Golf and Wind: The Physics of Playing Golf in Wind [1st ed.] 9789811597190, 9789811597206

Table of contents :
Front Matter ....Pages i-xxi
Introduction (Shantanu Malik, Sandeep Saha)....Pages 1-14
Understanding the Motion and the Environment (Shantanu Malik, Sandeep Saha)....Pages 15-41
Simulating the Motion in a Synthesised Environment (Shantanu Malik, Sandeep Saha)....Pages 43-62
How Does Wind Impact Gameplay? (Shantanu Malik, Sandeep Saha)....Pages 63-86
Gameplay, the Course and Wind (Shantanu Malik, Sandeep Saha)....Pages 87-111
Concluding Thoughts (Shantanu Malik, Sandeep Saha)....Pages 113-114

Citation preview

Shantanu Malik Sandeep Saha

Golf and Wind The Physics of Playing Golf in Wind

Golf and Wind

Shantanu Malik Sandeep Saha •

Golf and Wind The Physics of Playing Golf in Wind

123

Shantanu Malik Department of Aerospace Engineering Indian Institute of Technology Kharagpur Kharagpur, West Bengal, India

Sandeep Saha Department of Aerospace Engineering Indian Institute of Technology Kharagpur Kharagpur, West Bengal, India

ISBN 978-981-15-9719-0 ISBN 978-981-15-9720-6 https://doi.org/10.1007/978-981-15-9720-6

(eBook)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

“Golf is no longer a game of hitting the ball, finding it, and hitting it again. There is wind to be measured, whether that means tossing blades of grass in the air or studying the gentle movement of 60-foot high branches. There are caddie conferences for even the most routine shots. There are sports psychologists who tell players not to hit until they’re ready.” —Doug Ferguson, (PGA Lifetime Achievement Award in Journalism, 2019)

Preface

Golf as a sport is almost inevitably viewed as a game of the privileged and most often beyond the reach of the commoners. Though this might be true in practice, there is very little which prevents an enthusiastic physicist or an astute student of aerodynamics to wonder about the motion of a golf ball. There is no reason to suspect that such curiosity would be missing in an experienced golfer who perhaps is able to visualise the motion of the golf ball before playing the shot but knows little about the physics behind it. Perhaps a novice golfer would also feel the same urge to understand the golf ball’s trajectory as the subtleties of the club selection seem perplexing. To this end, the golf club and ball manufacturer would have a similar desire in order to design products which cater to the needs of golfers. Possibly the golf course architect has an intuitive understanding of all these perspectives but is left guessing about quantifying the uncertainties arising from the weather, for instance, the wind, while designing the course. The contents of this book concern all of them and has been presented in a style which should be comprehensible to all alike. The title, “Golf and Wind: The Physics of Playing Golf”, states quite clearly the subject of this book. At first sight, it might even appear that the effect of wind is too obvious and must have certainly been documented given the game dates back to the fifteenth century. However, as we dig a bit deeper, numerous questions spring to mind instantaneously without any significant literature to quell these queries. Most experienced golfers already know how wind induces a deviation and also make corrections to their game to play with the wind. Despite this intuitive knowledge, there is a lack of a scientific investigation into this problem which distils the effects in a systematic manner. Indeed, the fact that wind-induced uncertainties in golf could be three times that of inconsistencies in a golfer’s shot would be nerve-wracking. This is especially true because approach shots and pitch shots, which require great precision, are also the ones affected by wind more by a factor of 50–100% compared to drive shots, wherein distance holds more importance than accuracy. Speaking of distance, a ball that travels further and higher is often a preferred choice by most golfers. But is it necessarily the better choice while playing on a windy day? No, we find that such a vii

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ball is considerably less robust to the prevailing wind in the course. This is an important point for beginners to keep in mind, untill they develop a good judgement about playing in wind. Although counter-intuitive, we explain why a ball with inferior aerodynamic characteristics could be a safer choice when there is stronger wind. We had begun on this quest of exploring golf merely out of curiosity. As students of aerodynamics, the motion of a golf ball was not only fun to discuss, but also contained enough challenge for us to investigate. Trips to golf courses, conversations with golf players and coaches, reading blogs and articles and numerous brainstorming sessions helped identify important questions on the subject and hence use our engineering and analytical skills to come up with answers to these questions. A project which started with a simple motive to determine the aerodynamics characteristics of golf balls has come a long way to become an extensive study. We hope that this book would be of help to golf players, golf course architects and club/ball manufacturers. At the same time, we hope this would be an interesting and informative read for students, as well as golf enthusiasts. We invite all such enthusiasts to join our quest and welcome you on this journey to unravel what golf and wind together conspire on the course. Along with this book, we also provide our MATLAB-based tool allowing users to plot the trajectories of golf balls while varying the launch conditions, wind conditions, aerodynamic and material characteristics of the golf ball. This tool is freely available for use on our GitHub page (link below). https://github.com/golfandwind/trajectory-plotter. Kharagpur, India August 2020

Shantanu Malik Sandeep Saha

Acknowledgements

We would like to extend our thanks to Mr. Manu Gulati and the staff at The Tollygunge Club, Kolkata, who helped us get some insights about the game of golf, and thus understand the kind of problems golfers face during the game. We also acknowledge Mr. Subodh Ranjan, who assisted us in the preliminary stages of this work and Mr. Siddhant Gupta for aiding us with text formatting. We would also like to thank Ms. Priya Vyas for supporting us all throughout the publication process. Finally, a shout out to our family and friends for backing us with constant encouragement to finish this book, which seemed only a dream a few months earlier.

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Contents

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2 Understanding the Motion and the Environment 2.1 Trajectory of a Golf Ball . . . . . . . . . . . . . . . . 2.1.1 Flight . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Ground Run—Bouncing and Rolling . 2.2 Golf Course Environment . . . . . . . . . . . . . . . 2.2.1 Terrain . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Features and Layout . . . . . . . . . . . . . . 2.2.3 Wind . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Types of Golf Course Designs . . . . . . 2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Simulating the Motion in a Synthesised Environment . 3.1 Simulation Programme . . . . . . . . . . . . . . . . . . . . . 3.1.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Numerical Schemes . . . . . . . . . . . . . . . . . . 3.2 Constructing a Model Golf Course . . . . . . . . . . . .

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1 Introduction . . . . . . . . . . . . . . . . . . 1.1 The Game of Golf . . . . . . . . . . 1.2 Golf and Physics . . . . . . . . . . . 1.2.1 Launching a Golf Ball . . 1.2.2 A Golf Ball in Flight . . . 1.2.3 Landing on the Fairway . 1.3 The Golfing Environment . . . . . 1.3.1 Golf Course Architecture 1.3.2 Wind . . . . . . . . . . . . . . 1.4 Golf and Wind? . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . .

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3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 How Does Wind Impact Gameplay? . 4.1 Impact of Wind on Gameplay . . . 4.2 Analysing Wind’s Effect . . . . . . . 4.2.1 Heading and Wind . . . . . . 4.2.2 Clubs and Wind . . . . . . . 4.2.3 Aerodynamics and Wind . 4.3 Summary . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .

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5 Gameplay, the Course and Wind . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Playing Golf in Windy Conditions . . . . . . . . . . . . . . . . . . . . . 5.1.1 Compensating for Headwind and Tailwind . . . . . . . . . 5.1.2 Compensating for Crosswind . . . . . . . . . . . . . . . . . . . 5.1.3 Challenges and Limitations . . . . . . . . . . . . . . . . . . . . 5.2 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Penal: Hole 17, Tournament Players Club, Florida . . . . 5.2.2 Strategic: Hole 8, Muirfield Golf Club, Scotland . . . . . 5.2.3 Heroic: Hole 18, Pebble Beach Golf Links, California . 5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Concluding Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

About the Authors

Shantanu Malik has obtained his bachelor’s and master’s degrees in Aerospace Engineering from IIT Kharagpur in 2020. He has been working on golf ball aerodynamics and flight simulation for the past three years. While at IIT Kharagpur, he has also worked on projects related to dynamics and control of UAVs, as well as gravity and drag simulation for satellites in orbit. Shantanu holds technical skills in the areas of Programming & Simulation (MATLAB, C/C++, Simulink, Simscape, Arduino IDE, Solidworks), flight mechanics & control (Control theory, Aircraft Spacecraft & Helicopter dynamics) and flight testing (Hardware implementation, PID tuning, Piloting UAVs, In-flight experiments). Dr. Sandeep Saha is an Assistant Professor at the Department of Aerospace Engineering, IIT Kharagpur. He obtained his bachelors and masters degrees in Mechanical Engineering from IIT Kharagpur. He completed his PhD in Mechanical Engineering from Imperial College, London. Before joining IIT Kharagpur, he was working as Marie-Curie Experienced Researcher, C.N.R.S. (laboratoire FAST), Orsay, France. He worked as Aerodynamics Engineer, ALSTOM Power, Rugby, UK; as Research Scientist (Fluids), Schlumberger Gould Research, Cambridge, UK; and as academics Staff member, Mechanical Engineering, Univerisity of Duisburg-Essen, Germany (in collaboration with SIEMENS AG). Dr. Saha has published various articles in reputed journals and presented papers in many prestigious conferences. His area of specialisation is fluid mechanics.

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List of Figures

Fig. 1.1

Fig. 1.2

Fig. 1.3

Fig. 2.1

Fig. 2.2 Fig. 2.3 Fig. 2.4 Fig. 2.5

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Fig. 2.7

Hole 18, Pebble Beach Golf Links, California. Image courtesy Google Earth, Data MVARI, Data SIO, NOAA, US Navy, NGA, GEBCO Landsat/Copernicus [1] . . . . . . . . . . . . . . . . . . a Hole 17, Tournament Players Club in Florida [2]; b Hole 12, Augusta National Golf Club in Georgia [3]. Image courtesy Google Earth, Data SIO, NOAA, US Navy, NGA, GEBCO Landsat/Copernicus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nine Bridges Golf Course in South Korea [4]. Image courtesy Google Earth, CNES/Airbus, Data SIO, NOAA, US Navy, NGA, GEBCO, SK telecom, Landsat/Copernicus . . . . . . . . . . Parts of a golf ball trajectory: flight (black), bouncing (red), rolling (blue); a Complete trajectory and b ground run (enlarged); Launch conditions: u0 ¼ 80 m/s, h ¼ 10 and N0 ¼ 1500 RPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Forces acting on a golf ball . . . . . . . . . . . . . . . . . . . . . . . . . . Coordinate system and trajectory characteristics . . . . . . . . . . . Impact between the golf club and golf ball in a side-view and b top-view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison between a golf ball trajectory (red) and a regular parabolic projectile (black dashed) having u0 ¼ 60 m/s, N ¼ 4000 RPM and launch angles a h ¼ 15 and b h ¼ 45 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a Qualitative variation of drag with airspeed for a smooth sphere; b Qualitative comparison of drag forces acting on a smooth sphere (black) and a golf ball (red) as the airspeed varies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ground runs of a typical drive shot (red) and approach shot (blue) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Fig. 2.16 Fig. 2.17 Fig. 2.18

Fig. 2.19 Fig. 2.20 Fig. 3.1 Fig. 3.2 Fig. 3.3 Fig. Fig. Fig. Fig. Fig. Fig.

3.4 3.5 3.6 3.7 3.8 3.9

a Impact of the golf ball and ground when viewed in the xg  yg (left) and zg  yg (right); b Resultant forces acting on a golf during the impact with the ground. (Figures adapted from Penner [22]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transition of a ball’s motion from bouncing to pure rolling . . Ground run of a golf ball in three dimensions . . . . . . . . . . . . Hole 18, Pebble Beach Golf Links, California. Image courtesy Google Earth, Data MVARI, Data SIO, NOAA, US Navy, NGA, GEBCO Landsat/Copernicus [24] . . . . . . . . . . . . . . . . . Model of a hill using a general paraboloid having cx ¼ 160 m, cy ¼ 100 m, a ¼ 20 m, b ¼ 15 m, a ¼ 90 and h ¼ 5 m . . . . Model of a valley using a general paraboloid having cx ¼ 120 m, cy ¼ 100 m, a ¼ 20 m, b ¼ 15 m, a ¼ 90 and h ¼ 5 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model of an elevated fareway using two paraboloids having: (1) cx ¼ 75 m, cy ¼ 60 m, a ¼ 50 m, b ¼ 15 m, a ¼ 0 , h ¼ 5 m and (2) cx ¼ 75 m, cy ¼ 60 m, a ¼ 40 m, b ¼ 12 m, a ¼ 0 , h ¼ 3:2 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Visualisation of a putting green having cx ¼ 140 m, cy ¼ 140 m, a ¼ 20 m, b ¼ 10 m, a ¼ 0 , and the flag positioned at x ¼ y ¼ 140 m . . . . . . . . . . . . . . . . . . . . . . . . . . Visualisation of a pond having cx ¼ 200 m, cy ¼ 0 m, a ¼ 70 m, b ¼ 70 m and a ¼ 0 . . . . . . . . . . . . . . . . . . . . . . . Visualisation of a sand bunker having cx ¼ 160 m, cy ¼ 160 m, a ¼ 7 m, b ¼ 7 m, a ¼ 0 and h ¼ 1 m . . . . . Schematic of a tree positioned at x ¼ 120 m and y ¼ 165 m, surrounded by cylinder of obstacle interpretation (red) having diameter 6 m and height 10 m . . . . . . . . . . . . . . . . . . . . . . . . Visualisation of out-of-bounds regions . . . . . . . . . . . . . . . . . . Logarithmic wind velocity profile . . . . . . . . . . . . . . . . . . . . . . Computer simulation of a golf ball’s motion in a sythesized golf course environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flowchart of the overall algorithm of the simulation programme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Visualisation of a golf ball’s motion in a sythesized golf course environment using MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . Flowchart of flight subroutine . . . . . . . . . . . . . . . . . . . . . . . . . Flowchart of bounce subroutine . . . . . . . . . . . . . . . . . . . . . . . Flowchart of roll subroutine . . . . . . . . . . . . . . . . . . . . . . . . . . Flowchart of validity check. . . . . . . . . . . . . . . . . . . . . . . . . . . CD evaluated using thin plane spine interpolation . . . . . . . . . . CL evaluated using thin plane spine interpolation . . . . . . . . . .

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List of Figures

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Fig. 3.11

Fig. 3.12 Fig. 3.13 Fig. 3.14 Fig. 3.15 Fig. 4.1 Fig. 4.2

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Fig. 4.4

Fig. 4.5

Fig. 4.6 Fig. 4.7

Comparison between the trajectories: (1) reported by Bearman and Harvey (blue), (2) computed using RK-4 scheme with a time step (dt) of 1 ms (black) and (3) computed using Euler scheme with dt ¼ 10 ms (red) . . . . . . . . . . . . . . . . . . . . . . . . Hole 18, Pebble Beach Golf Links, California. Image courtesy Google Earth, Data MVARI, Data SIO, NOAA, US Navy, NGA, GEBCO Landsat/Copernicus [2] . . . . . . . . . . . . . . . . . . Defining the a domain, b tee and green region and c ellipse modelling the ocean in the Pebble Beach golf course . . . . . . . Modelling the a bunkers, b trees and c out-of-bounds region in the Pebble Beach golf course . . . . . . . . . . . . . . . . . . . . . . . . . a All features to be modelled and b resultant compiled model of the Pebble Beach golf course in MATLAB . . . . . . . . . . . . Simulation of multiple shots in the synthesised environment of the Pebble Beach golf course in MATLAB . . . . . . . . . . . . Types of shots in a typical game of golf. . . . . . . . . . . . . . . . . Comparison between the effects of logarithmic (dashed lines) and uniform (solid lines) wind velocity profiles of a headwind (red) and tailwind (blue) of 10 m/s on a typical approach shot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison between the deflection in the ball’s trajectory caused by the effect of wind, and the deviation in trajectory due to variation in launch speed; Legend: original trajectory (black-dotted), deflected by wind (red-solid), deviated due to variation in launch speed (black solid) . . . . . . . . . . . . . . . . . . Comparison between the deflection in the ball’s trajectory caused by the effect of wind, and the deviation in trajectory due to variation in launch spin rate; Legend: original trajectory (black-dotted), deflected by wind (red-solid), deviated due to variation in launch spin rate (black solid) . . . . . . . . . . . . . . . . Comparison between the deflection in the ball’s trajectory caused by the effect of wind, and the deviation in trajectory due to variation in launch angle; Legend: original trajectory (black-dotted), deflected by wind (red-solid), deviated due to variation in launch angle (black solid) . . . . . . . . . . . . . . . . . . Launch heading, wind heading and relative wind heading . . . Deflection in a golf ball’s trajectory due to wind incident along various directions (b ranging from 0 to 360 ) in: a 3D, b top and c side views; the solid red curve depicts the original trajectory and dotted red curves depict the deflected trajectories; curves on the x-y plane show the loci of first landing positions (dashed-red) and final halt positions (dotted-red) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Fig. 4.8

Fig. 4.9 Fig. 4.10

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Fig. 4.14

Fig. 4.15 Fig. 4.16 Fig. 4.17

Fig. 4.18

Fig. 4.19

Fig. 4.20

Fig. 4.21

Fig. 5.1

List of Figures

Variation in wind-induced deviations in: a landing position, b peak height, c landing angle and d ground run of a golf ball’s trajectory with respect to br . . . . . . . . . . . . . . . . . . . . . . . . . . Variation in time of flight with respect to br . . . . . . . . . . . . . . Deflection in a golf ball’s trajectory due to wind speeds of 3 m/s (red), 5 m/s (green) and 10 m/s (blue) in a 3D and b top views . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flight trajectories for various launch headings w and with no wind (black curves) and a 5 m/s wind having b ¼ 0 (red curves) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Loci of first bounce (solid curves) and final halt positions (dotted curves) for various launch heading w and wind heading: a b ¼ 0 and b b ¼ 30 . . . . . . . . . . . . . . . . . . . . . . Loci of final halt positions after launching from origin (blue) and launch positions to halt at origin (red) for various launch headings w and wind heading b ¼ 30 . . . . . . . . . . . . . . . . . . Golf ball trajectories of standard club shots (longest to shortest): (1) Driver, (2) 3-Iron, (3) 7-Iron and (4) Pitching wedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectories characteristics a Rf , b hM and c c of average club shots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Club-wise relative wind-induced deviations in a Rf and b hM due to a tailwind of 5 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . Relative wind-induced deviations in Rf a wind of 5 m=s for b ¼ 0 (tailwind) to 180 (headwind) for clubs: (1) Driver (blue), (2) 3-Iron (red), (3) 7-Iron (yellow) and (4) Pitching wedge (green) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectories of a conventional golf ball (red) and a hexagonally-dimpled golf ball (blue) for the clubs: a Drive, b 3-Iron, c 7-Iron and d Pitching wedge. . . . . . . . . . . . . . . . . Club-wise trajectory characteristics a Rf , b hM and c Tf for a conventional golf ball (red) and a hexagonally-dimpled golf ball (blue) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deflections in the flight trajectories of a 7-Iron shot due to a 5 m=s wind with respect to br , for a conventional golf ball (red) and a hexagonally-dimplied golf ball (blue) . . . . . . . . . . Variation in dRf of a drive (solid curves) and a 7-Iron shot (dashed curves) due to a 5 m=s wind with respect to br , for a conventional golf ball (red) and a hexagonally-dimplied golf ball (blue) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compensating for headwind; trajectories of a regular 7 Iron shot [2], without wind (red solid), 7 Iron shot in presence of a 5 m/s headwind (red dashed) and a 5 Iron shot [2], in presence of a 5 m/s headwind (black solid) . . . . . . . . . . . . . . . . . . . . . . . .

