ 1, we obtain Ds (𝜒, 𝜑, 𝛼) ≈

𝜕f (𝜑, 0, 𝛼) i . 2k sin 𝛾0 𝜕𝜑0

311

k𝜒 sin 𝛾0

(10.28)

Substitution of the quantities (10.20) and (10.28) into Equation (10.24) leads to the ray asymptotic (10.23). As shown in Problem 10.3, exactly at the shadow boundary the field (10.24) equals u(t) s

𝜕f (0, 𝜑01 , 𝛼1 ) i 1 = u01 2 𝜕𝜑1 2𝜋k sin 𝛾01 R10 + R

√

eik(R10 +R) R . √ R10 1 + (R + R)∕𝜌 10

(10.29)

1

In the far zone where R ≫ R10 , this expression is simplified as u(t) s = u01

𝜕f (0, 𝜑01 , 𝛼1 ) eik(R10 +R) i . √ √ 2 𝜕𝜑1 sin 𝛾01 2𝜋kR10 2𝜋kR(1 + R∕𝜌1 )

(10.30)

Because the incident wave (10.15) equals zero in the region 𝜋 ≤ 𝜑 ≤ 2𝜋, one concludes that the total field u1s + u(t) s is finite and continuous at the shadow boundary 𝜑 = 𝜋. In Problem 10.4 it is shown that the normal derivative 𝜕u(t) s ∕𝜕n is discontinuous at the shadow boundary (𝜑1 = 0, 𝜑 = 𝜋 ∓ 0), where it is equal to 𝜕u(t) (R, 𝜋 ∓ 0) s 𝜕n

1 𝜕u1s (R10 + R, 0) 2 𝜕n 𝜕f (0, 𝜑01 , 𝛼1 ) 1 eik(R10 +R)+i𝜋∕4 1 . = ∓ u01 √ 2 2 𝜕𝜑1 sin 𝛾01 (R10 + R) 2𝜋k(R10 + R)(1 + (R10 + R)∕𝜌1 ) =∓

(10.31) Therefore, at the shadow boundary the normal derivative of the total field is continuous and equal to one-half of the normal derivative of the incident wave, | 𝜕(u1s + u(t) s )| | | 𝜕n |R

=

1 =R10 +R,𝜑1 =0,𝜑=𝜋

1 𝜕u1s || . 2 𝜕n ||R1 =R10 +R,𝜑1 =0,

(10.32)

In the far zone where R ≫ R10 , this expression is simplified as | 𝜕(u1s + u(t) s )| | | 𝜕n |R

1 =R10 +R,𝜑1 =0,𝜑=𝜋

≈

𝜕f (0, 𝜑01 , 𝛼1 ) 1 eik(R10 +R)+i𝜋∕4 1 . u01 √ 2 𝜕𝜑1 sin 𝛾01 R 2𝜋kR(1 + R∕𝜌 ) 1 (10.33)

312

DIFFRACTION INTERACTION OF NEIGHBORING EDGES ON A RULED SURFACE

Thus, Equations (10.24) indeed provides a uniform approximation for the field everywhere along the diffraction cone. It is also simple and convenient for estimation. Here it is pertinent to mention several publications (Molinet, 2005; Molinet et al., 2005) related to the excitation of two-dimensional edge waves by creeping waves and whispering-gallery modes propagating over convex and concave scattering surfaces, respectively.

10.3

DIFFRACTION OF ELECTROMAGNETIC WAVES

In a general case, a wave diffracted at edge L1 can be considered to be the sum of two waves with basic polarizations, that is, with components Ht and Et . Because they play the role of the waves incident on edge L, we denote them as Htinc and Etinc . The wave with component Htinc can be represented as Equation (10.1), and its diffraction at edge L is calculated in the same way as in Section 10.1. The wave with component Etinc can be represented as Equation (10.15), and its diffraction at edge L is calculated in the same way as in Section 10.2. Taking into account these observations and the theory of diffracted electromagnetic waves presented in Section 8.1.2, one can obtain the following ray asymptotics:

E𝜑(t) = −Z0 H𝜗(t) =

eikR+i𝜋∕4 1 , Z0 Htinc (𝜁st )g(𝜑, 0, 𝛼) √ 2 2 sin 𝛾0 2𝜋kR(1 + R∕𝜌) eq

E𝜗(t) = Z0 H𝜑(t) = −Et (𝜁st )

(10.34)

𝜕f (𝜑, 0, 𝛼) eikR+i𝜋∕4 , √ 𝜕𝜑0 sin2 𝛾0 2𝜋kR(1 + R∕𝜌)

(10.35)

where Htinc (𝜁st ) = u01 g(0, 𝜑01 , 𝛼1 ) eq

Et (𝜁st ) = −

eikR10 +i𝜋∕4 , √ sin 𝛾01 2𝜋kR10 (1 + R10 ∕𝜌1 )

𝜕Etinc (R1 , 𝜑1 ) || 1 | | 2ikR10 sin 𝛾01 sin 𝛾0 𝜕𝜑1 |R

(10.36)

,

(10.37)

1 =R10 ,𝜑1 =0

Etinc (R10 , 0) = u01 f (0, 𝜑01 , 𝛼1 )

eikR10 +i𝜋∕4 . √ sin 𝛾01 2𝜋kR10 (1 + R10 ∕𝜌1 )

(10.38)

It follows from Equation (10.37) that 𝜕Etinc (𝜁st ) 𝜕n

eq

= −i2k sin 𝛾0 Et (𝜁st ).

(10.39)

DIFFRACTION OF ELECTROMAGNETIC WAVES

313

In view of Equation (7.159), one can rewrite the ray asymptotics in terms of the components parallel to the tangent to the edge at diffraction point 𝜁st : Ht(t) =

eikR+i𝜋∕4 1 inc , Ht (𝜁st )g(𝜑, 0, 𝛼) √ 2 sin 𝛾0 2𝜋kR(1 + R∕𝜌) eq

Et(t) = Et (𝜁st )

(10.40)

𝜕f (𝜑, 0, 𝛼) eikR+i𝜋∕4 . √ 𝜕𝜑0 sin 𝛾0 2𝜋kR(1 + R∕𝜌)

(10.41)

Their comparison with Equations (10.4) and (10.23) reveals the following relationships between acoustic and electromagnetic diffracted rays:

u(t) = Ht(t) h

if

uinc = Htinc , h

(t) u(t) s = Et

if

u0 (𝜁st ) = Et (𝜁st ),

eq

(10.42)

eq

that is, if

𝜕uinc s 𝜕n

=

𝜕Etinc 𝜕n

(10.43)

at the diffraction point on the scattering edge.

The uniform asymptotics for the diffracted waves valid for all directions along the diffraction cone (𝜗 = 𝜋 − 𝛾0 and 0 ≤ 𝜑 ≤ 𝛼), including the shadow boundary 𝜑 = 𝜋, are described by the electromagnetic versions of Equations (10.6) and (10.24): E𝜑(t) = −Z0 H𝜗(t) = Z0 Htinc (𝜁st )Dh (𝜒, 𝜑, 𝛾0 ) E𝜗(t) = Z0 H𝜑(t) = −

𝜕Etinc (𝜁st ) 𝜕n

Ds (𝜒, 𝜑, 𝛾0 )

eikR+i𝜋∕4 , (10.44) √ sin 𝛾0 2𝜋kR(1 + R∕𝜌) 2

eikR+i𝜋∕4 . √ sin2 𝛾0 2𝜋kR(1 + R∕𝜌)

(10.45)

Here, the superscript “t” indicates that these waves (diffracted at edge L) are radiated by the surface current ⃗j(t) = ⃗j(0) + ⃗j(1) . Together with the incident wave (diverging from edge L1 ) they produce the total field. In terms of the components parallel to the tangent to the edge at the diffraction point 𝜁st , these waves have the form Ht(t) = Htinc (𝜁st )Dh (𝜒, 𝜑, 𝛾0 ) Et(t) =

𝜕Etinc (𝜁st ) 𝜕n

eikR+i𝜋∕4 , √ sin 𝛾0 2𝜋kR(1 + R∕𝜌)

Ds (𝜒, 𝜑, 𝛾0 )

eikR+i𝜋∕4 . √ sin 𝛾0 2𝜋kR(1 + R∕𝜌)

(10.46)

(10.47)

Their comparison with Equations (10.6) and (10.24) confirm the equivalence relationships (10.42) and (10.43) existing between the acoustic and electromagnetic waves.

314

DIFFRACTION INTERACTION OF NEIGHBORING EDGES ON A RULED SURFACE

It follows from this equivalency that the sum of the incident and diffracted electromagnetic waves and the sum of their normal derivatives are continuous at the shadow boundary 𝜑 = 𝜋. Away from this boundary the uniform approximations (10.44) and (10.45) transform to the ray asymptotics (10.40) and (10.41). This transformation is straightforward with the application of Equations (10.10), (10.28), and (10.39).

10.4

TEST PROBLEM: SECONDARY DIFFRACTION AT A STRIP

In this section we test the theory described above on an example of the strip diffraction problem. This is a classic and well-studied problem (Bowman et al., 1987). Its exact solution in the form of infinite series in terms of Mathieu functions is not suitable for analysis of high-frequency diffraction, due to slow convergence. There is an alternative exact solution in the form of a fast convergent series appropriate to study the high-frequency diffraction (Ufimtsev, 1969, 1970, 2003, 2009). These references contain comprehensive asymptotic analysis: The uniform high-frequency asymptotics with arbitrary high precision have been developed there for the scattered field and for the surface currents on the strip. In this section we employ the first-order highfrequency approximations for the scattered field (Ufimtsev, 2003, 2009) presented in Section 5.1.3. 10.4.1

Diffraction at a Hard Strip

The geometry of the problem is shown in Figure 10.2. The incident wave uinc = e−ik(z cos 𝜑01 +y sin 𝜑01 )

(10.48)

excites the primary edge waves running from edge L1 along the strip to edge L, where they generate the secondary diffracted waves. The primary waves are described by Equation (10.1), where one should set u01 = exp(ikl cos 𝜑01 ), 𝛼1 = 2𝜋, 𝛾01 = 𝜋∕2, and 𝜌1 = ∞. Due to Equation (8.23), the parameter 𝜌1 is infinite because edge L1 y

n

φ01 = ϑ0

R

r

ϑ

φ

L1

L

z

n Figure 10.2 Location of edges L1 and L is determined by the coordinates z = −l and z = +l, respectively. The width of the strip equals R10 = 2l. The distance from edge L to the observation point in the far zone (R ≫ kl2 ) equals R = r − l cos 𝜗 = r + l cos 𝜑.

TEST PROBLEM: SECONDARY DIFFRACTION AT A STRIP

315

is a straight line and the incident wave is a plane wave propagating in the direction normal to the edge (𝛾01 = 𝜋∕2). Besides, we set 𝜑1 = 0 for the top primary wave and 𝜑1 = 2𝜋 for the bottom wave. Note that u1h (R1 , 2𝜋) = −u1h (R1 , 0); that is, the primary field is antisymmetric with respect to the strip. The secondary wave generated by the primary edge wave running along the top face of the strip was found in Section 10.1 and is determined by Equation (10.4), where one should set 𝛼 = 2𝜋, 𝛾0 = 𝜋∕2, and 𝜌 = ∞. It is clear that the secondary wave generated by the primary wave running along the bottom face is also determined by Equation (10.4), where one should set u1h = −u1h (R10 , 0) and replace 𝜑 by 2𝜋 − 𝜑 in the function g(𝜑, 0, 2𝜋); that is, set g(2𝜋 − 𝜑, 0, 2𝜋) = −g(𝜑, 0, 2𝜋) there. In view of these comments it is clear that the secondary wave generated by the primary bottom wave exactly equals the secondary wave created by the top primary wave. Therefore, the total secondary wave is double that of the wave (10.4); that is, eikR+i𝜋∕4 = u (R , 0)g(𝜑, 0, 2𝜋) u(t) √ 1h 10 h 2𝜋kR

(10.49)

or u(t) = u01 g(0, 𝜑01 , 2𝜋)g(𝜑, 0, 2𝜋) h

i eik(R10 +R) . √ 2𝜋k R10 R

(10.50)

According to Equation (2.64), this asymptotic can be written as = u01 u(t) h

1 i eik(R10 +R) . √ cos (𝜑01 ∕2) cos (𝜑∕2) 2𝜋k R10 R

(10.51)

Next we present the secondary wave predicted by the exact asymptotic theory. The first-order asymptotic approximation of the field scattered by the strip is given by Equations (5.47) and (5.49). The error of this approximation is determined by Equation (5.56). We are looking for the secondary wave diverging from the right edge (z = l). It can be identified by the presence of the following phase factors: ei𝜒𝛼0 ,

eiq ,

e−i𝜒𝛼 ,

(10.52)

𝛼0 = cos 𝜗0 = cos 𝜑01 ,

𝛼 = cos 𝜗 = − cos 𝜑. (10.53)

where 𝜒 = kl,

q = 2𝜒 = R10 ,

One should not confuse these quantities 𝛼0 and 𝛼 with the external angles 𝛼1 = 2𝜋 and 𝛼 = 2𝜋 of the strip edges. The first factor in Equation (10.52) determines the incident wave at edge L1 , that is, eikl cos 𝜑01 = u01 . The factor eiq = eikR10 shows that the wave running from edge L1 has arrived at edge L. The factor e−i𝜒𝛼 together with the factor eikr determines the phase of the edge wave acquired on its way from edge L to the observation

316

DIFFRACTION INTERACTION OF NEIGHBORING EDGES ON A RULED SURFACE

point, eik(r−l cos 𝜗) . These comments suggest that the secondary wave we are looking ̃ 1 (−𝛼, −𝛼0 ) in Equation (5.47). Specifically, it is for is described by the function Φ determined as

uexact h

] [ √ √ 1 + 𝛼0 1 1−𝛼 = −u01 𝜑(q, 𝛼0 ) − (1 + 𝛼) 𝜑(q, −𝛼) (1 − 𝛼0 ) 𝛼 + 𝛼0 2 2 eikR+i𝜋∕4 , × eiq √ 2𝜋kR

(10.54)

where ∞

2 2 eix dx 𝜑(q, 𝛼) = √ e−i𝜋∕4 e−iq(1+𝛼) √ ∫ q(1+𝛼) 𝜋

We are interested in the ray asymptotics valid in the region where √ kR10 (1 − cos 𝜑) ≫ 1, that is, away from direction 𝜑 = 𝜋. Here 𝜑(q, 𝛼) = √

(10.55) √ q(1 + 𝛼) =

ei𝜋∕4

ei𝜋∕4 + O[(kR10 )−3∕2 ] + O(q−3∕2 ) = √ 𝜋kR10 (1 + 𝛼) 𝜋q(1 + 𝛼)

(10.56)

and

uexact h

[ ] √ √ 1 + 𝛼0 i eik(R10 +R) 1 1−𝛼 = −u01 − (1 + 𝛼) (1 − 𝛼0 ) √ 2𝜋k 1 + 𝛼0 1−𝛼 R10 R 𝛼 + 𝛼0 = −u01

i eik(R10 +R) 1 1 [(1 − 𝛼0 )(1 − 𝛼) √ √ √ 2𝜋k R10 R 𝛼 + 𝛼0 1 − 𝛼 1 + 𝛼 0

− (1 + 𝛼)(1 + 𝛼0 )].

(10.57)

In view of the identity (1 − 𝛼0 )(1 − 𝛼) − (1 + 𝛼)(1 + 𝛼0 ) = −2(𝛼 + 𝛼0 ), formula (10.57) takes the form = u01 uexact h

eik(R10 +R) i 1 . √ √ √ 𝜋k 1 + 𝛼 1 − 𝛼 R10 R 0

(10.58)

TEST PROBLEM: SECONDARY DIFFRACTION AT A STRIP

317

√ √ √ √ Next, substitutions 1 + 𝛼0 = 2 cos(𝜑01 ∕2) and 1 − 𝛼 = 2 cos(𝜑∕2) lead to the final expression: uexact = u01 h

1 i eik(R10 +R) , √ cos(𝜑01 ∕2) cos (𝜑∕2) 2𝜋k R10 R

(10.59)

which completely coincides with approximation (10.51) and validates the theory in Section 10.1. 10.4.2

Diffraction at a Soft Strip

The geometry of the problem is shown in Figure 10.2. The incident wave is given by Equation (10.48). The primary edge wave excited at edge L1 is described by Equation (10.15), where one should set u01 = exp(ikl cos 𝜑01 ), 𝛼1 = 2𝜋, 𝛾01 = 𝜋∕2, and 𝜌1 = ∞. Traveling along the top face of the strip, this wave reaches edge L and generates the secondary wave described by Equation (10.23), where one should set 𝛼 = 2𝜋, 𝛾0 = 𝜋∕2, and 𝜌 = ∞. Referring to Equation (2.62), we can verify that the primary wave (10.15) and the secondary wave (10.23) are symmetric with respect to the strip. Hence, the primary edge wave traveling along the bottom face of the eq strip generates the same secondary wave (10.23), where u0 is defined by Equation (10.22). Thus, the total secondary wave diffracted at the right edge L is double that of the wave (10.23); that is, = −u01 uexact s

𝜕f (0, 𝜑01 , 2𝜋) 𝜕f (𝜑, 0, 2𝜋) eik(R10 +R) 1 , √ 3∕2 𝜕𝜑1 𝜕𝜑0 2𝜋(kR10 ) kR

(10.60)

where 𝜕f (0, 𝜑01 , 2𝜋) 𝜕f (𝜑, 0, 2𝜋) 1 sin(𝜑∕2) sin(𝜑01 ∕2) . = 𝜕𝜑1 𝜕𝜑0 4 cos2 (𝜑∕2) cos2 (𝜑01 ∕2) (10.61) According to Equations (5.46) and (5.48), the exact secondary wave is determined as √ √ ] ik(R +R)+i𝜋∕4 [ 1 + 𝛼 1 − 𝛼0 1 + 𝛼 1 − 𝛼0 e 10 exact , us = −u01 𝜓(q, −𝛼) − 𝜓(q, 𝛼0 ) √ 𝛼 + 𝛼0 2 2 2𝜋kR (10.62) where the parameters q, 𝛼, and 𝛼0 are defined in Equation (10.53) and 𝜕𝜑(q, 𝛼) i 𝜓(q, 𝛼) = √ 2(1 + 𝛼) 𝜕q

(10.63)

318

DIFFRACTION INTERACTION OF NEIGHBORING EDGES ON A RULED SURFACE

with function 𝜑(q, 𝛼) given by Equations (10.55) and (10.56). We are interested in the diffracted wave in the ray region where q(1 + 𝛼) = kR10 (1 − cos 𝜑) ≫ 1. Here 𝜓(q, 𝛼) =

1 (2q)3∕2 (1 + 𝛼)

e−i𝜋∕4 −5∕2 ) √ + O(q 𝜋

(10.64)

and the error of approximation (10.62) is on the order of q−5∕2 . Within this approximation, Equation (10.62) can be written as uexact s

1 1 = −u01 √ 3∕2 (kR )3∕2 2 𝜋 10

√ √ ( ) 1 + 𝛼 1 − 𝛼0 1 + 𝛼 1 − 𝛼0 eik(R10 +R) − √ 2(𝛼 + 𝛼0 ) 1 − 𝛼 1 + 𝛼0 2𝜋kR (10.65)

or uexact s

√ √ 1 + 𝛼 1 − 𝛼0 eik(R10 +R) 1 1 . = −u01 √ √ 𝜋 23∕2 (kR10 )3∕2 (1 − 𝛼)(1 + 𝛼0 ) 2𝜋kR

(10.66)

In view of the definitions (10.53), this equation reduces to uexact = −u01 s

eik(R10 +R) 1 1 sin(𝜑∕2) sin(𝜑01 ∕2) . √ 2 2 3∕2 8𝜋 cos (𝜑∕2) cos (𝜑01 ∕2) (kR10 ) kR

(10.67)

This expression coincides exactly with approximation (10.60) and confirms the asymptotic theory developed in Section 10.2.

PROBLEMS 10.1

Show that the field u1h + u(t) of acoustic waves is continuous at the shadow h boundary (𝜑 = 𝜋). The geometry of the problem is shown in Figure 10.1. The are described by Equations (10.1) and (10.6), respecfunctions u1h and u(t) h tively. Solution The function (10.7) contains the factor [(2∕N) sin(𝜋∕N) cos(𝜑∕2)]∕[cos(𝜋∕N) − cos(𝜑∕N)]. Its nominator and denominator equal zero when 𝜑 = 𝜋. Utilizing L’Hospital’s rule, one finds that this factor equals “−1”. Now consider other details of function (10.6): r Its first multiplier, that is, the function u1h (R10 , 0), contains the factor 1

√ 𝜌1

=√ . √ R10 (𝜌1 + R10 ) R10 (1 + R10 ∕𝜌1 )

PROBLEMS

319

r In view of Equations (10.8) and (10.11), the second and third multipliers of Equation (10.6) generate the factors √

√ R10 R R10 + R

and

𝜌1 + R10 . R(𝜌1 + R10 + R)

r The product of these three factors equals 1 . √ (R10 + R)(1 + (R10 + R)∕𝜌1 ) Taking into account that ∞ cos(𝜑∕2)

∫√2s cos(𝜑∕2)

it2

e

dt = ±

√ 𝜋 2

ei𝜋∕4

with 𝜑 = 𝜋 ∓ 0,

we find that the diffracted field (10.6) is discontinuous at the shadow boundary 𝜑 = 𝜋 ∓ 0 and equal to eik(R10 +R)+i𝜋∕4 1 1 , = ∓ g(0, 𝜑 , 𝛼 ) u u(t) √ 01 01 1 h 2 sin 𝛾01 2𝜋k(R + R)(1 + (R + R)∕𝜌 ) 10 10 1 (R, 𝜋 ∓ 0) = ∓ 12 u1h (R10 + R, 0). Thus, the sum of the diffracted and that is, u(t) h incident waves is continuous at the shadow boundary 𝜑 = 𝜋: = 12 u1h (R10 + R, 0). u1h + u(t) h 10.2

Show that the normal derivative of the diffracted field (10.6) is continuous at the shadow boundary 𝜑 = 𝜋. The geometry of the problem is shown in Figure 10.1. ∕𝜕n = −𝜕u(t) ∕R sin 𝛾0 𝜕𝜑 at the shadow boundary Solution To find 𝜕u(t) h h 𝜑 = 𝜋, one should analyze the derivative ∞ cos(𝜑∕2)

𝜑 2 2 𝜋 𝜕 𝜕 eit dt] = sin [g(𝜑, 0, 𝛼) cos e−is(1+cos 𝜑) √ (a ⋅ b ⋅ c), ∫ 2s cos(𝜑∕2) 𝜕𝜑 2 N N 𝜕𝜑 N=

𝛼 𝜋

320

DIFFRACTION INTERACTION OF NEIGHBORING EDGES ON A RULED SURFACE

with a=

cos(𝜑∕2) , cos(𝜋∕N) − cos(𝜑∕N)

∞ cos(𝜑∕2)

b = e−is(1+cos 𝜑) ,

c=

√

∫

2

eit dt, 2s cos(𝜑∕2)

s = k𝜒 sin 𝛾0 . Here 𝜕(a ⋅ b ⋅ c)∕𝜕𝜑 = (a ⋅ b ⋅ c)′ = a′ bc + ab′ c + abc′ . Note that the functions a(𝜑) and a′ (𝜑) are fractions whose nominators and denominators are equal to zero when 𝜑 = 𝜋. To disclose their indeterminacy, we use L’Hospital’s rule. Verify that √ 1 , a(𝜋) = − (2∕N) sin(𝜋∕N) 1 cos(𝜋∕N) a (𝜋) = , 4 sin2 (𝜋∕N) ′

and under the condition have

√

b(𝜋) = 1,

c(𝜋 ∓ 0) = ± √

′

′

b (𝜋) = 0,

c (𝜋) =

𝜋

2

ei𝜋∕4 ,

s , 2

√ k𝜒 sin 𝛾0 ≫ 1 and N > 1 (i.e., 𝛼 > 𝜋), we

s=

𝜋 𝜕 1 𝜋 √ i𝜋∕4 2 𝜋e − sin (a ⋅ b ⋅ c) = ± cot N N 𝜕𝜑 4N N

√

√ s s ≈− 2 2

Hence, 𝜕Dh (𝜒, 𝜋, 𝛾0 )∕𝜕𝜑) ≈ ik𝜒 sin 𝛾0 . This estimation and Equations (10.5), (10.6), and (10.11) lead to the expression that contains the product √

1

1

√ √ 1 + R10 ∕𝜌1 1 +

R 𝜌1 +R10

𝜌1

√

𝜌1 + R10 =√ √ 𝜌1 + R10 𝜌1 + R10 + R

1 =√ . 1 + (R10 + R)∕𝜌1 Finally, at the shadow boundary 𝜑 = 𝜋, 𝜕u(t) h

(t)

1 𝜕uh =− 𝜕n R sin 𝛾0 𝜕𝜑

1 1 = u01 g(0, 𝜑01 , 𝛼1 ) 2𝜋 sin 𝛾01 R10 + R

√

R10 eik(R10 +R) . √ R 1 + (R + R)∕𝜌 10

1

Thus, the normal derivative of the field (10.6) is finite and asymptotically continuous.

321

PROBLEMS

10.3

Show that the field u1s + u(t) s of acoustic waves is continuous at the shadow boundary (𝜑 = 𝜋). The geometry of the problem is shown in Figure 10.1. The functions u1s and u(t) s are described by Equations (10.15) and (10.24), respectively. Solution It is clear that that the incident wave (10.15) equals zero in the direction 𝜑1 = 0 as well as in the shadow region (R1 > R10 , 𝜑 ≥ 𝜋). Therefore, it is continuous at the shadow boundary 𝜑 = 𝜋. Next, one should evaluate the field √ (10.24) at the shadow boundary. According to Equation (10.26), 𝔽 [ 2𝜒 sin 𝛾0 cos(𝜑∕2] = 1 when 𝜑 = 𝜋. Next, Equation (10.24) contains the factor 𝜕f (𝜑, 0, 𝛼) 𝜑 2 sin(𝜋∕n) sin(𝜑∕N) cos2 (𝜑∕2) , cos2 = 𝜕𝜑 2 N 2 [cos(𝜋∕N) − cos(𝜑∕N)]2

N=

𝛼 . 𝜋

Its numerator and denominator have zeros of the second order at 𝜑 = 𝜋. Two subsequent applications of L’Hopital’s rule show that this factor equals “1/2”. Hence, u(t) s =

(𝜁 ) 𝜕uinc 1s st 𝜕n

𝜒

eikR+i𝜋∕4 . √ sin 𝛾0 2𝜋kR(1 + R∕𝜌)

∕𝜕n from Equations (10.20) and (10.15) and obtain Next, we substitute 𝜕uinc 1S the expression that contains the factor 1 3∕2 R10

R10 R 1 1 . √ √ R + R 1 + R10 ∕𝜌1 10 R(1 + R∕𝜌)

Taking into account that 𝜌 = 𝜌1 + R10 in accordance with Equation (10.11), one can transform this factor as √ √ √ 𝜌1 𝜌1 + R10 1 R 1 = √ √ R10 R10 + R 𝜌1 + R10 𝜌1 + R10 + R R10 + R √ R 1 . × √ R10 1 + (R + R)∕𝜌 10 1 As a result, we arrive at Equation (10.29): u(t) s

𝜕f (0, 𝜑01 , 𝛼1 ) i 1 = u01 2 𝜕𝜑1 R 2𝜋k sin 𝛾01 10 + R

√

eik(R10 +R) R , √ R10 1 + (R + R)∕𝜌 10

and confirm that the total field at the shadow boundary is continuous.

1

322

10.4

DIFFRACTION INTERACTION OF NEIGHBORING EDGES ON A RULED SURFACE

Show that the normal derivative of the field u1s (R1 , 𝜑1 ) + u(t) s (R, 𝜑) is continuous at the shadow boundary where 𝜑1 = 0 and 𝜑 = 𝜋. The geometry of the problem is shown in Figure 10.1. The functions u1s (R1 , 𝜑1 ) and u(t) s (R, 𝜑) are defined by Equations (10.15) and (10.24), respectively. Solution

First note that in view of Equation (10.15) the derivative

𝜕u1s (R10 + R, 0) 𝜕u1s (R10 + R, 0) 𝜕f (0, 𝜑01 , 𝛼1 ) 1 = u01 = 𝜕n (R10 + R) sin 𝛾01 𝜕𝜑1 𝜕𝜑1 ×

eik(R10 +R)+i𝜋∕4 1 √ sin 𝛾01 (R10 + R) 2𝜋k(R10 + R)(1 + (R10 + R)∕𝜌1 ) 2

is finite at the shadow boundary 𝜑1 = +0 and 𝜑 = 𝜋 − 0. However, it is discontinuous here, because in the shadow region (𝜑1 < 0, 𝜑 > 𝜋) it equals zero. Next, calculate the derivative of the field (10.24): (t) 𝜕u(t) 1 𝜕us (R, 𝜑) s =− . 𝜕n R sin 𝛾0 𝜕𝜑

Consider the quantity 𝜕f (𝜑, 0, 𝛼) 𝜑 ) 2 sin(𝜋∕N) 𝜑 (√ cos2 𝔽 2k𝜒 sin 𝛾0 cos B(𝜑), = 𝜕𝜑0 2 2 N2 where B(𝜑) = a ⋅ b ⋅ c and

a = sin

𝜑 , N

b=

cos2 (𝜑∕2) , [cos(𝜋∕N) − cos(𝜑∕N)]2

c=𝔽

(√

2k𝜒 sin 𝛾0 cos

𝜑) . 2

We need to calculate the function dB(𝜑)∕d𝜑 = a′ bc + ab′ c + abc′ ] at the point 𝜑 = 𝜋. To calculate the quantities b(𝜋) and b′ (𝜋) it is useful to find the function b(𝜑) in the vicinity of the point 𝜑 = 𝜋. For this goal we utilize the Taylor expansions: 1 1 (𝜑 − 𝜋)4 (𝜑 − 𝜋)2 − +⋯ , 4 2 4! ( 𝜑 )2 𝜋 1 1 𝜋 𝜋 𝜋 cos − cos = 2 sin2 ⋅ (𝜑 − 𝜋)2 + 3 sin cos (𝜑 − 𝜋)3 +⋯ N N N N N N N cos2 𝜑 =

PROBLEMS

323

Then

b(𝜑) = =

1 (𝜑 − 𝜋)2 {1 + O[(𝜑 − 𝜋)2 +⋯} 4 (1∕N 2 ) sin2 (𝜋∕N)(𝜑 − 𝜋)2 [1 + (1∕N) cot(𝜋∕N)(𝜑 − 𝜋) +⋯]

N2 4 sin2 (𝜋∕N)

{

1−

} 1 𝜋 cot (𝜑 − 𝜋) + O[(𝜑 − 𝜋)2 ] +⋯ . N N

It follows from this expansion that b(𝜋) =

N2 4 sin2 (𝜋∕N)

It is clear that a(𝜋) = sin(𝜋∕N), can also verify that

and

b′ (𝜋) = −

N cos(𝜋∕N) . 4 sin3 (𝜋∕N)

a′ (𝜋) = (1∕N) cos(𝜋∕N), and c(𝜋) = 1. We √

′

c (𝜋 ∓ 0) = ±

𝜋k𝜒 sin 𝛾0 −i𝜋∕4 e . 2

Now recall that the function (10.24) is a high-frequency approximation valid √ under the condition k𝜒 sin 𝛾0 ≫ 1. Within this approximation, 𝜕B(𝜋 ∓ 0) ≈ a(𝜋)b(𝜋)c′ (𝜋 ∓ 0), 𝜕𝜑 √ 𝜋k sin 𝛾0 3∕2 −i𝜋∕4 𝜕 . Ds (𝜒, 𝜋 ∓ 0, 𝛾0 ) ≈ ± 𝜒 e 𝜕𝜑 2 Utilizing this estimation, we obtain 𝜕f (0, 𝜑01 , 𝛼1 ) 𝜕u(t) 1 1 s (R, 𝜋 ∓ 0) = ∓ u01 𝜕n 2 𝜕𝜑1 sin2 𝛾01 (R10 + R)3∕2 ×√

eik(R10 +R)+i𝜋∕4 . √ 1 + R10 ∕𝜌1 2𝜋kR(1 + R∕𝜌)

Here 𝜌 = 𝜌1 + R10 according to Equation (10.11). Because of that, √ 𝜌1 + R10 =√ √ √ √ 𝜌1 + R10 𝜌1 + R10 + R 1 + R10 ∕𝜌1 1 + [R∕(𝜌1 + R10 )] √

1

=√

1 1 + (R10 + R)∕𝜌1

𝜌1

324

DIFFRACTION INTERACTION OF NEIGHBORING EDGES ON A RULED SURFACE

and 𝜕f (0, 𝜑01 , 𝛼1 ) 𝜕u(t) 1 s (R, 𝜋 ∓ 0) = ∓ u01 𝜕n 2 𝜕𝜑1 ×

eik(R10 +R)+i𝜋∕4 1 . √ sin2 𝛾01 (R10 + R) 2𝜋k(R10 + R)(1 + (R10 + R)∕𝜌1 )

Comparison with 𝜕u1s (R10 + R, 0)∕𝜕n given at the beginning of Problem 10.4 shows that 𝜕u(t) 1 𝜕u1s (R10 + R, 0) s (R, 𝜋 ∓ 0) =∓ . 𝜕n 2 𝜕n Therefore, the normal derivative of the total field u1s + u(t) s at the shadow boundary is continuous and equal to 𝜕(u1s + u(t) 1 𝜕u1s (R10 + R, 0) s ) = . 𝜕n 2 𝜕n

11 Focusing of Multiple Acoustic Edge Waves Diffracted at a Convex Body of Revolution with a Flat Base The theory presented below is based on papers by Ufimtsev (1989, 1991).

11.1 STATEMENT OF THE PROBLEM AND ITS CHARACTERISTIC FEATURES This problem is illustrated in Figure 11.1, which shows a convex body of revolution excited by the axisymmetrical incident wave constant along the edge. i

uinc = u0 eik𝜙 .

(11.1)

The axis of symmetry (z-axis) is a focal line for elementary edge waves and rays. With respect to the observation points P on this axis, each diffraction point at the edge is a point of the stationary phase. The elementary rays propagate in the directions of the edge-diffraction cones, which transform (in this particular case) into the meridian planes. Because of that, the directivity pattern of elementary edge rays are expressed

Fundamentals of the Physical Theory of Diffraction, Second Edition. Pyotr Ya. Ufimtsev. © 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

325

326

FOCUSING OF MULTIPLE ACOUSTIC EDGE WAVES DIFFRACTED

φ

Q

R

φ0

kˆi

P P

kˆi

Z

φ

ψ

R

R

ψ Figure 11.1 Body of revolution excited by the source Q. Focusing of diffracted edge waves occurs at points P on the z-axis. [Reprinted from Ufimtsev (1989) with the permission of the Journal of the Acoustical Society of America.]

in terms of the Sommerfeld functions f and g, as shown in Sections 7.6 and 8.1. The functions f and g are defined in Equations (2.62) and (2.264), and they describe the = j(0) + j(1) induced near the edge. field generated by the total scattering sources jtot s,h s,h s,h Analysis of this field is the main objective of this chapter. Here we ignore the exponentially small multiple edge waves created by the creeping waves [running over the front part (z < 0) of the object] and take into account only the multiple diffraction of edge waves propagating over the flat base. The denotation will be used for the field of multiple edge waves, where the index m = 2, 3, … u(m) s,h indicates the order of diffraction. The first-order (primary) edge waves excited directly by the incident wave (11.1) are determined by the integral expression (8.4) applied to the circular edge. Due to the symmetry of the problem, the integrand in Equation (8.4) is constant and the integration over the edge results in the expression pr

us

pr

uh

} i

= u0 eik𝜙 a

[

] f (𝜑, 𝜑0 , 𝛼) eikR , g(𝜑, 𝜑0 , 𝛼) R

(11.2)

where a is the radius of the edge and 𝛼 is the external angle between the faces of the edge. This asymptotic expression is valid for any point of observation on the focal line outside the scattering object, under the condition kR ≫ 1. with m = 2, 3, … (arising due to the diffraction of waves Multiple edge waves u(m) s,h running over the flat base) can be found using Equations (10. 3) and (10.19), where the tot transform into the functions f and g. For the calculation of edge waves functions Fs,h running over a flat base, one can utilize the ray asymptotics (10.4) and (10.23), where one should set 𝛾0 = 𝜋∕2, R = 2a, and 𝜌 = −a. On their way to the opposite point of the edge, these waves intersect the focal line and √acquire a phase√shift equal to −𝜋∕2, which is a direct consequence of the factor 1∕ 1 + R∕a = 1∕ 1 − 2a∕a = −i. We can now proceed to calculate multiple edge waves.

MULTIPLE HARD DIFFRACTION

11.2

327

MULTIPLE HARD DIFFRACTION

According to Equation (10.3), the field created by the (m + 1)th-order edge waves on the focal line can be represented in the form u(m+1) (P) = h

1 a eikR eikR ū (m) (𝜁 ) d𝜁 = g(𝜓, 0, 𝛼)̄u(m) g(𝜓, 0, 𝛼) . h h 4𝜋 R ∫L 2 R

(11.3)

Here ū (m) denotes the mth-order edge wave propagating along the flat base to the h opposite point 𝜁 at the edge, where it undergoes diffraction and creates elementary is the field of the mth-order wave at waves of (m + 1)th-order. More precisely, ū (m) h the diffraction point 𝜁 at the edge. One can show that pr

≡ uh = ū (1) h

1 ik𝜙i ei(2ka−𝜋∕4) , u0 e g(𝛼, 𝜑0 , 𝛼) √ 2 𝜋ka

(11.4)

ei(2ka−𝜋∕4) , √ 4 𝜋ka

(11.5)

ū (m) = ū (m−1) g(0, 0, 𝛼) h h

m = 2, 3, 4, … .

These relationships lead to [ ū (m) h

ik𝜙i

= u0 e

⋅2g(𝛼, 𝜑0 , 𝛼) [g(0, 0, 𝛼)]

m−1

ei(2ka−𝜋∕4) √ 4 𝜋ka

]m ,

m = 1, 2, 3, … . (11.6)

Therefore, the total field of all edge waves on the focal line equals pr

uew (P) = uh (P) + h

∞ ∑

u(m) (P) h

m=1

lk𝜙i

= u0 e

eikR a R

⎧ ⎪ ei(2ka−𝜋∕4) ⎨g(𝜑, 𝜑0 , 𝛼) + g(𝜓, 0, 𝛼)g(𝛼, 𝜑0 , 𝛼) √ 4 𝜋ka ⎪ ⎩

+ g(𝜓, 0, 𝛼)g(𝛼, 𝜑0 , 𝛼)

∞ ∑ m=3

[ [g(0, 0, 𝛼)]

m−2

ei(2ka−𝜋∕4) √ 4 𝜋ka

]m−1 ⎫ ⎪ ⎬. ⎪ ⎭

(11.7)

Here the series is the geometric progression that can be converted to its sum. The physical meaning of Equation (11.7) is clear. The first term in braces relates to the

328

FOCUSING OF MULTIPLE ACOUSTIC EDGE WAVES DIFFRACTED

primary edge waves, the second to the secondary waves, and the third term represents the sum of all multiple edge waves of order 3 and higher. The total scattered field on the focal line also includes the reflected rays (6.187) in front of the object (z < 0) and the shadow radiation (6.228) behind the object (z > 0). This approximation for the scattered field actually represents an incomplete asymptotic expansion, because it includes only the first term in the individual asymptotic expansion for each multiple edge wave. Also Equation (6.187) is only the first term in the asymptotic expansion for the reflected field. Expression (11.7) can be used to calculate the total scattering cross-section. In the case of the incident plane wave, uinc = eikz , this quantity is defined as 𝜎h,s =

) ( 4𝜋 Im utot ⋅ Re−ikR h,s k

(11.8)

where utot is the total field scattered in the forward direction (𝜓 = 𝜋∕2). This field h,s consists of the following components:

r The shadow radiation, which is equivalent to the PO field (for the forward direction) and determined by (6.228), where one should set u0 = 1.

r The primary edge waves generated by the nonuniform/fringe scattering sources j(1) and determined by Equations (6.41) and (6.42), where one should set u0 = 1. h,s r The sum of all multiple edge waves of order 2 and higher. The substitution of this total field into Equation (11.8) results in the following asymptotic expression: { 𝜎h = 2𝜋a

2

×

∞ ∑ m=1

1+

( ) 𝜋 2 g , 0, 𝛼 g(𝛼, 𝜑0 , 𝛼) ka 2

sin[m(2ka − 𝜋∕4)] [g(0, 0, 𝛼)]m−1 4m (𝜋ka)m∕2

} ,

(11.9)

which is incomplete in the sense mentioned above. Notice that the series in (11.9) equals zero when 2ka − 𝜋∕4 = l𝜋 (l = 1, 2, 3, …). In this case, all corrections to the first term in (11.9) are determined by the higher-order terms in the individual asymptotic expansions for each multiple edge wave.

11.3

MULTIPLE SOFT DIFFRACTION

The primary edge wave excited by the incident wave (11.1) is determined by (11.2). The higher-order edge waves arise due to the slope diffraction of waves running along the flat base of the scattering object. These higher-order edge waves are calculated

MULTIPLE SOFT DIFFRACTION

329

on the basis of Equation (10.19). For the (m + 1)th-order edge wave arriving at point P on the focal line, it can be written in the form (P) = u(m+1) s

ikR 1 𝜕f (𝜓, 0, 𝛼) eikR (m) 𝜕f (𝜓, 0, 𝛼) e ū (m) d𝜁 = ā u . s 2𝜋 𝜕𝜑0 R ∫L s 𝜕𝜑0 R

(11.10)

Here ū (m) s is the mth-order edge wave coming to edge point 𝜁 from its opposite point 𝜁̄ = 𝜁 − 𝜋a. This quantity is calculated using Equations (10.22) and (10.23), where one should set 𝛾0 = 𝛾01 = 𝜋∕2, R = 2a, and 𝜌 = −a. These calculations result in i

ik𝜙 ū (1) s = −u0 e

̄ (m−1) ū (m) s =u s

𝜕f (𝛼, 𝜑0 , 𝛼) ei(2ka+𝜋∕4) , √ 𝜕𝜑 8ka 𝜋ka

(11.11)

𝜕 2 f (0, 0, 𝛼) ei(2ka+𝜋∕4) √ 𝜕𝜑 𝜕𝜑0 8ka 𝜋ka

(11.12)

or ū (m) s

ik𝜙i 𝜕f (𝛼, 𝜑0 , 𝛼)

[

𝜕 2 f (0, 0, 𝛼) 𝜕𝜑 𝜕𝜑0

]m−1 [

ei(2ka+𝜋∕4) √ 8ka 𝜋ka

]m .

(11.13)

[ ]m−1 [ i(2ka+𝜋∕4) ]m 𝜕f (𝜓, 0, 𝛼) 𝜕f (𝛼, 𝜑0 , 𝛼) 𝜕 2 f (0, 0, 𝛼) e × . √ 𝜕𝜑0 𝜕𝜑 𝜕𝜑 𝜕𝜑0 8ka 𝜋ka

(11.14)

= −u0 e

𝜕𝜑

After the substitution of (11.13) into Equation (11.10), we obtain i

(P) = −u0 eik𝜙 a u(m+1) s

eikR R

The total field of all edge waves on the focal lines equals pr

uew s (P) = us (P) +

∞ ∑

u(m) s (P)

m=1

ik𝜙i

= u0 e

eikR a R

⎧ ⎪ 𝜕f (𝜓, 0, 𝛼) 𝜕f (𝛼, 𝜑0 , 𝛼) ⎨f (𝜑, 𝜑0 , 𝛼) − 𝜕𝜑0 𝜕𝜑 ⎪ ⎩

]m−2 [ i(2ka+𝜋∕4) ]m−1 ⎫ ∞ [ 2 ∑ ⎪ 𝜕 f (0, 0, 𝛼) e × √ ⎬. 𝜕𝜑 𝜕𝜑 0 8ka 𝜋ka ⎪ m=2 ⎭ The series in Equation (11.15) is a geometrical progression.

(11.15)

330

FOCUSING OF MULTIPLE ACOUSTIC EDGE WAVES DIFFRACTED

Now we apply Equation (11.15) to calculate of the total scattering cross-section. In the case of the incident plane wave uinc = eikz , it is defined by Equation (11.8), where sh (1) utot s = u + us +

∞ ∑

u(m) s .

(11.16)

m=2

Here ush is the shadow radiation (6.228) (where one should set u0 = 1), the quantity (1) u(1) s = af (𝜑, 𝜑0 , 𝛼)

eikR R

(11.17)

is the primary edge wave generated by the nonuniform scattering sources j(1) s , and the series represents the sum of all edge waves of order 2 and higher. Thus, the total scattering cross-section of the acoustically soft object equals { 2 𝜕f (𝜋∕2, 0, 𝛼) 𝜕f (𝛼, 𝜑0 , 𝛼) 2 1− 𝜎s = 2𝜋a ka 𝜕𝜑0 𝜕𝜑 } ]m−1 ∞ [ 2 ∑ sin[m(2ka + 𝜋∕4)] 𝜕 f (0, 0, 𝛼) × . (11.18) 𝜕𝜑 𝜕𝜑0 (8ka)m (𝜋ka)m∕2 m=1 We emphasize again that approximations (11.15) and (11.18) are incomplete asymptotic expansions in the sense discussed in Section 11.2. The series in Equation (11.18) equals zero when 2ka + 𝜋∕4 = l𝜋 (l = 1, 2, 3, …). In this case, all corrections to the first term in (11.18) are determined by the higher-order terms in the individual asymptotic expansions for each multiple edge wave. PROBLEMS 11.1

Derive multiple edge waves (11.3) of the second and third order (m = 2, 3). Solution These waves are generated by the grazing diffraction of the primary and ū (2) at edge point 𝜁 . They are calculated accordand secondary waves ū (1) h h ing to Equations (8.29) and (8.23), setting R = 2a and 𝜌 = −a. The quantity 𝜌 = −a is a consequence of the fact that the edge waves propagate in a direction perpendicular to the edge (𝛾0 = const = 𝜋∕2), and the unit propagation vector k̂ s coincides with the principal normal 𝜈̂ to the edge . In view of these comments, i

ū (1) = u0 eik𝜙 g(𝛼, 𝜑0 , 𝛼) h i

u(2) (P) = u0 eik𝜙 a h

i ei(2ka+𝜋∕4) ei(2ka−𝜋∕4) = u0 eik𝜙 g 𝛼, 𝜑0 , 𝛼) √ , √ i2 𝜋ka 2 𝜋ka

eikR ei(2ka−𝜋∕4) . g(𝛼, 𝜑0 , 𝛼)g(𝜓, 0, 𝛼) √ R 4 𝜋ka

331

PROBLEMS

To calculate u(3) , one should find ū (2) . It is determined by Equation (10.4) h h

, 𝜑 = 0, 𝛾0 = 𝜋∕2, R = 2a, and 𝜌 = −a. with the following settings: u1h = ū (1) h They lead to = ū (2) h

i 1 (1) ei(2ka−𝜋∕4) = u0 eik𝜙 ū h g(0, 0, 𝛼) √ 2 2 𝜋ka )2 ( ei(2ka−𝜋∕4) × 2g(𝜑, 𝜑0 , 𝛼)g(0, 0, 𝛼) . √ 4 𝜋ka

Substitution of this quantity into Equation (11.3) yields (P) u(3) h

ik𝜙i

= u0 e

eikR a g(𝜓, 0, 𝛼)g(𝛼, 𝜑0 , 𝛼)g(0, 0, 𝛼) R

(

ei(2ka−𝜋∕4) √ 4 𝜋ka

)2 .

and u(3) agree completely with Equation (11.7). The quantities u(2) h h 11.2

Derive multiple edge waves (11.10) of the second and third order (m = 2, 3). Solution These waves are generated by the slope diffraction of the primary ̄ (2) and secondary edge waves ū (1) s and u s at edge point 𝜁 . They are calculated according to Equations (10.21) to (10.23) adapted to the geometry of the represent the problem shown in Figure 11.1. Note that the quantities ū (m) s eq magnitudes u0 (𝜁st ) of the equivalent waves (10.17) incident at the edge. They are determined in terms of the normal derivative of the actual incident wave by adjusted Equations (10.20) and (10.21). eq First derive the quantity ū (1) s = u0 generated by the primary edge wave at the opposite point of the flat base. In the arbitrary direction belonging to the diffraction cone, the primary wave is described by Equation (8.29) as i eik(R+𝜋∕4) pr , us (R, 𝜑) = u0 eik𝜙 f (𝜑, 𝜑0 , 𝛼) √ 2𝜋kR(1 + R∕𝜌)

where it is assumed that 𝛾0 = 𝜋∕2. Recall that the angles 𝜑 and 𝜑0 are measured from the illuminated face of the edge as shown in Figure 11.1. Because of that, the normal derivative of this wave at the flat base is defined as pr

pr

𝜕us (R, 𝛼) 1 𝜕us (R, 𝛼) =− , 𝜕n R 𝜕𝜑 and in view of Equation (10.21), eq

i

ik𝜙 u0 (𝜁st ) = ū (1) s = −u0 e

i 𝜕f (𝛼, 𝜑0 , 𝛼) ei(kR+𝜋∕4) . √ 2kR 𝜕𝜑 2𝜋kR(1 + R∕𝜌)

332

FOCUSING OF MULTIPLE ACOUSTIC EDGE WAVES DIFFRACTED

Setting R = 2a and 𝜌 = −a here, one obtains i

eq

ik𝜙 u0 (𝜁st ) = ū (1) s = −u0 e

𝜕f (𝛼, 𝜑0 , 𝛼) ei(kR+𝜋∕4) . √ 𝜕𝜑 8ka 𝜋ka

Substitution of this quantity into Equation (11.10) results in i

ik𝜙 u(2) a s (P) = −u0 e

eikR 𝜕f (𝛼, 𝜑0 , 𝛼) 𝜕f (𝜓, 0, 𝛼) ei(kR+𝜋∕4) . √ R 𝜕𝜑 𝜕𝜑0 8ka 𝜋ka

Next, calculate a multiple wave of the third order. First it is necessary to find the secondary diffracted wave in arbitrary direction 𝜓. This wave is determined according to adjusted Equation (10.23) as ̄ (1) u(2) s =u s

𝜕f (𝜓, 0, 𝛼) ei(kR+𝜋∕4) . √ 𝜕𝜑0 2𝜋kR(1 + R∕𝜌)

In the vicinity of the flat base it is approximated by an equivalent wave of the type of Equation (10.17), where (2)

i 𝜕us ei(kR+𝜋∕4) i (1) 𝜕 2 f (0, 0, 𝛼) = ū s √ 2kR 𝜕𝜓 2kR 𝜕𝜓 𝜕𝜑0 2𝜋kR(1 + R∕𝜌)

eq

u0 =

Setting R = 2a and 𝜌 = −a here, one obtains eq

̄ (1) u0 = ū (2) s =u s

𝜕 2 f (0, 0, 𝛼) ei(2ka+𝜋∕4) √ 𝜕𝜓 𝜕𝜑0 8ka 𝜋ka

or ik𝜙i

ū (2) s = −u0 e

𝜕f (𝛼, 𝜑0 , 𝛼) 𝜕 2 f (0, 0, 𝛼) 𝜕𝜑 𝜕𝜓 𝜕𝜑0

[

ei(2ka+𝜋∕4) √ 8ka 𝜋ka

]2 .

Next, substitution of ū (2) s into Equation (11.10) provides a multiple wave of the third order: ik𝜙i

u(3) s (P) = −u0 e

eikR 𝜕f (𝜓, 0, 𝛼) 𝜕f (𝛼, 𝜑0 , 𝛼) 𝜕 2 f (0, 0, 𝛼) a R 𝜕𝜑0 𝜕𝜑 𝜕𝜓 𝜕𝜑0

[

ei(2ka+𝜋∕4) √ 8ka 𝜋ka

]2 .