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List of Figures

Fig. 5.2

Fig. 5.3

Fig. 5.4

Fig. 5.5

Fig. 5.6

Fig. 5.7

Fig. 5.8

Fig. 5.9

Fig. 5.10

Fig. 5.11

Fig. 5.12

Compensating for tailwind; trajectories of a regular 7 Iron shot [2], without wind (red solid), 7 Iron shot in presence of a 5 m/s tailwind (red dashed) and an 8 Iron shot [2], in presence of a 5 m/s tailwind (black solid) . . . . . . . . . . . . . . . . . . . . . . . . . . Compensating for crosswind; trajectories of a regular 7 Iron shot [2] without wind (red solid), in presence of a 5 m/s crosswind (red dashed), having w ¼ 7:5 in presence of a 5 m/s crosswind (black dashed) and having a sidespin of 1250 RPM (black solid) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hole 17, Tournament Players Club in Florida. Image courtesy Google Earth, Data SIO, NOAA, US Navy, NGA, GEBCO Landsat/Copernicus [6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a Identifying the features to be including in the model and b–c compiled model of Tournament Players Club in MATLAB in top view and 3D view respectively . . . . . . . . . . . . . . . . . . . . . Playing a regular 9 Iron shot [2]: a in the absence of wind, b in the presence of a 2 m/s wind and c in the presence of a 5 m/s wind along various directions; black solid curve: original trajectory, black dashed curve: deflected and valid trajectory, red dashed curve: deflected and invalid trajectory . . . . . . . . . . Mean deviation in first bounce location due to winds incident along various directions v/s magnitude of wind speeds ðm=sÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hole 8, Muirfield Golf Clubs, Scotland. Image courtesy Google Earth, CNES/Airbus, Data SIO, NOAA, US Navy, NGA, GEBCO, IBCAO, Landsat/Copernicus, Mexam Technologies [9] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a Identifying the features to be included in the model and b–c compiled model of Muirfield Golf Club in MATLAB in top view and 3D view, respectively . . . . . . . . . . . . . . . . . . . . . . . Playing in the absence of wind: a 3D view and b top view; red trajectories: greater total distance covered but less accuracy required (1st strategy); blue trajectories: shorter total distance but more accuracy required (2nd strategy) . . . . . . . . . . . . . . . Playing in the presence of: a a 5 m/s crosswind inwards (b ¼ 60 ) and b a 5 m/s crosswind outwards (b ¼ 120 ); red trajectories: route 1, blue trajectories: route 2, purple trajectory: alternative drive shot; solid curves: wind absent, dashed curves: wind present . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a Hole 18, Pebble Beach Golf Links, California. Image courtesy Google Earth, Data MVARI, Data SIO, NOAA, US Navy, NGA, GEBCO Landsat/Copernicus [10]; b Regular playing strategy of three shots (black curves) and heroic playing strategy of two shots (red curves) . . . . . . . . . . . . . . .

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Fig. 5.14

List of Figures

Playing in the absence of wind: a 3D view and b top view; red trajectories: driver shots played from tee, black trajectories: 3 Wood shots played to the putting green, magenta band: landing region of driver shots, cyan band: launching region of 3 Wood shots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Playing in the presence of: a a 5 m/s tailwind (b ¼ 10 ), b a 5 m/s headwind (b ¼ 170 ) and c a 5 m/s sea breeze (b ¼ 100 ); magenta band: landing region of possible first shots, cyan band: launching region of possible second shots . . . 106

List of Tables

Table 3.1 Table 4.1 Table 4.2 Table 4.3 Table 5.1 Table 5.2

Parameters of the bunkers in the Pebble Beach golf course . . Average launch conditions of standard clubs from PGA Tours reported by TrackMan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectory characteristics of average club shots . . . . . . . . . . . Club-wise relative wind-induced deviations in Rf , hM and c due to a tailwind of 5 m=s . . . . . . . . . . . . . . . . . . . . . . . . . . . Parameters of the course features in hole 17 of TPC Sawgrass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parameters of the course features in hole 8 of Muirfield Golf Club . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Introduction

I get to play golf for a living. What more can you ask for, getting paid for doing what you love. —Tiger Woods

1.1 The Game of Golf Golf, one of the most high-speed ball sports, is popularly referred to as the “Gentleman’s Game”. The objective of the game is simple: the players need to start from the tee (which lies in the teeing box) and deliver the ball into the hole (which lies in the putting green) by hitting a series of shots using a club. A golf course consists of several holes each having a unique layout design. Once all the competing golfers successfully play all the holes, the player who played the least number of shots wins the game. Despite its simple objective, golf is one of the most difficult ball sports due to the challenges it poses on the players. Apart from the teeing box and the hole, a golf course contains several other features that make up its architecture. These include hills, valleys, trees, sand bunkers and water bodies like lakes, ponds and sometimes even seas and ocean (see Fig. 1.1). While these features provide an aesthetic touch to the golf course, their primary role is to act as obstacles for the golfer during a game. In order to gain a competitive edge, the golfers need to minimise the number of shots they play. Hence, they need to properly plan their route taking into account the distance to be covered as well the obstacle and hazards that may come in the way. If a poorly planned (or poorly delivered) shot lands the ball in any of the hazards, it yields additional shots to the player which, in turn, manifests as a competitive disadvantage. Golfers achieve this by using a wide variety of golf clubs to hit the ball, each resulting in a different kind of shot. These shots all differ in terms of their range, maximum heights and landing angle. As a result, some shots are flat and long, while others are lobbed and short. While flat and long shots help in quickly getting closer © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. Malik and S. Saha, Golf and Wind, https://doi.org/10.1007/978-981-15-9720-6_1

1

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1 Introduction

Fig. 1.1 Hole 18, Pebble Beach Golf Links, California. Image courtesy Google Earth, Data MVARI, Data SIO, NOAA, US Navy, NGA, GEBCO Landsat/Copernicus [1]

to the hole, lobbed shots allow a golfer to directly shoot over obstacles like trees without the need to go around them through multiple shots. Moreover, lobbed shots give a player greater control over the ball once it lands on the ground since the ball tends to land steep and thus rolls less. Depending on the kind of shot a golfer needs, he chooses an appropriate golf club. Knowing his own capacity, he plans a sequence of shots over the golf course such that he gets to the hole in the most optimum way. A player who can study the course well, judge his abilities accurately, and hence execute his playing strategy as planned, often performs well. However, mastering these skills demands practice, experience and last but not the least, creativity. The situation becomes much more complicated when environmental effects like wind come into the picture. Since the golf ball travels at speeds as high as 90 m/s while in the air, the aerodynamics forces, and hence the wind, have a major influence over the flight trajectory of the ball. As a result, wind adds to the challenge and greatly impacts the gameplay strategies. Further, accounting for the deflection caused by wind is a challenge in itself. Hence, incorporating this practice in gamepay is mostly limited to the abilities of experts. Moreover, the wind speed and direction often vary in both space and time and the exact wind profile is beyond the golfer’s knowledge during a game. This, in turn, introduces an element of uncertainty in the game, making even experts nervous at times.

1.1 The Game of Golf

3

When the golf course architecture is designed in a way that the arrangement of hazards complements the local wind conditions over the course, a coupling occurs between them and both the challenges amalgamate, thus making even an ordinary course iconic. In a well designed course which uses the environmental conditions to its advantage, one ill-executed shot can make the subsequent shots even more difficult. This way, the wind not only affects individual shots, but also the entire game. This shifts the focus of the game from the course to the mind of the golfer, wherein experience, skill and confidence make the real difference. Such a situation provides for an effective means to clearly distinguish between an excellent and an intermediate player, hence letting the best one win the game. Let’s explore some real examples, wherein the golf course layout together with the local wind conditions challenge even the expert players. 1. Russell Knox, a top-ranking PGA Tour player, hit three shots into water at the iconic hole 17 of Tournament Players Club (TPC Sawgrass) in Florida [6]. This course has a unique layout consisting of an island green with nothing but water between the teeing box and the green (see Fig. 1.2a). Due to spatially varying wind speeds at the ground level and at the apex of the trajectory, it was difficult to judge how the ball would get deflected by wind. Hence, unable to correctly account for its effect, even an expert player like Russell Knox lost three shots into water and consequently went down on the leader board. 2. Hole 12 of Augusta National Course in Georgia, is famous for its notorious winds that swirl around the tree surrounding the putting green (see Fig. 1.2b). As a result, golfers standing at the teeing box have no idea about how the wind is blowing near the green. This makes it impossible to judge how the ball would be deflected giving golfers no choice but to play blind. About this hole, the golfing legend Tiger Woods once said, “One of the hardest par-3s in the world; you don’t know where the wind is coming from” [7]. 3. Golf course designers David Dale and Ronald Fream of Golfplan designed the Nine Bridges golf course at Jeju island in South Korea (see Fig. 1.3), in such a way that the presence of the local Jeju Trade Winds complemented the course layout, thus greatly enhancing the difficulty of the course. The influence of wind is so prominent that during the PGA tour of 2017, the player Justin Thomas played the course on a day when the wind was calm and obtained the lowest score (i.e. the best score) of the day. The next day, when the wind started blowing, not a single player was able to score better than Justin, hence leading him to victory [8]. These three examples show how significantly wind can impact the game of golf. In fact, a moderate wind speed of 5 m/s deflects a golf ball’s flight trajectory by up to 20%. This is more than three times the deviations caused by inconsistency in any other factors, like launch speed, launch spin rate and launch angle. This raises an important question: why does wind have a significant effect on the flight trajectories of golf balls? The answer to this question leads us on a quest which forms the theme of the book. A brief clue to the answer is the fact we mentioned earlier that golf is a very high-speed ball sport. Due to this, a golf ball’s flight trajectory is very sensitive to aerodynamic forces that act on the ball while it travels through the air. These forces

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1 Introduction

Fig. 1.2 a Hole 17, Tournament Players Club in Florida [2]; b Hole 12, Augusta National Golf Club in Georgia [3]. Image courtesy Google Earth, Data SIO, NOAA, US Navy, NGA, GEBCO Landsat/Copernicus

1.1 The Game of Golf

5

Fig. 1.3 Nine Bridges Golf Course in South Korea [4]. Image courtesy Google Earth, CNES/Airbus, Data SIO, NOAA, US Navy, NGA, GEBCO, SK telecom, Landsat/Copernicus

act as a function of the relative airspeed of the ball which, in turn, is directly related to the wind speed. As a result, the flight trajectories become very sensitive to wind itself. Furthermore, the effect of wind greatly varies according to the situation. The contributing factor that first comes to mind is the aerodynamic behaviour of the ball. The dimples on a golf ball’s surface help in reducing the drag force on the ball and hence let the ball fly over longer distances. Any variation in the shape, size and density of these dimples brings about a change in the lift and drag characteristics of the ball. As a result, balls having different dimple geometries are affected differently by the same wind conditions. While aerodynamic behaviour affects the ball during its flight, the launch conditions of the shots determine the shape of the flight trajectories (that is, how flat or lobbed it will be). As a result, the effect of wind also varies as we change the launch conditions of the ball. Finally, the geographical setting (latitude, location, landscape) determines the magnitude and direction of the wind velocity, hence having a direct effect on how the wind affects golf balls’ flight trajectories. All these factors, and their contributions to the effect of wind, hold more meaning when viewed from the eye of a golfer. The gameplay strategies in golf courses having

6

1 Introduction

different types of design layouts are affected by wind differently. The same windinduced deviation in a shot’s trajectory can have very different implications when put into the context of the course layout. Moreover, the situation evolves over the course of a day as the wind’s direction changes as, for instance, sea-breeze changes to land-breeze. In this chapter, we provide a glimpse of various physical phenomena that are significant in the game of golf (Sect. 1.2). Next, in Sect. 1.3, we briefly describe the golf course environment, which includes course architecture, as well as wind conditions. This, in turn, leads us to the main theme of this book, “Golf and Wind” (Sect. 1.4). We identify specific questions that help elucidate the role of wind in strategic golfing through the eyes of a physicist. Herein, we also give an overview of our work and explain how we approach each specific question to discover the answers. Finally, we conclude this chapter by describing how this book is structured.

1.2 Golf and Physics There is a lot of physics involved in the game of golf ranging from concepts of biomechanics, collision mechanics, rolling motion to aerodynamics. Understanding these concepts in the context of golf is an essential step in demystifying the role of wind in golf. The understanding provides an insight as to why wind affects golf ball flight trajectories in a certain way, and hence enables us to appreciate the implication in gameplay strategies. Wesson explains all these concepts in a very interactive way in his book The Science of Golf [9]. Since we focus on the effect of wind in trajectories, we do not need to dive into the detailed physical models of the biomechanics involved in a golfer’s swing. However, the reader can refer to [10–12], for understanding the swing mechanics before launch. In this book, we begin our discussion from the moment it is launched till the point it halts on the ground. Thus, we split its motion into three phases based on the phenomena involved: (1) the launch, (2) the flight and (3) the ground run. The following subsections briefly introduce these phases.

1.2.1 Launching a Golf Ball The short 6 s journey of a golf ball begins when the golfer hits it with his golf club. As the golfer swings his club to hit the ball, the impact between the clubface and the ball sends the ball flying at speeds up to 90 m/s making an angle ranging from 9◦ to 25◦ from the horizontal. The impact also imparts a back-spin to the ball which helps the ball go higher and further. Both, the golf ball and the golf club are designed in a way such that the energy transfer from the club to the ball is optimal. This depends of several factors like clubface size and weight, coefficient of restitution, as well as friction between the ball and clubface. Apart from this, the length of club shaft, loft

1.2 Golf and Physics

7

of clubface and the height of the golf ball from the ground also plays a vital role in imparting the suitable launch conditions to the ball (i.e. the launch speed, launch angle and launch spin rate). We describe the swing and the launch in greater detail later in Sect. 2.1.1. The launch is a very crucial phase of the ball’s motion since it is the only aspect of a shot that a golfer has direct control over. Once the ball leaves the club, it faces the wrath of the wind conditions (during flight) and the course terrain (during ground run), neither of which the golfer has any control over. Naturally, a lot of research has been done on designing the golf clubs (material properties, shape, size and ergonomics [13, 26]) and the golf balls (material properties [27, 31]), in order to optimise the launch of a golf ball. In practice, a golfer uses various kinds of golf clubs to alter the launch conditions and hence deliver a suitable type of shot (flat and long or lobbed and short) as per his requirement. In this book, we carry out our analysis using the average launch conditions of each club from PGA Tours, as reported by TrackMan [32].

1.2.2 A Golf Ball in Flight After the launch, the golf ball enters the flight phase. This is the most important phase for the scope of this book because: (1) this is generally the longest part of a golf ball’s trajectory and (2) this is the only phase in which the ball acts under the effect of aerodynamics force, and hence the wind. The aerodynamics force is a result of the ball’s interaction with the air while it travels through at very high speeds. Owing to these high speeds, the role of aerodynamics force, along with the gravitational force is vital in governing the projectile motion of the ball. The dimples present on the golf ball’s surface aid in reducing the drag and enhancing the lift force on the ball. The reduced drag allows the ball to cover greater distances. On the other hand, the Magnus lift force due to the back-spin of the ball negates the gravitation force, hence keeping the ball in the air for longer and letting it flying higher and further. The dimple geometry and pattern plays a crucial role in defining the aerodynamic characteristics of the ball, and hence is the most vital feature of a golf ball. Defining the aerodynamic characteristics is a major part of conducting any study involving golf balls. There have been several studies focusing on determining these characteristics through various means, such as through wind tunnel tests, through computational methods like computational fluid dynamics (CFD), as well as observational techniques, involving tracking a golf balls trajectory to estimate the forces acting on it [33, 41]. Some studies have also investigated the trends of how dimple the geometry affects the aerodynamic performance of the golf balls [42–44]. As described in detail later in Sect. 2.1.1, we use the aerodynamic characteristics data reported by Bearman and Harvey [33], to simulate our flight trajectories.

8

1 Introduction

1.2.3 Landing on the Fairway The final phase of a golf ball’s motion, the ground run, begins when it lands on the ground for the very first time. The ground run, in turn, consists of a bouncing phase followed by a rolling phase. Here, the motion of the ball is governed by the local terrain, the ball’s landing conditions (landing angle, speed and spin rate) alongside its material and surface properties, as well as the ground’s hardness and type of grass. Depending on the coefficients of restitution and friction between the surfaces of the ground and the ball, the ball bounces several times. With each bounce, the ball dissipates some energy due to the inelastic collision, thus diminishing the bounce height subsequently. Eventually, the bounce height becomes so small that the ball’s motion transitions from bouncing to pure rolling. Here too, the friction between the ground and ball governs the retarding motion along the local terrain profile. Finally, when the ball comes to a halt, its final phase of motion ends. Even though the ground run distance is generally much smaller when compared to the flight range, it becomes very crucial when the situation demands high accuracy and control. Since the golfer has no control over how the ball would move along the terrain profile after landing on the fairway, the ground run is desired to be as short as possible. For this reason, surface hardness is often a decisive factor while selecting a ball. When accuracy is prime, soft surface balls are a good choice. There have been several studies to model the ground run of a golf ball. Daish [45] and Cross [46] describe the physics involved in bouncing motion of general balls. Later, Penner [47] and Cross [48], described the model specific to the bouncing and rolling of a golf ball. In 2010, Cross [49], also studied how the bouncing behaviour of balls could be enhanced by modifying the material properties. Haake [50] and Hubbard and Alaways [51] used experimental means to quantify the parameters that were useful in creating the mathematical models of a golf ball bouncing and rolling of turf. Recently, more advanced techniques like image processing and sensor data have also been used to determine these parameters [52, 53]. Penner [54, 55] have also studied the motion of a ball in putt shots, which includes rolling and falling into the hole. In this book, we simulate the ground run of golf based on the model described by Penner [47]. We first explain his model for two-dimensional bouncing and then extend the model to incorporate three-dimensional motion.

1.3 The Golfing Environment While physics helps us comprehend and hence model how the golf ball behaves during different phases of its motion, it is equally important to understand the environment in which the motion takes place. The golfing environment comprises of: (1) the golf course and (2) the local wind conditions. These factors provide the muchneeded context of golf as a game, so as to investigate how wind impacts strategic gameplay.

1.3 The Golfing Environment

9

1.3.1 Golf Course Architecture We discussed in Sect. 1.1, a typical golf course consists of a teeing box (starting point), a putting green (containing the hole) and several other features which make every golf course unique. These features are placed in the course to act as obstacles or hazards which the golfers must avoid in order to finish the game optimally. They include hills and valleys, sand bunkers, trees and water bodies like ponds, lakes or even oceans. A careful placement of such hazards compels the player to plan the way through the course based on the strengths, so that he reaches the hole in the minimum number of shots while avoiding all hazards. Primarily there are three types of layout design styles: (1) penal, (2) strategic and (3) heroic. A penal course offers only one route to reach the putting green and poses a heavy penal on the players if they miss their target due to a poor shot. On the other hand, a strategic course offers multiple routes to reach the putting green. All of these routes consist of the same number of shots and none offer a heavy penalty for misplayed shots. Depending on his comfort zone, strengths and confidence, a player can choose either of the routes. A heroic course is an amalgamation of a penal and strategic course. It offers two or more possible routes to the green which significantly differ in terms of reward and risk. For instance, by opting for a risky route (known as heroic route), the player may need less number of shots to reach the green. However, if he fails, the consequence could be disastrous. Hence, each type of golf course design challenges the player differently.

1.3.2 Wind Finally, the main focus of this book, the wind, is an important part of the golfing environment and the game itself. When a course is designed in such a way that there occurs a coupling between the course elements and the local wind conditions, the situation becomes far more complicated. In reality, the wind flowing through a golf course varies with respect to both time and space. The temporal variation occurs as a result of gusts, while the spatial variation is a result of swirling motion of wind due to wakes formed behind hill, trees, buildings. Due to the high speeds of a golf ball, it covers very large distances (up to around 300 m). As a result, even minor variation in the contributing factors (launch conditions, aerodynamic forces, wind conditions) bring about a significant deviation in the entire flight trajectory. Such dependence on launch conditions and aerodynamic characteristics have been studied by [43, 56, 57]. In the presence of wind, the relative airspeed of a golf ball in flight changes. As a result, it changes the magnitude and direction of aerodynamic forces that act on the ball. Since the ball’s flight trajectory is primarily governed by these forces, it experiences a significant deflection as compared to the case when there is no wind. The effects of wind on individual flight trajectories have been studied by McPhee and Andrews [58], wherein they explored

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1 Introduction

how sidespin can be used to counter the effect of crosswind. Naruo and Mizota [59] also studied how the trajectory behaves under the effect of an atmospheric boundary layer. In this book, we consider two simplified wind models as described in detail in Sect. 2.2.3: (1) uniform wind profile and (2) logarithmic wind profile. Keeping the models simple helps us to elucidate the most essential features of wind’s impact. In the simplest case of a uniform wind model, the wind has no variation with respect to space and time. The logarithmic wind profile is a closer description of the wind velocity profile due to the boundary layer effect which causes the wind speed to show a logarithmic increase as the ball soars higher. The presence of stronger wind with increasing height has a marked impact and is evaluated in greater detail in the following chapters.

1.4 Golf and Wind? The deflection in a golf ball’s flight trajectory becomes especially critical in situations which demand highly accurate shots. For instance, accuracy is key in the final approach shot which delivers the ball onto the putting green. An unaccounted wind-induced deviation at such a stage leaves no window to cover up in future shots. Moreover, golf courses are often designed so as to penalise the golfer (be it heavily or mildly). Therefore, a shot which does not land as planned may cost the golfer extra shots and, in turn, result in a competitive disadvantage. Such a case was studied by Yaghoobian and Mittal [7], for hole 12 of Augusta National Golf Course in Georgia. On the other hand, the motive in a tee-shot (the first shots of the game), is often just to get as close as possible to the hole. In this case, even if the ball is unexpected deflected by wind, the golfer generally has an opportunity to make up for the loss in the subsequent shots. Additional complexity in shot selection is introduced by the location of the golf course hazards and whether the wind-induced deflection creates the possibility of the ball rolling into one. Therefore, the wind-induced deviation has varying degrees of importance in different situations. This makes it more important to understand the impact of wind on the entire game, rather than just individual trajectories. In this book, we explore this problem from the eye of a golfer, while using physics as our guiding tool. That is, we raise our questions while thinking as a golfer would, and then address them in the due course of this book using physics. Finally, we embark upon the impact on strategic gameplay by applying our technical findings to case studies of real golf courses. The insights help us to bridge the gap between golfers and physicists which is one of the objectives of the book. More specifically the questions that we aim to answer in the following chapters of this book are: 1. What are the factors that affect a golf ball’s trajectory in windy conditions? • Launch conditions imparted by the golf club? • Dimple pattern of the ball?

1.4 Golf and Wind?

11

2. How significant is the effect of wind-induced deviation? • • • • •

How does the deviation compare to other parameters affecting the trajectory? What is the impact of the direction and velocity profile of wind? What happens when the club is changed? What happens when the ball is changed? What role does the golf course design play?

3. How can the problem of wind be addressed in a realistic game of golf? • How can the launch conditions be adapted? • How is the gameplay strategy affected by different types of golf courses? In order to find the answers to these questions, we solve the equations of motion (derived from the Newton’s laws associated with physics of flying, bouncing and rolling) and simulate the golf ball’s trajectory in a synthesised golf course environment in MATLAB. Using the simulations, we re-create the situations that a golfer would experience during a game. This includes simple tools like using different golf clubs and balls, as well as more complex concepts like playing a sequence of shots in the presence of various wind conditions in an environment which models a realistic game of golf. In Chap. 2, we explain the details of all the physical models that we use in our simulations. Thereafter in Chap. 2, we describe the simulation algorithm itself. In Chap. 4, we use our simulations to carry out a quantitative analysis which helps us get a deeper insight as to how exactly wind affects the golf ball’s trajectories. We first establish the importance of wind’s impact on gameplay by comparing the windinduced deviations in the trajectory with the deviation due to errors in other factors like launch speed, launch angle and launch spin rate. Thereafter, we explore how these deviations vary as we change the physical setting (wind conditions, launch conditions and aerodynamics) and also investigate the underlying causes for such variations. In Chap. 5, we steer the discussion back to a golfer’s perspective and explore how the presence of wind affects the gameplay strategy, all the while keeping in mind the results from Chap. 4. First, we discuss some common techniques that golfer use to adapt their launch conditions so as to minimise, and at times nullify, the effect of wind. Thereafter, we use our simulation tool to re-create three real golf courses having distinct design styles. Then, we simulate a sequence of aerial shots from the tee to the putting green just as a golfer would during a real game. On comparing the scenario with and without wind, we distil the situations, when the wind plays a more critical role in the game. This, in turn, leads us to the discussion on how wind affects a golfer’s game strategy differently in golf courses having distinct design styles. Finally in Chap. , we summarise our key findings and end the discussion with some limitations and future prospects of our work.

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References 1. Google Earth, Data MVARI, Data SIO, NOAA, US Navy, NGA, GEBCO Landsat/Copernicus. Pebble Beach Golf Links, California. http://earth.google.com. 2. Google Earth, Data SIO, NOAA, US Navy, NGA, GEBCO Landsat/Copernicus. Tournament Players Club, Florida. http://earth.google.com. 3. Google Earth, Data SIO, NOAA, US Navy, NGA, GEBCO Landsat/Copernicus. Augusta National Golf Club, Georgia. http://earth.google.com. 4. Google Earth, CNES/Airbus, Data SIO, NOAA, US Navy, NGA, GEBCO, SK telecom, Landsat/Copernicus. Nine Bridges Golf Club, South Korea. http://earth.google.com. 5. Google Earth, CNES/Airbus, Data SIO, NOAA, US Navy, NGA, GEBCO, IBCAO, Landsat/Copernicus, Mexam Technologies. Muirfield Golf Club. http://earth.google.com. 6. PGA Tour, Russell Knox implodes on No. 17 at THE PLAYERS https://youtu.be/ BRJdDD0fj9c. 7. Yaghoobian, N., & Mittal, R. (2018). Experimental determination of baseball spin and lift. Sports Engineering, 21, 1–10. https://doi.org/10.1007/s12283-017-0239-9. 8. Humphreys, R. (2018), The precarious balance of designing for strong Jeju winds. Golf Course Architecture https://www.golfcoursearchitecture.net/content/the-precarious-balanceof-designing-for-strong-jeju-winds. 9. Wesson, J. (2009). The Science of Golf. Oxford University Press. 10. Meister, D., Ladd, A., Butler, E., Zhao, B., Rogers, A., Ray, C., et al. (2011). Rotational Biomechanics of the Elite Golf Swing: Benchmarks for Amateurs. Journal of applied biomechanics, 27, 242–51. https://doi.org/10.1123/jab.27.3.242. 11. Chu, Y., Sell, T. C., & Lephart, S. M. (2010). The relationship between biomechanical variables and driving performance during the golf swing. Journal or Sports Sciences, 28(11), 1251–1259. https://doi.org/10.1080/02640414.2010.507249. 12. Hume, P. A., Keogh, J., & Reid, D. (2005). The Role of Biomechanics in Maximising Distance and Accuracy of Golf Shots. Sports Med, 35, 429–449. https://doi.org/10.2165/00007256200535050-00005. 13. Newman, S., Clay, S., & Strickland, P. (2002). The dynamic flexing of a golf club shaft during a typical swing. Proceedings Fourth Annual Conference on Mechatronics and Machine Vision in Practice, Toowoomba, Queensland, Australia, 1997, 265–270. https://doi.org/10. 1109/MMVIP.1997.625343. 14. Worobets, J., & Stefanyshyn, D. (2012). The influence of golf club shaft stiffness on clubhead kinematics at ball impact. Sports Biomechanics, 11(2), 239–248. https://doi.org/10.1080/ 14763141.2012.674154. 15. Milne, R. D. & Davis, J. P. (1992). The role of the shaft in the golf swing. Journal of Biomechanics, 25(9), 975–983. https://doi.org/10.1016/0021-9290(92)90033-W. 16. MacKenzie, S. J., & Sprigings, E. J. (2009). Understanding the role of shaft stiffness in the golf swing. Sports Eng, 12, 13–19. https://doi.org/10.1007/s12283-009-0028-1. 17. Monk, S. A., Wallace, E. S & Otto, S. R. (2012). Effects of golf shaft stiffness on strain, clubhead presentation and wrist kinematics. Sports Biomechanics, 11(2), 223–228. https://doi. org/10.1080/14763141.2012.681796. 18. Iwatsubo, T., Kawamura, S., Miyamoto, K., & Yamaguchi, T. (2000). Numerical analysis of golf club head and ball at various impact points. Sports Engineering, 3(4), 195–204. https:// doi.org/10.1046/j.1460-2687.2000.00055.x. 19. Nesbit, S. M., Hartzell, T. A., Nalevanko, J. C., Starr, R. M., White, M. G., Anderson, J. R., et al. (1996). A Discussion of Iron Golf Club Head Inertia Tensors and Their Effects on the Golfer. Journal of Applied Biomechanics, 12(4), 449–469. https://doi.org/10.1123/jab.12.4.449. 20. Sweeney, M., Mills, P., Alderson, J., & Elliott, B. (2013). The influence of club-head kinematics on early ball flight characteristics in the golf drive. Sports Biomechanics, 12(3), 247–258. https://doi.org/10.1080/14763141.2013.772225.

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21. Feng, Z. M., Chiu, Y. J., & Chen, H. H. (2013). Design and Simulate of Golf Wood Club. Applied Mechanics and Materials, 419, 438–441. https://doi.org/10.4028/www.scientific.net/ AMM.419.438. 22. Chiu, J., & Shen, C. (2005). Analysis of the restitution characteristics of a golf ball colliding with a club-head. Japan J. Indust. Appl. Math., 22, 429. https://doi.org/10.1007/BF03167493. 23. Nakai, K., Wu, Z., Sogabe, Y., & Arimitsu, Y. (2004). A study of thickness optimization of golf club heads to maximize release velocity of balls. Communications in Numerical Methods in Engineering, 20(10), 747–755. https://doi.org/10.1002/cnm.698. 24. Matsumoto, K., Tsujiuchi, N., Koizumi, T., Ito, A., Ueda, M., & Okazaki, K. (2015). Analysis of Shaft Movement Using FEM Model Considering Inertia Effect of Club Head. Procedia Engineering, 112, 10–15. https://doi.org/10.1016/j.proeng.2015.07.168. 25. Mackenzie, S. J., & Boucher, D. E. (2016). The influence of golf shaft stiffness on grip and clubhead kinematics. Journal of Sport Sciences, 35(2), 105–111. https://doi.org/10.1080/02640414. 2016.1157262. 26. Kakiuchi, H., Inoue, A., Onuki, M., Takano, Y., & Yamaguchi, T. (2001). Applicaion of Zrbased bulk glassy alloys to golf clubs. Material Transactions, 42(4), 678–681. https://doi.org/ 10.2320/matertrans.42.678. 27. Ismail, K., & Stronge, B. (2008). Calculated golf ball performance based on measured viscohyperelastic material properties. Researchgate,. https://doi.org/10.1007/978-2-287-09411-82. 28. Tanaka, K., Sato, F., Oodaira, H., Teranishi, Y., Sato, F., & Ujihashi, S. (2006). Construction of the Finite-Element Models of Golf Balls and Simulations of Their Collisions. Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications, 220(1), 13–22. https://doi.org/10.1243/14644207JMDA80. 29. Monk, S. A., Davis, C. L., Otto, S. R., & Strangewood, M. (2005). Material and surface effects on the spin and launch angle generated from a wedge/ball interaction in golf. Sports Eng, 8, 3–11. https://doi.org/10.1007/BF02844127. 30. Axe, J. D., Brown, K., & Shannon, K. (2002). The vibrational mode structure of a golf ball. Journal of Sports Sciences, 20(8), 623–627. https://doi.org/10.1080/026404102320183185. 31. Arakawa, K., Mada, T., Komatsu, H., Shimizu, T., Satou, M., Takehara, K., et al. (2009). Dynamic Deformation Behavior of a Golf Ball during Normal Impact. Experimental Mechanics, 49, 471–477. https://doi.org/10.1007/s11340-008-9156-y. 32. TrackManGolf, TrackMan Average Tour Stats https://blog.trackmangolf.com/trackmanaverage-tour-stats/. 33. Bearman, P., & Harvey, J. (1976). Golf Ball Aerodynamics. Aeronautical Quarterly, 27(2), 112–122. https://doi.org/10.1017/S0001925900007617. 34. Davies, J. M. (1949). The aerodynamics of golf balls. Journal of Applied Physics, 20, 821–828. https://doi.org/10.1063/1.1698540. 35. Tsuji, Y., Morikawa, Y., & Mizuno, O. (1985). Experimental Measurement of the Magnus Force on a Rotating Sphere at Low Reynolds Numbers. Journal of Fluids Engineering, 107, 484–488. https://doi.org/10.1115/1.3242517. 36. Smits, A. J., & Smith, D. R. (1994). A new aerodynamic model of a golf ball in flight. Science and Golf, II, 340–347. 37. Choi, J., Jeon, W., & Choi, H. (2006). Mechanism of drag reduction by dimples on a sphere. Physics of Fluids, 18, 041702. https://doi.org/10.1063/1.2191848. 38. Lyu, B., Kensrud, J., Smith, L., & Toyasa, T. (2018). Aerodynamics of Golf Balls in Still Air. The 12th Conference of the International Sports Engineering Association, 2, 238. https://doi. org/10.3390/proceedings2060238. 39. Aoki, K., Muto, K., & Okanaga, H. (2010). Aerodynamic Characteristics and Flow Pattern of a Golf Ball with Rotation. Procedia Engineering, 2, 2431–2436. https://doi.org/10.1016/j. proeng.2010.04.011. 40. Ting, L. L. (2002). Application of CFD technology analyzing the three-dimensional aerodynamic behavior of dimpled golf balls. ASME 2002 International Mechanical Engineering Congress and Exposition, 725–733. https://doi.org/10.1115/IMECE2002-32349.