(3) The quantities u(2) s (P) and us (P) found coincide with the terms in Equation (11.15) associated with the indexes m = 2 and m = 3.

12 Focusing of Multiple Edge Waves Diffracted at a Disk The theory presented in this chapter is based on papers by Ufimtsev (1989, 1991). It represents an extension of Chapter 11 to the disk diffraction problem, where it is necessary to take into account the edge waves propagating along both faces of a disk (Fig. 12.1). This problem is complicated by the fact that the wave traveling along one face of the disk generates (due to diffraction at the edge) higher-order waves not only on the same face but also on the other face. However, its solution can be lightened if we utilize the symmetry of the scattered field. Let us consider scattering at an arbitrary plate located in the plane z = 0. It follows from Equation (1.10) that sc usc s (−z) = us (z),

sc usc h (−z) = −uh (z).

(12.1)

Here, the first equality is obvious and the second is caused by the factor eikr eikr 𝜕 eikr d eikr d eikr = ∇′ ⋅ n̂ = −∇ ⋅ n̂ = − (∇r ⋅ n̂ ) = − (̂r ⋅ n̂ ), 𝜕n r r r dr r dr r

(12.2)

where r̂ (−z) ⋅ n̂ = −̂r(z) ⋅ n̂ . The geometry of the problem is shown in Figure 12.1. The incident wave is given by uinc = eikz . The scattered field is investigated at points P on the z-axis, which is the focal line of the edge-diffracted waves.

Fundamentals of the Physical Theory of Diffraction, Second Edition. Pyotr Ya. Ufimtsev. © 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

333

334

FOCUSING OF MULTIPLE EDGE WAVES DIFFRACTED AT A DISK

y

y

R z

φ– ψ

R

P

P

z

Figure 12.1 Disk projection on the plane y0z (bold solid lines) and edge waves (dashed lines). The angles 𝜑 and 𝜓 = 2𝜋 − 𝜑 (measured from the left and right faces of the disk) determine the directions to the observation point P.

12.1

MULTIPLE HARD DIFFRACTION

The primary edge waves excited by the incident wave directly are given by Equation (11.2), where one should set u0 exp(ik𝜙i ) = 1. The (m+1)th-order waves are determined by Equation (11.3) adjusted to the disk problem as [ ] ikR e 1 (m,t) ̄ ū (m,t) (𝜁 )g(𝜑, 0, 𝛼) + u (𝜁 )g(𝜓, 0, 𝛼) d𝜁 r l ∫ 4𝜋 L R [ ] ikR e a (m,t) ū l g(𝜑, 0, 𝛼) + ū (m,t) g(𝜓, 0, 𝛼) , m = 1, 2, 3, …. (12.3) = r 2 R

(P) = u(m+1) h

is the total Here a is the radius of the disk, 𝜓 = 2𝜋 − 𝜑, 𝛼 = 2𝜋. The quantity ū (m,t) l mth-order edge wave arriving along the left face of the disk to the edge point 𝜁 , = ū (m) + ū (m) . ū (m,t) l ll rl

(12.4)

denotes the mth-order wave on the left face (at the edge point 𝜁 ) The quantity ū (m) ll generated (at the edge point 𝜁 − 𝜋a) by the (m − 1)th-order wave ū (m−1,t) that arrived l is the mth-order wave at the edge along the left face. Analogously, the quantity ū (m) rl on the left face (at the point 𝜁 of the edge) generated (at the point 𝜁 − 𝜋a) by the (m − 1)th-order wave ū r(m−1,t) that arrived there along the right face. With this type of denotation, the sense of the function becomes clear: ū (m,t) = ū (m) + ū (m) r rr . lr

(12.5)

Due to Equation (12.1), = −̄u(m,t) , ū (m,t) r l

ū (m) = −̄u(m) , ll lr

ū (m) u(m) . rr = −̄ rl

(12.6)

MULTIPLE HARD DIFFRACTION

335

In addition, according to Equation (2.64), g(𝜓, 0, 𝛼) = −g(𝜑, 0, 𝛼) =

1 . cos(𝜑∕2)

(12.7)

In view of these relationships, Equation (12.3) can be rewritten as eikR R

(12.8)

eikR . R

(12.9)

(P) = a ū (m,t) g(𝜑, 0, 𝛼) u(m+1) h l or u(m+1) (P) = a ū (m,t) g(𝜓, 0, 𝛼) r h pr

pr

The quantities ū (1,t) ≡ ū l and ū (1,t) ≡ ū r are the primary edge waves at the disk face, r l which passed through the focal line and arrived at the edge point 𝜁 . They can be found by use of Equation (8.29), where one should set 𝛾0 = 𝜋∕2, R = 2a, 𝜌 = −a, pr pr and g = g(0, 𝜋∕2, 𝛼) for ū l , and g = g(𝛼, 𝜋∕2, 𝛼) for ū r . In the same way, one can , ū (m) , and ū (m) with m = 2, 3, ...; only the function g will be find the quantities ū (m) ll lr rl different: namely, g = g(0, 0, 𝛼) = −1. We omit all intermediate manipulations and obtain: √ pr pr ū l = −̄ur = g(0, 𝜋∕2, 𝛼)𝜆 = − 2 𝜆, √ ū (m,t) = −̄ul(m−1,t) 𝜆 = (−1)m 2 𝜆m , l

(12.10) (12.11)

where 𝜆=

ei(2ka−𝜋∕4) . √ 2 𝜋ka

(12.12)

Hence, √ ikR m me u(m+1) (P) = a 2 g(𝜑, 0, 𝛼)(−1) 𝜆 h R

(12.13)

and the total focal field of all edge waves equals [

] ∞ √ ikR ∑ m m e ue𝑤 (P) = a g(𝜑, 𝜋∕2, 𝛼) + 2g(𝜑, 0, 𝛼) (−1) 𝜆 . h R m=1

(12.14)

336

FOCUSING OF MULTIPLE EDGE WAVES DIFFRACTED AT A DISK

The function g(𝜑, 𝜋∕2, 𝛼) is singular in the directions 𝜑 = 𝜋∕2 and 𝜑 = 3𝜋∕2. Because of this, the total scattered field in the far zone should be represented in traditional PTD form: usc h

=

u(0) h

+ ufrh

+

∞ ∑

u(m) . h

(12.15)

m=2

Here u(0) =± h

ika2 eikR 2 R

{ in the directions

𝜑=

3𝜋∕2 𝜋∕2

(12.16)

, or, in other words, it is is the field generated by the uniform scattering sources j(0) h the PO approximation. The quantity ufrh = ag(1) (𝜑, 𝜋∕2, 𝛼)

eikR , R

(12.17)

with g(1) (𝜋∕2, 𝜋∕2, 𝛼) = −g(3𝜋∕2, 𝜋∕2, 𝛼) = − 12 , is the field generated by the caused by primary diffraction of the incinonuniform/fringe scattering sources j(1) h dent wave at the disk. The series in Equations (12.14) and (12.15) represents the contributions generated by that part of j(1) that is caused by the multiple diffraction. h After substitution of Equation (12.15) into Equation (11.8), one finds the total scattering cross-section of the disk: { 𝜎h = 2𝜋a2

∞ sin[m(2ka − 𝜋∕4)] 2 ∑ (−1)m 1+ ka m=1 2m−1 (𝜋ka)m∕2

} .

(12.18)

This is an incomplete asymptotic approximation, which includes only the first term of the total asymptotic expansion (with ka → ∞) for each multiple edge wave. Comparison with the exact asymptotic solution (Witte and Westpfahl, 1970), which contains six first terms for the total cross-section, confirms that Equation (12.18) is correct.

12.2

MULTIPLE SOFT DIFFRACTION

The focal field created by the primary edge waves [excited by the incident wave uinc = exp(ikz)] is determined according to Equation (11.2) as pr

us = af (𝜑, 𝜋∕2, 𝛼)

eikR , R

(12.19)

MULTIPLE SOFT DIFFRACTION

337

where 𝛼 = 2𝜋 and f (𝜑, 𝜑0 , 𝛼) =

1 2

{ −

1 1 + cos[(𝜑 − 𝜑0 )∕2] cos[(𝜑 + 𝜑0 )∕2]

} .

(12.20)

(0) (1) This is an edge wave generated by the total scattering sources jtot s = js + js . The focal field of the primary edge waves created only by the nonuniform component j(1) s is also described by (12.19), where one should replace the function f by

f (1) (𝜑, 𝜑0 , 𝛼) = f (𝜑, 𝜑0 , 𝛼) − f (0) (𝜑, 𝜑0 )

(12.21)

with f (0) (𝜑, 𝜑0 ) =

sin 𝜑0 . cos 𝜑 + cos 𝜑0

(12.22)

The higher-order edge waves arise due to the slope diffraction of waves running along the flat faces of the disk. They are calculated on the basis of Equation (10.19). For diffraction at a solid convex body of revolution, the related technique was developed in Section 11.3. In the present section, this technique is extended for investigation of diffraction at an acoustically soft disk. According to Equation (10.19), the focal field generated by the (m + 1)th-order edge waves is determined by (P) = u(m+1) s

[ ] ikR 𝜕f (𝜑, 0, 𝛼) 1 (m,t) 𝜕f (𝜓, 0, 𝛼) e ̄ ū (m,t) + u d𝜁 , r 2𝜋 ∫L l 𝜕𝜑0 𝜕𝜑0 R

(12.23)

where d𝜁 = a d𝜃 is the differential arc length of disk edge L. The geometry of the problem is shown in Figure 12.1. The angles 𝜑 and 𝜓 are measured from different faces of the disk and 𝜓 = 2𝜋 − 𝜑. Due to the axial symmetry of the problem, Equation (12.23) is reduced to [ (P) u(m+1) s

=a

𝜕f (𝜑, 0, 𝛼) ū (m,t) l 𝜕𝜑0

𝜕f (𝜓, 0, 𝛼) + ū (m,t) r 𝜕𝜑0

]

eikR . R

(12.24)

Here ū (m,t) is the amplitude factor of the wave, which is equivalent to the total mthl,r order edge wave coming (to the edge point 𝜁 from its opposite point 𝜁̄ = 𝜁 − 𝜋a) along the left or right face, as indicated by the subscripts “l” and “r”. In accordance with Equation (12.1), the scattered field is also symmetric with respect to the disk plane; therefore, = ū (m,t) . ū (m,t) r l

(12.25)

338

FOCUSING OF MULTIPLE EDGE WAVES DIFFRACTED AT A DISK

Also, sin(𝜑∕2) 𝜕f (𝜓, 0, 𝛼) 𝜕f (𝜑, 0, 𝛼) . = = 𝜕𝜑0 𝜕𝜑0 2 cos2 (𝜑∕2)

(12.26)

Hence, (P) = 2a ū (m,t) u(m+1) s l

𝜕f (𝜑, 0, 𝛼) eikR . 𝜕𝜑0 R

(12.27)

The quantity = ū (m) + ū (m) = 2̄u(m) ū (m,t) l ll ll rl

(12.28)

relates to the mth-order edge wave on the consists of two equal terms. The term ū (m) ll left face of the disk at point 𝜁 . This wave is generated by the (m − 1)-order edge wave at the opposite point 𝜁̄ = 𝜁 − 𝜋a, which arrived there along the left side of relates to the mth-order edge wave (at the same point 𝜁 on the disk. The term ū (m) rl the left face of the disk) created by the (m − 1)th-order edge wave at the opposite point 𝜁̄ = 𝜁 − 𝜋a and arrived there along the right side of the disk. Because of the symmetry of the field, these terms are equal. and ū (m,t) are calculated using Equations (10.21) and (10.22), The quantities ū (m) ll l where one should set 𝛾0 = 𝛾01 = 𝜋∕2, R10 = 2a, and 𝜌1 = −a. These calculations result in ū (1,t) ≡ ū (1) =𝜇 l l

𝜕f (0, 𝜋∕2, 𝛼) 𝜕𝜑

ū (m,t) = 2𝜇 ū l(m−1,t) l

𝜕 2 f (0, 0, 𝛼) 𝜕𝜑 𝜕𝜑0

with 𝜇 =

ei(2ka+𝜋∕4) , √ 8ka 𝜋ka

with m = 2, 3, 4, ….

(12.29)

(12.30)

= ū (1) is used to emphasize that only one primary edge wave The denotation ū (1,t) l l exists on each side of the disk, but two edge waves of any higher order are on every side. Thus, [

]m−1 𝜕 2 f (0, 0, 𝛼) =2 𝜇 𝜕𝜑 𝜕𝜑 𝜕𝜑0 [ 2 ]m−1 𝜕f (𝜑, 0, 𝛼) eikR m m 𝜕f (0, 𝜋∕2, 𝛼) 𝜕 f (0, 0, 𝛼) (P) = a2 𝜇 u(m+1) s 𝜕𝜑 𝜕𝜑 𝜕𝜑0 𝜕𝜑0 R ū (m,t) l

m−1 m 𝜕f (0, 𝜋∕2, 𝛼)

(12.31)

(12.32)

with m = 1, 2, 3, … and 𝜕f (0, 𝜋∕2, 𝛼) 1 =√ , 𝜕𝜑 2

𝜕 2 f (0, 0, 𝛼) 1 = . 𝜕𝜑 𝜕𝜑0 4

(12.33)

MULTIPLE SOFT DIFFRACTION

339

The total focal field created by the all edge waves together equals uew s (P) = a

eikR R

{ f (𝜑, 𝜋∕2, 𝛼)

𝜕f (0, 𝜋∕2, 𝛼) 𝜕f (𝜑, 0, 𝛼) 𝜕𝜑 𝜕𝜑0 [ ]m−1 } ∞ ∑ eim(2ka+𝜋∕4) 𝜕 2 f (0, 0, 𝛼) × 2m . √ 𝜕𝜑 𝜕𝜑0 (8ka 𝜋ka)m m=1 +

(12.34)

This expression can be used to calculate the total cross-section (11.8), where utot s is the total field scattered in the forward direction 𝜑 = 3𝜋∕2. In the present case, {

𝜕f (0, 𝜋∕2, 𝛼) 𝜕f (3𝜋∕2, 0, 𝛼) ka + f (1) (3𝜋∕2, 𝜋∕2, 𝛼) + 2 𝜕𝜑 𝜕𝜑0 [ ] } ∞ m−1 ∑ eim(2ka+𝜋∕4) 𝜕 2 f (0, 0, 𝛼) × 2m (12.35) √ 𝜕𝜑 𝜕𝜑0 (8ka 𝜋ka)m m=1

u(t) s =a

eikR R

i

with f (1) (3𝜋∕2, 𝜋∕2, 𝛼) =

1 , 2

𝜕f (3𝜋∕2, 0, 𝛼) 1 =√ . 𝜕𝜑0 2

(12.36)

Substitution of Equation (12.35) into Equation (11.8) determines the total crosssection: { 𝜎s = 2𝜋a

2

∞ 2 ∑ sin[m(2ka + 𝜋∕4)] 1+ √ ka m=1 2m−1 (8ka 𝜋ka)m

} .

(12.37)

This is an incomplete asymptotic expression, which includes only the first term of the total asymptotic expansion for every edge wave. It can be verified by comparison with the exact asymptotic expression (14.54) in Bowman et al. (1987)], which contains asymptotic terms up to the order of(ka)−4 . According to Equation (12.37), { 𝜎s = 2𝜋a

2

cos(2ka − 𝜋∕4) cos(4ka) + + O[(ka)−11∕2 ] 1+ √ 4 5∕2 64𝜋(ka) 4 𝜋(ka)

} . (12.38)

All three of these terms are identical to the exact terms. Thus, a comparison of the asymptotics (12.18) and (12.37) with the known exact results proves that PTD correctly predicts the first term in the total asymptotic expansion for every multiple edge wave.

340

12.3

FOCUSING OF MULTIPLE EDGE WAVES DIFFRACTED AT A DISK

MULTIPLE DIFFRACTION OF ELECTROMAGNETIC WAVES

Here, we investigate the diffraction of a plane wave Exinc = Z0 Hyinc = eikz

(12.39)

at a perfectly conducting disk (Fig. 12.1). The basic features of this problem are essentially the same as those in the acoustical problems above. For this reason we do not repeat them here and discuss only briefly a new specific feature caused by the vector nature of electromagnetic waves. Due to this nature and to the axial symmetry of the problem, one can separate the diffracted waves (of second and higher order) ⃗ ⋅ 𝜑̂ and into two independent groups with E𝜑 - and H𝜑 -polarizations, where E𝜑 = E ⃗ H𝜑 = H ⋅ 𝜑̂ are the field components parallel to the tangent ̂t = 𝜑̂ of the edge. This separation is possible because all multiple waves hit the edge in the normal direction (𝛾0 = 𝜋∕2) and do not create cross-polarized components. Multiple diffraction of the E𝜑 -waves (H𝜑 -waves) is calculated just like the diffraction of acoustic waves at a soft (hard) disk. These observations facilitate the investigation significantly, resulting in the following approximations for the focal field on the z-axis (z ≫ kR2 , 𝜗 = 0 or 𝜗 = 𝜋). The focal fields generated by all the multiple E𝜑 - and H𝜑 -waves are equal to

Ex(e) = Z0 Hy(e) =

∞ a ikz ∑ eim(2ka+𝜋∕4) e 4m m∕2 (ka)3m∕2 z m=1 2 𝜋

Ex(h) = Z0 Hy(h) =

∞ eim(2ka−𝜋∕4) a ikz ∑ e (−1)m z 2m (𝜋ka)m∕2 m=1

(E𝜑 -waves),

(12.40)

(H𝜑 -waves).

(12.41)

The focal field includes these fields plus the contributions generated by the current ⃗j(0) (PO contribution) and the fringe current ⃗j(1) (related to the primary edge diffraction). According to Equation (1.134), the PO contribution equals

Ex(0) = Z0 Hy(0) =

ika2 eikz . 2 z

(12.42)

Equations (8.30) and (8.32), together with Equations (7.148), (7.150), and (7.151), (7.153), define the contribution of the fringe current (for the direction 𝜗 = 0) as Ex(1) = Z0 Hy(1) =

eikz a (1) [f (3𝜋∕2, 𝜋∕2, 2𝜋) + g(1) (3𝜋∕2, 𝜋∕2, 2𝜋)] . (12.43) 2 z

PROBLEMS

341

Note also that this equation represents a particular case of the more general formula (2.3.18) of Ufimtsev (2003, 2009). However, 1 f (1) (3𝜋∕2, 𝜋∕2, 2𝜋) = −g(1) (3𝜋∕2, 𝜋∕2, 2𝜋) = − , 2

(12.44)

so Ex(1) = Hy(1) = 0. Therefore, the total focal field equals Ex(t) = Z0 Hy(t) = a +

[ ∞ eim(2ka−𝜋∕4) eikz ika ∑ (−1)m + z 2 2m (𝜋ka)m∕2 m=1

] eim(2ka+𝜋∕4) . 4m m∕2 (ka)3m∕2 m=1 2 𝜋 ∞ ∑

(12.45)

Now, according to Equation (11.8), which is also valid for electromagnetic waves (with the replacement of u(t) by Ex(t) ), one obtains the total scattering cross-section: { 𝜎 = 2𝜋a

2

1+

∞ sin[m(2ka − 𝜋∕4)] 2 ∑ (−1)m ka m=1 2m (𝜋ka)m∕2

∞ 2 ∑ sin[m(2ka + 𝜋∕4)] + ka m=1 24m 𝜋 m∕2 (ka)3m∕2

} .

(12.46)

It turns out that this quantity is connected by the relation 𝜎 = 12 (𝜎h + 𝜎s )

(12.47)

with the similar quantities (12.18) and (12.37) found for acoustic waves.

PROBLEMS 12.1

Explain the physical sense, origination, and structure of Equation (12.14). Explanation This equation describes the focal field generated by the edge waves arising due to primary and multiple hard diffraction. r The first term represents the contribution of primary edge waves and is determined by Equation (11.2). r Other terms represent the contribution of multiple edge waves. They are calculated according to Equation (10.3). The focal line belongs to the diffraction cone, where the function Fh(t) is determined by the function g(𝜑, 𝜑0 , 2𝜋). In view of the axial symmetry of the problem, the integral (10.3) is calculated in closed form.

342

FOCUSING OF MULTIPLE EDGE WAVES DIFFRACTED AT A DISK

r All multiple diffracted waves arise due to the grazing diffraction of the edge waves running from the opposite edge along the disk. Because of that, the multiple diffracted waves acquire the factor 12 . They also acquire the phase shift −𝜋∕2 because the lower-order incident edge wave intersects the focal line. r In addition, the amplitude of each multiple edge wave of order m decreases √ by a factor proportional to 1∕ ka compared to the edge wave of order m − 1. This factor is caused by attenuation of the incident edge wave due to its geometrical divergence. 12.2

Explain the physical sense, origination, and structure of Equation (12.34). Explanation This equation describes the focal field generated by edge waves arising due to primary and multiple soft diffraction. r The first term represents the contribution of primary edge waves and is determined by Equation (11.2). r Other terms represent the contribution of multiple edge waves. They arise due to slope diffraction and are calculated according to Equation (10.19). The focal line belongs to the diffraction cone, where the function Fs(t) is determined by the function f (𝜑, 𝜑0 , 2𝜋). That is why all multiple edge waves are determined by the derivatives of this function. r In view of the axial symmetry of the problem, the integral (10.19) is calculated in closed form. r All multiple diffracted waves acquire the phase shift −𝜋∕2 because the incident edge wave intersects the focal line. r In addition, the amplitude of each multiple edge wave of order m decreases by a factor proportional to (ka)−3∕2 compared to an edge wave of order m − 1. This factor is caused by the slope diffraction and attenuation of the incident lower-order edge wave due to its geometrical divergence.

13 Backscattering at a Finite-Length Cylinder 13.1

ACOUSTIC WAVES

The geometry of the problem is shown in Figure 13.1. A solid circular cylinder with flat bases is illuminated by the incident plane wave uinc = u0 eik(y sin 𝛾+z cos 𝛾)

with

0 ≤ 𝛾 ≤ 𝜋∕2.

(13.1)

The total length of the cylinder and its diameter are denoted as L = 2l and d = 2a, respectively. The scattered field is evaluated for the backscattering direction 𝜗 = 𝜋 − 𝛾, 𝜑 = 3𝜋∕2 in the far zone (R ≫ ka2 , R ≫ kl2 ). 13.1.1

PO Approximation

According to Equation (1.37), the PO far fields backscattered by acoustically hard and soft objects differ from each other only in sign. Hence, it is sufficient to exhibit the PO calculations only for the case of scattering at a hard cylinder. First we calculate the far field scattered by the left base/disk of the hard cylinder. In this case, use of Equation (1.37) leads to the expression = u0 u(0)disk h

a

eikR ik r′ dr′ cos 𝜗 ei2kl cos 𝜗 ∫0 2𝜋 R ∫0

2𝜋

ei2kr

′ sin 𝜗 sin 𝜓

d𝜓,

(13.2)

Fundamentals of the Physical Theory of Diffraction, Second Edition. Pyotr Ya. Ufimtsev. © 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

343

344

BACKSCATTERING AT A FINITE-LENGTH CYLINDER

y 1

4

γ

z

2

3

Figure 13.1 Cross-section of the cylinder by the y0z-plane. The dots at 1, 2, and 3 are the stationary-phase points visible in the region 𝜋∕2 < 𝜗 < 𝜋, 𝜑 = 3𝜋∕2.

where 𝜗 = 𝜋 − 𝛾, r′ = tion (6.55), we have

√ x′2 + y′2 , x′ = r′ cos 𝜓, and y′ = r′ sin 𝜓. In view of Equa-

= u0 u(0)disk h

eikR ia cos 𝜗 J1 (2ka sin 𝜗)ei2kl cos 𝜗 . 2 sin 𝜗 R

(13.3)

The application of Equation (1.37) to the field scattered by the cylindrical part of the object leads to the integral expression (0)cyl

uh

l

= u0

′ eikR ika e−i2kz cos 𝜗 dz′ sin 𝜗 ∫ ∫𝜋 2𝜋 R −l

2𝜋

ei2ka sin 𝜗 sin 𝜓 sin 𝜓 d𝜓.

(13.4)

Here the integration is performed over the illuminated part of the scattering surface where 𝜋 ≤ 𝜓 ≤ 2𝜋 under the condition 𝜋∕2 + 0 ≤ 𝜗 ≤ 𝜋 − 0. In the limiting case when 𝜗 = 𝜋, the integration encompasses the entire cylindrical surface (0 ≤ 𝜓 ≤ 2𝜋) and results in a zero scattered field. However, Equation (13.4) is also valid in this case, as it equals zero due to the factor sin 𝜗. Thus, (0)cyl

uh

=0

if

𝜗 = 𝜋.