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41. Aoyama, S. (1990). A Modern Method for the Measurement of Aerodynamic Lift and Drag on Golf Balls. Science and Golf, 199–204. 42. Ting, L. L. (2003). Effects of dimple size and depth on golf ball aerodynamic performance. ASME/JSME 2003 4th Joint Fluids Summer Engineering Conference, 811–817. https://doi. org/10.1115/FEDSM2003-45081. 43. Sajima, T., Yamaguchi, T., Yabu, M., & Tsunoda, M. (2006). The aerodynamic influence of dimple design on flying golf ball. The Engineering of Sport, 6, 143–148. https://doi.org/10. 1007/978-0-387-46050-5_26. 44. Naruo, T., & Mizota, T. (2014). The influence of golf ball dimples on aerodynamic characteristics. Procedia Engineering, 72, 780–785. https://doi.org/10.1016/j.proeng.2014.06.132. 45. Daish, C. B. (1972). The physics of ball games. London: English Universities. 46. Cross, R. (1999). The bounce of a ball. American Journal of Physics, 67, 222. https://doi.org/ 10.1119/1.19229. 47. Penner, A. R. (2002). The run of a golf ball. Canadian Journal of Physics, 80, 931–941. https:// doi.org/10.1139/P02-035. 48. Cross, R. (2002). Grip-slip behaviour of a bouncing ball. American Journal of Physics, 70, 1093. https://doi.org/10.1119/1.1507792. 49. Cross, R. (2010). Enhancing the bounce of a ball. The Physics Teacher, 48, 450. https://doi. org/10.1119/1.3488187. 50. Haake, S.J. (1989). An apparatus for measuring the physical properties of golf turf https:// research.aston.ac.uk/en/studentTheses/apparatus-and-test-methods-for-measuring-theimpact-of-golf-balls. 51. Hubbard, M. and Alaways, L.W. (1999). Mechanical interaction of the golf ball with putting greens. In Proceedings of the 1998 World Scientific Congress of Golf. Edited by M.R. Farrally and A.J. Cochran. Human Kinetics, Leeds. 52. Baek, S. and Kim M. (2013). Golf Ball Bouncing Model Based on Real Images. Ubiquitous Information Technologies and Applications. Lecture Notes in Electrical Engineering, vol 214. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5857-575. 53. Chaisuwan, P., Khemmani, S., Wicharn, S., Plaipichit, S., Pipatpanukul, C., & Puttharugsa, C. (2019). Measuring the coefficient of restitution for tennis and golf balls using smartphone sensors. Physics Education, 54, 065011. https://doi.org/10.1088/1361-6552/ab3c08. 54. Penner, A. R. (2002). The physics of putting. Canadian Journal of Physics, 80, 83–96. https:// doi.org/10.1139/P01-137. 55. Holmes, B. W. (1991). Putting: How a golf ball and hole interact. American Journal of Physics, 59, 129. https://doi.org/10.1119/1.16592. 56. Erlichson, H. (1983). Maximum projectile range with drag and lift, with particular application to golf. American Journal of Physics, 51, 357–362. https://doi.org/10.1119/1.13248. 57. Stengel, R. F. (1992). On the flight of a golf ball in the vertical plane. Dynamics and Control, 2, 147–159. https://doi.org/10.1007/BF02169495. 58. McPhee, J. J., & Andrews, G. C. (1988). Effect of sidespin and wind on projectile trajectory, with particular application to golf. American Journal of Physics, 56, 933–939. https://doi.org/ 10.1119/1.15363. 59. Naruo, T., & Mizota, T. (2006). Experimental verification of trajectory analysis of golf ball under atmospheric boundary layer. The Engineering of Sport, 6, 149–154. https://doi.org/10. 1007/978-0-387-46050-5_27.

Chapter 2

Understanding the Motion and the Environment

A hole in one is amazing when you think of the different universes this white mass of molecules has to pass through on its way to the hole. —Mac O’Grady

Overview The game of golf is associated with numerous physical phenomena. From the moment the golfer swings his club to hit his first shot, to the time when the golf ball falls into the hole, all actions are governed by physical laws. The swing of a golfer’s club is a biomechanical problem, which is characterised by the physical properties of the club, the muscle strength and body stance of the golfer. The impact of the golf club and golf ball is influenced by the material properties and surface roughness of the ball and club. The ball’s trajectory is an aerodynamic problem, followed by the ball’s bouncing and rolling behaviour on the ground. The primary goal of this book is to analyse the effect of wind on golf balls trajectories, and hence the gameplay. However, in order to facilitate a detailed quantitative analysis, it is important first to understand the physical laws governing the motion of the ball, develop approximate physical models and then simulate them using a computer. For the analysis, we focus on modelling the trajectory of a golf ball and the golf course environment in this chapter. Section 2.1 describes the physical models for the ball’s flight and ground run (bouncing and rolling) and Sect. 2.2, discusses the schemes to model the golf course architecture and the wind conditions.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. Malik and S. Saha, Golf and Wind, https://doi.org/10.1007/978-981-15-9720-6_2

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(a)

(b)

Fig. 2.1 Parts of a golf ball trajectory: flight (black), bouncing (red), rolling (blue); a Complete trajectory and b ground run (enlarged); Launch conditions: u0 = 80 m/s, θ = 10◦ and N0 = 1500 RPM

2.1 Trajectory of a Golf Ball The trajectory of a golf ball has three parts: flight, bouncing and rolling (see Fig. 2.1). Bouncing and rolling are collectively referred to as the ground run. When a golfer hits the ball with his club, the impact imparts a velocity and a spin to the ball. In a general three-dimensional case, the spin comprises of backspin, as well as sidespin, which generates upward and sideward Magnus forces. This initial momentum, along with the aerodynamic characteristics of the ball, describes the ball’s flight. After the first impact with the ground, the next phase of the trajectory begins, where the ball bounces. Here, the velocity and spin rate of the ball just before landing, along with its material properties, determine its behaviour right after impact. After the ball bounces several times, its motion transitions from bouncing to pure rolling, before it finally comes to a halt. This section describes the physical models of each part of the trajectory, along with the governing equations and a discussion on the realistic values of the parameters required.

2.1.1 Flight The motion of a projectile is determined by the launch conditions and the forces that act on it while it is in the air (see Fig. 2.2). These forces are gravity (W ), and the resultant aerodynamic force split into RD and RM representative of the drag and

2.1 Trajectory of a Golf Ball

17

Fig. 2.2 Forces acting on a golf ball

Magnus forces, respectively. In the case of short and low-speed flights, one often neglects the effect of aerodynamic forces. Thus, the motion of such a projectile is given by a perfect parabola. However, in case of a golf ball, both the velocity and the distance covered are quite large. The typical launch velocity for a drive shot is around 75 m/s, and the ball covers a range of 250 m [1]. Therefore, the aerodynamic forces play an essential roll in determining the flight trajectory of a golf ball. A number of investigations have been carried out to determine the aerodynamic forces acting on golf balls using experiments [2–7], and simulations [8–11]. Davies [3] calculated the aerodynamic forces by dropping a spinning golf ball in a wind tunnel, assuming a straight-line trajectory. Later, Aoyama [12], used short exposure videos to track the golf ball trajectory and computed the forces more accurately. Bearman and Harvey [2] determined the aerodynamic force over a range of Reynolds numbers and spin rates using wind tunnel tests on enlarged golf ball models. They used the aerodynamic data to compute the trajectories and showed the effects of launch conditions. Later, the spin rate decay of golf balls due to the aerodynamic moment was found to be logarithmic by Smits and Smith [5]. Choi [6] verified the reduction in drag by measuring the separation angle using a surface oil flow visualisation on a half-dimpled sphere. He also reported the velocity profile on individual dimples using smoke flow visualisation. More recently, image processing techniques have been used to track the flight of sports balls [7, 13–15], which could equally well be applied to golf balls. Research on golf balls using computational methods has steadily increased in the last two decades for multiple reasons like the elimination of physical models, interference effects in experimental measurements [12]. However, the data reported by Bearman and Harvey [2], has been widely used [5, 16, 17] and is perhaps well described by Mehta [18], “measurements due to Bearman and Harvey [2], are probably more representative and accurate”. Hence, in this book too, we evaluate the aerodynamic forces using the data reported by Bearman and Harvey [2].

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Fig. 2.3 Coordinate system and trajectory characteristics

Our physical model captures the aerodynamic behaviour of a golf ball throughout its flight while incorporating wind. It is suitable not only for two-dimensional motion, but also for three-dimensional motion, as is most often the case with real golf ball trajectories. We begin by describing the coordinate system and the parameters used to characterise the flight trajectories. We chose the z-axis along the direction opposite to gravity, pointing to the sky, thus making the x-y plane the zero-elevation local horizontal plane, as shown in Fig. 2.3. The flight trajectory depends on the initial conditions of the ball. We refer to them as the “launch conditions”, given by • • • • •

launch velocity (u0 ), launch angle (θ ), heading angle (ψ), launch spin rate (N0 ) and spin axis tilt (φ) and

as shown in Fig. 2.3. These launch conditions are a result of the impact between the golf club and golf ball at launch. The impact characteristics, in turn, are determined by the physical properties of the golf club, and the golfer’s “swing”. How a golfer swings the golf club is a complex biomechanical problem. Typically, it consists of five primary stages [19]: 1. Address: This is the starting position and orientation of the golf club. The golfer sets the club exactly where and how he wants it to be when it strikes the ball. The stance of the golfer plays a critical role in how well the swing is executed. 2. Backswing: After setting the “address” of the club, the golfer swings the club backwards, so as to gain some room to accelerate the club forward later. Typically, the clubface is swung back by 14 feet [20]. 3. Downswing: Next, the golfer begins the main stage of the swing, wherein he accelerates the club forward so that clubface hits the ball at the required speed. 4. Impact: When the accelerating clubface returns to the position where it makes contact with the ball, the impact stage begins. Figure 2.4 shows the parameters used to describe the impact stage. Here,

2.1 Trajectory of a Golf Ball

19

Fig. 2.4 Impact between the golf club and golf ball in a side-view and b top-view

• uclub is the velocity of the clubface • θL (called dynamic loft) is the angle between the normal to clubface and the horizontal along the side-view • θA (called attack angle), is the angle between uclub and the horizontal along the side-view • ψF (called face angle) is the angle between the normal to clubface and the target line along the top-view and, • ψP (called path angle), is the angle between uclub and the target line along the top-view. The impact imparts the launch conditions (u0 , θ , ψ, N0 and φ) to the ball. Here, the resultant launch spin vector N0 comprises of backspin and sidespin. The spin is

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imparted to the ball due to the tangential component of the impact force, caused by the misalignment between uclub and normal to the clubface. Naturally, the club’s shape, material properties and orientation play a critical role in the impact. Ideally, the position and orientation of the club is identical to the address set by the golfer at the beginning. However, imperfections in the back and downswings cause inconsistencies in the impact, and thus the ball’s trajectory. 5. Follow-through: The motion of the golf club after the impact is called the followthrough. Even though this stage does not have a direct effect on the impact, a proper and balanced follow-through helps the golfer ensure that his swings are consistent. Apart from the launch conditions, the flight trajectory exhibits a major dependence on the local wind velocity profile (uw ) represented as a function of the ball height z. We discuss the role of wind at length in Sect. 2.2, as part of the golf course environment. The shape of the trajectory is of immense interest to the golf players, golf ball and golf club manufacturers. A typical game of golf sees two kinds of aerial shots: (1) long and flat and (2) short and lobbed. A primary objective of a long and flat shot (drive) is to maximise the range. Whereas, a short and lobbed shot (approach and pitch) gives the player greater control over the final landing point of the ball, by minimising the ground run of the ball. There are various types of golf clubs (like a driver, woods, irons and wedges) which yield trajectories of varying shapes and sizes. We will discuss more about golf clubs in Chap. 4. In order to distinguish between these trajectories based on their shape and behaviour, we characterise them using the following parameters as shown in Fig. 2.3: • • • •

the range (Rf = xf i + yf j), the peak height (hM ), for obstacle avoidance and uphill shots, the landing angle (γ ) and landing velocity (vi ), for bouncing behaviour and the final deviation from the initial heading, for the purpose of accuracy.

The other factor that influences the flight trajectory of a golf ball is its aerodynamic behaviour. The aerodynamic forces acting on a golf ball are commonly resolved into two components, drag D, and Magnus lift L. Figure 2.5a compares a typical trajectory of a golf ball with that of a hypothetical parabolic projectile (as would be observed in a perfect vacuum) with identical launch conditions. Both depart the origin with u0 = 60 m/s, N = 4000 RPM and θ = 15◦ . Similarly, Fig. 2.5b, compares the two trajectories launched at θ = 45◦ . It is interesting to note that when θ = 45◦ , which is the optimum launch angle for a regular parabolic projectile, there is a significant difference between the ranges of the two trajectories because of the absence of the aerodynamic force. Whereas, when θ = 15◦ (typical launch angle for a golf shot), the range is nearly the same, but the peak height is enhanced in the presence of the aerodynamic force. Figure 2.5a, b elucidate that the aerodynamic force is important and must be modelled accurately in any golf ball trajectory predictions. As the ball gains height, the speed and the spin rate decay, while the resulting Magnus force creates a proportional lift and side force. The lift force partially nullifies the acceleration due to gravity, thus enabling the golf ball to travel even further. Upon

2.1 Trajectory of a Golf Ball

21

(a)

(b)

Fig. 2.5 Comparison between a golf ball trajectory (red) and a regular parabolic projectile (black dashed) having u0 = 60 m/s, N = 4000 RPM and launch angles a θ = 15◦ and b θ = 45◦

reaching the peak, the ball travels a shorter distance in the descent phase compared to the ascent. The underlying reasons are: 1) the reduced lift because of a decay in spin rate and 2) the momentum lost due to the pressure drag. The drag force acting on a golf ball exhibits a strong dependence on its airspeed, as well as the dimple geometry [2, 21]. Golf balls take advantage of a phenomenon known as the “drag crisis”. It is observed that the drag force acting on a smooth sphere initially increases with its airspeed until the ball reaches a critical airspeed (uc ). As the airspeed is increased further, the drag abruptly starts decreasing until it reaches a minimum value, and later continues to increase with airspeed (see Fig. 2.6a). This abrupt drop in the drag force is known as the drag crisis. It happens because turbulence is triggered in the boundary layer at the surface of the sphere above the critical airspeed. The transition to turbulence, in turn, delays flow separation and hence reduces the pressure drag. For a smooth sphere, the critical airspeed is uc ≈ 110 m/s [20]. However, the drag crisis for a golf ball is intentionally introduced at a lower critical airspeed of uc ≈ 15 m/s [20] (see Fig. 2.6b), to enhance the range. The drag crisis is triggered by the dimpled geometry on the ball’s surface, which introduces turbulence in the boundary layer earlier as compared to a smooth surface. In this manner, the range is greatly enhanced by lowering the critical airspeed using various dimple geometries.

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Fig. 2.6 a Qualitative variation of drag with airspeed for a smooth sphere; b Qualitative comparison of drag forces acting on a smooth sphere (black) and a golf ball (red) as the airspeed varies

2.1.1.1

Equations Governing the Flight of the Golf Ball

The aerodynamic force acting on the ball depends on air density ρ, viscosity μ, ball diameter d , the instantaneous spin vector N and the instantaneous velocity u. Hence, the resultant aerodynamic force R can be written as Eq. 2.1 R = f (u, N, ρ, μ, d ).

(2.1)

where u is the instantaneous velocity and N is the instantaneous spin vector of the golf ball. In our formulation, we need to incorporate the side force due to Magnus lift as well along with lift and drag, and therefore, express R ≡ RD + RM which are drag and Magnus force components, respectively. The trajectory of a golf ball is governed by the vector equation 2.2 m¨x = RD + RM +mg   

(2.2)

R

where m denotes mass of the ball (46 g), g is the acceleration due to gravity, x denotes the position vector of the ball. The components RD , RM are given by Eqs. 2.3 and 2.4

2.1 Trajectory of a Golf Ball

23

1 RD = − CD ρS(|u − uw |)(u − uw ) 2 RM =

1 CL ρS(|N × (u − uw )|)(N × (u − uw )) 2

(2.3) (2.4)

Here, u is the ball’s instantaneous velocity, uw is the wind velocity, N is the instantaneous spin vector, ρ is the density of air (taken as 1.225 Kg/m3 ), S is the projected area of the golf ball (computed from the radius 21.4 mm) along the airspeed (u − uw ), respectively. The coefficients of lift and drag (CL and CD , respectively) are computed using Eq. 2.5 CL = fL (Re, ); CD = fD (Re, ),

(2.5)

is the Reynolds number,  ≡ πdN is the nondimensional spin where, Re ≡ ρud μ 60ub parameter. Here, u ≡ |u| is the instantaneous speed and N ≡ N instantaneous spin rate of the golf ball in rotations per minute (r.p.m.). The spin vector N is assumed to be fixed by the initial orientation of the axis throughout the flight. However, the magnitude decays with respect to time due to the frictional drag acting on the ball. The spin rate decay is estimated from an exponential decay law of 4 % per second [1], which corresponds to a time-scale for decay τ ≈ 24.5 s, given by N = N0 exp(−t/τ )

(2.6)

We evaluate the aerodynamic forces RD , RM using functional dependence of CL and CD (Eq. 2.5), on Re,  as computed from the data reported by Bearman and Harvey [2]. They obtained these values by performing wind tunnel experiments on enlarged golf ball models, having (1) hexagonal and (2) circular dimples. The coefficients have been reported for several ball speeds and spin rates, covering the entire operating range of a typical golf ball. The proposed model is, however, generic and the aerodynamic characteristics of any golf ball, if known, could be used to predict the in-flight performance.

2.1.2 Ground Run—Bouncing and Rolling When the flight of the ball ends and it lands on the ground for the first time, it enters the next phase of the trajectory—the ground run. In this phase, the golf ball bounces on the ground several times, each time losing some energy due to inelastic collisions between the ball and ground. This happens because neither the ball nor the ground are perfectly rigid bodies. The loss in energy causes the peak height of the ball to reduce with every bounce. When the ball’s maximum height falls below a certain threshold value, its motion transitions from bouncing to pure rolling.

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2 Understanding the Motion and the Environment

Fig. 2.7 Ground runs of a typical drive shot (red) and approach shot (blue)

The ground run of the ball depends on three factors—landing conditions of the ball (i.e. the velocity and spin vectors vi and N), material characteristics of the ball and the ground (i.e. the coefficients of restitution e, kinetic friction μ and rolling friction μr ) and the local gradient of the ground with respect to the horizontal. Typically, the terrain of a golf course is not flat; it has many slopes, hills and valleys. As a result, unlike the flight trajectory, it is difficult for a golfer to anticipate the motion of a golf ball after it lands on the ground. Therefore, the ground run of a golf ball is preferably kept short. This is all the more important in case of a lobbed approach shot, wherein the final landing point of the ball is vital. This objective is achieved by having a larger landing angle and backspin. Figure 2.7 compares the ground runs of a typical drive shot (having a lower landing angle and backspin) and an approach shot (having a higher landing angle and backspin). To model the bounce in three dimensions, we define a local ground frame (xg , yg , zg ), where xg points along the component of landing velocity parallel to the local terrain plane, yg is perpendicular to the local terrain pointing towards the sky, and zg can be determined by the right-hand rule. When the golf ball hits the ground, it tends to penetrate and slip on the ground (see Fig. 2.8a). The deformation of the ground’s surface causes multiple points of contact between the ball and the ground, each contributing an elemental normal force n(θ ), perpendicular to the contact surface, and tangential force f (θ ), along the contact surface. The resultant of these forces acts on the ball as an impulse, thus changing the ball’s velocity from vi to vr and spin from ωi to ωr . The tangential force f (θ ), being due to kinetic friction, can be modelled as μn(θ ). Therefore, the resultant contact forces acting on the ball are given by Eqs. 2.7 and 2.8  n = n(θ )d θ (2.7) 

and f = −μ

n(θ )d θ

(2.8)

2.1 Trajectory of a Golf Ball

25

(a)

(b)

Fig. 2.8 a Impact of the golf ball and ground when viewed in the xg − yg (left) and zg − yg (right); b Resultant forces acting on a golf during the impact with the ground. (Figures adapted from Penner [22])

This simplification reduces the ground run to a problem of rigid body inelastic collision between the golf ball and a surface inclined at an angle θc with respect to the local terrain plane (see Fig. 2.8b). Here, n and f still act perpendicular and tangential to the ball’s surface, respectively. However, instead of acting at multiple contact points along the ball’s surface (Fig. 2.8a), they act at a single resultant point located at θc (Fig. 2.8b). The value of θc depends on the distributions of n(θ ) and f (θ ) along the contact surface. These, in turn, depend on the degree of penetration into the ground and the tendency of the ball to slide along the contact surface during impact.

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2.1.2.1

Equations Governing the Ground Run of the Golf Ball

In this book, we model the three-dimensional ground run of a golf ball based on the scheme described by Penner [22], for two-dimensional motion and extending it to incorporate three-dimensional motion. As explained by Penner [22], the ball’s penetration into the ground is determined by how fast the ball hits the ground (vi ) and how hard are the ball’s and ground’s surfaces. Whereas, the sliding tendency depends on the roughness of the ball and the ground, and also on how fast the ball was spinning (ωi ) just before the impact. However, for simplicity, we can neglect any significant variation in the hardness and roughness of the ball and the ground. Penner [22] empirically determined θc using experimental data reported by Haake [23], and is given by the relation Eq. 2.9 θc = 15.4◦

 φ

 v i m/s 18.6 44.4◦

(2.9)

where, φ is the angle between yg and vi . Note that the constants 15.4◦ , 18.6 m/s and 44.4◦ correspond to the specific ball and ground in Haake’s [23] experiment, and will be different when the surface properties change. Since there is no component of vi along the zg -axis, the critical angle in Eq. 2.9, is in the xg − yg plane as shown in Fig. 2.8b. For ease of calculation, we can thus define a new set of coordinate axes (xg , yg , zg ) where xg is parallel and yg is perpendicular to the equivalent inclined surface (see Fig. 2.8b). The impact velocity components vix and viy along the xg and yg axes are given by Eq. 2.10   v vix = C ix (2.10) viy viy where, the rotation matrix C is given by Eq. 2.11  cosθc sinθc C= . −sinθc cosθc

(2.11)

In the model described by Penner [22], there are two possible cases. The first case is when the net frictional force is not large enough, and the ball slides along the contact surface during impact. The second case is when the net frictional force is large enough to stop the ball during impact and causes it to roll along the contact surface during impact. The ball behaves in either one of these ways, depending on the value of the net frictional force given by Eq. 2.8, where μ = 0.4 [22]. Similar to the case of θc , the critical value of frictional force required to cause the ball to roll along the contact surface depends on the impact velocity vi and impact spin rate ωi . It can be determined using coefficients of critical kinetic friction μcz and μcx for the xg − yg and zg − yg planes, respectively, given by Eqs. 2.12 and 2.13

2.1 Trajectory of a Golf Ball

27

μcz =

2 (vix + rωiz ) 7 (1 + e)|viy |

(2.12)

rωix 2 7 7(1 + e)|viy |

(2.13)

and μcx = −

where, r is the radius of the ball and e is the coefficient of restitution between the ball and ground. The coefficient of restitution e signifies the decrease in the effective value as the normal component of impact velocity (viy ) increases. This is because a stronger impact causes the ball compress more, which, in turn, heats the ball during impact. The extra energy induced into the ball due to heat, causes it to bounce back faster, thus reducing the effective value of e. Thus, e can be evaluated using the empirical relation Eqs. 2.14 and 2.15 e = 0.510 − 0.0375|viy | + 0.000903|viy |2 ; |viy | ≤ 20 m/s

(2.14)

e = 0.120;

(2.15)

|viy | > 20 m/s.

Case 1: Sliding When μ < μcz , the ball slides in the xg − yg plane. In this case, the frictional force μn acts on the ball during impact opposite to the xg direction, thus reducing the tangential velocity vix and impact spin rate along zg −axis ωiz . The normal velocity viy is simply acted upon by the normal force n and sees a reduction by the factor of e. The rebound quantities vrx , vry and ωrz are thus given by (2.16) vrx = vix − μ|viy |(1 + e) vry = e|viy | and ωrz = ωiz − (

5μ )|viy |(1 + e) 2r

(2.17)

(2.18)

Similarly, when μ < μcx (sliding in the zg − yg plane), the rebound quantities vrz and ωrx are given by vrz = −μ|viy |(1 + e) and ωrx = ωix − (

5μ )|viy |(1 + e) 2r

(2.19)

(2.20)

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2 Understanding the Motion and the Environment

Case 2: Rolling When μ > μcz , the ball rolls in the xg − yg plane. In this case, the ball conserves the momentum of its centre of mass along the tangential direction while adhering to the pure rolling constraint applied at the contact point. The normal velocity viy still sees a reduction by the factor of e, similar to the previous case of sliding. The rebound quantities vrx , vry and ωrz are thus given by 5 2 (2.21) vrx = vix − rωiz 7 7 vry = e|viy |

(2.22)

vrx r

(2.23)

and ωrz = −

Finally, when μ > μcx (rolling in the zg − yg plane), the rebound quantities vrz and ωrx are given by 5 2 (2.24) vrz = viz − rωix 7 7 and ωrx = −

vrz r

(2.25)

Thereafter, similar to Eq. 2.10, the rebound velocity components along xg and yg can be determined by Eq. 2.26   vrx v  = C T rx (2.26) vry vry

Fig. 2.9 Transition of a ball’s motion from bouncing to pure rolling

2.1 Trajectory of a Golf Ball

29

This rebound velocity vr and spin ωr are the launch conditions of the next phase of projectile motion. The landing conditions of the second phase projectile are then used to compute the second rebound quantities, and thus the launch conditions of the third phase of the projectile motion and so on. Hence, each subsequent impact generates a bounce, each one reaching a maximum height lower than the previous bounce as shown in Fig. 2.9. In an ideal scenario of a rigid surface collision, the ball would bounce infinitely many times, with the maximum heights of the resultant projectiles exhibiting an exponential and asymptotic decay. In such a case, the ball would never come to a halt. However, since the real case is not a rigid surface collision, the ball tends to penetrate the ground at every bounce. When the maximum achievable height of a projectile becomes less than the penetrated distance during impact, the ball never leaves the ground and its motion transitions from bouncing to pure rolling (see Fig. 2.9). Here, path p is the trajectory of the centre of mass o of the ball and hc = 5 mm is the threshold height of projectile, below which the ball is assumed to be in the state of pure rolling [22]. The motion of the golf ball while it rolls is given by m¨x = mgt + fr

(2.27)

where, gt is the component of acceleration due to gravity tangential to the local terrain surface and fr and rolling friction that acts opposite to the local tangential velocity direction xg , and is given by 5 |fr | = mμr gn (2.28) 7 where, gn is the component of acceleration due to gravity perpendicular to the local terrain surface and μr = 0.131 is the coefficient of rolling friction [22]. Figure 2.10 shows the bouncing and rolling phases of a golf ball in three dimensions. The ball tends to follow the local terrain profile and settle into a valley as it loses its energy.

2.2 Golf Course Environment The gameplay in golf greatly depends on the architecture of the golf course and the environmental conditions like wind direction, speed and profile. A well-designed golf course can help distinguish between an amateur and an expert player, and at the same time makes the game more captivating. Typically, a golf course consists of certain general features and a few special ones. All courses have a teeing box, a fairway, a putting green and a hole. However, what makes every course unique is the presence of other special features. These features include the local terrain profile, obstacles and hazards like trees, water bodies and sand bunkers. The size and number of such features vary from one course to another. Figure 2.11 shows a Google Earth

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Fig. 2.10 Ground run of a golf ball in three dimensions

view of hole 18 at the Pebble Beach Golf Links, California. This is one of the courses for which we study the impact of wind in the gameplay later in Chap. 4. Here we can see the course features like teeing box, fairway, putting green, waterbody, sand bunkers and trees. The game becomes even more complex when the local wind conditions come into the picture. A golf ball’s flight trajectory is very sensitive to wind, which makes it a significant part of the golf course itself. When the golf course layout complements the local wind conditions, the golf course environment as a whole compels the golfer to carefully plan his way across the course. The plan needs an accurate diagnosis of his own capacity to get the ball from the tee to the hole in the minimum number of shots to win the game. This way, the golf course environment often takes the game away from the course to the mind of the player. Hence, it is as important to model the environment, as it is to model the golf ball’s trajectory. This section describes the mathematical relations used to model the environment and incorporate the ball’s interaction with individual course features in the analysis.

2.2 Golf Course Environment

31

Fig. 2.11 Hole 18, Pebble Beach Golf Links, California. Image courtesy Google Earth, Data MVARI, Data SIO, NOAA, US Navy, NGA, GEBCO Landsat/Copernicus [24]

2.2.1 Terrain The terrain of a golf course often has an uneven terrain comprising of slopes, varying elevations, as well as small hills and valleys. This subsection describes the relations used to model hills and valleys within the course. We approximate both hills and valleys using the equation of a general paraboloid, given by

X2 Y2 (2.29) zT = h 1 − 2 − 2 a b which intersects with the assumed z = 0 plane at an ellipse having semimajor axis a and semi-minor axis b. Here, h is the maximum height of the hill, zT is the height of the local terrain with respect to the zero-elevation plane. It is represented as a function of the position (X , Y ) in a transformed frame of reference, which relates to the coordinate system defined in Sect. 2.1, as

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2 Understanding the Motion and the Environment

Fig. 2.12 Model of a hill using a general paraboloid having cx = 160 m, cy = 100 m, a = 20 m, b = 15 m, α = 90◦ and h = 5 m

Fig. 2.13 Model of a valley using a general paraboloid having cx = 120 m, cy = 100 m, a = 20 m, b = 15 m, α = 90◦ and h = −5 m

X = (x − cx )cosα + (y − cy )sinα

(2.30)

Y = (x − cx )sinα − (y − cy )cosα

(2.31)

where (cx , cy ) is the centre and α is the orientation of the semimajor axis of the ellipse of intersection with respect to the x−axis. Hence, the height of the ball with respect to the ground is determined by the difference between the z coordinate of the ball’s position vector x and the local terrain height zT . Figure 2.12 shows a hill modelled using Eq. 2.29, where the ellipse of intersection is indicated by the red dashed line. The same scheme could be used to model valleys (see Fig. 2.13), as well as elevated sections of the fairway (see Fig. 2.14).

2.2 Golf Course Environment

33

2.2.2 Features and Layout We model the golf course layout using some of the most common features present in real golf courses. These include—tee, hole, green, ponds, sand bunkers and trees. Finally, the playing area of the course is highlighted by defining the complementary out-of- bounds regions. This subsection describes the representation of each of these features in the physical modelling scheme, as later required in Chap. 3.

2.2.2.1

Tee, Hole and Green

In a golf course, the tee represents the position from which the golfer hits his first shot. The exact position of the tee can be chosen by the golfer within a region called “teeing box”. The hole, on the other hand, is the position on the course, which hosts the “hole” into which the golfer must put the ball in order to finish the game. The hole is surrounded by a region known as the “putting green”, or simply the green. This is a relatively levelled ground having well maintained short turf grass. Once the ball reaches the green, the golfer rolls it into the hole using a “putter”. For the purpose of our analysis, we assign the tee’s position as the launch position of the first shot of the game, and visualise it using a white triangular marker on the terrain surface (see Fig. 2.14). The position of the hole is indicated using an orange flag, and the putting green region is defined as an ellipse containing the hole (see Fig. 2.15), given by Y2 X2 + =1 (2.32) a2 b2 where, a and b are the semimajor and semi-minor axes, and X and Y are given by Eqs. 2.30 and 2.31. We do not include putt shots in our analysis. Therefore, shots in which the final resting position of the ball lies within the green region are considered as valid final shots, and their trajectories are marked “green”.

Fig. 2.14 Model of an elevated fareway using two paraboloids having: (1) cx = 75 m, cy = 60 m, a = 50 m, b = 15 m, α = 0◦ , h = 5 m and (2) cx = 75 m, cy = 60 m, a = 40 m, b = 12 m, α = 0◦ , h = −3.2 m

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2 Understanding the Motion and the Environment

Fig. 2.15 Visualisation of a putting green having cx = 140 m, cy = 140 m, a = 20 m, b = 10 m, α = 0◦ , and the flag positioned at x = y = 140 m

2.2.2.2

Ponds

Golf courses often contain ponds and other water bodies (like lakes, seas and oceans) to serve as a challenge to the golfer. Any ball that falls into the water hazard yields a penalty. For the purpose of our analysis, we consider all shots in which the ball falls in a pond as invalid trajectories, and are marked “red”. Similar to the green region, we model ponds using one or more ellipses as defined by Eq. 2.32. Within these ellipses, we assign the local terrain height as −5 m (see Fig. 2.16), which helps to detect the ponds, as discussed later in Chap. 3.

2.2.2.3

Sand Bunkers

Sand bunkers are another common hazard found in golf courses. In practice, when a ball lands in the bunker, it has to be played from its final resting position. However,

Fig. 2.16 Visualisation of a pond having cx = 200 m, cy = 0 m, a = 70 m, b = 70 m and α = 0◦

2.2 Golf Course Environment

35

Fig. 2.17 Visualisation of a sand bunker having cx = 160 m, cy = 160 m, a = 7 m, b = 7 m, α = 0◦ and h = −1 m

in our analysis, we treat the bunker just like ponds, and trajectories causing a ball to end up in a bunker are considered invalid. Bunkers are visualised as brown coloured valleys defined by Eq. 2.29 (see Fig. 2.17).

2.2.2.4

Trees

The third hazard we include in our model is trees. They are very commonly found in golf courses, and like ponds and bunkers, are often strategically positioned to increase the difficulty of the course. However, unlike ponds and sand bunkers, trees are extruded out of the terrain surface, and hence offer a three-dimensional hindrance. That is, not only should the ball not land on the tree, but it also must not hit the tree in its entire trajectory, which includes the flight. We model the obstacle caused by the tree as a cylinder having diameter 6 m and height 10 m (see Fig. 2.18). Similar to the bunker, we consider all trajectories that hit a tree as invalid.

2.2.2.5

Out-of-Bounds Regions

An out-of-bounds region is the final feature in our model of a golf course. This is the region that the ball must not enter at any height. Similar to ponds, we assign the local terrain height as −10 m everywhere within these regions and visualise them as black surfaces (see Fig. 2.19).

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2 Understanding the Motion and the Environment

Fig. 2.18 Schematic of a tree positioned at x = 120 m and y = 165 m, surrounded by cylinder of obstacle interpretation (red) having diameter 6 m and height 10 m

Fig. 2.19 Visualisation of out-of-bounds regions

2.2.3 Wind The simplest approach to modelling the effect of wind blowing in a certain direction over the golf course is to assume a uniform velocity profile with magnitude determined by the average wind speed. However, the effect of wind shear could be significant, which we must account for in the trajectory prediction. In reality, the wind velocity profile depends on the precise location of various roughness elements like plant canopies and hillocks in the golf course. In order to accurately model the wind conditions, one needs to simulate the atmospheric flow over the golf course, and thus evaluate the wind velocity as a function of both time and space. Yaghoobian and Mittal [16] carried out such simulations for the hole 12 of Augusta National Golf Club.

2.2 Golf Course Environment

37

Fig. 2.20 Logarithmic wind velocity profile

In this book, however, we analyse the effect of wind on gameplay for several golf course layouts. Hence, we neglect the effect of local elements and approximate the wind velocity profile by the surface layer of the atmospheric boundary layer. The following two wind models are considered in our analysis to highlight the effect of wind shear along the wind speed: 1. Uniform: The wind velocity profile is assumed to be uniform in space and time. 2. Logarithmic: The wind velocity profile is assumed to be invariant in time, but changes with height z (see Fig. 2.20), according to [25] uw (z) = uw (zref )

ln z − ln z0 ln zref − ln z0

(2.33)

where, z0 is the roughness length-scale (taken as 0.4 m) suitable for a golf course [25], and the reference height zref is chosen as 10 m to define the wind-speed. The wind velocity influences the flight trajectory of a golf ball by changing its relative airspeed, and consequently the aerodynamic forces RD and RM (see Eqs. 2.3 and 2.4). The effects of the uniform and logarithmic profiles have been compared later in Chap. 4.

2.2.4 Types of Golf Course Designs Golf courses are divided into various categories on basis of several parameters like playing length, landscape style, design layout, etc. The design layout and architecture of a golf course directly affects the gameplay strategy of the player. As a result, each design layout offers a distinct challenge to the player. However, courses are broadly categorised into three design styles on the basis of the nature of challenge that they offer. These are: (1) penal designs, (2) strategic designs and (3) heroic designs [26]).

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Moreover, depending on the playing length of a course, it is given a par rating or 3, 4, 5, etc. In this subsection, we describe the three design styles and also explain what is meant by par rating of a golf course.

2.2.4.1

Penal Design

A golf course is said to have a penal design if a misplayed shot results in a heavy penalty on the player. Such a design may require the player to deliver either a single multiple aerial shots in order to reach the putting green. In either case, there must be at least one such critical shot, which the player cannot avoid. The hazards in the course are arranged in such a way that it allows a very little margin of error in this critical shot. Often, a bad shot results in losing the ball into the water, out-of-bounds or dense forest. This way, it is not recoverable and the player must retry the shot until he gets it right.

2.2.4.2

Strategic Design

In a strategic design style, the player has several possible routes to reach the putting green. All the routes have a similar level reward, risk and difficulty, but differ in the kind of challenge each of them poses. For instance, out of the two 3-shot routes that a course offers, one could have a slightly shorter total distance than the other. At the same time, it could demand a higher level of accuracy from the player due to well placed sand bunkers. Depending on the confidence and comfort of a player, he could choose any one of the strategies to make his way to the green.