(13.5)

The integral in Equation (13.4) over the variable z′ is calculated in closed form. The integral over the variable 𝜓 is calculated (under the condition 2ka sin 𝜗 ≫ 1) by the stationary-phase technique (Copson, 1965; Murray, 1984). The details of this technique were considered in Sections 6.1.2 and 8.1. The stationary point 𝜓st = 3𝜋∕2 is found from the equation d 2ka sin 𝜗 sin 𝜓 = 2ka sin 𝜗 cos 𝜓 = 0. d𝜓

(13.6)

ACOUSTIC WAVES

345

The asymptotic expression for the integral is given by 2𝜋

∫𝜋

ei2ka sin 𝜗 sin 𝜓 sin 𝜓 d𝜓 ∼ −

√

𝜋 e−i2ka sin 𝜗+i𝜋∕4 . ka sin 𝜗 (0)cyl

Therefore, under the condition 2ka sin 𝜗 ≫ 1, the scattered field uh asymptotically as (0)cyl

uh

∼ −u0

(13.7)

is determined

e−i2ka sin 𝜗+i𝜋∕4 eikR ia sin 𝜗 sin(2kl cos 𝜗) √ . 2 cos 𝜗 𝜋ka sin 𝜗 R

(13.8)

Having Equations (13.5) and (13.8), one can construct an approximation valid in the entire region 𝜋∕2 ≤ 𝜗 ≤ 𝜋. This can be done in a manner similar to that demonstrated in Section 6.1.4. We use asymptotic expressions for Bessel functions with large arguments (x ≫ 1) and observe that e−i2x+i𝜋∕4 ≈ J0 (2x) − iJ 1 (2x) √ 𝜋x

(13.9)

e−i2x+i𝜋∕4 1 ≈ [J1 (2x) − iJ 2 (2x)], √ i 𝜋x e−i2x+i𝜋∕4 ≈ e−in𝜋∕2 [Jn (2x) − iJ n+1 (2x)], √ 𝜋x

(13.10)

n = 0, 1, 2, 3, …

(13.11)

Each of these combinations can be used to construct the approximation for the field in the region 𝜋∕2 ≤ 𝜗 ≤ 𝜋. We apply and analyze the simplest ones, (13.9) and (13.10). With these approximations the total PO field can be represented in the two forms { ia eikR cos 𝜗 J (2ka sin 𝜗)ei2kl cos 𝜗 2 R sin 𝜗 1 } sin 𝜗 − sin(2kl cos 𝜗)[J0 (2ka sin 𝜗) − iJ 1 (2ka sin 𝜗)] , cos 𝜗

= u0 u(0) h,1

{ ia eikR cos 𝜗 J (2ka sin 𝜗)ei2kl cos 𝜗 2 R sin 𝜗 1 } sin 𝜗 +i sin(2kl cos 𝜗)[J1 (2ka sin 𝜗) − iJ 2 (2ka sin 𝜗)] . cos 𝜗

(13.12)

u(0) = u0 h,2

(0)cyl

(13.13)

We can see that the second terms here, related to the field uh , indeed equal zero for the direction 𝜗 = 𝜋 and, therefore, agree with Equation (13.5).

346

BACKSCATTERING AT A FINITE-LENGTH CYLINDER

Figure 13.2 Backscattering at a finite cylinder according to the physical optics approximations PO-1 [Equation (13.12)] and PO-2 [Equation (13.13)].

The scattering cross-section 𝜎 is defined by Equation (1.26). We have calculated the normalized scattering cross-section 𝜎norm =

𝜎 𝜎d

(13.14)

where the quantity 𝜎d = 𝜋a2 (ka)2

(13.15)

is the PO scattering cross-section of the disk under normal incidence (𝜗 = 𝜋). The results are shown in Figures 13.2 and 13.3. The curves PO-1 and PO-2 relate to Equations (13.12) and (13.13), respectively. The small discrepancy between the two curves in Figure 13.2 is caused by the different higher-order terms in the asymptotic expressions (13.9) and (13.10). It takes place when the cylinder diameter is not sufficiently large and equals only one wavelength (d = 2a = 𝜆) when the argument of the Bessel functions does not exceed 2𝜋. The discrepancy between the approximations PO-1 and PO-2 becomes practically negligible in the case when d = 3𝜆, as is shown clearly in Figure 13.3. Because of that, in the following calculations we use the simplest approximation, (13.12). There

ACOUSTIC WAVES

347

Figure 13.3 Backscattering at a finite cylinder according to the physical optics approximations PO-1 [Equation (13.12)] and PO-2 [Equation (13.13)].

is another reason in favor of using Equation (13.12). It matches an analogous type of approximations for the fields (13.22) and (13.23) generated by the nonuniform/fringe source j(1) . This field is investigated in the following section.

13.1.2

Backscattering Produced by the Nonuniform Component j(1)

There are three types of nonuniform components j(1) of the scattering sources on a finite cylinder. The edge/fringe component j(1) concentrates near the edges. The comfr associated with creeping waves concentrates near the shadow boundary ponent j(1) cr on the cylindrical part of the object and attenuates away from this boundary expois caused by the transverse diffusion of the wave nentially. The third component j(1) dif field between the adjacent rays reflected from the cylindrical surface. This component exists on the illuminated part of the cylindrical surface away from the shadow j(1) dif boundary. Among these components of j(1) , a main contribution to the backscattering is provided by the fringe component j(1) , and this contribution is investigated here. fr is comparable with that produced by j(1) , Notice also that the field generated by j(1) dif fr but this situation occurs only in the direction of the specular rays reflected from the cylindrical surface. This topic is considered in Chapter 14.

348

BACKSCATTERING AT A FINITE-LENGTH CYLINDER

First we analyze the far field (R ≫ ka2 , √ R ≫ kl2 ) generated by the fringe component located near the left edge (z = −l, x2 + y2 = a, x = a cos 𝜓, y = a sin 𝜓). According to Equations (7.89) and (7.90), it is determined as = u0 u(1)left s,h

a i2kl cos 𝜗 eikR e 2𝜋 R ∫0

2𝜋

(1) Fs,h [𝜃(𝜓), 𝜙(𝜓)]ei2ka sin 𝜗 sin 𝜓 d𝜓. (13.16)

Here we denote the local coordinates of EEWs at the edge points as 𝜃 and 𝜙 instead of 𝜗 and 𝜑, according to the rule formulated in the boxes in Sections 7.2 and 7.8. However, we admit to deviation from this rule by choosing the clockwise arc-length coordinate as 𝜁 = a𝜓 instead of the counterclockwise coordinate 𝜁 ′ = a𝜓 ′ = a(𝜋 − 𝜓). This deviation in the equation, 2𝜋

∫0

(1) Fs,h (𝜓)ei2ka sin 𝜗 sin 𝜓 d𝜁 =

𝜋

∫−𝜋

(1) Fs,h (𝜓 ′ )ei2ka sin 𝜗 sin 𝜓 d𝜁 ′ , ′

(1) (1) (1) is admissible since functions Fs,h are symmetric: Fs,h (𝜋 − 𝜓) = Fs,h (𝜓). (1) In the direction 𝜗 = 𝜋, the functions Fs,h transform into the functions f (1) and g(1) as shown in Equation (7.115). Recall that these functions are defined in Section 4.1. Hence, for this direction, } [ (1) ] u(1)left f eikR s (𝜗 = 𝜋). (13.17) = u0 a (1) e−i2kl (1)left g R uh

For other directions 𝜗, which satisfy the condition 2ka sin 𝜗 ≫ 1, the integral in Equation (13.16) is evaluated asymptotically using the stationary-phase technique. There are two stationary points (𝜓st,1 = 𝜋∕2 and 𝜓st,2 = 3𝜋∕2) in the integrand (1) of Equation (13.16). At these points, the functions Fs,h also transform into the (1) (1) functions f and g . The resulting asymptotic approximations for the field (13.16) are given as eikR a 1 = u0 ei2kl cos 𝜗 u(1)left √ s 2 R 𝜋ka sin 𝜗 [ (1) ] × f (1)ei2ka sin 𝜗−i𝜋∕4 + f (1) (2)e−i2ka sin 𝜗+i𝜋∕4 ,

(13.18)

eikR a 1 u(1)left = u0 ei2kl cos 𝜗 √ h 2 R 𝜋ka sin 𝜗 [ (1) ] i2ka sin 𝜗−i𝜋∕4 × g (1)e + g(1) (2)e−i2ka sin 𝜗+i𝜋∕4 .

(13.19)

Here the functions f (1) (1) and g(1) (1) relate to stationary point 1 (𝜓st,1 = 𝜋∕2), and the functions f (1) (2) and g(1) (2) relate to stationary point 2 (𝜓st,2 = 3𝜋∕2). These points are shown in Figure 13.1.

ACOUSTIC WAVES

349

To construct field approximations in the entire region 𝜋∕2 ≤ 𝜗 ≤ 𝜋, we employ an idea suggested in Sections 6.1.4 and 13.1.1. In other words, we substitute the asymptotic approximations ei2ka sin 𝜗−i𝜋∕4 ≈ J0 (2ka sin 𝜗) + iJ 1 (2ka sin 𝜗), √ 𝜋ka sin 𝜗

(13.20)

e−i2ka sin 𝜗+i𝜋∕4 ≈ J0 (2ka sin 𝜗) − iJ 1 (2ka sin 𝜗) √ 𝜋ka sin 𝜗

(13.21)

into Equations (13.18 and (13.19)) and extend the field expressions obtained to the entire region 𝜋∕2 ≤ 𝜗 ≤ 𝜋, 𝜑 = 𝜓 = 3𝜋∕2. It turns out that the resulting expressions, eikR a = u0 ei2kl cos 𝜗 u(1)left s 2 R { (1) × f (1)[J0 (2ka sin 𝜗) + iJ1 (2ka sin 𝜗)] } + f (1) (2)[J0 (2ka sin 𝜗) − iJ1 (2ka sin 𝜗)] ,

(13.22)

a i2kl cos 𝜗 eikR e = u u(1)left 0 h 2 R { (1) × g (1)[J0 (2ka sin 𝜗) + iJ 1 (2ka sin 𝜗)] } + g(1) (2)[J0 (2ka sin 𝜗) − iJ 1 (2ka sin 𝜗)]

(13.23)

transform into Equation (13.17) exactly when 𝜗 → 𝜋. Therefore, these expressions can be considered as appropriate approximations to the scattered field in all directions 𝜋∕2 ≤ 𝜗 ≤ 𝜋, 𝜑 = 𝜓 = 3𝜋∕2. √ The contribution of the right edge (z = +l, x2 + y2 = a) to the field in the region 𝜋∕2 < 𝜗 < 𝜋 is described by the expression (1)right

us,h

= u0

a −i2kl cos 𝜗 eikR e 2𝜋 R ∫𝜋

2𝜋

(1) Fs,h (𝜓)ei2ka sin 𝜗 sin 𝜓 d𝜓,

(13.24)

analogous to Equation (13.16). Its asymptotic approximation found using the stationary-phase technique is determined as (1)right

us (1)right uh

}

[ ] a f (1) (3) e−i2ka sin 𝜗+i𝜋∕4 −i2kl cos 𝜗 eikR e = u0 , √ 2 g(1) (3) R 𝜋ka sin 𝜗

(13.25)

where 2ka sin 𝜗 ≫ 1 and the functions f (1) (3) and g(1) (3) relate to stationary point 3 (𝜓st,3 = 3𝜋∕2) shown in Figure 13.1. With the application of Equation (13.21),

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BACKSCATTERING AT A FINITE-LENGTH CYLINDER

this expression is extended to all directions 𝜋∕2 < 𝜗 < 𝜋 and provides the following approximations to the field (13.24): (1)right

us

(1)right

uh

eikR a = u0 f (1) (3)[J0 (2ka sin 𝜗) − iJ 1 (2ka sin 𝜗)]e−i2kl cos 𝜗 , 2 R

(13.26)

eikR a = u0 g(1) (3)[J0 (2ka sin 𝜗) − iJ 1 (2ka sin 𝜗)]e−i2kl cos 𝜗 . 2 R

(13.27)

Thus, in the first approximation, the total field produced by the component j(1) fr equals a eikR { (1) f (1)[J0 (2ka sin 𝜗) + iJ 1 (2ka sin 𝜗)]ei2kl cos 𝜗 2 R [ ]} + f (1) (2)ei2kl cos 𝜗 + f (1) (3)e−i2kl cos 𝜗 ][J0 (2ka sin 𝜗) − iJ 1 (2ka sin 𝜗) ,

u(1) s = u0

u(1) h

(13.28) a eikR { (1) g (1)[J0 (2ka sin 𝜗) + iJ 1 (2ka sin 𝜗)]ei2kl cos 𝜗 = u0 2 R [ ]} + g(1) (2)ei2kl cos 𝜗 + g(1) (3)e−i2kl cos 𝜗 ][J0 (2ka sin 𝜗) − iJ 1 (2ka sin 𝜗) . (13.29)

The functions f (1) and g(1) are determined according to Section 4.1 as f (1) (1) =

sin(𝜋∕n) n

(

1 1 − cos(𝜋∕n) − 1 cos(𝜋∕n) − cos[(𝜋 − 2𝜗)∕n]

) +

1 cos 𝜗 , 2 sin 𝜗 (13.30)

sin(𝜋∕n) g (1) = n

(

(1)

1 1 + cos(𝜋∕n) − 1 cos(𝜋∕n) − cos[(𝜋 − 2𝜗)∕n]

) −

1 cos 𝜗 , 2 sin 𝜗 (13.31)

f

(1)

sin(𝜋∕n] (2) = n

(

1 1 − cos(𝜋∕n) − 1 cos(𝜋∕n) − cos(2𝜗∕n)

) −

1 cos 𝜗 1 sin 𝜗 − , 2 sin 𝜗 2 cos 𝜗 (13.32)

g(1) (2) =

sin(𝜋∕n) n

(

1 1 + cos(𝜋∕n) − 1 cos(𝜋∕n) − cos(2𝜗∕n)

) +

1 cos 𝜗 1 sin 𝜗 + , 2 sin 𝜗 2 cos 𝜗 (13.33)

351

ACOUSTIC WAVES

f (1) (3) =

g(1) (3) =

sin(𝜋∕n) n sin(𝜋∕n) n

(

(

1 1 − cos(𝜋∕n) − 1 cos(𝜋∕n) − cos[(𝜋 + 2𝜗)∕n]

1 1 + cos(𝜋∕n) − 1 cos(𝜋∕n) − cos[(𝜋 + 2𝜗)∕n]

) +

1 sin 𝜗 , 2 cos 𝜗 (13.34)

) −

1 sin 𝜗 2 cos 𝜗 (13.35)

with n = 32 . Although certain terms in these functions are singular in the directions 𝜗 = 𝜋∕2 and 𝜗 = 𝜋, these singularities always cancel each other, and the functions f (1) and g(1) remain finite. In the direction 𝜗 = 𝜋∕2 they have the following values: f (1) (1) = 0, g(1) (1) =

f (1) (2) = f (1) (3) =

(2∕n) sin(𝜋∕n) , cos(𝜋∕n) − 1

(1∕n) sin(𝜋∕n) 1 𝜋 + cot , cos(𝜋∕n) − 1 2n n

g(1) (2) = g(1) (3) =

(13.36)

(1∕n) sin(𝜋∕n) 1 𝜋 − cot , cos(𝜋∕n) − 1 2n n (13.37)

and in the direction 𝜗 = 𝜋 they are determined as f (1) (1) = f (1) (2) =

(1∕n) sin(𝜋∕n) 1 𝜋 + cot , cos(𝜋∕n) − 1 2n n

f (1) (3) = 0,

g(1) (1) = g(1) (2) =

(1∕n) sin(𝜋∕n) 1 𝜋 − cot , cos(𝜋∕n) − 1 2n n

g(1) (3) =

(13.38)

(2∕n) sin(𝜋∕n) . cos(𝜋∕n) − 1 (13.39)

We also obtain expressions for the functions f (1) (4) and g(1) (4) related to stationary point 4 (Fig. 13.1), which becomes visible in the directions 𝜗 = 𝜋∕2 and 𝜗 = 𝜋. For both directions, the functions f (1) (4) and g(1) (4) have the same values, f (1) (4) = 0 and

g(1) (4) =

(2∕n) sin(𝜋∕n) . cos(𝜋∕n) − 1

(13.40)

As the contribution from point 4 equals zero for the soft cylinder, the approximation (13.28) can be used in the entire region 𝜋∕2 ≤ 𝜗 ≤ 𝜋. For expression (13.29) for the hard cylinder, it is valid (strictly speaking) for the directions 𝜋∕2 + 0 ≤ 𝜗 ≤ 𝜋 − 0 when point 4 is invisible. However, the contribution of point 4 for the hard cylinder is ka (or kl) times less than the PO field in the direction 𝜗 = 𝜋∕2 (or 𝜗 = 𝜋), and it can be neglected for the case of large cylinders. on the backscattering is illustrated graphA quantitative influence of the field u(1) s,h ically in the next section.

352

13.1.3

BACKSCATTERING AT A FINITE-LENGTH CYLINDER

Total Backscattered Field

The total field is the sum u(t) = u(0) + u(1) , s,h s,h s,h

(13.41)

(0) (0) (1) where u(0) s = −uh and the terms uh and us,h are determined by Equations (13.12), (13.28), and (13.29). Utilizing these approximations, we have calculated the normalized scattering cross-section (13.14) and demonstrated the individual contribution is produced by the uniof each term in Equation (13.41). Recall that the field u(0) s,h and represents the PO approximation. The field u(1) is form scattering source j(0) s,h s,h (1) produced by the nonuniform source js,h concentrated near the edges and is denoted below as the fringe component of the backscattering. The numerical results are presented in Figures 13.4 to 13.7 for two sets of geometrical parameters of the cylinder: (a) d = 2a = 𝜆, L = 2l = 3𝜆; (b) d = 3𝜆, L = 9𝜆. Here d is the diameter and L is the length of the cylinder. An interesting observation follows from Figures 13.4 to 13.7. Most of the maximums in the soft fringe field are located in the vicinity of the angular positions of the minimums of the PO field. The opposite situation is observed for the hard fringe field; its maximums are positioned near the maximums of the PO field. This observation

Figure 13.4 Backscattering at a soft cylinder. According to Equation (13.46), the PO curve here also displays the backscattering of electromagnetic waves (with Ex -polarization) from a perfectly conducting cylinder.

ACOUSTIC WAVES

353

Figure 13.5 Backscattering at a hard cylinder. According to Equation (13.73), the PO curve here also displays the backscattering of electromagnetic waves (with Hx -polarization) from a perfectly conducting cylinder.

Figure 13.6 Backscattering at a soft cylinder. According to Equation (13.46), the PO curve here also displays the backscattering of electromagnetic waves (with Ex -polarization) from a perfectly conducting cylinder.

354

BACKSCATTERING AT A FINITE-LENGTH CYLINDER

Figure 13.7 Backscattering at a hard cylinder. According to Equation (13.73), the PO curve here also displays the backscattering of electromagnetic waves (with Hx -polarization) from a perfectly conducting cylinder.

explains why the minimums of the field scattered by soft cylinders are not as deep as in the case of hard cylinders.

13.2

ELECTROMAGNETIC WAVES

The original PTD of electromagnetic waves scattered from a finite perfectly conducting cylinder was described by Ufimtsev (1958a, 1962b). Below we present in brief a revised version based on the concept of EEWs. 13.2.1

E-polarization

The incident wave is defined as Exinc = E0x eik(z cos 𝛾+y sin 𝛾)

Eyinc = Ezinc = Hxinc = 0.

(13.42)

The uniform component (1.113) of the induced surface current is determined by j(0)disk = 2Y0 E0x cos 𝛾e−ikl cos 𝛾 eik𝜌 sin 𝜗 sin 𝜓 , x j(0)disk = j(0)disk =0 y z

(13.43)

ELECTROMAGNETIC WAVES

355

on the left base of the cylinder (Fig. 13.1), and by (0)cyl

= −2Y0 E0x sin 𝛾 sin 𝜓eik(z cos 𝛾+a sin 𝛾 sin 𝜓) ,

(0)cyl

= 2Y0 E0x sin 𝛾 cos 𝜓eik(z cos 𝛾+a sin 𝛾 sin 𝜓)

(0)cyl

= 2Y0 E0x cos 𝛾 cos 𝜓eik(z cos 𝛾+a sin 𝛾 sin 𝜓)

jx jy and

jz

(13.44)

on the cylindrical part of the surface (−l ≤ z ≤ l, 𝜋 ≤ 𝜓 ≤ 2𝜋). Here Y0 = 1∕Z0 is the admittance of free space (vacuum). The field Ex(0) generated by the current ⃗j(0) is found with the help of Equations ⃗] = , because ⃗jm = −[̂n × E (1.108) and (1.109), where one should drop the terms Am 𝜑,𝜗 0 due to the boundary condition on a perfectly conducting surface. The nonuniform (fringe) currents ⃗j(1) concentrate near the left (z = −l) and right (z = l) edges. The field Ex(1) radiated by these currents is calculated in accordance with the theory developed in Section 7.8. The total scattered field is the sum (0)cyl

Ex = Ex(0)disk + Ex

(1)right

+ Ex(1)left + Ex

.

(13.45)

One can show that , Ex(0)disk = u(0)disk s (0)cyl

(0)cyl

Ex

(0)cyl

= us

.

(13.46)

The quantities u(0)disk and us are defined in Section 13.1.1, where one should set s u0 = E0x .Therefore, the PO curves in Figures 13.4 and 13.6 for the backscattering of acoustic waves from a soft cylinder also display the backscattering of electromagnetic waves with Ex -polarization from a perfectly conducting cylinder. (1)right are calculated by the integration of EEWs given by The fields Ex(1)left and Ex Equations (7.136) and (7.137), where R, 𝜗, 𝜑 are local coordinates with the origin at edge point 𝜁 . Now we re-denote them as 𝜌, 𝜃, 𝜙 and introduce the local cylindrical coordinates rloc , 𝜙, zloc (̂zloc ≡ ̂t). The symbols R, 𝜗, 𝜑 will be used for the basic system of coordinates shown in Figure 13.1. The scattered field is calculated at the observation point P(0, y, z), where y = −R sin 𝜗, z = R cos 𝜗 and 𝜗 = 𝜋 − 𝛾, 𝜑 = 3𝜋∕2. The integration point on the circular edge is determined by the coordinates x = a cos 𝜓, y = a sin 𝜓, and 𝜁 = a𝜓. We emphasize that the local coordinates are introduced according to the rule formulated in the boxes in Sections 7.2 and 7.8. ̂ the angle 𝜙 Their unit vectors are connected by the vector product ẑ loc = r̂loc × 𝜙, is measured from the illuminated face of the edge, and the unit vector 𝜃̂ relates to the spherical coordinate 𝜃 measured from the direction ẑ loc ≡ ̂t. The necessary preliminary work is to define the local coordinates in terms of the basic coordinates R, 𝜗, 𝜓.

356

BACKSCATTERING AT A FINITE-LENGTH CYLINDER

First, note that we can use the approximations 𝜌left = R + a sin 𝜗 sin 𝜓 + l cos 𝜗, 𝜌right = R + a sin 𝜗 sin 𝜓 − l cos 𝜗

(13.47)

for the distance 𝜌 between the integration and observation points, assuming that the latter are in the meridian plane 𝜑 = 3𝜋∕2 in the far zone (R ≫ ka2 , R ≫ kl2 ). In this approximation, the unit vector directed from the integration point to the observation point is determined as 𝜌̂left = 𝜌̂right = R̂ = −̂y sin 𝜗 + ẑ cos 𝜗.

(13.48)

Then we introduce the unit vectors 𝜃̂ = x̂ 𝜃x + ŷ 𝜃y + ẑ 𝜃z ,

𝜙̂ = x̂ 𝜙x + ŷ 𝜙y + ẑ 𝜙z ,

(13.49)

and find their components from the equations R̂ ⋅ 𝜃̂ = 0,

√ ̂𝜃 ⋅ [R̂ × ̂t] = 0, ̂t ⋅ 𝜃̂ = − 1 − sin2 𝜗 cos2 𝜓,

̂ 𝜙̂ = R̂ × 𝜃, (13.50)

where ̂t = x̂ sin 𝜓 − ŷ cos 𝜓 is the tangent to the edge (Fig. 13.8). According to these equations, sin 𝜓 , 𝜃x = − √ 2 2 1 − sin 𝜗 cos 𝜓

cos 𝜓 cos2 𝜗 𝜃y = √ , 2 2 1 − sin 𝜗 cos 𝜓

cos 𝜓 sin 𝜗 cos 𝜗 , 𝜃z = √ 2 2 1 − sin 𝜗 cos 𝜓 cos 𝜓 cos 𝜗 , 𝜙x = − √ 2 2 1 − sin 𝜗 cos 𝜓

(13.51)

sin 𝜓 cos 𝜗 𝜙y = − √ , 2 2 1 − sin 𝜗 cos 𝜓

sin 𝜓 sin 𝜗 𝜙z = − √ . 1 − sin2 𝜗 cos2 𝜓

(13.52)

These equations for the unit vectors 𝜃̂ and 𝜙̂ are valid for the left and right edges. The angle 𝜃 is defined by the equation R̂ ⋅ ̂t = cos 𝜃 = sin 𝜗 cos 𝜓.