2.2.4.3

Heroic Design

A heroic design style is a combination of a penal and strategic design styles, which makes it perhaps the most exciting out of the three. Similar to a strategic design, a heroic course design offers multiple possible routes to reach the putting green. However, unlike a regular strategic style, in case of a heroic layout, these routes significantly vary in the level of reward, risk and difficulty. For instance, in a particular heroic golf course, the regular route could involve hitting three aerial shots to reach the green. Whereas, another possible route could let the player reach the green in just two shots, which is a significant reward. However, the course is designed in such a way, that in the later case the level of difficultly and the risk involved is significantly higher. As a result, a misplayed shot can land a heavy penalty on the player, as in the case of a penal design. Therefore, only an expert player would try to opt for this route. Such a design style makes the game more interesting by introducing an element of gambling, since the player must take a significant risk in order to earn a significant reward.

2.2 Golf Course Environment

2.2.4.4

39

Par Rating of a Course

The par rating of a golf course indicates the number of shots that an average golfer needs to execute in order to finish the game in regulation. The par rating only depends on the playing distance between the tee and the hole and does not account for the difficulty of the course. By convention, it is assumed that players would need two putt shots to hit the ball into the hole after they reach the putting green. Therefore, the minimum par rating a course can have is par-3, i.e. one approach shot followed by two putt shots. Whereas, the more regular case, that involves a drive shot followed by an approach shot to reach the green, is called a par-4 course. Larger courses have a rating of par-5 and in rare cases, par-6 (i.e. four aerial shots and two putt shots).

2.3 Summary A game of golf involves multiple physical phenomena like aerodynamics, collision mechanics, rolling and sliding mechanics and also biomechanics. When a golf ball is hit using a golf club, the initial momentum imparted to it sets it into motion. This motion involves three components: flight, bouncing and rolling. The flight is governed by the launch conditions, gravity and aerodynamic forces that act on the ball due to its airspeed and spin. The net aerodynamic force comprises of drag and Magnus force. The dimples on a golf ball’s surface help in reducing the pressure drag and consequently enhance the upward Magnus force due to the ball’s backspin. This helps the ball to cover a much larger distance while remaining airborne for a longer time. The magnitude and direction of the aerodynamic force at any given instant of time depends on the instantaneous airspeed and spin rate of the ball, which, in turn, undergoes an exponential decay due to the frictional drag acting on the ball. When the ball hits the ground for the first time after the launch, it continues to bounce several times. Each bounce is modelled as an inelastic collision between the ball and the ground. The ball’s rebound velocity and spin rate depend on three factors: (1) the angle at which it lands, (2) the velocity with which it was travelling just before the impact and (3) the spin vector just before the impact. Additionally, the material properties of the ball and the ground govern the collision. After several bounces, the ball’s motion transitions from bouncing to a pure rolling phase until it comes to a halt. The rolling motion is governed by the local terrain profile of the ground and rolling friction between the ball’s and ground’s surfaces. Along with the ball’s motion, the golf course environment and the ball’s interaction with it plays a major role in the game of golf. The environment includes the golf course architecture, as well as the wind conditions. The golf course architecture, in turn, comprises of various elements like terrain, positions of the tee, putting green and hole, and several hazards like ponds, sand bunkers, trees and out-of-bounds regions. Each of these features is modelled with the intention of providing a visual aid, as well as the means to simulate the interaction of the ball with the course itself. The

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2 Understanding the Motion and the Environment

wind is modelled using a logarithmic velocity profile to incorporate the effect of the atmospheric boundary layer. Finally, we describe three design styles of a golf course: (1) penal, (2) strategic and (3) heroic, and explain what is meant by par rating of a golf course.

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References

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18. Mehta, R. D. (1985). Aerodynamics of sports balls. Annual Review of Fluid Mechanics, 17, 151–189. https://doi.org/10.1146/annurev.fl.17.010185.001055. 19. Meister, D., Ladd, A., Butler, E., Zhao, B., Rogers, A., Ray, C., et al. (2011). Rotational Biomechanics of the Elite Golf Swing: Benchmarks for Amateurs. Journal of applied biomechanics, 27, 242–51. https://doi.org/10.1123/jab.27.3.242. 20. Wesson, J. (2009). The Science of Golf. Oxford University Press. 21. Achenbach, E. (1974). The effects of surface roughness and tunnel blockage on the flow past spheres. Journal of Fluid Mechanics, 65(1), 113–125. https://doi.org/10.1017/ S0022112074001285. 22. Penner, A. R. (2002). The run of a golf ball. Canadian Journal of Physics, 80, 931–941. https:// doi.org/10.1139/P02-035. 23. Haake, S. J. (1989). An apparatus for measuring the physical properties of golf turf. https://research.aston.ac.uk/en/studentTheses/apparatus-and-test-methods-for-measuringthe-impact-of-golf-balls. 24. Google Earth, Data MVARI, Data SIO, NOAA, US Navy, NGA, GEBCO Landsat/Copernicus. Pebble Beach Golf Links, California. http://earth.google.com. 25. World Meteorological Organization (2008). Guide to Meteorological Instruments and Methods of Observation. https://www.weather.gov/media/epz/mesonet/CWOP-WMO8.pdf. 26. Smith, W. G. (1998). Teaching golf course design in a landscape architecture curriculum. Thesis for Master of Landscape Architecture: The University of Georgia.

Chapter 3

Simulating the Motion in a Synthesised Environment

A golf ball is like a clock. Always hit it at 6 o’clock and make it go toward 12 o’clock. But make sure you’re in the same time zone. —Chi Chi Rodriguez

Overview In Chap. 2, we described various models to capture the physics associated with a golf ball’s trajectory and its interaction with various elements of the golf course environment. Before we proceed towards the quantitative analysis of the effect of wind in golf (Chap. 4), we need to discuss the algorithms and computational schemes that facilitate this analysis. In order to perform any scientific study, we require data. This data includes information about the inputs provided to the system and the corresponding results that follow. The data can be obtained through different means like experiments, computer simulations, as well as observational studies. In the case of observational studies, one does not have control over the inputs to the system, and inferences are made only based on readily available data. Whereas, in the case of experiments and simulations, one can provide custom inputs, which help in conducting a complete and detailed analysis, while eliminating any bias. However, the fact that golf is a game played outdoors in vast fields makes it difficult to arrange experiments for studying the effects of wind. Moreover, they do not provide control over environmental conditions like wind, which is a crucial factor for us. For this reason, computer simulations are insightful and the backbone of our analysis. This gives us complete control over all parameters of the system, like aerodynamic and material properties of the ball, wind conditions and the golf course architecture. Moreover, it ensures that the inputs to the system are precisely what we desire while maintaining the required degree of consistency. Figure 3.1 shows our © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. Malik and S. Saha, Golf and Wind, https://doi.org/10.1007/978-981-15-9720-6_3

43

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3 Simulating the Motion in a Synthesised Environment

Fig. 3.1 Computer simulation of a golf ball’s motion in a sythesized golf course environment

computer simulation of a golf ball’s motion in a synthesised golf course environment. Here, the ball automatically perceives the environment and suitably interacts with the golf course features (see Sect. 2.2) like terrain profile and obstacles. For instance, the solid curve is a valid trajectory, in which the golf ball does not hit any obstacle or fly out-of-bounds. The black segment denotes the flight trajectory, while the red and blue segments denote the bouncing and rolling phases, respectively. As can be seen, the ball tends to bounce/roll down the hill during its ground run. On the other hand, the black-dotted curve is an invalid trajectory, since the ball collides with a tree. In this chapter, Sect. 3.1, explains the algorithm and the numerical schemes used in the integrated programme and individual subroutines that simulate the motion of a golf ball in a synthesised environment, based on the models in Chap. 2. Section 3.2 describes the process that we follow to model real golf courses for the case studies in Chap. 5.

3.1 Simulation Programme Simulation of a golf ball’s motion in a golf course environment consists of four major subroutines, that are linked in an integrated manner via logical conditions. These are flight, bounce, roll and validity check. In this section, we explain the algorithm of the overall programme, followed by each of the four individual subroutines.

3.1 Simulation Programme

45

3.1.1 Algorithm Similar to the real world, the simulation of a golf ball’s motion begins with the assignment of launch conditions (see Fig. 2.3) to the ball. The user enters the values of launch speed (u 0 ), launch spin rate (N0 ), launch heading (ψ), launch angle (θ ) and spin axis tilt (φ). Additionally, the user enters information about the wind velocity profile (uw (z)), which includes the wind’s heading direction (β), reference speed (u w (zr e f )) and the choice between the uniform and logarithmic profiles (see Sect. 2.2.3). These launch conditions are used to determine the initial velocity and spin vectors, which along with the initial position vector (pre-defined position of the tee) is fed to the flight subroutine. The flight subroutine uses the initial state of the ball (position and velocity vectors) and computes the flight trajectory by numerical integration of Eq. 2.2, over time. After the ball lands on the ground, the validity check is performed to determine whether or not the impact position is inside a hazard (pond, bunker, tree or out-ofbounds). If the check detects that the ball lies inside a hazard, then the programme is terminated, and a flag is raised to indicate the invalidity of the trajectory. If not, then the maximum height of the ball with respect to terrain surface throughout the projectile is determined. If this maximum height is greater than 5 mm, then the bounce subroutine is initiated, which evaluates the rebound velocity (vr ) and rebound spin (ωr ) based on the scheme described in Sect. 2.1.2. These rebound quantities are then used to determine the new initial conditions (initial spin rate, position and velocity vectors) for the next projectile. The flight subroutine is called again, and the loop continues. On the other hand, if the maximum height of the ball in the previous projectile is less than 5 mm, then the loop breaks and the rolling subroutine is initiated. This subroutine computes the rolling motion of the ball by numerically integrating Eq. 2.27, over time. When the ball comes to a halt, the programme checks whether the final resting position lies inside the pre-defined putting green. If so, another flag is raised to indicate the same. Finally, the trajectory is plotted in three-dimensional space along with the golf course environment for visualisation (see Fig. 3.3). Figure 3.2 shows the flowchart of the overall algorithm of the simulation programme.

3.1.1.1

Flight Subroutine

The flight subroutine computes the flight trajectory by numerical integration of Eq. 2.2, over time. Two schemes are commonly used to time-stepping—(1) Euler method and (2) 4th order Runge-Kutta. Here, we explain the algorithm using the Euler scheme. However, the same methodology is equally applicable to the RK-4 scheme.

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3 Simulating the Motion in a Synthesised Environment

Fig. 3.2 Flowchart of the overall algorithm of the simulation programme

3.1 Simulation Programme

47

Fig. 3.3 Visualisation of a golf ball’s motion in a sythesized golf course environment using MATLAB

Before we explain the Euler scheme, it is important to understand where it comes from. Any mathematical function ( f (x)) can be estimated in the proximity of a known point (x = a) using Taylor’s series, given by f (x) =

∞  f (n) (a) (x − a)n n! n=0

(3.1)

where, f (n) (a) is the n th derivative of f (x) at x = a. Note that when x is very close to a, the higher order terms can be ignored. In a general case of second order approximation, f (x) can be rewritten as f (a + h) = f (a) +

f  (a) 2 f  (a) h+ h + O(h 3 ) 1! 2!

(3.2)

where, h = x − a and O(h 3 ) denotes the terms having order higher than or equal to h 3 . Ignoring the higher order terms, we get f (a + h) ≈ f (a) +

f  (a) 2 f  (a) h+ h 1! 2!

(3.3)

Similarly, first order approximation is given by f (a + h) ≈ f (a) +

f  (a) h 1!

(3.4)

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3 Simulating the Motion in a Synthesised Environment

Hence, we get f  (a) ≈

f (a + h) − f (a) h

(3.5)

Now, consider an ordinary differential equation (ODE) in time (t) f  (t) = g(t, f (t))

(3.6)

Using first order approximation at t = tn+1 , we get f n+1 = f n + g(tn , f n ) × dt

(3.7)

where, f n ≡ f (tn ) and dt = tn+1 − tn . This is called the Euler time-stepping method. When the function’s initial state ( f 0 ) is known, then the ODE can be solved using numerical integration as shown in Eq. 3.7. Such a case is known as an initial value problem (IVP). The same is equally applicable when the function is given in vector form (i.e. f(t)). A golf ball’s flight is also described as an ODE (Eq. 2.2), which can be solved as an IVP. The subroutine requires the initial position, velocity and spin vectors as inputs. The position and velocity vectors and then concatenated to form the 6 × 1 state vector (Y) of the ball. The z component of the ball’s position, along with the user-defined wind profile, is used to determine the wind velocity vector uw . This then determines the relative airspeed of the ball, given by (ub − uw ). Meanwhile, the decay in the spin rate is evaluated at the current time using Eq. 2.6. Therefore, the instantaneous airspeed and spin rate are used to estimate the C L and C D , as explained later in Sect. 3.1.2. Thereafter, the aerodynamic forces R D and R M are determined using Eqs. 2.3 and 2.4, and thus the instantaneous acceleration x¨ is obtained from Eq. 2.2. Hence, ˙ is defined by concatenating the the time derivative of the ball’s state vector (Y) velocity and acceleration vectors. Finally, the state vector of the ball is updated as per ˙ × dt Ynew = Yold + Y

(3.8)

where dt is the time step of integration. Next, the validity check is performed on this updated state vector. If the ball is found to lie inside a hazard, then the programme is terminated, and a flag is raised to indicate the invalidity of the trajectory. If not, then the height of the ball with respect to terrain surface is found as per z b = (z − z T ). If z b ≤ 0, i.e. the ball is on or below the terrain surface, the subroutine is terminated. If z b > 0, i.e. the ball is in the air, then the updated state vector Ynew is fed back to the routine, and the loop continues. Figure 3.4 shows the flowchart of the flight subroutine.

3.1 Simulation Programme

Fig. 3.4 Flowchart of flight subroutine

49

50

3.1.1.2

3 Simulating the Motion in a Synthesised Environment

Bounce Subroutine

The bounce subroutine computes the rebound velocity and spin rate vectors (vr and ωr ) using the relations described in Sect. 2.1.2. It begins by extracting the impact velocity and impact spin vectors (vi and ωi ) from the trajectory of the previous projectile. It transforms these vectors into the (x g , yg , z g ) frame. Then, θc is determined using Eq. 2.9. Hence, the impact quantities are transformed to the (x g , yg , z g ) frame using Eq. 2.10. Thereafter, μcz and μcx are determined from Eqs. 2.12 and 2.13. Therefore, after comparing μ with μc for both the planes (x g − yg and z g − yg ), the rebound quantities (vr and ωr ) are calculated using either the sliding or the rolling relations. Finally, vr and ωr are transformed to the original (x, y, z) frame and the subroutine is terminated. Figure 3.5 shows the flowchart of the bounce subroutine.

3.1.1.3

Roll Subroutine

Similar to the flight subroutine, the roll subroutine computes the rolling motion of the golf ball by numerical integration of Eq. 2.27, over time. The routine begins with the extraction of the component of vi that is tangential to the local terrain surface. Alongside, the acceleration of the ball (¨x is calculated using Eq. 2.27). The tangential velocity vi x and the acceleration x¨ are concatenated to form the time derivative of ˙ Thus, the state vector of the ball is updated as per the ball’s state vector (Y). ˙ × dt Ynew = Yold + Y

(3.9)

where dt is the time step of integration. Thereafter, the validity check is performed on the ball’s position. As before, if the ball is found to lie inside a hazard, then the programme is terminated, and a flag is raised to indicate the invalidity of the trajectory. If not, then vi x and x¨ are compared to certain pre-defined tolerance values v and a . If both are lower than their respective tolerances, the subroutine terminated normally. If not, then Ynew is fed back to the subroutine, and the loop continues. Figure 3.6 shows the flowchart of the bounce subroutine.

3.1.1.4

Validity Check

The validity check is performed to determine if a position lies inside any hazards discussed in Sect. 2.2.2 (i.e. pond, bunker, tree or out-of-bounds). The first check is performed to detect an out-of-bounds region. As mentioned before in Sect. 2.2.2, the ball must not enter these regions at any height with respect to the local terrain. Also, for distinction, the local terrain height is assigned a value of −10 m inside these regions. As a result, at a given position, if the terrain height

3.1 Simulation Programme Fig. 3.5 Flowchart of bounce subroutine

51

52

Fig. 3.6 Flowchart of roll subroutine

3 Simulating the Motion in a Synthesised Environment

3.1 Simulation Programme

53

z T = −10 m then the ball lies out-of-bounds. This raises a flag to indicate invalidity, and the subroutine terminates. The second check is to detect ponds. Similar to out-of-bounds regions, the local terrain height is assigned a value of −5 m inside ponds. However, in case of ponds, the hazard is only detected if the ball’s z coordinate lies on or below the local terrain surface. In such a case, the invalidity flag is raised, and the subroutine is terminated. Next, the check to detect a bunker is performed. Again, like in the case of ponds, ellipses defining bunkers are only considered an invalid region if the ball lies on or below the local terrain height z T . Finally, the check for tree detection is performed. Here, the invalidity flag is raised if the ball lies anywhere inside the 10 m high, 6 m diameter cylinders centred at the trees’ locations (see Fig. 2.18). Figure 3.7 shows the flowchart of the validity check.

3.1.2 Numerical Schemes A golf ball’s flight trajectory, unlike a regular projectile, depends not only on earth’s gravity, but also on the aerodynamic forces that act on the ball throughout its flight. The magnitudes and directions of the aerodynamic forces vary as a function of the ball’s instantaneous velocity and spin vectors, which, in turn, change throughout the flight. All-the-more, in case of a complicated geometry like a golf ball, the drag and Magnus forces are not known in the form of simple analytical or empirical relations. This level of complication (see Chap. 2), makes it difficult to calculate the trajectory of a golf ball analytically. Therefore, we solve this problem by adopting numerical techniques in our programme’s subroutines. Here, we discuss the specifications of the same.

3.1.2.1

Interpolation Schemes

Interpolation schemes are useful when a certain quantity needs to be evaluated over a continuous domain, but the associated data is available only at discrete points. This is required at two instances in the programme: (1) estimating the aerodynamic coefficients (C L and C D ) at a given ball speed and spin rate and (2) evaluating the local terrain height (z T ) and gradient at a given location. Aerodynamic coefficients As discussed in Sect. 2.1.1, the aerodynamic forces acting on a golf ball as evaluated using Eqs. 2.4 and 2.3, where the values of C L and C D are extracted from the paper published by Bearman and Harvey [1]. They have reported the coefficient values at around 50 discrete combinations of ball speeds and spin rates covering the entire operating range of a typical golf ball. We evaluate them over the continuous domain by using in thin plate spline interpolation in the MATLAB curve-fitting toolbox. Figures 3.9 and 3.9 show the surfaces of C L and C D over the u b − N plane.

54

Fig. 3.7 Flowchart of validity check

3 Simulating the Motion in a Synthesised Environment

3.1 Simulation Programme

55

Fig. 3.8 C D evaluated using thin plane spine interpolation

Fig. 3.9 C L evaluated using thin plane spine interpolation

Terrain and gradient The terrain surface of the golf course comprises of hills and valleys, as well as other features of the course, like water, bunkers and the out-ofbounds regions. As discussed in Sect. 2.2, the hills, valleys and bunker surfaces are approximated as paraboloids. Whereas, the water bodies and out-of-bounds regions are assigned the elevations of −5 m and −10 m, respectively. In the backend, the entire terrain surface is evaluated at discrete points in the x-y plane with a resolution of 1 m, referred to as terrain matrix. The gradient of discretised terrain is computed using the in-built MATLAB function gradient on the terrain matrix. This thus forms the gradient matrix. Finally, both the terrain and gradient is evaluated over the continuous domain using linear interpolation in the MATLAB curve-fitting toolbox.

3.1.2.2

Time-Stepping Schemes

The flight and rolling motions of the golf ball are governed by ordinary differential equations (ODEs). We solve these ODEs using Euler and 4th order Runge-Kutta (RK-4) numerical integration schemes. The RK-4 scheme is known to be more accu-

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3 Simulating the Motion in a Synthesised Environment

Fig. 3.10 Comparison between the trajectories: (1) reported by Bearman and Harvey (blue), (2) computed using RK-4 scheme with a time step (dt) of 1 ms (black) and (3) computed using Euler scheme with dt = 10 ms (red)

rate than the Euler scheme for the same time step. However, the computation time associated with it is longer. Therefore, there needs to be a trade-off between accuracy and computation time. In order to validate the integration routine, we compare the trajectory reported by Bearman and Harvey [1], with the one generated in our the programme using their aerodynamic data, for the same launch conditions. Figure 3.10 shows the comparison between three trajectories: (1) reported by Bearman and Harvey [1] (blue), (2) computed using RK-4 scheme with a time step (dt) of 1 ms (black) and (3) computed using Euler scheme with dt = 10 ms (red). We see that the three trajectories match well. However, the difference between reported trajectory (1) and the computed trajectories (2) and (3) is more than that between the two computed trajectories themselves. The underlying cause of this difference may be errors in the digital extraction of C L and C D from Bearman and Harvey’s article [1], and the difference in the interpolation schemes used. This further shows that for our analysis, the trajectory computed using Euler method with dt = 10 ms is as good as the one computed using RK-4 scheme with dt = 1 ms. Since the later is computationally much more expensive, we use the Euler scheme everywhere for our analysis.

3.2 Constructing a Model Golf Course In this section, we explain the process we use to model a golf course for the case studies discussed later in Chap. 5. As an example, we describe the modelling process of a real golf course, Pebble Beach Golf Links (Hole 18), in California (see Fig. 3.11). This model is later used in Chap. 5 to analyse the impact of wind on the gameplay at the Pebble Beach course. A similar process can be used to model any other golf course, or even an arbitrary one (see Fig. 3.3). The process of constructing a model golf course involves eight steps: Step 1: Defining the domain We begin the construction by defining a rectangular region in the top view of the course map. This region must include the entire playing area of the course, along with the course features (see Sect. 2.2.2), to be considered in the analysis. For the Pebble Beach course, we chose a (520 × 300) m, with x−axis point about 70o west of north, and origin at the bottom-left corner (see Fig. 3.12a).

3.2 Constructing a Model Golf Course

57

Fig. 3.11 Hole 18, Pebble Beach Golf Links, California. Image courtesy Google Earth, Data MVARI, Data SIO, NOAA, US Navy, NGA, GEBCO Landsat/Copernicus [2]

Hereafter, the positions and orientation of all the features are defined in the same coordinate system. Step 2: Terrain features After defining the domain, we recommend capturing the major characteristics of the golf course’s terrain. These may include hills, valleys and elevated fairways (see Sect. 2.2.1). In this example, however, we assume the terrain to be flat throughout the playing area (see Fig. 3.11). Step 3: Tee, hole and green Next, the positions of the tee and hole are set, and the ellipse approximating the green is defined as per Eq. 2.32. For the Pebble Beach course, the tee is positioned at (34, 186) and the hole at (487, 113). The green region is approximated by an ellipse centred at the hole, with a = 12, b = 10 andα = 30◦ . Figure 3.12b shows the tee position and the ellipse modelling the putting green. Step 4: Water bodies Water bodies like ponds, lakes and in this case oceans, often occupy a significant portion of the golf course’s architecture. Hence, we model them before any other hazard in the course. In this case, we model the ocean as per Eq. 2.32, using a single ellipse centred at (273, 207), with a = 250, b = 100 andα = −9◦ (see fig. 3.12c). More complicated cases can be modelled using multiple ellipses.

58

3 Simulating the Motion in a Synthesised Environment

(a)

(b)

(c)

Fig. 3.12 Defining the a domain, b tee and green region and c ellipse modelling the ocean in the Pebble Beach golf course

3.2 Constructing a Model Golf Course

59

(a)

(b)

(c)

Fig. 3.13 Modelling the a bunkers, b trees and c out-of-bounds region in the Pebble Beach golf course

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Table 3.1 Parameters of the bunkers in the Pebble Beach golf course S. No. x (m) y (m) a (m) b (m) 1 2 3 4 5 6 7

173 234 373 417 498 500 477

66 60 95 98 97 99 96

27 19 29 34.5 8 8 13.5

10 8 6.5 5 4.5 4.5 4

α (◦ ) −30 0 −15 20 0 90 −30

Step 5: Sand bunkers As discussed in Sect. 2.2.2, sand bunkers are modelled similar to how hills and valleys are modelled as per 2.29. The Pebble Beach golf course contains seven sand bunkers. Table 3.1 lists the parameters of the bunkers, (see Fig. 3.13a). Step 6: Trees The Pebble Beach golf course contains two trees within the playing area. These are indicated by the green spots in Fig. 3.13b. The two trees are defined in the model by specifying their locations, (475, 85) and (250, 75). Step 7: Out-of-bounds The last feature used to model golf courses are the out-ofbounds regions. This provides a simple way to eliminate the portions that play no role in the analysis. For example, neighbouring holes, and buildings at the periphery of the playing area Figure 3.13c, shows the region that has been defined as out-of-bounds in the Pebble Beach golf course model. Step 8: Compiling the model Finally, the model is compiled, which ensures that all the features are integrated properly. The process involves • ambiguity due to the intersection of two or more features of different types, • interpolating the terrain matrix and evaluating the gradients over the entire domain, as per the scheme described in Sect. 3.1.2, and • assigning the z coordinate of the positions of tee, hole and trees. Figure 3.14a shows all the features that are to be modelled in the Pebble Beach golf course, and Fig. 3.14b, shows the top view of the compiled model in MATLAB. Finally, Fig. 3.15, shows the simulation of multiple shots in the synthesised golf course environment.

3.3 Summary We conduct the analysis presented in this book simulating the golf ball’s motion in a synthesised golf course environment. All the physical models of the motion and the environment described in Chap. 2 are integrated together in the form of a computer

3.3 Summary

61

(a)

(b)

Fig. 3.14 a All features to be modelled and b resultant compiled model of the Pebble Beach golf course in MATLAB

algorithm and are solved using numerical techniques. The simulation algorithm is broken down into four subroutines: flight, bounce, rolling and validity check. These subroutines together evaluate the complete trajectory of the golf ball and also model the interaction of the ball with the golf course environment. The flight trajectory is computed as an initial value problem, using popular numerical schemes: Euler method and 4th order Runge-Kutta method. During the flight, the aerodynamic forces are evaluated using the instantaneous airspeed and spin rate of the ball. Upon impact with the ground, the ball enters the bouncing phase. This is solved as a series of ground-impacts and subsequent projectiles. When the peak height of the projectile becomes very small, the rolling subroutine is executed. The

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3 Simulating the Motion in a Synthesised Environment

Fig. 3.15 Simulation of multiple shots in the synthesised environment of the Pebble Beach golf course in MATLAB

rolling motion is also solved as an initial value problem, like in the case of the flight trajectory. Here, the gradient of the local terrain profiles helps in determining the inclination of the ground with respect to the local horizontal, which, in turn, determines the gravitational force that the ball experiences while rolling. In order to incorporate the golf course environment in our analysis, we describe a scheme to model any real or arbitrary golf course in the simulation programme. This scheme involves eight steps which are performed in a particular sequence. First, the size of the golf course is defined in the form of a rectangle. This is followed by indicating the location of the primary elements of any golf course: tee, hole, and putting green. After that, the terrain of the course is modelled in the form of hills, valleys and elevated fairways. Next, the sizes, locations and orientations of the hazards present in the golf course are defined. These may include ponds, sand bunkers, tree and out-of-bounds regions. Finally, the environment is compiled to integrate all features properly and hence evaluate the terrain gradients.

References 1. Bearman, P., & Harvey, J. (1976). Golf Ball Aerodynamics. Aeronautical Quarterly, 27(2), 112–122. https://doi.org/10.1017/S0001925900007617. 2. Google Earth, Data MVARI, Data SIO, NOAA, US Navy, NGA, GEBCO Landsat/Copernicus. Pebble Beach Golf Links, California. http://earth.google.com.

Chapter 4

How Does Wind Impact Gameplay?

With that green and the way the wind swirls in that corner of the golf course, playing that hole in even par over four days is going to be really good. —Charles Howell

Overview We discussed the influence of aerodynamic forces on the flight trajectories of golf balls in Chap. 2. The aerodynamic forces (Eqs. 2.3 and 2.4), in turn, depend on the relative airspeed of the ball, and thus the wind speed (u w ). As a result, wind has a direct impact on the flight trajectory of golf balls. In a sport like golf, where the ball travels at very high speeds and covers large distances, the aerodynamic forces and the resulting effect of wind become a crucial part of the analysis. The book aims to narrow the gap between golfers and scientists, so that one may benefit the other. We do this by presenting a quantitative analysis of wind’s effect on the golf ball’s trajectory from the viewpoint of a golf player. The trajectory of a golf ball is of great importance in the game [1]. Therefore, several studies have been conducted on the subject. Elrichson [2] determined the optimal loft for maximising range, using lift and drag models that linearly varied with ball speed. Thereafter, Mcphee and Andrews [3], derived the analytical solution of the trajectory and studied the effects of crosswind and sidespin. Stengel [4] examined the sensitivity of the trajectory with respect to launch conditions, wind and aerodynamic coefficients. Sajima et al. [5] showed the effect of dimple depth on the aerodynamic coefficients, and subsequently the range of the trajectory of driver club shots. Naruo and Mizota [6] computed the trajectory for Skyway (SD432) golf ball using wind tunnels tests and flight under the influence of a logarithmic wind profile for drive and pitch shots. Later the same authors showed the effect of change in depth of dimples and the presence of additional tiny dimples on the flight trajectory using data from wind-tunnel tests of custom-made golf balls [7]. Recently, Yaghoobian and Mittal © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. Malik and S. Saha, Golf and Wind, https://doi.org/10.1007/978-981-15-9720-6_4

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[8], investigated the uncertainty in landing point due to plant canopy in the hole 12 of Augusta National Golf Club. They performed large-eddy simulations of the local atmospheric flow and used aerodynamic data reported by Bearman and Harvey [9]. Through this monograph, we aim to quantitatively analyse how important the effect of wind is in the game of golf. We also study the influence of launch conditions and aerodynamic characteristics of the golf ball. Later, in Chap. 5, we also discuss how launch conditions can be adapted to counter the effect of wind. Finally, we provide a realistic perspective on the results through case studies of three golf courses. In the game of golf, the primary objective is to hit the ball from the tee into the hole in the minimum number of shots. Typically, this is done by means of mainly three types of shots: (1) drive, (2) approach and (3) putt (see Fig. 4.1). In case of a drive shot, the golfer hits the ball placed while it is placed on the tee. This is often the first shot of the game, and is also called “teeing off”. The motive here is to cover the maximum possible distance and hence get the ball as close as possible to the hole. This is achieved by providing launch conditions such that the range of the trajectory is maximised. For a golf ball, such a trajectory is quite flat, having a launch angle of around 10◦ . A drive shot is often followed by an approach shot, wherein the golfer aims to deliver the ball onto the putting green. This requires precise control over the ball’s trajectory. Moreover, it is desirable to minimise the ground run of the ball, since it is difficult to predict without detailed information about the local terrain profile at the landing site (see Sect. 2.1.2). In order to reduce the number of shots, a golfer may prefer to aim for the putting green despite the presence of obstacles (like trees, sand bunkers and ponds). As a result, a lobbed trajectory is most suitable for approach shots, since it facilitates obstacle avoidance, as well as minimises the ground run of the ball, owing to higher landing angles (γ ). Finally, when the ball lands onto the putting green, a putt shot is used to roll it into the hole. Additionally, depending on the golf course, a golfer may need to hit other types of shots like lay-up, chip, punch and flop [10]. However, in this book, we restrict our focus on the three main types—drive, approach and putt. Wind holds different levels of importance in each kind of shot. For a drive shot, the increment or decrement in the range is of primary interest. While a tailwind aids the golfer by increasing the range, a headwind reduces the range, hence costing the golfer an additional shot or two, in order to get closer to the hole. Nevertheless, in the case of an approach shot, wind plays a much more critical role [11]. This is because the golfer needs fine control not only on the landing position, but also the shape of

Fig. 4.1 Types of shots in a typical game of golf

4.1 Impact of Wind on Gameplay

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the trajectory. This precision is compromised by the action of wind. A putt shot, on the other hand, does not have a flight component. So, naturally it shows negligible influence of wind and is primarily governed by the local terrain profile. Therefore, we exclude putt shots from our analysis hereon. In order to analyse the impact of wind in gameplay, we raise the following questions, which we aim to answer in this chapter: • • • •

How important is the effect of wind? How does it vary as the wind conditions change? How do launch conditions influence the effect of wind? What role do the aerodynamic characteristics of the ball play?

We begin the discussion with Sect. 4.1, wherein we establish the significance of wind’s effect in the game of golf. Following this, in Sect. 4.2, we present a parametric study on the effect of wind, wherein we study the dependence of wind-induced deviations on three factors that define the physical setting (wind conditions, launch conditions and aerodynamic characteristics of the ball).

4.1 Impact of Wind on Gameplay In Sect. 2.1.1, we described the physical model of the flight of a golf ball. We showed how the trajectory depended on various factors like launch conditions, aerodynamic characteristics and wind conditions. Before we begin the discussion on wind’s effect on the ball’s flight trajectory, we need to understand the importance of wind in the game of golf as compared to the other factors. Even before that, we must identify which model of wind velocity profile should be used throughout the analysis. As discussed in Sect. 2.2.3, we have considered two profiles: (1) uniform and (2) logarithmic (Eq. 2.33) [12]. Figure 4.2 shows the comparison between the effects of these two profiles: logarithmic (solid lines) and uniform (dashed lines). The black curve is the trajectory of a typical 7-Iron approach shot [13], without the presence of wind, the red trajectories are shots in the presence of a 10 m/s headwind (along negative x-axis,) and the blue ones are in the presence of a 10 m/s tailwind (along positive x-axis). A wind of 10 m/s magnitude is typically considered a moderately high wind speed in the context of golf. In the case of the headwind, the trajectories generated under the effect of the two profiles are significantly different in terms of range and landing angle. Also, we know that the logarithmic profile more correctly models the actual wind conditions due to the development of the atmospheric boundary layer [12]. Therefore, we use the logarithmic wind profile everywhere in the analysis hereon. Another observation in Fig. 4.2, is that the tailwind deflects the original trajectory by more than 25 m thus increasing the range by 18%. Moreover, the landing angle changes from 50◦ to 31◦ and consequently the ground run increases from 3.4 to 7.6 m.

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Fig. 4.2 Comparison between the effects of logarithmic (dashed lines) and uniform (solid lines) wind velocity profiles of a headwind (red) and tailwind (blue) of 10 m/s on a typical approach shot

On the other hand, the deflection due to the headwind is even higher. The range is reduced by 65 m, bringing it down to almost 60% of the original value. While, the landing angle increases from 50◦ to nearly 85◦ , thus causing the ball to roll backwards by 3 m due to its backspin. Such a deflection, if not accounted for, can potentially change the entire game for a golfer. In order to put these numbers in the perspective of the game, we compare the deflection due to a moderate wind of 5 m/s with the variation in flight trajectories due to inconsistency in the launch conditions (i.e. launch speed u 0 , launch spin rate N0 and launch angle θ ). As discussed in Sect. 2.1.1, a golfer aims to deliver his swings such that they consistently yield the launch conditions he wants. However, imperfections in the swings cause deviations from the desired launch conditions. We refer to these deviations as launch inconsistency. We model it using data of “consistency numbers” in the launches of a junior golfer, as reported by TrackMan [14], on their website. The consistency number of a launch parameter reflects its standard deviation among multiple shots executed by the golfer. TrackMan [14] reports the consistency numbers and average launch conditions for three kinds of shots, having ranges: 60 yar ds, 70 yar ds and 80 yar ds. For instance, the consistency numbers for launch speed (u 0 ) corresponding to 60 yar ds, 70 yar ds and 80 yar ds are 2.1 mph, 2.1 mph and 1.0 mph, respectively. While, the mean launch speeds are 57.6 mph, 65.6 mph and 71.7 mph, respectively. We thus get relative standard deviation as the ratio of consistency number to corresponding mean speed. That is 2.1 1 2.1 , , 57.6 65.6 71.7

(4.1)

Thereafter, we define the inconsistency in launch speed (δu 0 ) as the mean of the relative standard deviations, given by δu 0 =

2.1 1 u 0 1 2.1 + + ) = 0.027 = ( u0 3 57.6 65.6 71.7

(4.2)

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Fig. 4.3 Comparison between the deflection in the ball’s trajectory caused by the effect of wind, and the deviation in trajectory due to variation in launch speed; Legend: original trajectory (blackdotted), deflected by wind (red-solid), deviated due to variation in launch speed (black solid)

Fig. 4.4 Comparison between the deflection in the ball’s trajectory caused by the effect of wind, and the deviation in trajectory due to variation in launch spin rate; Legend: original trajectory (black-dotted), deflected by wind (red-solid), deviated due to variation in launch spin rate (black solid)

This means that u 0 deviates by up to 2.7% of its intended value. Thus, the inconsistent ball speed (u˜b ) is given by (1 − δu 0 )u 0 ≤ u˜0 ≤ (1 + δu 0 )u 0

(4.3)

Figure 4.3 shows the comparison between the effects of wind and ball speed on the trajectory of a golf ball. The dotted black curve is the trajectory of a typical approach shot without any wind. The solid red curves are the trajectories of the same shot under the effect of a 5 m/s headwind and tailwind, respectively. Whereas, the solid black curves depict the deviations from the original trajectory when it is launched with modified ball speeds of u˜b = (1 − δu 0 )u 0 and u˜0 = (1 + δu 0 )u 0 . Clearly, the deflection due to wind is much higher than the deviation induced by inconsistency in launch speed. Similarly, for δ N0 = 0.13 and δθ = 0.09, we get Figs. 4.4 and 4.5. Here too, we see a similar trend. Therefore, the effect of wind is much more profound than the effects of either of the launch conditions. Hence, we conclude that wind has a major impact on the flight of golf balls, and thus gameplay itself.