(13.53)

357

ELECTROMAGNETIC WAVES

To define the angles 𝜙 and 𝜙0 , one should note that they are measured from the illuminated face of the edge in the plane perpendicular to the tangent ̂t to the edge. ̂ = −k̂ i = −̂y sin 𝛾 − ẑ cos 𝛾 on By projecting the vectors R̂ = −̂y sin 𝜗 + ẑ cos 𝜗 and Q this plane, we obtain sin 𝜗 sin 𝜓 cos 𝜙l = √ , 1 − sin2 𝜗 cos2 𝜓

cos 𝜗 , sin 𝜙l = − √ 1 − sin2 𝜗 cos2 𝜓 cos 𝛾 , sin 𝜙l0 = √ 2 2 1 − sin 𝛾 cos 𝜓

sin 𝛾 sin 𝜓 cos 𝜙l0 = √ 1 − sin2 𝛾 cos2 𝜓

(13.54)

(13.55)

for the left edge (z = −l), and sin 𝜗 sin 𝜓 sin 𝜙r = − √ , 2 2 1 − sin 𝜗 cos 𝜓

cos 𝜗 cos 𝜙r = − √ , 2 2 1 − sin 𝜗 cos 𝜓

(13.56)

sin 𝛾 sin 𝜓 , sin 𝜙r0 = − √ 2 2 1 − sin 𝛾 cos 𝜓

cos 𝛾 cos 𝜙r0 = √ 1 − sin2 𝛾 cos2 𝜓

(13.57)

for the right edge (z = l). Now, according to Equations (7.136) and (7.137), we obtain Ex(1)left = E0x

a eikR i2kl cos 𝜗 e ∫0 2𝜋 R

2𝜋

{

sin 𝜓F𝜃(1) (𝜓, 𝜃, 𝜙l ) ⋅ 𝜃x + cos 𝜗 cos 𝜓

]} [ (1) l l ei2ka sin 𝜗 sin 𝜓 d𝜓, (𝜓, 𝜃, 𝜙 ) ⋅ 𝜃 + G (𝜓, 𝜃, 𝜙 ) ⋅ 𝜙 × G(1) x x 𝜃 𝜙 (13.58) (1)right

Ex

= E0x

a eikR −i2kl cos 𝜗 e ∫𝜋 2𝜋 R

2𝜋

{

sin 𝜓F𝜃(1) (𝜓, 𝜃, 𝜙r ) ⋅ 𝜃x + cos 𝜗 cos 𝜓

]} [ (1) r r ei2ka sin 𝜗 sin 𝜓 d𝜓. (𝜓, 𝜃, 𝜙 ) ⋅ 𝜃 + G (𝜓, 𝜃, 𝜙 ) ⋅ 𝜙 × G(1) x x 𝜃 𝜙 (13.59) Note that the details of the derivation of Equation (13.58) are presented in Problem 13.1. Equations (13.58) and (13.59) need additional comments. They were derived assuming that the arc coordinate equals 𝜁 = a𝜓, with the angle 𝜓 shown in Figure 13.8. However, according to the rules for local coordinates, which are formulated in the boxes at the beginning of Sections 7.2 and 7.3, the local coordinate 𝜁 should have been defined as 𝜁 ′ = a𝜓 ′ (with 𝜓 ′ = 𝜋 − 𝜓), to be consistent with the direction of the tangent ̂t = d⃗r∕d𝜁 ′ (Fig. 13.8). Nevertheless, it turns out that the equations

358

BACKSCATTERING AT A FINITE-LENGTH CYLINDER

y

ψ

ψ′

t x

Figure 13.8 Angles 𝜓 and 𝜓 ′ and tangent ̂t.

above are totally equivalent to those derived with 𝜁 ′ = a𝜓 ′ . Indeed, one can prove the following relationships: F𝜃(1) (𝜋 − 𝜓) = F𝜃(1) (𝜓), G(1) (𝜋 − 𝜓) = G(1) (𝜓), G(1) (𝜋 − 𝜓) = −G(1) (𝜓). 𝜙 𝜙 𝜃 𝜃 (13.60) Together with Equations (13.51) and (13.52), they show that the integrands u(𝜓) in Equations (13.58) and (13.59) are symmetric; that is, u(𝜋 − 𝜓) = u(𝜓). Now represent Equation (13.58) in the form Ex(1)left = E0x

𝜋

a eikR i2kl cos 𝜗 e u(𝜓) d𝜓. ∫−𝜋 2𝜋 R

Changing the integration variable by 𝜓 = 𝜋 − 𝜓 ′ , we obtain 𝜋

∫−𝜋

u(𝜋 − 𝜓) d𝜓 = −

0

∫2𝜋

2𝜋

u(𝜓 ′ ) d𝜓 ′ =

∫0

u(𝜓 ′ ) d𝜓 ′ .

A similar procedure is valid for the field (13.59). Hence, due to symmetry of the integrands, Equations (13.58) and (13.59) are invariant with respect to the choice of the arccoordinate 𝜁 ; it can be 𝜁 = a𝜓 or 𝜁 ′ = a𝜓 ′ . We prefer the definition 𝜁 = a𝜓 associated with rectangular coordinates x,y,z arranged according to the right-hand rule, where x = a cos 𝜓 and y = a sin 𝜓. Notice that these comments are also valid for Equations (13.74) and (13.75) and in the bistatic problem for Equations (14.68), (14.69) and (14.79), (14.80). Another important consequence of Equation (13.60) consists of the absence of cross-polarized components in the scattered field: (1)left = Hx(1)left = 0, Ey,z

(1)right

Ey,z

(1)right

= Hx

= 0.

(13.61)

From a physical point of view, this result is a consequence of the symmetry of the scattering problem with respect to the plane x = 0.

ELECTROMAGNETIC WAVES

359

In the direction 𝜗 = 𝜋 − 0, which is the focal line for EEWs, Equations (13.58) and (13.59) predict the field [ ) )] ikR ( ( e 𝜋 𝜋 3𝜋 a (1) 𝜋 𝜋 3𝜋 e−i2kl , f − g(1) , , , , 2 2 2 2 2 2 2 R [ ) )] ikR ( ( e 3𝜋 3𝜋 a ei2kl . − g(1) 0, 0, = E0x f (1) 0, 0, 4 2 2 R

Ex(1)left = E0x

(13.62)

(1)right

(13.63)

Ex

According to the definitions of functions f (1) and g(1) , f (1)

) ) ( 1 𝜋 𝜋 3𝜋 𝜋 𝜋 3𝜋 𝜋 − g(1) = cot , , , , , 2 2 2 2 2 2 n n ( ( ) ) 3𝜋 3𝜋 2 sin(𝜋∕n) f (1) 0, 0, − g(1) 0, 0, =− , 2 2 n cos(𝜋∕n) − 1 (

(13.64)

and therefore a 𝜋 eikR −i2kl e , cot 2n n R a sin(𝜋∕n) eikR i2kl e . = −E0x 2n cos(𝜋∕n) − 1 R

Ex(1)left = E0x

(13.65)

(1)right

(13.66)

Ex

It is assumed in Equation (13.66) that for 𝜗 = 𝜋 − 0, only the lower part of the right edge (𝜋 + 𝛿 < 𝜓 < 2𝜋 − 𝛿 with 𝛿 → 0) is illuminated. For 𝜗 = 𝜋 the entire right edge becomes illuminated and its field is exactly doubled. The equations above allow complete calculation of the total scattered field (13.45). They involve elementary functions, Bessel functions, and the one-dimensional integrals (13.58) and (13.59), which can be calculated numerically. However, one can avoid this direct integration by introducing approximations similar to (13.22) and (13.23). The idea of these approximations is as follows. Away from the focal line (ka sin 𝜗 ≫ 1), the asymptotic evaluation of the integrals (13.58) and (13.59) leads to the following ray asymptotics: [ (1) a 1 f (1)ei2ka sin 𝜗−i𝜋∕4 Ex(1)left = E0x √ 2 𝜋ka sin 𝜗 + f (1) (2)e−i2ka sin 𝜃+i𝜋∕4

(1)right

Ex

] eikR i2kl cos 𝜗 e , R

e−i2ka sin 𝜗+i𝜋∕4 eikR −i2kl cos 𝜗 a e = E0x f (1) (3) √ . 2 𝜋ka sin 𝜗 R

(13.67)

(13.68)

360

BACKSCATTERING AT A FINITE-LENGTH CYLINDER

These are the electromagnetic versions of the acoustic asymptotics (13.18) and (13.25). They reveal again the equivalence relationships existing between the acoustic and electromagnetic diffracted rays, Ex(1) = u(1) s

if

u0 = E0x .

(13.69)

As shown above, the focal asymptotics (13.17) and (13.62) for acoustic and electromagnetic waves are different. However, they are small quantities of the order (0)disk scattered from the left base (disk). (ka)−1 compared to the basic component E0x Therefore, the approximation (13.28) derived for acoustic waves can also be used for the calculation of electromagnetic waves but with the relative error on the order of (ka)−1 in the vicinity of the focal line 𝜗 = 𝜋. In this case, it is just sufficient to replace (1) the quantity u0 in Equation (13.28) by E0x and u(1) s by Ex . 13.2.2

H-polarization

The incident wave Hxinc = H0x eik(z cos 𝛾+y sin 𝛾) ,

inc Hy,z = Exinc = 0

(13.70)

generates the uniform currents j(0)disk = −2H0x e−ikl cos 𝛾 eik𝜌 sin 𝛾 sin 𝜓 , y

j(0)disk = j(0)disk =0 x z

(13.71)

on the left base of the cylinder (Fig. 13.1), and (0)cyl

jz

= −2H0x sin 𝜓eik(z cos 𝛾+a sin 𝛾 sin 𝜓) , jx

(0)cyl

(0)cyl

= jy

=0

(13.72)

on its cylindrical part (−l ≤ z ≤ l, 𝜋 ≤ 𝜓 ≤ 2𝜋). One can show that these currents radiate the field Hx(0) = u(0) h

(13.73)

under the condition u0 = H0x , where the function u(0) is determined in Section 13.1.1 h and represents the acoustic field scattered at a rigid cylinder. Therefore, the PO curves in Figure 13.5 and 13.7 for the backscattering of acoustic waves from a rigid cylinder also display the backscattering of electromagnetic waves with Hx -polarization from a perfectly conducting cylinder. The nonuniform currents induced in the vicinity of the left (z = −l) and right (z = l) (1)right , which is determined according to edges radiate the field Hx(1) = Hx(1)left + Hx Equations (7.136) and (7.137) as a eikR i2kl cos 𝜗 e 2𝜋 R [ ] 2𝜋 { (1) l l sin 𝜓 G(1) × (𝜓, 𝜃, 𝜙 ) ⋅ 𝜙 − G (𝜓, 𝜃, 𝜙 ) ⋅ 𝜃 x x 𝜃 𝜙 ∫0 } − cos 𝜗 cos 𝜓F𝜃(1) (𝜓, 𝜃, 𝜙l ) ⋅ 𝜙x ei2ka sin 𝜗 sin 𝜓 d𝜓, (13.74)

Hx(1)left = H0x

ELECTROMAGNETIC WAVES

(1)right

Hx

361

a eikR −i2kl cos 𝜗 e 2𝜋 R [ ] 2𝜋 { sin 𝜓 G(1) × (𝜓, 𝜃, 𝜙r ) ⋅ 𝜙x − G(1) (𝜓, 𝜃, 𝜙r ) ⋅ 𝜃x 𝜃 𝜙 ∫𝜋 } − cos 𝜗 cos 𝜓F𝜃(1) (𝜓, 𝜃, 𝜙r ) ⋅ 𝜙x ei2ka sin 𝜗 sin 𝜓 d𝜓. (13.75)

= H0x

Due to the symmetry of this problem, the cross-polarized component of the backscat(1) tered field equals zero, Hy,z = 0. The local spherical coordinates 𝜃 and 𝜙 of EEWs involved in these integrals are defined in Section 13.2.1. Note also that the details of derivation of Equation (13.74) are presented in Problem 13.3. For the observation point on the focal line (𝜗 = 𝜋 − 0), these integrals are calculated in closed form: [ ) )] ikR ( ( e 𝜋 𝜋 3𝜋 a (1) 𝜋 𝜋 3𝜋 e−i2kl , g − f (1) , , , , 2 2 2 2 2 2 2 R ( ( [ ) )] ikR e 3𝜋 3𝜋 a ei2kl . = H0x g(1) 0, 0, − f (1) 0, 0, 4 2 2 R

Hx(1)left = H0x (1)right

Hx

(13.76) (13.77)

Taking into account Equation (13.64), we can represent the focal field as a 𝜋 eikR −i2kl e , cot 2n n R a sin(𝜋∕n) eikR i2kl e . = H0x 2n cos(𝜋∕n) − 1 R

Hx(1)left = −H0x (1)right

Hx

(13.78) (13.79)

These expressions for the magnetic field are similar to Equations (13.65) and (13.66) for the electric field and differ from them only in sign. It is also assumed in Equation (13.79) that for 𝜗 = 𝜋 − 0, only the lower part of the right edge (𝜋 + 𝛿 < 𝜓 < 2𝜋 − 𝛿 with 𝛿 → 0) is illuminated. For 𝜗 = 𝜋 exactly, the entire right edge becomes illuminated and its field is doubled. The ray asymptotics of the field Hx(1) are given by [ (1) a 1 g (1)ei2ka sin 𝜗−i𝜋∕4 Hx(1)left = H0x √ 2 𝜋ka sin 𝜗 + g(1) (2)e−i2ka sin 𝜗+i𝜋∕4 (1)right

Hx

] eikR i2kl cos 𝜗 e , R

eikR −i2kl cos 𝜗 a 1 e g(1) (3)e−i2ka sin 𝜗+i𝜋∕4 = H0x √ . 2 𝜋ka sin 𝜗 R

(13.80) (13.81)

in Section These asymptotics are identical to those found for the acoustic waves u(1) h 13.1.2 if we replace u0 by H0x .

362

BACKSCATTERING AT A FINITE-LENGTH CYLINDER

For electromagnetic waves with this polarization, one can also suggest the approximation, which follows from Equation (13.29) with the replacements u0 → H0x , → Hx(1) . The expression obtained in this way transforms exactly into the ray u(1) h asymptotics (13.80) and (13.81), but it leads to approximations for the total scattered field which possess a relative error on the order of (ka)−1 in the vicinity of the focal line, 𝜗 = 𝜋.

PROBLEMS 13.1

Use the general Equations (7.136) and (7.137) for EEWs and derive Equation (13.58) for the field scattered by the left edge. Solution r Read the paragraph above Equation (13.47) about local coordinates R, 𝜗, 𝜑 used in Equations (7.136) and (7.137) and about coordinates 𝜌, 𝜃, 𝜙, which we utilize here. r Transform Equations (7.136) and (7.137) for the field in the far zone (R ≫ ka2 , R ≫ kl2 ), taking into account approximation (13.47). Equation (7.136) reduces to ⃗ (1) = dE

d𝜁 ⃗(1) eikR ik(−z′ cos 𝜗+a sin 𝜗 sin 𝜓) e ,  (𝜁 , 𝜃, 𝜙) 2𝜋 R

where z′ = −l for the left edge, z′ = l for the right edge, and 𝜁 = a𝜓. r The edge points are determined by the coordinates x = a cos 𝜓 and y = a sin 𝜓. The tangent to the edge is defined as ̂t = x̂ sin 𝜓 − ŷ cos 𝜓. Figure P13.1 clarifies the geometry of the problem. r One should specify some quantities in Equation (7.137). According to Equation (13.42), exp[ik𝜙i (𝜁 )] = exp[ik(−z′ cos 𝜗 + a sin 𝜗 sin 𝜓)] y

y

ψ t x

ϑ

z

ϑˆ (a)

(b)

̂ Figure P13.1 (a) Illuminated base of the cylinder; (b) Orientation of the unit vector 𝜗.

PROBLEMS

363

and E0t = E0x sin 𝜓, H0t = −H0y cos 𝜓 = H0𝜗 cos 𝜗 cos 𝜓 = Y0 E0x cos 𝜗 cos 𝜓.

r Then, the field scattered from the left edge can be represented in the form ikR ⃗ (1)left = E0x a e e2ikl cos 𝜗 E ∫0 2𝜋 R

2𝜋

⃗ (1) (𝜓, 𝜃, 𝜙l ) [sin 𝜓 F

⃗ (1) (𝜓, 𝜃, 𝜙l )]ei2ka sin 𝜗 sin 𝜓 d𝜓 + cos 𝜗 cos 𝜓 G

r According to Equations (7.139) to (7.141), the vector F⃗ (1) contains only the

⃗ (1) contains two components, G(1) ⋅ 𝜃̂ ̂ and the vector G component F𝜃(1) ⋅ 𝜃, 𝜃 ̂ Hence, the x-component of the scattered field is determined and G(1) ⋅ 𝜙. 𝜙

⃗ (1) exactly, as shown in Equation ⃗ (1) and G by the x-components of vectors F (13.58). 13.2

Show that the field (13.58) for the direction 𝜗 = 𝜋 transforms to the focal form (13.62). Solution The focal line 𝜗 = 𝜋 belongs to the diffraction cone, where the are determined by Equations (7.148), (7.150), and functions F𝜃(1) and G(1) 𝜃,𝜙 (7.151), (7.153). The angle 𝛾0 in these equations equals 𝜋∕2, as the direction of the incident wave is normal to the edge. Hence, F𝜃(1) = −f (1)

(

) ) ( 𝜋 𝜋 3𝜋 𝜋 𝜋 3𝜋 , G(1) , G(1) = g(1) = 0. , , , , 𝜙 𝜃 2 2 2 2 2 2

Also, it follows from Equations (13.51) and (13.52) that 𝜃x = − sin 𝜓 and 𝜙x = cos 𝜓 when 𝜗 = 𝜋. Thus, Ex(1) = E0x

a eikR −i2kl e ∫0 2𝜋 R

= E0x

2𝜋

a eikR −i2kl e ∫0 2𝜋 R

[

] −𝜃x sin 𝜓f (1) − 𝜙x cos 𝜓g(1) d𝜓

2𝜋

[

] sin2 𝜓f (1) − cos2 𝜓g(1) d𝜓.

Integration here leads to Equation (13.62). 13.3

Use the general Equations (7.136) and (7.137) for EEWs and derive Equation (13.74) for the field scattered by the left edge.

364

BACKSCATTERING AT A FINITE-LENGTH CYLINDER

Solution r This solution is similar to that in Problem 13.1. Equations (7.136) and (7.137) are transformed for the field in the far zone and integrated over the left edge: ikR ⃗ (1)left = a e ei2kl cos 𝜗 E ∫0 2𝜋 R

2𝜋

⃗ (1) (𝜓, 𝜃, 𝜙l )] ⃗ (1) (𝜓, 𝜃, 𝜙l ) + Z0 H0t G [E0t F

× ei2ka sin 𝜗 sin 𝜓) d𝜓.

r The magnetic field is found according to Equation (7.136) as H⃗ (1)left = Y0 [R̂ × Ê (1)left ]. r Utilizing the equations R̂ × 𝜃̂ = 𝜙̂ and R̂ × 𝜙̂ = −𝜃, ̂ we obtain

2𝜋 { ikR ⃗ (1)left = a e ei2kl cos 𝜗 Y0 E0t F𝜃(1) (𝜓, 𝜃, 𝜙l ) ⋅ 𝜙̂ H ∫0 2𝜋 R [ ]} l ) ⋅ 𝜙̂ − G(1) (𝜓, 𝜃, 𝜙l ) ⋅ 𝜃̂ ei2ka sin 𝜗 sin 𝜓 d𝜓. + H0t G(1) (𝜓, 𝜃, 𝜙 𝜃 𝜙

r According to box at the beginning of Section 7.2, the tangent ̂t to the edge is directed counterclockwise, ̂t = x̂ sin 𝜓 − ŷ cos 𝜓. Then, in view of Equation (13.70) and Figure P13.1, H0t = H0x sin 𝜓. r Utilizing the relationship between the electric and magnetic vectors of the incident wave (E𝜗inc = −Z0 Hxinc ) and referring to Figure P13.1, we find that E0y = −E0𝜗 cos 𝜗 = Z0 H0x cos 𝜗

and

E0t = −E0y cos 𝜓

= −Z0 H0x cos 𝜗 cos 𝜓. ⃗ (1)left Substitution of these quantities into the equation above for the field H shows that its x-component is described by Equation (13.74).

14 Bistatic Scattering at a Finite-Length Cylinder 14.1

ACOUSTIC WAVES

The geometry of the problem is shown in Figure 14.1. The diameter of the cylinder equals d = 2a, and its length equals L = 2l. The incident wave is given by uinc = u0 eik(y sin 𝛾+z cos 𝛾) ,

0 < 𝛾 < 𝜋∕2.

(14.1)

The scattered field is evaluated in the plane y0z (𝜑 = 𝜋∕2 and 𝜑 = 3𝜋∕2) in the far zone (R ≫ ka2 , R ≫ kl2 ). It is convenient to indicate the scattering direction by the angle Θ (0 ≤ Θ ≤ 2𝜋): { Θ=

𝜗 2𝜋 − 𝜗

if if

𝜑 = 𝜋∕2, 𝜑 = 3𝜋∕2,

(14.2)

where 𝜗 (0 ≤ 𝜗 ≤ 𝜋) is the ordinary spherical coordinate of the field point (R,𝜗,𝜑). One should not confuse this angle Θ with the local angle 𝜃 used for description of an EEW diverging from the diffraction point 𝜁 = a𝜓 at the edge. The relevant local coordinates r,𝜃,𝜙 are introduced below by Equations (14.14) to (14.18).

Fundamentals of the Physical Theory of Diffraction, Second Edition. Pyotr Ya. Ufimtsev. © 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

365

366

BISTATIC SCATTERING AT A FINITE-LENGTH CYLINDER

y 1 Θ

γ

z

2

3

Figure 14.1 Cross-section of the cylinder by the y0z-plane. Dots 1, 2, and 3 are the stationary-phase points.

14.1.1

PO Approximation

The incident wave (14.1) generates the uniform component j(0) of the scattering s,h sources only on the left base (disk) and on the lower lateral part (𝜋 ≤ 𝜓 ≤ 2𝜋) of the cylindrical surface. The scattered field is determined by Equations (1.33) and (1.34). We omit all intermediate calculations and bring the final expressions for the far field, = u0 Φ(0) (Θ, 𝛾) u(0) s,h s,h

eikR , R

(14.3)

where 2 Φ(0) s = ika cos 𝛾

J1 (p) iq ikal sin q 2𝜋 ip sin 𝜓 e − e sin 𝜓 d𝜓, sin 𝛾 p 𝜋 q ∫𝜋

(14.4)

J1 (p) iq ikal sin q 2𝜋 ip sin 𝜓 e − e sin 𝜓 d𝜓, sin Θ p 𝜋 q ∫𝜋

(14.5)

= ika2 cos Θ Φ(0) h and

p = ka(sin 𝛾 − sin Θ),

q = kl(cos Θ − cos 𝛾).

(14.6)

, which contain the Bessel function J1 (p), relate The first terms in the functions Φ(0) s,h to the field scattered by the disk, and the second terms describe the scattering at the lateral (cylindrical) surface. These terms were evaluated numerically and their magnitudes are plotted in Figures 14.2 and 14.4. It follows from the field expressions above that the total scattering cross-section (1.53) equals tot = 2A, 𝜎s,h

(14.7)

ACOUSTIC WAVES

367

Figure 14.2 Scattering at the individual parts of a soft cylinder. According to Equation (14.66), this figure also demonstrates the PO approximation for electromagnetic waves (with Ex -polarization) scattered from the parts of a perfectly conducting cylinder.

where A = 𝜋a2 cos 𝛾 + 4al sin 𝛾

(14.8)

is the area of the shadow beam cross-section. According to Equations (14.4) and (14.5), the normalized scattering cross-section (13.14) is defined as (0) (Θ, 𝛾) = 𝜎norm

(

) 2 | (0) | 2 (Θ, 𝛾) . Φ | | | ka2 | s,h

(14.9)

368

BISTATIC SCATTERING AT A FINITE-LENGTH CYLINDER

Figure 14.3 Scattering at a soft cylinder. According to Equation (14.66), this figure also demonstrates the PO approximation for electromagnetic waves (with Ex -polarization) scattered from a perfectly conducting cylinder.

This quantity was calculated numerically and the results are presented (in the decibel scale) in Figures 14.2 to 14.5 for the incident wave direction 𝛾 = 45◦ . Figures 14.2 and 14.3 show the scattering at the individual parts of the cylinder and demonstrate how the total scattered field is formed. 14.1.2

Shadow Radiation as a Part of the Physical Optics Field

In Section 1.3.4 it was shown that the shadow radiation is the constituent part of the PO field. It was noted there that this field concentrates in the vicinity of the shadow region. We can verify this property by numerical investigation of the shadow radiation generated by the finite-length cylinder. The most appropriate procedure for

ACOUSTIC WAVES

369

Figure 14.4 Scattering at the individual parts of a hard cylinder. According to Equation (14.78), this figure also demonstrates the PO approximation for electromagnetic waves (with Hx -polarization) scattered from the parts of a perfectly conducting cylinder.

doing this work would be the direct application of the shadow contour theorem proven in Section 1.3.5. However, we can facilitate our work by utilizing the relationship given in Equation (1.74): (0) ush = 12 [u(0) s + uh ]

(14.10)

and the numerical results obtained in the preceding section for the PO fields u(0) s and u(0) . Figure 14.6 shows the spatial distribution of the shadow radiation found in this h way. The normalized scattering cross-section (14.9) is plotted here on the decibel scale.

370

BISTATIC SCATTERING AT A FINITE-LENGTH CYLINDER

Figure 14.5 Scattering at a hard cylinder. According to Equation (14.78), this figure also demonstrates the PO approximation for electromagnetic waves (with Hx -polarization) scattered from a perfectly conducting cylinder.

It is shown clearly in Figure 14.6 that the shadow radiation really represents the nature of the scattered field in the forward sector (0◦ ≤ Θ ≤ 90◦ ). According to Equation (14.78), this figure also demonstrates the PO approximation for electromagnetic waves with Hx -polarization scattered from a perfectly conducting cylinder.

14.1.3

PTD for Bistatic Scattering at a Hard Cylinder

Here we consider only scattering at a hard or rigid cylinder. This problem is more important from a practical point of view. According to PTD, the scattered field is

ACOUSTIC WAVES

371

Figure 14.6 Shadow radiation as part of the PO field.

generated by the uniform (j(0) ) and nonuniform (j(1) ) scattering sources induced by h h the incident wave on the cylinder. The field created by j(0) represents the PO field h investigated in earlier sections. Now we calculate the field generated by that part that concentrates near the circular edges of the cylinder and which is also of j(1) h called the fringe source. Then we combine both components of the scattered field and provide the results of numerical calculation. is found by integrating the elementary edge waves The field generated by j(1) h introduced in Chapter 7. This field can be written in the form

(1)right

= u(1)left + uh u(1) h h

,

(14.11)

372

BISTATIC SCATTERING AT A FINITE-LENGTH CYLINDER

where = u0 u(1)left h (1)right

uh

= u0

a eikR iq e ∫0 2𝜋 R

2𝜋

a eikR − iq e ∫𝜋 2𝜋 R

Fh(1) (𝜓, 𝜃, 𝜙l )eip sin 𝜓 d𝜓,

2𝜋

Fh(1) (𝜓, 𝜃, 𝜙r )eip sin 𝜓 d𝜓.

(14.12) (14.13)

Here the quantities p and q are defined by Equation (14.6), and the angle Θ (0 ≤ Θ ≤ 2𝜋) is determined by Equation (14.2). (1)right and uh represent the components related to the left and right The terms u(1)left h edges, respectively. Only half of the right edge is illuminated by the incident wave (14.1), which is why the integration limits in Equation (14.13) are 𝜋 and 2𝜋. To avoid the grazing singularity in the direction Θ = 𝛾, which is caused by the function g(1) at points 𝜓 = 0, 𝜋, and 2𝜋, one should skip integration in a certain vicinity of these points in the integral (14.12). This singularity relates to the situation indicated in Equation (4.34). Note that the theory of EEWs presented in Section 7.9 (which is free of the grazing singularity) cannot treat the singularity above in the direction Θ = 𝛾 because this theory is only applicable to objects with planar faces. The functions Fh(1) (𝜓, 𝜃, 𝜙l,r ) are described by Equation (7.92). In section 13.2.1 we show how one defines the local angles 𝛾0 , 𝜃, 𝜙, and 𝜙0 for the backscattering direction 𝜗 = 𝜋 − 𝛾. In the same way, we can introduce these angles for arbitrary scattering directions in the y0z plane. The relationships cos 𝛾0 = sin 𝛾 cos 𝜓,

cos 𝜃 = − sin Θ cos 𝜓,

(14.14)

are valid for both edges. Recall that the local angles 𝛾0 , 𝜃, and 𝜙 are shown in Figure 7.3, where the angles 𝜃 and 𝜙 are denoted as 𝜗 and 𝜑, respectively. In accordance with the box following Figure 7.3, the angle 𝜑 (𝜙) is measured from the illuminated face of the edge. For the left edge one should use the expressions cos 𝛾 , sin 𝜙l0 = √ 1 − sin2 𝛾 cos2 𝜓 cos Θ , sin 𝜙l = − √ 1 − sin2 Θ cos2 𝜓

sin 𝛾 sin 𝜓 cos 𝜙l0 = √ , 1 − sin2 𝛾 cos2 𝜓 sin Θ sin 𝜓 cos 𝜙l = − √ ; 1 − sin2 Θ cos2 𝜓

(14.15)

(14.16)

however, for the right edge one should use the definitions sin 𝛾 sin 𝜓 sin 𝜙r0 = − √ , 2 2 1 − sin 𝛾 cos 𝜓 sin Θ sin 𝜓 , sin 𝜙r = √ 2 2 1 − sin Θ cos 𝜓

cos 𝛾 cos 𝜙r0 = √ , 2 2 1 − sin 𝛾 cos 𝜓

(14.17)

cos Θ cos 𝜙r = − √ ; 2 2 1 − sin Θ cos 𝜓

(14.18)

ACOUSTIC WAVES

373

Notice that these expressions also follow from Equations (13.54) to (13.57) with replacement of 𝜗 by 2𝜋 − Θ. The unit vectors 𝜃̂ and 𝜙̂ (the same for both edges) are found from Equations (13.50) with ̂t = x̂ sin 𝜓 − ŷ cos 𝜓 and are defined as sin 𝜓 , 𝜃x = − √ 2 2 1 − sin Θ cos 𝜓

cos 𝜓 cos2 Θ 𝜃y = √ , 2 2 1 − sin Θ cos 𝜓

cos 𝜓 sin Θ cos Θ , 𝜃z = − √ 2 2 1 − sin Θ cos 𝜓

(14.19)

and cos 𝜓 cos Θ , 𝜙x = − √ 1 − sin2 Θ cos2 𝜓 sin 𝜓 sin Θ . 𝜙z = √ 2 2 1 − sin Θ cos 𝜓

sin 𝜓 cos Θ 𝜙y = − √ 1 − sin2 Θ cos2 𝜓 (14.20)

The numerical results found using Equations (14.12) and (14.13) for the normalized scattering cross-section (13.14) are presented below (in the decibel scale) for two cylinders with parameters L = 3d = 3𝜆 and L = 3d = 9𝜆. The direction of the incident wave (14.1) is given by the angle 𝛾 = 45◦ . Figures 14.7 and 14.9 demonstrate the individual contributions by the PO field and by the field generated by j(1) . The h sum of these fields and its comparison with the PO field is shown in Figures 14.8 and 14.10. These figures clearly show the influence of a field generated by the nonuni. In particular, this field fills in the deep minima in form/fringe scattering sources j(1) h the PO field. More accurate PTD approximation can be obtained through calculation of the high-order edge waves. However, in contrast to thin dipoles, thick cylinders are not resonant bodies and all high-order edge waves can be neglected when the size of cylinders exceeds three to five wavelengths. The larger the cylinders, the greater the accuracy of the PTD expressions (14.12) and (14.13). The following comments explain some details of the numerical calculations:

r The functions F(1)left,right are determined using Equations (7.92), and (7.94) h

through the functions Vt (𝜎1 , 𝜙0 ) and Vt (𝜎2 , 𝛼 − 𝜙0 ), which contain the factors 1∕ sin 𝜎1,2 . These factors become singular when 𝜎1,2 → 0 or 𝜎1,2 → 𝜋. When 𝜎1 → 0, the functions Vt remain finite. They can be transformed into the more convenient form

Vt (𝜎1 , 𝜙0 ) =

cos(𝜎1 ∕3) 1 4 . (14.21) 2 9 sin 𝛾 1 − (4∕3) sin2 (𝜎1 ∕3) cos (2𝜙0 ∕3) − cos (2𝜎1 ∕3)

374

BISTATIC SCATTERING AT A FINITE-LENGTH CYLINDER

Figure 14.7 Bistatic scattering at a hard or rigid cylinder. According to Equation (14.78), the PO curve here also demonstrates the scattering of electromagnetic waves (with Hx -polarization) from a perfectly conducting cylinder.

The replacements 𝜎1 by 𝜎2 and 𝜙0 by 𝛼 − 𝜙0 in (14.21) lead to a transformed expression for Vt (𝜎2 , 𝛼 − 𝜙0 ). r However, these expressions are still singular when 𝜎1.2 → 𝜋. In this case one should calculate the products Vt (𝜎1 , 𝜙0 ) ⋅ sin 𝜙 and Vt (𝜎2 , 𝛼 − 𝜙0 ) ⋅ sin(𝛼 − 𝜙). They remain finite when 𝜎1.2 → 𝜋 because the ratios sin 𝜙∕ sin 𝜎1 and sin(𝛼 − 𝜙)∕ sin 𝜎2 are equal to plus or minus unity. r The function V(𝜎2 , 𝛼 − 𝜙0 ) related to the left edge is singular at the points 𝜓 = 0, 𝜋, and 2𝜋 for the observation direction Θ = 𝛾. This is the grazing singularity that was mentioned in Equation (4.34). It is removed by the exclusion of a certain

ACOUSTIC WAVES

375

Figure 14.8 Scattering at a hard cylinder. According to Equation (14.78), the PO curve here also demonstrates the scattering of electromagnetic waves (with Hx -polarization) from a perfectly conducting cylinder.

vicinity of the singular points in the integral (14.12). This exclusion is done only for that part of the integral (14.12) that contains the function V(𝜎2 , 𝛼 − 𝜙0 ). The function V(𝜎1 , 𝜑0 ) is not singular, and it is integrated in (14.12) in the entire region 0 ≤ 𝜓 ≤ 2𝜋. Notice that the theory of EEWs presented in Section 7.9 (which is free of the grazing singularity) cannot treat the singularity above in the direction Θ = 𝛾 because this theory is applicable only for objects with planar faces. r Finally, we note that when 𝜎1 → 𝜙0 , or 𝜎2 → 𝛼 − 𝜙0 , one should use Equation (7.107) for the functions V(𝜙0 , 𝜙0 ) and Equation (7.109) for the function V(𝛼 − 𝜙0 , 𝛼 − 𝜙0 ).

376

BISTATIC SCATTERING AT A FINITE-LENGTH CYLINDER

Figure 14.9 Scattering at a hard cylinder. According to Equation (14.78), the PO curve here also demonstrates the scattering of electromagnetic waves (with Hx -polarization) from a perfectly conducting cylinder.

14.1.4

Beams and Rays of the Scattered Field

In the preceding section a numerical investigation of the scattered field was presented. Here we consider its physical structure and present simple high-frequency asymptotics for the directivity pattern Φ(Θ, 𝛾) defined by this equation: usc s,h = u0 Φs,h (Θ, 𝛾)

eikR . R

(14.22)

From a physical point of view, a scattered field consists of the following basic components: beams and rays. Beams are the fields scattered by those parts of a reflecting

ACOUSTIC WAVES

377

Figure 14.10 Scattering at a hard cylinder. According to Equation (14.78), the PO curve here also demonstrates the scattering of electromagnetic waves (with Hx -polarization) from a perfectly conducting cylinder.

object that contain continuous distribution of an infinite number of stationary-phase points. In the optical spectrum such points form shining lines and spots visible on the object illuminated by light (Ufimtsev, 1996). Rays are produced by discrete stationary points. The following beams are present in the scattered field:

r The reflected beam in the vicinity of the direction Θ = 𝜋 − 𝛾. This beam appears due to the transverse diffusion of the field in ordinary rays reflected from the left base of the cylinder (Fig. 14.1). It is described by the first terms in Equations (14.4) and (14.5), which contain the Bessel function J1 (p). Exactly in this direction its value equals 1 1 = −Φbeam = Φbeam s h

ika2 cos 𝛾e−i2kl cos 𝛾 . 2

(14.23)

378

BISTATIC SCATTERING AT A FINITE-LENGTH CYLINDER

r The reflected beam in the vicinity of the direction Θ = 2𝜋 − 𝛾. It appears due to the transverse diffusion of the field in ordinary rays reflected from the lower lateral part of the cylinder. This beam is described by the high-frequency asymptotics for the second terms in Equations (14.4) and (14.5): 2 = ikal sin 𝛾 Φbeam s

2 Φbeam h

sin q q

√

sin q = ikal sin Θ q

2 −ip i𝜋∕4 e e , 𝜋p

√

2 −ip i𝜋∕4 e e , 𝜋p

(14.24)

(14.25)

where p = ka(sin 𝛾 − sin Θ) and q = kl(cos Θ − cos 𝛾). In exactly the direction Θ = 2𝜋 − 𝛾, its value equals √ 2 Φbeam s

=

2 −Φbeam h

=l

ka sin 𝛾 −i⋅2ka sin 𝛾 i3𝜋∕4 e . e 𝜋

(14.26)

r The beam of the shadow radiation in the vicinity of the direction Θ = 𝛾. It is described by both terms in Equations (14.4) and (14.5). In exactly this direction, its value equals = Φshad,beam = Φshad,beam s h

ika2 i2kal cos 𝛾 + sin 𝛾 2 𝜋

(14.27)

r The beam of edge-diffracted rays generated by the fringe scattering sources, which are located near the left edge. It propagates in the direction Θ = 𝜋 − 𝛾 and supplements the reflected beam (14.23). This beam is described by Equation (14.12). In the case of the soft cylinder, the corresponding expression follows from Equation (14.12) with the obvious replacement of Fh(1) by Fs(1) . In exactly the direction Θ = 𝜋 − 𝛾, all elementary edge waves belong to the diffraction cone (𝜃 = 𝜋 − 𝛾0 ) and the functions Fs(1) and Fh(1) transform into the functions f (1) , and g(1) according to Equation (7.115). The beam of these waves is determined by a −i2kl cos 𝛾 e ∫0 2𝜋

2𝜋

1 = Φfr,beam s

1 = Φfr,beam h

a −i2kl cos 𝛾 e ∫0 2𝜋

2𝜋

f (1) (𝜋 − 𝜙l0 , 𝜙l0 , 𝛼) d𝜓,

(14.28)

g(1) (𝜋 − 𝜙l0 , 𝜙l0 , 𝛼) d𝜓.

(14.29)

Here 𝛼 = 3𝜋∕2 and the angle 𝜙l0 is determined by Equation (14.15).

r The beam of edge-diffracted rays generated by the fringe scattering sources, which are located near both edges. It propagates in the direction Θ = 𝛾 and supplements the shadow beam. This beam is described by Equations (14.12) and

ACOUSTIC WAVES

379

(14.13) for a hard cylinder. For a soft cylinder, the corresponding field expressions follow from Equations (14.12) and (14.13) with obvious replacement of Fh(1) by Fs(1) . In exactly the direction Θ = 𝛾, all elementary edge waves belong to the diffraction cone 𝜃 = 𝜋 − 𝛾0 . It follows from Equations (7.115) and (7.116) that Fs(1)left = f (1) and Fh(1)left = g(1) for the edge points 0 ≤ 𝜓 ≤ 𝜋, which are visible

from the shadow direction Θ = 𝛾, and Fs(1)left = −f (0) , Fh(1)left = −g(0) for the invisible edge points 𝜋 < 𝜓 < 2𝜋. Because of that, the beam of fringe waves is determined as Φfr,shad,beam s

[ 𝜋 2𝜋 a = f (1) (𝜋 + 𝜙l0 , 𝜙l0 , 𝛼) d𝜓 − f (0) (𝜋 + 𝜙l0 , 𝜙l0 , 𝛼) d𝜓 ∫𝜋 2𝜋 ∫0 2𝜋

+

Φfr,shad,beam = h

f

∫𝜋

(1)

] (𝜋 + 𝜙r0 , 𝜙r0 , 𝛼) d𝜓 ,

(14.30)

[ 𝜋 2𝜋 a g(1) (𝜋 + 𝜙l0 , 𝜙l0 , 𝛼) d𝜓 − g(0) (𝜋 + 𝜙l0 , 𝜙l0 , 𝛼) d𝜓 ∫𝜋 2𝜋 ∫0 2𝜋

+

∫𝜋

] g(1) (𝜋 + 𝜙r0 , 𝜙r0 , 𝛼) d𝜓 .

(14.31)

The functions f (0) (𝜋 + 𝜙0 , 𝜙0 , 𝛼) and g(0) (𝜋 + 𝜙0 , 𝜙0 , 𝛼) are defined according to Equations (7.118a) and (7.119a). For a cylinder (where 𝛼 = 3𝜋∕2) they are given by f (0) (𝜋 + 𝜙l0 , 𝜙l0 , 𝛼) = − 12 cot 𝜙l0 − g(0) (𝜋 + 𝜙l0 , 𝜙l0 , 𝛼) =

1 2

cot 𝜙l0 +

1 2

tan 𝜙l0 ,

(14.32)

1 2

tan 𝜙l0

(14.33)

and they are singular at the points 𝜓 = 0, 𝜋, and 2𝜋, where 𝜙l0 = 𝜋∕2. According to Equation (4.31), the functions f (1) (𝜋 + 𝜙l0 , 𝜙l0 , 3𝜋∕2) and g(1) (𝜋 + 𝜙l0 , 𝜙l0 , 3𝜋∕2) are also singular at the same points. All these singularities are grazing singularities associated with Equation (4.34). In contrast, as follows from Equation (4.31), the functions f (1) (𝜋 + 𝜙r0 , 𝜙r0 , 3𝜋∕2) and g(1) (𝜋 + 𝜙r0 , 𝜙r0 , 3𝜋∕2) for the right edge are free of the grazing singularity at the points 𝜓 = 0, 𝜋, and 2𝜋, where 𝜙r0 = 0. Indeed, one can verify with Equation (4.31) that in these particular points 𝜓, the functions f (1) and g(1) are finite, f (1) (𝜋, 0, 𝛼) = 0 and g(1) (𝜋, 0, 𝛼) = −(1∕n) cot(𝜋∕n). In view of these comments, it is reasonable to disregard the integrals over the left edge and calculate the fringe field in the direction Θ = 𝛾 only by the integrals over the right edge.

380

BISTATIC SCATTERING AT A FINITE-LENGTH CYLINDER

The theory of EEWs presented in Section 7.9 (which is free of the grazing singularity) cannot treat the singularities above in the direction Θ = 𝛾 because it is applicable only for objects with planar faces. Away from these beams, the scattered field contains the three edge-diffracted + j(1) . They can be determined by rays generated by the total surface current j(0) s,h (0) (1) s,h asymptotic estimation of the field us,h + us,h . However, a simpler way is to replace (t) the function Fh(1) in Equations (14.12) and (14.13) by the functions Fs,h and then use the stationary-phase technique under the condition |p| ≫ 1. (t) transform to the functions f and g. As a At the stationary points the functions Fs,h result, one obtains the following expressions for these rays:

r Ray 1: ray

Φs (1) = √

2𝜋 |p| a

ray

Φh (1) = √

a

2𝜋 |p|

f (1)ei(p+q) e∓i𝜋∕4 ,

(14.34)

g(1)ei(p+q) e∓i𝜋∕4 .

(14.35)

It propagates from stationary point 1 (Fig. 14.1) and exists in the regions 0 ≤ Θ < 𝛾, 𝛾 < Θ < 𝜋 − 𝛾, and 𝜋 − 𝛾 < Θ ≤ 3𝜋∕2. The factor exp(−i𝜋∕4) is taken for positive values of p, and the factor exp(+i𝜋∕4) is valid for negative values of p. r Ray 2: ray

Φs (2) = √ ray

Φh (2) = √

a 2𝜋 |p| a

2𝜋 |p|

f (2)ei(q−p) e±i𝜋∕4 ,

(14.36)

g(2)ei(q−p) e±i𝜋∕4 .

(14.37)

It propagates from stationary point 2 and exists in the region 𝜋∕2 ≤ Θ < 𝜋 − 𝛾, 𝜋 − 𝛾 < Θ < 2𝜋 − 𝛾, and 2𝜋 − 𝛾 < Θ ≤ 2𝜋. The factor exp(+i𝜋∕4) is taken for positive values of p, and the factor exp(−i𝜋∕4) is valid for negative values of p. r Ray 3: a ray Φs (3) = √ f (3)e−i(q+p) e±i𝜋∕4 , 2𝜋 |p| ray

Φh (3) = √

a 2𝜋 |p|

g(3)e−i(q+p) e±i𝜋∕4

(14.38)

(14.39)

propagates from stationary point 3 (Fig. 14.1) and exists in the regions 0 ≤ Θ < 𝛾, 𝛾 < Θ ≤ 𝜋∕2, 𝜋 ≤ Θ ≤ 2𝜋 − 𝛾, and 2𝜋 − 𝛾 < Θ ≤ 2𝜋. The factor

ACOUSTIC WAVES

381

exp(+i𝜋∕4) is taken for positive values of p, and the factor exp(−i𝜋∕4) is valid for negative values of p. It is clear that the rays generated by nonuniform/fringe sources are described also by Equations (14.34) to (14.39), where one should replace the functions f and g by the functions f (1) and g(1) . In the expressions above, the functions f ,g and f (1) , g(1) are defined according to Chapters 2, 3, and 4: f (m) = f (𝜙m , 𝜙0m , 𝛼),

g(m) = g(𝜙m , 𝜙0m , 𝛼),

f (1) (m) = f (1) (𝜙m , 𝜙0m , 𝛼),

g(1) (m) = g(1) (𝜙m , 𝜙0m , 𝛼)

(14.40) (14.41)

with m = 1, 2, 3. Here, 𝛼 = 3𝜋∕2 and 𝜙1 = 3𝜋∕2 − Θ,

𝜙01 = 𝜋∕2 − 𝛾,

𝜙2 = Θ − 𝜋∕2,

(14.42)

𝜙02 = 𝜋∕2 + 𝛾,

(14.43)

𝜙3 = Θ − 𝜋

if

𝜋 ≤ Θ ≤ 2𝜋,

(14.44)

𝜙3 = 𝜋 + Θ

if

0 ≤ Θ ≤ 𝜋∕2,

(14.45)

𝜙03 = 𝛾.

(14.46)

All diffracted rays undergo a phase shift equal to ±𝜋∕2 when they cross the focal lines Θ = 𝛾 and Θ = 𝜋 − 𝛾, which are the axes of the field beams. 14.1.5

PO Shooting-Through Rays and Their Cancellation by Fringe Rays

In the preceding section we introduced diffracted rays generated by the total scatter= j(0) + j(1) . Now we consider the physics related to these rays. The ing sources jtot s,h s,h s,h PO/uniform sources j(0) and the fringe/nonuniform sources j(1) separately generate s,h s,h their own rays. For example, they create the rays diverging from edge point 1 (Fig. 14.1) in all directions in the y0z-plane. In particular, they freely penetrate through the perfectly reflecting cylinder and represent the shooting-through rays in the region 𝜑 = 3𝜋∕2, 0 < 𝜗 < 𝜋∕2 (Fig. 14.11). These individual rays from point 1 can be described by Equations (14.34) and (14.35), where one should replace the functions ), and by f (1), and g(1) by the functions f (0) and g(0) (for rays generated by j(0) s,h ). the functions Fs(1) (𝜋 − 𝛾0 , 𝜑) and Fh(1) (𝜋 − 𝛾0 , 𝜑) (for fringe rays generated by j(1) s,h However, it follows from Equation (7.116) that Fs(1) (𝜋 − 𝛾0 , 𝜑) = −f (0) (𝜑, 𝜑0 , 𝛼) and Fh(1) (𝜋 − 𝛾0 , 𝜑) = −g(0) (𝜑, 𝜑0 , 𝛼). Therefore, the fringe rays cancel the PO shooting-through rays. A similar situation happens with such individual rays diverging from point 3 and penetrating through the cylinder into the region 𝜑 = 𝜋∕2, 𝜋∕2 < 𝜗 < 𝜋.

382

BISTATIC SCATTERING AT A FINITE-LENGTH CYLINDER

y 1 STR-3 z

2

3

STR-1 Figure 14.11 Shooting-through rays STR-1 and STR-3.

As discussed in Section 7.8.2, the spurious shooting-through rays are the consequence of calculation of a diffracted field via the equivalent scattering sources introduced on a scattering object. Any inaccurate approximations of equivalent sources result in spurious shooting-through rays. The fundamental role of PTD consists, in particular, in elimination of PO spurious shooting-through rays. The same situation happens with electromagnetic shooting-through rays.

14.1.6 Refined Asymptotics for the Specular Beam Reflected from the Lateral Surface We refer again to Figure 14.1 and focus on the beam reflected from the lower lateral surface of the cylinder and propagating in the specular direction Θ = 2𝜋 − 𝛾. In the first-order approximation, this beam is as evaluated in the preceding section. Here we consider some fine features of the theory, which are beyond the first approximation. The results of this section were published earlier in a paper by Ufimtsev (1989). According to PTD, the scattered field is generated by the uniform j(0) and nonuniform j(1) components of the scattering sources induced by the incident wave on the object. Up to now we calculated the field radiated only by the basic part of the component j(1) that is caused by sharp bending (edges). This is the fringe component j(1)fr . The other component, j(1)sm , is caused by smooth bending of the scattering surface and is asymptotically small compared to j(0) and j(1)fr . In particular, on a circular cylinder, the ratio j(1)sm ∕j(0) is on the order of 1∕ka. That is why the component j(1)sm is usually neglected for thick cylinders. However, in the papers (Ufimtsev, 1979, 1981, 1989) it was shown that in the specular direction Θ = 2𝜋 − 𝛾, this small component distributed over the entire generatrix (−l ≤ z ≤ l,𝜓 = 3𝜋∕2, kl ≫ 1) creates co-phased radiation of the same order (ka)−1∕2 as the field generated by j(1)fr . For this reason one should include this additional radiation in the beam field. This is the first important feature of the theory. It was also shown (Ufimtsev, 1979, 1981, 1989) that the second term of the asymptotic expansion for the PO field (in the specular direction) is also a quantity

383

ACOUSTIC WAVES

on the order of (ka)−1∕2 and should be included in the beam field. Usually, highorder terms in the PO field are considered incorrect and neglected. However, in the framework of PTD, the PO field is the constituent part of the scattered field. Therefore, one should incorporate into the field expression the high-order asymptotic terms of the PO field, which are of the same order of magnitude as those taken from . This is asymptotic expansion of the field generated by the nonuniform sources j(1) s,h the second important feature of PTD. Here these observations are demonstrated in analytic form for the directivity pattern Φ(Θ, 𝛾) introduced in Equation (14.22). The scattered field is evaluated in the vicinity of the specular direction Θ = 2𝜋 − 𝛾. The PO field generated by j(0) is described by Equations (14.4) and (14.5). The first term there relates to the field scattered by the left base (disk) of the cylinder. Its contribution to the field in the region 3𝜋∕2 < Θ < 2𝜋 is created by the vicinity of point 2 (Fig. 14.1). By asymptotic evaluation of this contribution, the expressions (14.4) and (14.5) can be written as sin q 2𝜋 ip sin 𝜓 cos 𝛾 aei𝜋∕4 ikal i(q−p) e e = − − sin 𝜓 d𝜓, sin 𝛾 Φ(0) √ s 𝜋 q ∫𝜋 2𝜋p sin 𝛾 − sin Θ (14.47) sin q aei𝜋∕4 ikal cos Θ ei(q−p) − = −√ sin Θ Φ(0) h 𝜋 q ∫𝜋 2𝜋p sin 𝛾 − sin Θ

2𝜋

eip sin 𝜓 sin 𝜓 d𝜓, (14.48)

where p = ka(sin 𝛾 − sin Θ) and q = kl(cos Θ − cos 𝛾). Here, in accordance with the discussion above, we retain two first terms in the asymptotic expansion for the integrals and obtain cos 𝛾 ei(q−p)+i𝜋∕4 2𝜋p sin 𝛾 − sin Θ √ ( ) sin q 2𝜋 ikal 3 e−ip+i𝜋∕4 , − sin 𝛾 −1 + i 𝜋 q p 8p

Φ(0) s = − √

a

a cos Θ ei(q−p)+i𝜋∕4 Φ(0) = −√ h sin 𝛾 − sin Θ 2𝜋p √ ( ) sin q 2𝜋 ikal 3 e−ip+i𝜋∕4 . − sin Θ −1 + i 𝜋 q p 8p

(14.49)

(14.50)

The field radiated by j(1)fr is determined in accordance with Section 14.1.4 as a = √ Φ(1)fr [f (1) (2)eiq + f (1) (3)e−iq ]e−ip+i𝜋∕4 , s 2𝜋p a = √ [g(1) (2)eiq + g(1) (3)e−iq ]e−ip+i𝜋∕4 . Φ(1)fr h 2𝜋p

(14.51) (14.52)

384

BISTATIC SCATTERING AT A FINITE-LENGTH CYLINDER

The functions f (1) = f − f (0) and g(1) = g − g(0) are introduced in Chapters 2, 3, and 4. Their arguments are defined in Equations (14.40) to (14.46). The nonuniform component j(1)sm caused by the smooth bending of the cylindrical surface is found by extension of the results of a paper by Franz and Galle (1955) to the oblique direction of the incident wave. The asymptotic approximations found in this way are 1 eik(z cos 𝛾+a sin 𝛾 sin 𝜓) , a sin2 𝜓 i eik(z cos 𝛾+a sin 𝛾 sin 𝜓) . = u0 ka sin 𝛾 sin3 𝜓

= u0 j(1)sm s

(14.53)

j(1)sm h

(14.54)

According to Equations (1.16), (1.17), and (14.22), the field radiated by these sources is described by

Φ(1)sm s

l sin q =− 2𝜋 q ∫𝜋

= Φ(1)sm h

2𝜋

eip sin 𝜓

l sin q d𝜓 ∼ − 2 2𝜋 q sin 𝜓

l sin Θ sin q 2𝜋 sin 𝛾 q ∫𝜋

2𝜋

eip sin 𝜓 sin2 𝜓

d𝜓 ∼

√

2𝜋 −ip+i𝜋∕4 e , p

l sin Θ sin q 2𝜋 sin 𝛾 q

√

(14.55) 2𝜋 −ip+i𝜋∕4 e p (14.56)

under the condition p ≫ 1. The total field equals (0) (1)fr (1)sm Φtot s,h = Φs,h + Φs,h + Φs,h .

(14.57)

−1∕2 Away from √ the specular direction, this field contains terms on the order of (ka) and (kl ka)−1 . In exactly the specular direction Θ = 2𝜋 − 𝛾, its components are equal to

( ) 3 l 1 a e−i2ka sin 𝛾+i𝜋∕4 , ikl sin 𝛾 + = − cot 𝛾 Φ(0) √ s 16 a 4 𝜋ka sin 𝛾 = −√ Φ(0) h

a 𝜋ka sin 𝛾

(

ikl sin 𝛾 +

)

(14.58)

3 l 1 + cot 𝛾 e−i2ka sin 𝛾+i𝜋∕4 , 16 a 4 (14.59)

ACOUSTIC WAVES

{ a

385

[ ]−1 2(𝜋 − 2𝛾) 1 1 + cos √ 3 3 2 }

Φ(1)fr s

= −√ 𝜋ka sin 𝛾

Φ(1)fr h

1 1 + √ − cot 𝛾 e−i2ka sin 𝛾+i𝜋∕4 , 4 3 3 { [ ]−1 2(𝜋 − 2𝛾) 1 1 a = −√ + cos √ 3 𝜋ka sin 𝛾 3 2 } 1 1 − √ − cot 𝛾 e−i2ka sin 𝛾+i𝜋∕4 , 3 3 4

a l −i2ka sin 𝛾+i𝜋∕4 e Φ(1)sm = Φ(1)sm =− √ . s h 𝜋ka sin 𝛾 2a

(14.60)

(14.61) (14.62)

By summing these components, one obtains the total field in the specular direction: a Φs = √ 𝜋ka sin 𝛾

{

ikl sin 𝛾 +

l 1 − √ − 2a 3 3 a Φh = √ 𝜋ka sin 𝛾

}

{

l 1 + √ − 2a 3 3

[ ]−1 2(𝜋 − 2𝛾) 3 l 1 1 −√ + cos 16 a 3 3 2

e−i2ka sin 𝛾+i𝜋∕4 ,

− ikl sin 𝛾 −

}

(14.63)

[ ]−1 2(𝜋 − 2𝛾) 3 l 1 1 −√ + cos 16 a 3 3 2

e−i2ka sin 𝛾+i𝜋∕4 .

(14.64)

The origination and meaning of each term here is clear from the preceding expressions for field components. The first and second terms in braces relate, respectively, to the first and second terms in the asymptotic expansion for the PO field. The third and , and the last term fourth terms represent the contribution by the fringe sources j(1)fr s,h caused by the smooth bending of the shows the contribution by the sources j(1)sm s,h cylindrical surface. Notice that the primary edge waves propagating along the cylinder base from point 1 (Fig. 14.1) to point 2 undergo edge diffraction and generate the secondary diffracted rays in the region 3𝜋∕2 < Θ ≤ 2𝜋. These secondary rays can be calculated by application of the theory developed in Chapter 10. Their contributions to the field in the specular direction are on the order of (ka)−3∕2 for the soft cylinder and on the order of (ka)−1 for the hard cylinder. They are small compared to the fields (14.63) and (14.64) and can be neglected. The counterpart of the present theory developed for electromagnetic waves scattered at a perfectly conducting cylinder of finite size has been published in a paper by Ufimtsev (1981) and is considered in Section 14.2.3.

386

BISTATIC SCATTERING AT A FINITE-LENGTH CYLINDER

14.2

ELECTROMAGNETIC WAVES

14.2.1

E-Polarization

On the surface of a perfectly conducting cylinder (Fig. 14.1), the incident wave Exinc = E0x eik(z cos 𝛾+y sin 𝛾) ,

inc Ey,z = Hxinc = 0

(14.65)

generates the uniform currents given by Equations (13.43) and (13.44). The PO field radiated by these currents is calculated according to Equations (1.108) and (1.109). In the plane y0z (𝜑 = 𝜋∕2 and 𝜑 = 3𝜋∕2), this field is described by expressions totally identical to Equations (14.3) and (14.4), derived above for acoustic waves: Ex(0) = u(0) s

if

u0 = E0x .

(14.66)

In particular, this relationship means that the PO curves in Figures 14.2 and 14.3 for the acoustic waves scattered from a soft cylinder also demonstrate the electromagnetic waves with Ex -polarization scattered from a perfectly conducting cylinder of the same size. The field radiated by the nonuniform current ⃗j(1) is calculated according to Section 7.8 by integrating the elementary edge waves introduced in Chapter 7. This field can be written in the form (1)right

Ex(1)tot = Ex(1)left + Ex

,

(14.67)

where the function Ex(1)left = E0x

a eikR iq e ∫0 2𝜋 R

2𝜋

{sin 𝜓 F𝜃(1) (𝜓, 𝜃, 𝜙l ) ⋅ 𝜃x − cos 𝛾 cos 𝜓

(𝜓, 𝜃, 𝜙l ) ⋅ 𝜃x + G(1) (𝜓, 𝜃, 𝜙l ) ⋅ 𝜙x ]}eip sin 𝜓 d𝜓 × [G(1) 𝜃 𝜙

(14.68)

describes the field scattered from the left edge. Here p = ka(sin 𝛾 − sin Θ), q = kl(cos Θ − cos 𝛾). The derivation of Equation (14.68) is given in Problem 14.2. The function (1)right

Ex

= E0x

a eikR −iq e ∫𝜋 2𝜋 R

2𝜋

{sin 𝜓 F𝜃(1) (𝜓, 𝜃, 𝜙r ) ⋅ 𝜃x

− cos 𝛾 cos 𝜓 [G(1) (𝜓, 𝜃, 𝜙r ) ⋅ 𝜃x + G(1) (𝜓, 𝜃, 𝜙r ) ⋅ 𝜙x ]}eip sin 𝜓 d𝜓 𝜃 𝜙 (14.69)

387

ELECTROMAGNETIC WAVES

represents the field scattered from the right edge. Only half of the right edge is illuminated by the incident wave (14.1), which is why the integration limits in Equation (14.69) are 𝜋 and 2𝜋. In Equations (14.68) and (14.69), the local angles 𝛾0 and 𝜃 are defined by Equation (different for the left and right edges) are defined (14.14). The local angles 𝜙l,r and 𝜙l,r 0 by Equations (14.15) to (14.18). The unit vectors 𝜃̂ and 𝜙̂ (the same for the left and right edges) are determined according to Equations (14.19) and (14.20). Note that according to Equations (7.170) and (7.171), the field (14.68) contains fringe rays that cancel the PO shooting-through rays in the region 𝜑 = 3𝜋∕2, 0 < 𝜗 < 𝜋∕2 (Fig. 14.11). Analogously, the field (14.69) contains fringe rays that cancel the PO shooting-through rays in the region 𝜑 = 𝜋∕2, 0 < 𝜗 < 𝜋∕2. In calculating of the field in the forward/shadow direction Θ = 𝛾, we take into account the comments following Equations (14.32) and (14.33). According to them the integral over the left edge should be omitted (for this particular direction), due to the grazing singularity. The fringe field is calculated there only by the integral over the right edge: (1)right

Ex(1) = Ex

= E0x

a eikR 2𝜋 R

{

2𝜋

∫𝜋

[

sin2 𝜓 f (1) (𝜋 + 𝜙r0 , 𝜙r0 , 𝛼)

+ cos2 𝛾 cos2 𝜓g(1) (𝜋 + 𝜙r0 , 𝜙r0 , 𝛼) + sin 𝛾 cos 𝛾

2𝜋

∫𝜋

sin 𝜓 cos2 𝜓 sin2 𝛾0

] d𝜓 sin2 𝛾0 }

d𝜓

(14.70)

A valuable feature of this integral is the absence of the grazing singularity at the points 𝜓 = 𝜋 and 2𝜋. Next we use the results of Equation (14.68) when 𝛼 = 2𝜋 and the cylinder transforms to the disk. In this specific case, Equation (14.68) takes the form Ex(1)

[ ( ) 1 + cos2 𝛾 ( )] ikR e a 𝜋 𝜋 = E0x 2 cos 𝛾K , sin 𝛾 − E , sin 𝛾 . (14.71) 2 2 cos 𝛾 2 R 𝜋 sin 𝛾

This result was obtained earlier by Ufimtsev (1962b) and Butorin et al. (1987)]. Here 𝜋∕2

K(𝜋∕2, x) =

∫0

d𝜓 , √ 1 − x2 cos2 𝜓

E(𝜋∕2, x) =

𝜋∕2 √

∫0

1 − x2 cos2 𝜓 d𝜓 (14.72)

are complete elliptic integrals (Gradshteyn and Ryzhik, 1994). One can show that for the normal incidence (𝛾 = 0) the field (14.71) equals zero and agrees with Equation (2.3.18) of Ufimtsev (2009).

388

BISTATIC SCATTERING AT A FINITE-LENGTH CYLINDER

To simplify the comparative analysis of electromagnetic and acoustic waves scattered at a finite cylinder, it is convenient to represent the total scattered field in the form (1) Ex = Ex(0) + Ex(1) = E0x [Φ(0) ex + Φex ]

eikR eikR = E0x Φex (Θ, 𝛾) , R R

(14.73)

similar to Equation (14.22) for acoustic waves. According to Equation (14.66), all the beam and ray asymptotics of the electromagnetic field Ex(0) are identical to those presented in Section 14.1. 4 for the acoustic field u(0) s : (0) Φ(0) ex (Θ, 𝛾) = Φs (Θ, 𝛾).

(14.74)

The ray asymptotics of the field Ex(1) are also identical to those of the acoustic field u(1) s shown in Section 14.1.4: (1) Φ(1) ex (Θ, 𝛾) = Φs (Θ, 𝛾).

(14.75)

However, the beam asymptotics for the field generated in the directions Θ = 𝛾 and Θ = 𝜋 − 𝛾 by nonuniform/fringe currents are different for electromagnetic and acoustic waves: Φ(1)beam (Θ, 𝛾) ≠ Φfr.beam (Θ, 𝛾). ex s 14.2.2

(14.76)

H-Polarization

On the surface of a perfectly conducting cylinder (Fig. 14.1), the incident wave Hxinc = H0x eik(z cos 𝛾+y sin 𝛾) ,

inc Hy,z = Exinc = 0

(14.77)

excites the currents with the uniform components (13.71) and (13.72). The field radiated by these currents is calculated according to Equations (1.108) and (1.109). In the plane y0z (𝜑 = 𝜋∕2 and 𝜑 = 3𝜋∕2) it is described by expressions identical to (14.3) and (14.5) for acoustic waves: Hx(0) = u(0) h

if

H0x = u0 .

(14.78)

In particular, this relationship means that the PO curves in Figures 14.4 to 14.10 plotted for the acoustic waves scattered from a hard cylinder also demonstrate the electromagnetic waves with Hx -polarization scattered from a perfectly conducting cylinder of the same size.

389

ELECTROMAGNETIC WAVES

The field radiated by the nonuniform component of the current (⃗j(1) ) is calculated according to Section 7.8. The expression Hx(1)left = H0x

a eikR iq e ∫0 2𝜋 R

2𝜋

{cos 𝛾 cos 𝜓 F𝜃(1) (𝜓, 𝜃 l , 𝜙l ) ⋅ 𝜙x

+ sin 𝜓 [G(1) (𝜓, 𝜃 l , 𝜙l ) ⋅ 𝜙x − G(1) (𝜓, 𝜃 l , 𝜙l ) ⋅ 𝜃x ]}eip sin 𝜓 d𝜓 𝜃 𝜙 (14.79) describes the field scattered from the left edge, and the expression (1)right

Hx

= H0x

a eikR −iq e ∫𝜋 2𝜋 R

2𝜋

{cos 𝛾 cos 𝜓 F𝜃(1) (𝜓, 𝜃 r , 𝜙r ) ⋅ 𝜙x

+ sin 𝜓 [G(1) (𝜓, 𝜃 r , 𝜙r ) ⋅ 𝜙x − G(1) (𝜓, 𝜃 r , 𝜙r ) ⋅ 𝜃x ]}eip sin 𝜓 d𝜓. 𝜃 𝜙 (14.80) represents the field scattered from the right edge. Here p = ka(sin 𝛾 − sin Θ), q = kl(cos Θ − cos 𝛾). In addition, Equations (14.15) to (14.18) define the local angles , and Equations (14.19) and (14.20) determine the unit vectors 𝜃̂ and 𝜙̂ 𝜃, 𝜙l,r and 𝜙l,r 0 (the same for the left and right edges). Note also that the derivation of Equation (14.79) is given in Problem 14.3. For the forward scattering direction Θ = 𝛾 (belonging to the diffraction cone), these expressions lead to (1)right

Hx(1) = Hx

= H0x

a eikR 2𝜋 R

+ sin 𝜓g (𝜋 2

(1)

{

2𝜋

∫𝜋

[

cos2 𝛾 cos2 𝜓f (1) (𝜋 + 𝜙r0 , 𝜙r0 , 𝛼)

] d𝜓

+ 𝜙r0 , 𝜙r0 , 𝛼)

sin2 𝛾0

− sin 𝛾 cos 𝛾

2𝜋

∫𝜋

sin 𝜓 cos2 𝜓 sin2 𝛾0

} d𝜓 (14.81)

Here we omitted the integral over the left edge in view of the grazing singularity discussed in the comments following Equations (14.32) and (14.33). In contrast, the integral (14.81) for the right edge is free of such a singularity. Next we present the results of Equation (14.79) when 𝛼 = 2𝜋 and the cylinder transforms to the disk. In this particular case, Equation (14.79) takes the form Hx(1)

[ ) ( )] ikR 1 + cos2 𝛾 ( 𝜋 e a 𝜋 = H0x E , sin 𝛾 − 2 cos 𝛾K , sin 𝛾 , (14.82) 2 cos 𝛾 2 2 R 𝜋 sin 𝛾

where the functions E(𝜋∕2, x) and K(𝜋∕2, x) are as defined in Equation (14.72).

390

BISTATIC SCATTERING AT A FINITE-LENGTH CYLINDER

To continue the comparative analysis of electromagnetic and acoustic waves scattered at a finite cylinder, it is convenient to represent the total scattered field in the form + Φ(1) ] Hx = Hx(0) + Hx(1) = H0x [Φ(0) hx hx

eikR eikR = H0x Φhx (Θ, 𝛾) , R R

(14.83)

similar to Equation (14.22) for acoustic waves. Due to Equation (14.78), all of the beam and ray asymptotics for the electromagnetic field Hx(0) are the same as those for the acoustic field u(0) (presented in Section 14.1.4): h Φ(0) (Θ, 𝛾) = Φ(0) (Θ, 𝛾). hx h

(14.84)

It turns out that the ray asymptotics for the electromagnetic field Hx(1) and those in Section 14.1.4 for the acoustic fringe field u(1) are also identical: h Φ(1) (Θ, 𝛾) = Φ(1) (Θ, 𝛾). hx h

(14.85)

However, the beam asymptotics for the fields Hx(1) and u(1) generated by the nonunih form (fringe) components j(1) in the scattering directions Θ = 𝛾 and Θ = 𝜋 − 𝛾 are different, (Θ, 𝛾) ≠ Φ(1)beam (Θ, 𝛾), Φ(1)beam hx h

(14.86)

although they are of the same order of magnitude. Besides, the examination of Figures 14.7 and 14.9 reveals the following situations:

r For a cylinder with diameter d = 𝜆 and length L = 3𝜆, the quantity Φ(1) is about h

in the direction Θ = 𝛾 and Θ = 𝜋 − 𝛾, 18 dB less than for the PO beams Φ(0)beam h and it is about 25 dB less in the direction Θ = 2𝜋 − 𝛾. r For a cylinder with diameter d = 3𝜆 and length L = 9𝜆, the quantity Φ(1) is h in the directions Θ = 𝛾 and about 28 dB less than for the PO beams Φ(0)beam h Θ = 2𝜋 − 𝛾 and about 35 dB less in the direction Θ = 2𝜋 − 𝛾.

14.2.3 Refined Asymptotics for the Specular Beam Reflected from the Lateral Surface This section is the electromagnetic version of Section 14.1.5. It covers the scattered field in the vicinity of the specular direction Θ = 2𝜋 − 𝛾 in the plane 𝜑 = 3𝜋∕2 . This study is based on a paper by Ufimtsev (1981).

ELECTROMAGNETIC WAVES

391

The incident waves with Ex - and Hx -polarizations are defined by Equations (14.65) and (14.77). According to Sections 14.2.1 and 14.2.2, the following asymptotic relationships exist between the electromagnetic and acoustic scattered waves: (1)fr Ex = Ex(0) + Ex(1)fr = u(0) s + us

Hx = Hx(0) + Hx(1)fr = u(0) + u(1)fr h h

if

E0x = u0

(14.87)

if

H0x = u0 .

(14.88)

They are valid under the conditions ka sin 𝛾 ≫ 1 and kl ≫ 1 with an additional restriction pointed below. Here fields with the superscript “0” are generated by the uniform components of the surface current (j(0) ) and represent the PO fields. The fields with the superscript “(1)fr” are generated by the nonuniform/fringe components of the surface current (j(1)fr ), which concentrate in the vicinity of the edges. The additional restriction mentioned above relates specifically to these fields. The beams generated by acoustic and electromagnetic fringe sources in the directions Θ = 𝛾 and Θ = 𝜋 − 𝛾 are different in principle, as stressed by Equations (14.76) and (14.86). One should also include in the scattered field the contributions Ex(1)sm and Hx(1)sm generated by that part of the nonuniform component j(1)sm that is caused by smooth bending of the cylindrical surface. As shown in Section 14.1.5 for acoustic waves, such, contributions (in the vicinity of the specular direction) are quantities of the same order of magnitude as those radiated by the fringe currents. To calculate them for electromagnetic waves is a principal objective of the present section. First it necessary to determine the nonuniform currents ⃗j(1)sm . We use the results of a paper by Franz and Galle (1955), which are reproduced in a work of Bowman et al. (1987). This paper contains the high-frequency asymptotics for the surface field induced on an infinite circular cylinder by an incident plane wave. It is assumed there that the incident wave propagates in the direction perpendicular to the cylinder axis. A quite subtle procedure is to extend the results of this paper to oblique incidence and obtain correct currents ⃗j(1)sm . Eventually, one obtains i eik(z cos 𝛾+a sin 𝛾 sin 𝜓) , ka sin2 𝜓 i cos 𝜓 ik(z cos 𝛾+a sin 𝛾 sin 𝜓) e = Y0 E0x , ka sin3 𝜓 i cos 𝛾 cos 𝜓 ik(z cos 𝛾+a sin 𝛾 sin 𝜓) e = Y0 E0x ka sin 𝛾 sin3 𝜓

= −Y0 E0x j(1)sm x

(14.89)

j(1)sm y

(14.90)

j(1)sm z

(14.91)

for E-polarization, and j(1)sm = H0x z

i eik(z cos 𝛾+a sin 𝛾 sin 𝜓) , 2 ka sin 𝛾 sin 𝜓

j(1)sm =0 x,y

(14.92)

for H-polarization. Then one employs Equations (1.108) and (1.109) and calculates the retarded vector-potential. The integrals over the variable 𝜁 are calculated in closed

392

BISTATIC SCATTERING AT A FINITE-LENGTH CYLINDER

form. Integrals over the variable 𝜓 are evaluated asymptotically using the stationaryphase technique. As a result, one finds that Ex(1)sm = E0x Φ(1)sm ex (Θ, 𝛾)

eikR , R

(14.93)

(Θ, 𝛾) Hx(1)sm = H0x Φ(1)sm hx

eikR , R

(14.94)

where Φ(1)sm ex (Θ, 𝛾) = l

sin q e−ip+i𝜋∕4 , √ q 2𝜋p

Φ(1)sm (Θ, 𝛾) = −l hx

sin Θ sin q e−ip+i𝜋∕4 √ sin 𝛾 q 2𝜋p

(14.95)

(14.96)

with p = ka(sin 𝛾 − sin Θ) and q = kl(cos Θ − cos 𝛾). Comparison of these quantities with their acoustic counterparts (14.55) and (14.56) shows that they differ only in sign. Now one can calculate the total scattered field, Extot = E0x Φtot ex (Θ, 𝛾)

eikR , R

(14.97)

Hxtot = H0x Φtot hx (Θ, 𝛾)

eikR , R

(14.98)

where (0) (1)fr (1)sm Φtot ex = Φex + Φex + Φex ,

(14.99)

(0) (1)fr (1)sm Φtot hx = Φhx + Φhx + Φhx .

(14.100)

According to Equations (14.74), (14.75) and (14.84), (14.85), (0) Φ(0) ex = Φs ,

(1)fr Φ(1)fr , ex = Φs

(14.101)

Φ(0) = Φ(0) , hx h

Φ(1)fr = Φ(1)fr hx h

(14.102)

where Φ(0) and Φ(1)fr are the acoustic functions defined in Section 14.1.5. In exactly s,h s,h the specular direction Θ = 2𝜋 − 𝛾, the scattered field is determined as Φtot ex

[ ]−1 { 2(𝜋 − 2𝛾) 3 l 1 1 ikl sin 𝛾 + = √ −√ + cos 16 a 3 𝜋ka sin 𝛾 3 2 } l 1 e−i2ka sin 𝛾+i𝜋∕4 , − √ + (14.103) 3 3 2a a

PROBLEMS

Φtot hx

393

[ ]−1 { 2(𝜋 − 2𝛾) 3 l 1 1 = √ − ikl sin 𝛾 − −√ + cos 16 a 3 𝜋ka sin 𝛾 3 2 } l 1 e−i2ka sin 𝛾+i𝜋∕4 . + √ + (14.104) 2a 3 3 a

The origination and meaning of each term here is clear. The first and second terms in braces relate, respectively, to the first and second terms in the asymptotic expansion for the PO field. The third and fourth terms represent the contribution by the fringe sources j(1)fr , and the last term shows the contribution by the sources j(1)sm caused by smooth bending of the cylindrical surface. Thus calculation of the specular reflected beam of electromagnetic waves has been completed.

PROBLEMS 14.1

(0)cyl

scattered by the lateral part Derive the PO approximation for the field Ex (−l ≤ z ≤ l) of a finite cylinder (Fig. 14.1) illuminated by the incident wave (14.65). Verify the relationship (14.66) related to this field. Solution The field is calculated according to Equations (1.108), (1.109), and (1.112), (1.113). The magnetic field of the incident plane wave is found ⃗ inc ), where k̂ i = ẑ cos 𝛾 + ŷ sin 𝛾. On the ⃗ inc = Y0 [k⃗i × E using the equation H lateral cylindrical surface (−l ≤ 𝜁 ≤ l, y = a sin 𝜓), this field contains the components Hyinc = Y0 E0x cos 𝛾eik(𝜁 cos 𝛾+a sin 𝛾 sin 𝜓) ,

Hxinc = 0,

Hzinc = −Y0 E0x sin 𝛾eik(𝜁 cos 𝛾+a sin 𝛾 sin 𝜓) and generates the PO surface current (1.113): (0)cyl

= −2Y0 E0x sin 𝛾 sin 𝜓eik(𝜁 cos 𝛾+a sin 𝛾 sin 𝜓) ,

(0)cyl

= 2Y0 E0x sin 𝛾 cos 𝜓eik(𝜁 cos 𝛾+a sin 𝛾 sin 𝜓) ,

(0)cyl

= 2Y0 E0x cos 𝛾 cos 𝜓eik(𝜁 cos 𝛾+a sin 𝛾 sin 𝜓) .

jx jy jz

The retarding vector-potential (1.112) takes the form (0)cyl

Ax,y,z =

l

a eikR d𝜁 4𝜋 R ∫−l ∫𝜋

2𝜋

−ik(𝜁 cos 𝜗+a sin 𝜑 sin 𝜗 sin 𝜓) j(0) d𝜓. x,y,z (𝜁 , 𝜓)e

We calculate the field in the y0z–plane where 𝜑 = 𝜋∕2 or 𝜑 = 3𝜋∕2 and introduce the new coordinate Θ with Equation (14.2). The function cos 𝜓 is

394

BISTATIC SCATTERING AT A FINITE-LENGTH CYLINDER

(0)cyl

(0)cyl

antisymmetric with respect to 𝜓 = 3𝜋∕2; hence, Ay,z = 0 and only Ax generates the scattered field. In accordance with Equation (1.109), it equals (0)cyl (0)cyl Ex = ikZ0 Ax . Here the integral over the variable 𝜁 is calculated in closed form and leads to the final expression (0)cyl

Ex

( = −E0x

sin q ikal sin 𝛾 𝜋 q ∫𝜋

2𝜋

eip sin 𝜓 sin 𝜓 d𝜓

)

eikR , R

where p = ka(sin 𝛾 − sin Θ) and q = kl(cos Θ − cos 𝛾). Comparison with Equations (14.3), (14.4) confirms the relationship (14.66) between electromagnetic and acoustic PO fields. 14.2

Use the general Equations (7.136) and (7.137) for EEWs and derive Equation (14.68) for the field Ex(1)left scattered by the left edge of a finite cylinder. Solution r Re-denote the local coordinates R,𝜗,𝜑 used in Equations (7.136) and (7.137) by 𝜌, 𝜃, 𝜙. See Equations (14.15) to (14.20) for the coordinates 𝜃, 𝜙, 𝜙0 . r Retain the symbols R, 𝜗, 𝜑 for the basic system of coordinates with the origin in the center of a cylinder (Fig. 14.1). r Calculate the scattered field in the y0z-plane where 𝜑 = 𝜋∕2 or 𝜑 = 3𝜋∕2. Introduce a new coordinate Θ with Equation (14.2). r The incident field (14.65) is given on the left edge as Exinc = E0x exp(ik𝜙inc ) = E0x exp[ik(−l cos 𝛾 + a sin 𝛾 sin 𝜓)].

r For

the far zone (R ≫ ka2 ) 𝜌 = R + l cos Θ − a sin Θ sin 𝜓 and

use

the

approximations

eik𝜌 ik𝜙inc ikR i(q+p sin 𝜓) e e = , 𝜌 R where p = ka(sin 𝛾 − sin Θ), and q = kl(cos Θ − cos 𝛾).

r Now Equation (7.136) takes the form ⃗ (1)left = dE

d𝜁 ⃗(1) eikR ik𝜙inc e ,  (𝜁 , 𝜃 l , 𝜙l ) 2𝜋 R

and according to Equation (7.137) the field scattered by the left edge equals ikR ⃗ (1)left = a e eiq E ∫0 2𝜋 R

2𝜋

⃗ (1) (𝜓, 𝜃 l , 𝜙l ) [E0t (𝜁 )F

⃗ (1) (𝜓, 𝜃 l , 𝜙l )]eip sin 𝜓 d𝜓, + Z0 H0t (𝜁 )G where 𝜁 = a𝜓.

r In view of Figure P13.1, E0t = E0x sin 𝜓 and H0t = −Y0 E0x cos 𝛾 cos 𝜓.

PROBLEMS

395

r Hence, Ex(1)left = E0x

a eikR iq e ∫0 2𝜋 R

2𝜋

{sin 𝜓F𝜃(1) (𝜓, 𝜃 l , 𝜙l ) ⋅ 𝜃x

− cos 𝛾 cos 𝜓[G(1) (𝜓, 𝜃 l , 𝜙l ) ⋅ 𝜃x + G(1) (𝜓, 𝜃 l , 𝜙l ) ⋅ 𝜙x ]}eip sin 𝜓 d𝜓 𝜃 𝜙 in agreement with Equation (14.68). 14.3

Use the general Equations (7.136) and (7.137) for EEWs and derive Equation (14.79) for the field Hx91)left scattered by the left edge of a finite cylinder. Solution r Follow the first three bulleted items in the solution to Problem 14.2. r Represent the incident field (14.77) in the form Hxinc = H0x exp(ik𝜙i ) with 𝜙i = z cos 𝛾 + y sin 𝛾.

r Utilize the far-zone approximations shown in Problem 14.2. r Then Equations (7.136) and (7.137) lead to the electric field ⃗ (1)left = dE

[ a eikR iq ⃗ (1) (𝜓, 𝜃 l , 𝜙l ) e E0t (𝜁 )F 2𝜋 R ] ⃗ (1) (𝜁 , 𝜃 l , 𝜙l ) eip sin 𝜓 d𝜓, + Z0t H0t (𝜁 )G

where 𝜁 = a𝜓, p = ka(sin 𝛾 − sin Θ), and q = kl(cos Θ − cos 𝛾).

r The magnetic field of EEW is found as dH⃗ = Y0 [R̂ × dE⃗ ], ⃗ (1)left = dH

{ a eikR iq e Y0 E0t (𝜁 )F𝜃(1) (𝜓, 𝜃 l , 𝜙l ) ⋅ 𝜙̂ 2𝜋 R [ ]} (𝜓, 𝜃 l , 𝜙l ) ⋅ 𝜙̂ − G(1) (𝜓, 𝜃 l , 𝜙l ) ⋅ 𝜃̂ eip sin 𝜓 d𝜓. + H0t (𝜁 ) G(1) 𝜃 𝜙

Here in view of Figure P13.1, H0t = H0x sin 𝜓 and E0t = Z0 H0x cos 𝛾 cos 𝜓. The unit vectors 𝜃̂ and 𝜙̂ are given in Equations (14.19) and (14.20). r Integrating this field over the left edge and taking its x-component, one arrives at Equation (14.79). 14.4

(1)right

Derive Equation (14.70) for the field Ex the basis of Equation (14.69).

in the shadow direction 𝜗 = 𝛾 on

Solution Utilizing Equations (14.14), (14.17), and (14.18) one can show that for this specific scattering direction, we have 𝜃 r = 𝜋 − 𝛾0 and 𝜙r = 𝜋 + 𝜙r0 . Therefore, according to Equations (7.148) to (7.154), the expression (14.69) transforms to Equation (14.70).

Conclusion

The PTD developed in this book clarifies scattering physics. Via the shadow radiation, it elucidates the nature of Fresnel diffraction and forward scattering as well as the optical theorem. It also establishes the diffraction limit for reduction of the total power scattered by large (compared to the wavelength) objects covered by absorbing materials. This theory shows that even with the application of perfectly absorbing coatings on perfectly reflecting objects, their total scattered power can be reduced solely by a factor of 2. This means that against bistatic sonar and radar, it is impossible to mask the scattering object completely by any absorbing materials (Ufimtsev, 1996). A recent cloaking idea to construct invisible objects is discussed briefly in Section 1.4.3. It consists of the development of nonabsorbing coatings with special tensor permittivity and permeability which would allow the incident wave to bend an object smoothly without reflection and without shadow radiation. As a source-based theory, PTD allows the calculation of contributions to the scattered field which are generated by individual elements of the scattering surface. Such data are valuable in the design of antennas and of objects with given characteristics of radiation and scattering. PTD is a flexible theory amenable to further development and generalization. In combination with other analytic and numerical approaches, it can be used to create efficient hybrid techniques for the solution of complex diffraction problems. Some examples are presented in the papers listed in the reference section “Additional References Related to the PTD Concept: Applications, Modifications, and Developments.” PTD and other high-frequency asymptotic techniques describe the physical structure of diffracted fields and develop useful analytic procedures for their investigation. Fundamentals of the Physical Theory of Diffraction, Second Edition. Pyotr Ya. Ufimtsev. © 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

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However, they are typically limited to perfectly reflecting objects with simple geometry. Objects of practical interest often have more complex geometrical and material structures where it is impossible to find high-frequency solutions in analytical forms. Here a promising idea can be the development of computing algorithms (utilizing direct numerical methods such as MOM and FDTD) for calculation of fundamental diffraction items such as PTD differential diffraction coefficients for elementary fringe waves, GTD integral diffraction coefficients for diffracted rays, typical caustic forms, and so on. Papers focusing on such subjects are being published.

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Additional References Related to the PTD Concept: Applications, Modifications, and Developments de Adana, F.S., O. Gutierrez, I. Gonzales, M.F. Catedra, L. Lozano (2011). Practical Applications of Asymptotic Techniques in Electromagnetics, Artech House, Norwood, MA. Akashi, T., M. Ando, and T. Kinoshita (1989). Effects of multiple diffraction in PTD analysis of scattered field from a conducting disk, Trans. IEICE, E72, (4), 259–261. Altintas, A., and P. Russer (2001). Time-domain equivalent edge currents for transient scattering, IEEE Trans. Antennas Propagat., 49(4), 602–606. Altintas, A., O.M. Buyukdura, and P.H. Pathak (1994). An extension of the PTD concept for aperture radiation problems, Radio Sci., 29(6), 403–1407. Andersh, D.J., M. Hazlett, S.W. Lee, D.D. Reeves, D.P. Sullivan, and Y. Chu (1984). XPATCH: a high-frequency electromagnetic-scattering code and environment for complex three-dimensional objects, IEEE Antennas Propagat. Mag., 36(1), 65–69. Ando, M. (1990). Modified physical theory of diffraction, in E. Yamashita, Ed., Analysis Methods for EM Wave Problems, Artech House, Boston. Ando, M., and T. Kinoshita (1989a). PO and PTD analysis in polarization prediction for plane wave diffraction from large circular disk, Digests of 1989 IEEE AP/S Int. Symp., June 26–30, 1989, San Jose, CA.

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Ando, M., and T. Kinoshita (1989b). Accuracy comparison of PTD and PO for plane wave diffraction from large circular disk, Trans. IEICE, E72(11), 1212–1218. Asvestas, J.S. (1995). A class of functions with removable singularities and their application in the physical theory of diffraction, Electromagnetics, 15(2), 143–155. Balling, P. (1995). Fringe-currents effects on reflector antenna cross polarization, Electromagnetics, 15(1), 55–69. Bor, S.S., S.Y. Yang, S.M. Yeth, S.R. Hwang, and C.C. Hwang (1996). Electromagnetic backscattering of helicopter rotor, Electromagnetics, 16(1), 63–74. Bouche, D.P., J.J. Bouquet, H. Manene, and R. Mittra (1992). Asymptotic computation of the RCS of low observable axisymmetric objects at high frequency, IEEE Trans. Antennas Propagat., 40(10), 1165–1174. Bouche, D.P., F.A. Molinet, and R. Mittra (1995). Asymptotic and hybrid techniques for electromagnetic scattering, Proc. IEEE, 81(12), 1658–1684. Boutillier, M., and M.A. Blondell-Fournier (1995). CAD based high-frequency RCS computing code for complex objects: Sermat. Special issue on Radar Cross Section of Complex Objects, Ann. Telecomm., 50(5–6), 536–539. Breinbjerg, O., and E. Jorgansen (1999). Slope diffraction in the geometrical and physical theories of diffraction, USNC/URSI Meeting, July 11–16, Orlando, FL. Digests. Brown, R.T. (1984). Treatment of singularities in the physical theory of diffraction, IEEE Trans. Antennas Propagat., AP-32(6), 640–641. Cha, C.C., J. Michels, and E. Starczewski (1988). An analysis of airborne vehicles dependence on frequency and bistatic angle, Proc. 988 IEEE National Radar Conference, pp. 214–219, University of Michigan, Ann Arbor, MI. Corona, P., A. De Bonitatibus, G. Ferrara, and C. Gennarelli (1993). Accurate evaluation of backscattering by 90◦ dihedral corners, Electromagnetics, 13(1), 23–36. Cote, M.G., M.B. Woodworth, and A.D. Yaghjian (1988). Scattering from perfectly conducting cube, IEEE Trans. Antennas Propagat., 36(9), 1321–1329. Domingo, M., R.P., Torres, and M.F. Catedra (1994). Calculation of the RCS from the interaction of edges and faces, IEEE Trans. Antennas Propagat., 42(6), 885–898. Domingo, M., F. Rives, J. Perez, R.P. Torres, and M.F. Catedra (1995). Computation of the RCS of complex bodies modeled using NURBS surfaces, IEEE Trans. Antennas Propagat., 37(6), 36–47. Duan, D.W., Y. Rahmat-Samii, and J.P. Mahon (1991). Scattering from a circular disk: comparative study of PTD and GTD techniques, Proc. IEEE, 79(10), 1472–1480. Gabrielyan, D.D., O.M. Tarasenko, and V.V. Shatskyi (1991). Ispol’zovanie predstavleniya kraevykh voln v sochetanii s metodom integral’nykh uravnenyi pri reshenii zadach difraktsii na ideal’no provodyaschikh telakh slozhnoi formy [Usage of the edge-wave representation combined with the method of integral equations to solve problems of diffraction by ideally conducting bodies with a complicated shape], Radiotekhnika I Elektronika, 36(6), 1159– 1163. [English translation: J. Commun. Technol. Electron.] Guiraud, J.L. (1983). Une approche spectrale de la theorie physique de la diffraction, Ann. Telecomm., 38(3–4), 145–157. Hansen, T.B., and R.A. Shore (1998). Incremental length diffraction coefficients for the shadow boundary of a convex cylinder, IEEE Trans. Antennas Propagat., 46(10), 1458–1466. Hongo, K., and H. Kobayashi (2001). Evaluation of the surface field scattered by an impedance polygonal cylinder, Electromagnetics, 21, 319–339.

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Idemen, M., and A. Buyukaksoy (1984). High-frequency surface currents induced on a perfectly conducting cylindrical reflector, IEEE Trans. Antennas Propagat., AP-32(5), 501– 507. Jeng, S.K. (1998). Near-field scattering by PTD and shooting and bouncing rays, IEEE Trans. Antennas Propagat., AP-46(4), 551–558. Johansen, P.M. (1996). Uniform physical theory of diffraction equivalent edge currents for truncated wedge strips, IEEE Trans. Antennas Propagat., 44(7), 989–995. Johansen, P.M. (1999). Time-domain version of the PTD, IEEE Trans. Antennas Propagat., 47(2), 261–270. Jorgansen, E., S. Maci, and A. Toccafondi (2001). Fringe integral equation method for a truncated grounded dielectric slab, IEEE Trans. Antennas Propagat., 49(8), 1210– 1217. Kim, J.J., and O.B. Kesler (1996). Hybrid scattering analysis (PTD+UFIM) for large airframe with small details, USNC/URSI Radio Science Meeting, Digests, July 21–26, 1996, Baltimore, MD. Kim, S.Y., J.W. Ra, and S.Y. Shin (1991). Diffraction by an arbitrary-angled dielectric wedge: Part II. Corrections to physical optics solution, IEEE Trans. Antennas Propagat., 39(9), 1282–1292. Kobayashi, H., and K. Hongo (1997). Scattering of electromagnetic plane waves by conducting plates, Electromagnetics, 17(6), 573–587. Landsberg, I.L. (1974). O polarizatsionnoi structure izlucheniya osesymmetrichnogo zerkala vblizi osi [Polarization structure of radiation by an axisymmetric reflector close to the symmetry axis], Radiotekhnika i Elektronika, 19(9), 1817–1823 [English translation: Radio Eng. Electron. Phys]. Landsberg, I.L. (1979). Scattering of a plane wave at a metallic cone close to its symmetry axis (in Russian), Radiotekhnika i Elektronika, 24(5), 886 [English translation: Radio Eng. Electron. Phys]. Lee, S.W. (1977). Comparison of uniform asymptotic theory and Ufimtsev’s theory of electromagnetic edge diffraction, IEEE Trans. Antennas Propagat., AP-25(2), 162. Martinez-Burdalo, M., A. Martin, and R. Villar (1993). Uniform PO and PTD solution for calculating plane wave backscattering from a finite cylindrical shell of arbitrary cross section, IEEE Trans. Antennas Propagat., 41(9), 1336–1339. Michaeli, A. (1985). A new asymptotic high-frequency analysis of electromagnetic scattering by a pair parallel wedges: closed form results, Radio Sci., 20, 1537–1548. Michaeli, A. (1995). Incremental diffraction coefficients for the extended physical theory of diffraction, IEEE Trans. Antennas Propagat., 43(7), 732–734. Molinet, F.A. (1991). Modern high frequency techniques for RCS computation: a comparative analysis. Special issue on RCS, ACS J., Sept. Morita, N. (1971). Diffraction by arbitrary cross-sectional semi-infinite conductor, IEEE Trans. Antennas Propagat., AP-19(5), 358–364. Murthy, P.K., and G.A. Thiele (1986). Non-uniform currents on a wedge illuminated by a TE-plane wave, IEEE Trans. Antennas Propagat., AP-34(8), 1038–1045. Pelosi, G., S. Maci, R. Tiberio, and A. Michaeli (1992). Incremental length diffraction coefficients for an impedance wedge, IEEE Trans. Antennas Propagat., 40(10), 1201– 1210.

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Polycarpou, A.C., C.A. Balanis, and C.R. Bitcher (1995). Radar cross section of trihedral corner reflectors using PO and MEC. Special issue on Radar Cross Section of Complex Objects, Ann. Telecomm., 50(5–6), 510–516. Ruis, J.M., M. Ferrando, and L. Jofre (1993a). GRECO: graphical electromagnetic computing for RCS prediction in real-time, IEEE Antennas Propagat. Mag., 35(2), 7–17. Ruis, J.M., M. Ferrando, and L. Jofre (1993b). GRECO: high-frequency RCS of complex radar targets in real-time, IEEE Trans. Antennas Propagat., 41(9), 1308–1319. Rius, J.M., M. Vall-Lossera, and A. Cardama (1995). GRECO: graphycal processing methods for high-frequency RCS prediction. Special issue on Radar Cross Section of Complex Objects, Ann. Telecomm., 50(5–6), 551–556. Schretter, S.J., and D.M. Bolle (1969). Surface currents on a wedge under plane wave illumination: an approximation, IEEE Trans. Antennas Propagat., AP-17, 246–248. Shore, R.A. and A.D. Yaghjian (1993). Application of incremental length diffraction coefficients to calculate the pattern effects of the rim and surface cracks of the reflector antenna, IEEE Trans. Antennas Propagat., 41(1), 1–11. Shore, R.A., and A.D. Yaghjian (2004). A comparison of high-frequency scattering determined from PO, enhanced with alternative ILDC’s, IEEE Trans. Antennas Propagat., 52(1), 336– 341. Skyttemyr, S.S. (1986). Cross polarization in dual reflector antennas: a PO and PTD analysis, IEEE Trans. Antennas Propagat., AP-34(6), 849–853. Somov, V.A., and Vyaz’mitinova (1990). Primenenie metoda kraevykh voln pri chislennom analize zerkal’nykh antenn. [Application of the edge wave method for the numeric analysis of reflector antennas], Padiotekhnika, 1, 69–71 [English translation: Telecommunication and Radio Engineering, 45(2), 99–102, published by Scripta Technica]. Syed, H.H., and J.L. Volakis (1996). PTD analysis of impedance structures, IEEE Trans. Antennas Propagat., 44(7), 983–988. Tran, H.B., and T.J. Kim (1989). The interior wedge scattering, Chap. 4 in Monostatic and Bistatic RCS Analysis, vol. 1, Northrop Corporation, Aircraft Division, Report NOR-82215. Vasil’ev, E.N., V.V. Solodukhov, and A.I. Fedorenko (1991). The integral equation method in the problem of electromagnetic waves diffraction by complex bodies, Electromagnetics, 11(2), 161–182. Vermersch, S., M. Sesques, and D. Bouche (May-June 1995). Computation of the RCS of coated objects by a generalized PTD approach. Special issue on Radar Cross Section of Complex Objects, Ann. Telecomm., 50(5–6), 563–572. Vesnik, M.V., and P.Y. Ufimtsev (1992). An asymptotic feature of corner waves scattered by polygonal plates, Electromagnetics, 12(3), 265– 272. Wang, D.S., and L.N. Medgyesi-Mitschang (1985). Electromagnetic scattering from finite circular and elliptic cone, IEEE Trans. Antennas Propagat., AP-33(5), 488–497. Wang, S.Y., and S.K. Jeng (1998). A compact RCS formula for a dihedral corner reflector at arbitrary aspect angles, IEEE Trans. Antennas Propagat., AP-46(7), 1112–1113. Weinmann, F. (2006). Ray tracing with PO/PTD for RCS modeling of large complex objects, IEEE Trans. Antennas Propagat., 54(6), 1797–1806. Yaghjian, A.D., and R.V. McGahan (1985). Broadside RCS of the perfectly conducting cube, IEEE Trans. Antennas Propagat., AP-33(3), 321.

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Yarygin, A.P. (1972). Primenenie metoda kraevykh voln v zadachakh difraktsii na telakh nakhodyaschikhsya v plavno neodnorodnoi srede [Application of the edge waves method to problems of diffraction from bodies placed in smoothly inhomogeneous medium], Radiotekhnika i Elektronika, 17(10), 1601–1609 [English translation: Radio Eng. Electron. Phys.]. Youssef, N.N. (1989). Radar cross section of complex targets, Proc. IEEE, 77(5), 722–734. Additional References Related to Vertex and Edge Waves Babich, V.M., M.A. Lyalinov, and V.E. Grikurov (2007). Diffraction Theory: The Sommerfeld– Malyuzhinets Technique. Alpha Science, Oxford, UK. Bernard, J.M.L., M.A. Lyalinov, and N.Y. Zhu (2008). Analytical-numerical calculation of diffraction coefficients for a circular impedance cone, IEEE Trans. Antennas Propagat., 56(6), 1616–1623. Budaev, B.V., and D.B. Bogy, (2005). Two-dimensional diffraction by a wedge with impedance boundary conditions, IEEE Trans. Antennas Propagat., 53(6), 2073–2080. Daniele, V., and C. Lombardi (2011). Wiener–Hopf solution of the isotropic penetrable wedge problem, IEEE Trans. Antennas Propagat., 59(10), 3797–3818. Gennareli, G., and G. Riccio (2011). A uniform asymptotic solution for diffraction by a right-angle dielectric wedge, IEEE Trans. Antennas Propagat., 53(3), 898–903. Lyalinov, M.A., and N.Y. Zhu (2013). Scattering of Waves by Wedges and Cones with Impedance Boundary Conditions, SciTech Publishing, Raleigh, NC. Osipov, A.V., and T.B.A. Senior (2009). Diffraction and reflection of a plane wave by a right-angle impedance wedge, IEEE Trans. Antennas Propagat., 57(6), 1789–1797. Syed, H.H., and J.L. Volakis (1996). PTD analysis of impedance structures, IEEE Trans. Antennas Propagat., 44(7), 983–988. Ufimtsev P. Ya. (2013b). Diffraction at a wedge with one face electric and the other face magnetic: Exact and asymptotic solutions of the wave and parabolic equations, IEEE Antennas and Propagation Magazine, 55(5), 63–73. Vesnik, M.V. (2001). Analytical solution for electromagnetic diffraction on 2-D perfectly conducting scatterers of arbitrary shape, IEEE Trans. Antennas Propagat. 49(12), 1638– 1644. Vesnik, M.V. (2008). Analytic solution of the diffraction problem for the 2-D perfectly conducting half-plate using the generalized eikonal method, Radiotekhnika i Electronika, 53(1), 1–13 (English translation: J. Commun. Technol. Electron.).

Appendix to Chapter 4

MATLAB Codes for Two-Dimensional Fringe Waves and Figures Main Program % ----------------------------------------------------------------------------------------------------------------------------------% Name : main_fringe.m % Author : Feray Hacivelioglu, Levent Sevgi % Purpose : to calculate and compare normalized exact and asymptotic fringe fields % ----------------------------------------------------------------------------------------------------------------------------------% *************************************************************************************** % ********************************** Input parameters *************************************** % *************************************************************************************** Fringe = 1; Fringe_Asym = 1; alfamax = 300; %input('wedge angle [Deg] = '); angle0 = 45; %input('incident angle [Deg] = '); %**************************************************************************************** alfa = alfamax*pi/180; % change wedge angle degree to radians angle0 = angle0*pi/180; % change incident angle degree to radians x = 1; % r = x*lamda; r:observation distance kr = 2*x*pi; A = 0:2.0:alfamax; % observation angle [dd nA]=size(A); Ar=A*pi/180; % change observation angle degree to radians %**************************************************************************************** % --------------------------------------------------------- Exact Fringe Fields -------------------------------------------------if (Fringe == 1) for m = 1:nA fprintf(1, 'Calculating Exact Fringe Field = > Angle %5.3f : \n',A(m)); angle = Ar(m); % Absolute values of the normalized exact fringe fields: | u(1)/u0 | Us_Fringe(m) = abs(Int_calcFringe(alfa,kr,angle0,angle,'Soft')); % for soft boundary conditions Uh_Fringe(m) = abs(Int_calcFringe(alfa,kr,angle0,angle,'Hard')); % for hard boundary conditions end; end

Fundamentals of the Physical Theory of Diffraction, Second Edition. Pyotr Ya. Ufimtsev. © 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

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MATLAB CODES FOR TWO-DIMENSIONAL FRINGE WAVES AND FIGURES

% --------------------------------------------------- Asymptotic Fringe Fields ----------------------------------------------if (Fringe_Asym == 1) coeff = exp(1i*(kr+pi/4))/sqrt(2*pi*kr); for m = 1:nA fprintf(1, 'Calculating PWA Fringe Field = > Angle %5.3f : \n',A(m)); angle = Ar(m); [f1,g1] = fun_fg(angle,angle0,alfa); % directivity patterns % Absolute values of the normalized asymptotic fringe fields : | u(1)/u0 | Us_Fringe_PWA(m) = abs(f1.*coeff); % for soft boundary conditions Uh_Fringe_PWA(m) = abs(g1.*coeff); % for hard boundary conditions end; end % ---------------------------------------------------------------- END -------------------------------------------------------- ----% **************************************** FIGURES ************************************** figure (1) polar(Ar,Us_Fringe, 'k'); hold on; polar(Ar,Uh_Fringe_PWA,'k:'); hold on; legend('Exact','Asymp'); figure (2) polar(Ar,Uh_Fringe ,'k'); hold on; polar(Ar,Uh_Fringe_PWA,'k:'); hold on; legend('Exact','Asymp');

Codes for Exact Integrals (4.16), (4.17), (4.18), and (4.19) % ------------------------------------------------------------------------------------------------------------------------------------% Name : Int_calcFringe.m % Author : Feray Hacivelioglu, Levent Sevgi % Purpose : to calculate normalized fringe waves using (4.16), (4.17) for SSI and (4.18), (4.19) for DSI % ------------------------------------------------------------------------------------------------------------------------------------function result = Int_calcFringe(alfa,kr,angle0,angle,S_H) % S_H : Boundary condition: ‘Soft’ or ‘Hard’ eps1 = 1e-12; y_old = 1e6; y1_old = 1e6; y2_old = 1e6; Mmax = 500; eps = 1e-6; % 0