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Fig. 4.5 Comparison between the deflection in the ball’s trajectory caused by the effect of wind, and the deviation in trajectory due to variation in launch angle; Legend: original trajectory (blackdotted), deflected by wind (red-solid), deviated due to variation in launch angle (black solid)

4.2 Analysing Wind’s Effect In this section, we study various effects of wind in detail. Our objective is to analyse how the wind-induced deviation varies as we change the physical setting. For this, we consider three parameters which together define the physical conditions governing the flight trajectories of a golf ball. Based on the discussion in Sects. 2.1.1 and 3.1.1, we understand that the flight can be completely determined by the knowledge of the (1) launch conditions, (2) aerodynamic coefficients (C L and C D ) and (3) wind velocity profile (uw ). Therefore, we use the same parameters to define the physical setting. However, in order to provide the perspective of gameplay, we redefine these parameters as • relative heading of wind velocity with respect to the launch direction ψ, • standard golf clubs that impart various launch conditions (u 0 , N0 and θ ) to the ball and • dimple geometry of the golf ball that determines its aerodynamic characteristics. We analyse the effects of each of these parameters in this section.

4.2.1 Heading and Wind We begin this analysis by defining heading in the context of wind. One of the launch conditions of a golf shot is heading angle (ψ). This is the angle that the component of the ball’s initial velocity vector (u0 ) along the x-y plane makes with the x-axis. Similarly, for the wind velocity vector (uw ), we call this angle wind heading (β). Note that β is actually a function of altitude (z) since the wind curls around local elements like trees [8]. However, we neglect this effect in our analysis and treat β as

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Fig. 4.6 Launch heading, wind heading and relative wind heading

a constant in space. Thus, relative wind heading (βr ) is defined as the angle that the wind vector makes with the initial shot direction (see Fig. 4.6), and is given by βr = β − ψ

(4.4)

Here, we analyse how the effect of wind varies as βr changes. Since this is a relative quantity, it can be analysed from two stationary frames of reference: (1) launch frame of reference and (2) wind frame of reference.

4.2.1.1

Launch Frame of Reference

In this frame of reference, ψ is kept constant, while β is varied. This is suitable for visualising the deflection in a particular trajectory due to the wind that is incident along various directions. Figure 4.7a shows a typical 7-Iron approach shot trajectory [13] (solid red curve), with ψ = 0. When this trajectory is subjected to wind having β ranging from 0◦ to 360◦ , we obtain the dotted red curves depicting the deflected trajectories. These deflected trajectories naturally land at positions different from that of the original trajectory. We mark the loci of the first landing positions (beginning of the ground run) and the final halt positions with the thick dashed and dotted red curves on the x-y plane respectively. Figure 4.7b, c provides the top and side views of the same. There are certain characteristics and trends of the wind-induced deviations and their underlying causes 1. Shape of the trajectory In case of a tailwind (βr = 0◦ ), the trajectory becomes longer and flatter (see Fig. 4.7c). That is, there is an increment in the flight range (R f ) and a decrement in the peak height (h M ). Whereas, a headwind (βr = 180◦ ) makes the trajectory short (lower R f ) and lobbed (higher h M ). On the other hand, a pure crosswind (βr = 90◦ ) brings about almost no change in both R f and h M along the original heading direction (blue dashed curve in Fig. 4.7c).

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Cause The presence of a tailwind leads to a reduction in the relative airspeed (ub − uw ) of the ball. Due to this, the magnitudes, aerodynamic lift and drag forces also reduce. As the lift reduces but the gravity remains unaffected, the net downward force increases. As a result, the ball is not able to travel as much along the vertical direction as it did in the absence of wind, thus reducing h M . At the same time, due to the reduction in drag, the ball’s velocity does not undergo as much retardation as it did in the absence of wind. As a result, the ball travels relatively faster while it is in the air, and hence covers a larger distance along the horizontal direction, before it finally returns to the ground. This leads to an increase in R f . Similarly, the opposite effect is seen in case of a headwind since the airspeed of the ball increases, thus increasing the magnitudes of the aerodynamic lift and drag forces. In case of a crosswind, however, there is no change in the component of ball’s airspeed along the original heading direction, in this case along the x-axis. Therefore, the motion of the ball in the x-z plane remains unaffected. However, there is an additional component of the airspeed opposite to uw which causes the deflection the trajectory along (or opposite to) the y-axis (see Fig. 4.7b). 2. Length of the ground run In the presence of a tailwind, the ball tends to bounce and roll over a larger distance as compared to the case with no wind (see Fig. 4.8d). As βr increases, the ground run gradually becomes shorter, until it reaches its minimum value at β = 180◦ (heading). Cause As discussed in Sect. 2.1.2, the bouncing motion of a golf ball has a direct dependence on its landing conditions. Larger the tangential component of impact velocity (vi x ), larger will be the tangential component of rebound velocity (vr x ). This, in turn, increases the horizontal range of the subsequent projectile, thus increasing the overall length of the ground run. Therefore, when γ is smaller (as in the case of a tailwind), the ground run is longer. As βr increases, the relative airspeed of the ball, and thus the aerodynamic drag, increases. Due to this, the velocity of the ball at z = h M decreases. Hence, the ball falls back to the ground at a steeper angle. As a result, γ increases and the ground run becomes shorter (see Fig. 4.8a). 3. Direction of the ground run In the cases of pure tailwind and headwind (β = 0◦ and 180◦ ), as well as in the absence of wind, the ground run of the ball is along the direction of the ball’s velocity just before the first bounce (vi ). However, when there is a component of uw along the y−axis, the direction of ground differs from the direction of vi (see Fig. 4.7b).

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(a)

(b)

(c)

Fig. 4.7 Deflection in a golf ball’s trajectory due to wind incident along various directions (β ranging from 0◦ to 360◦ ) in: a 3D, b top and c side views; the solid red curve depicts the original trajectory and dotted red curves depict the deflected trajectories; curves on the x-y plane show the loci of first landing positions (dashed-red) and final halt positions (dotted-red)

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Fig. 4.8 Variation in wind-induced deviations in: a landing position, b peak height, c landing angle and d ground run of a golf ball’s trajectory with respect to βr

Fig. 4.9 Variation in time of flight with respect to βr

Cause We discussed previously in Sect. 2.1.2, in the case of a pure backspin with respect to the instantaneous velocity of the ball (i.e. ωi x = 0), its rebound velocity has no component along the z g direction (i.e. vr z = 0). Therefore, the direction of the ball’s motion in the x-y plane remains unchanged after the bounce. Such is the situation in case of no wind and pure tailwind/headwind when the ball is imparted a pure backspin at launch. This is because the direction of the ball’s motion in the x-y plane never changes during the entire flight (see Fig. 4.7b). Hence, the originally imparted backspin has no component along x g while landing.

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However, when u w has a component along the y−axis, the ball changes its direction of motion in the x-y plane (see Fig. 4.7b), and the spin develops a ωi x component. Due to this, vr z = 0, and the ground run’s direction differs from that of the ball’s motion just before the first bounce. 4. Magnitude of deviations Figure 4.8a–d show that the magnitude of wind-induced deviations in the trajectory characteristics (R f , h M and γ ) due to a headwind are considerably higher than those due to a tailwind having the same speed. Cause The increased lift force in the presence of a headwind leads to a decrease in the net downward force that acts of the ball during its flight. Due to this, the ball is able to stay airborne for a longer duration of time (see Fig. 4.9). This, in turn, gives both the aerodynamic forces more time to influence the flight trajectory. As a result, the effects of both lift and drag (i.e. decrease in R f and increase in h M ) are further enhanced. Whereas in the case of tailwind, the opposite happens. The reduced lift force leads to a lower time of flight (T f ), which, in turn, diminishes the effects of the aerodynamic forces. As a result, the wind-induced deviations are more profound in a headwind than in a tailwind. Moreover, due to the logarithmic nature of the wind velocity profile (see Sect. 2.2.3, the magnitude of wind speed (u w ) increases with height. Since the ball attains greater heights in the presence of a headwind, it is subjected to higher wind speeds during a part of its flight. This is not the case in the presence of a tailwind. It is perhaps for this reason that we saw a more significant difference between the effects of uniform and logarithmic wind profiles due to a headwind than due to a tailwind. Therefore, in the case of a headwind, the logarithmic wind profile further contributes to the deviations, making them even more profound.

The wind-induced deviations are naturally amplified as u w increases, while their trends remain the same. Figure 4.10 shows the loci of the first landing positions of the ball, subjected to wind speeds of 3, 5 and 10 m/s. Such might be the case during the course of the day. While sea breeze changes to land breeze, the magnitude, as well as the direction of wind changes. Thus, the players need to understand how the magnitude and direction of wind affects the ball’s trajectory. If such effects are not properly accounted for, the ball might land into hazards like sand bunkers, water bodies and trees. This would cost the player an additional shot, and he would need to plan an alternate path to reach the green.

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(a)

(b)

Fig. 4.10 Deflection in a golf ball’s trajectory due to wind speeds of 3 m/s (red), 5 m/s (green) and 10 m/s (blue) in a 3D and b top views

4.2.1.2

Wind Frame of Reference

In this frame of reference, β is kept constant, while ψ is varied. This helps in visualising the resultant range of the golf ball’s trajectory when launched along various headings while being subjected to wind incident along a particular direction. This is often the scenario in the real game, when a golfer needs to study the wind speed and direction, and thereby estimate the necessary corrective measures to account for its effect. Figure 4.11 shows the flight trajectories of a typical 7-iron launched from the origin along various directions. The black curves depict the trajectories in the absence of wind, while the red curves depict the trajectories in the presence of a 5 m/s wind having β = 0◦ . Figure 4.12a shows the x-y plane of Fig. 4.11. Here, the dashed-black curve is the locus of the first landing points of the 7-iron shots in the absence of wind. This curve is a perfect circle whose radius equals the flight range of the shot. Whereas, the solid and dotted red curves are the loci of the first bounces and the final halt positions

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Fig. 4.11 Flight trajectories for various launch headings ψ and with no wind (black curves) and a 5 m/s wind having β = 0◦ (red curves)

of the ball in the presence of wind. Naturally, the ball covers the longest distance when ψ = β (or βr = 0◦ , i.e. tailwind) and the shortest distance when βr = 180◦ , i.e. headwind. When the value of β is changed, the curves simply rotate about the origin by the same angle β (see Fig. 4.12b). This is because wind-induced deviations depend only on the relative direction of the wind, i.e. they are functions of βr . For instance, a headwind (i.e. βr = 180◦ ) will always cause the same deviation the trajectory characteristics, no matter what the launch heading is. The only difference is that the β corresponding to headwind (i.e. ψ + 180◦ ) changes as we vary ψ. Using this frame of reference, we can also visualise the locus of points from where the ball may be launched so that it finally halts at the origin (see Fig. 4.13). For a particular value of βr , this locus (red curve) is simply the mirror image of the landing locus (blue curve), about the plane perpendicular to uw and passing through the origin. We refer to this as the launching locus. These concepts of the launching and landing loci will be utilised later for the case studies in Chap. 5.

4.2.2 Clubs and Wind We discussed earlier in this chapter, a typical game of golf comprises of a sequence of various types of shots, namely drive, approach and putt. During a game, a golfer carries along a collection of standard golf clubs, each of which is designed to impart a distinct set of launch conditions to the ball. The resulting flight trajectory of the ball can then be categorised as either a drive or an approach shot. In practice, a golfer trains to deliver consistent shots with his golf clubs, and in the process, he finds out

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(a)

(b)

Fig. 4.12 Loci of first bounce (solid curves) and final halt positions (dotted curves) for various launch heading ψ and wind heading: a β = 0◦ and b β = 30◦

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Fig. 4.13 Loci of final halt positions after launching from origin (blue) and launch positions to halt at origin (red) for various launch headings ψ and wind heading β = 30◦

the typical flight trajectories his strokes generate. This knowledge thus helps him decide which club to use, depending on the requirement during a game. In this section, we analyse the variation in wind-induced deviation as the club is changed. For this purpose, we use the average launch conditions of various standard golf clubs from PGA Tours, as reported by TrackMan on their website [13] (see Table 4.1). Out of these standard clubs, the driver and woods are used for the long drive shots, and the Irons are used for the lobbed approach shots. A wedge is used for high-lofted

Table 4.1 Average launch conditions of standard clubs from PGA Tours reported by TrackMan [13] Club u0 N0 θ (m/s) (R.P.M) (◦ ) Driver 3 Wood 5 Wood Hybrid 15-18o 3 Iron 4 Iron 5 Iron 6 Iron 7 Iron 8 Iron 9 Iron Pitching Wedge

74.7 70.6 68.0 65.3 63.5 61.2 59.0 56.8 53.6 48.7 48.7 45.6

2686 3655 4350 4437 4630 4836 5361 6231 7097 7998 8647 9304

10.9 9.2 9.4 10.2 10.4 11.0 12.1 14.1 16.3 18.1 20.4 24.2

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Fig. 4.14 Golf ball trajectories of standard club shots (longest to shortest): (1) Driver, (2) 3-Iron, (3) 7-Iron and (4) Pitching wedge

pitch shots and finally, a hybrid, as the name suggests, provides the length of a drive shot and the accuracy of an approach shot. When these launch conditions are fed to our simulation programme, the resultant trajectories of the golf shots are computed. Figure 4.14 shows the typical trajectories of four golf clubs, (1) Driver, (2) 3-Iron, (3) 7-Iron and (4) Pitching wedge, in decreasing order of their ranges. A driver has the longest shaft among all the clubs. Therefore, when a golfer swings it, the clubface has the highest tangential velocity, thus imparting the highest u 0 to the ball. Moreover, its clubface is inclined at a lower angle with respect to the shaft (known as the loft) as compared to the other golf clubs. This, along with its surface roughness, imparts relatively lower θ and N0 to the ball. Such launch conditions are suitable to maximise the range of the golf ball. Thus, the purpose of a drive shot is met. As we move towards the lower clubs in Table 4.1, the shaft length decreases and the loft increases. As a result, u 0 decreases and θ and N0 decrease, ultimately making the trajectory shorter and more lobbed (decreasing R f and increasing h M and γ ). The club-wise trajectory characteristics (R f , h M and γ ) are listed in Table 4.2 and Fig. 4.15, shows the trends in their magnitudes across different clubs.

Table 4.2 Trajectory characteristics of average club shots Club Rf hM (m) (m) Driver 3 Wood 5 Wood Hybrid 15–18o 3 Iron 4 Iron 5 Iron 6 Iron 7 Iron 8 Iron 9 Iron Pitching Wedge

236.7 221.6 209.0 199.6 191.6 182.0 171.1 161.3 149.7 138.8 126.9 113.1

29.8 27.5 28.3 27.5 26.4 25.6 26.1 28.9 31.1 33.3 33.7 34.3

γ (◦ ) 39.8 40.6 42.1 42.1 41.9 42.0 44.0 47.4 49.7 51.5 52.5 54.0

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Fig. 4.15 Trajectories characteristics a R f , b h M and c γ of average club shots Table 4.3 Club-wise relative wind-induced deviations in Rf , h M and γ due to a tailwind of 5 m/s Club δRf δh M δγ Driver 3 Wood 5 Wood Hybrid 15–18◦ 3 Iron 4 Iron 5 Iron 6 Iron 7 Iron 8 Iron 9 Iron Pitching Wedge

6.9 6.5 6.6 6.4 6.0 5.9 5.9 6.6 8.6 10.0 11.7 14.7

8.7 11.1 12.4 12.5 12.8 12.9 5361 13.0 12.7 13.1 13.8 13.3

20.9 22.8 23.3 23.1 23.3 23.1 23.1 14.1 22.9 22.8 22.5 22.1

To analyse the variation in wind’s effect as the launch conditions are changed, we evaluate the relative wind-induced deviations in the trajectory characteristics. These are given by |Q| × 100%; (4.5) δQ = Q where Q is the trajectory characteristic (R f , h M or γ ) of in the absence of wind and |Q| is the magnitude of deviation in Q when the ball is subjected to wind. For a tailwind or 5 m/s, these relative deviations (δRf , δh M and δγ ) are listed in table 4.3 for each of the golf clubs. While there is no clear trend in the case of δγ , δRf and δh M show tend to increase as one moves towards higher lofted clubs (see Fig. 4.16).

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Fig. 4.16 Club-wise relative wind-induced deviations in a Rf and b h M due to a tailwind of 5 m/s

Cause As the loft of the club increases, the spin imparted to the ball at launch (N0 ) also increases (see Table 4.1). Since the spin rate decays exponentially from the initial value throughout the flight, it implies that a higher initial spin rate will result in higher instantaneous spin rates through the entire duration of the flight. As per the values of C L and C D reported by Bearman and Harvey [9], both the coefficients have an increasing trend with respect to the instantaneous spin rate of the ball (see Figs. 3.9 and 3.8). As a result, a trajectory that launches with a higher spin rate is influenced more by both the aerodynamic forces (lift and drag). As established in sec. 4.2.1, wind has a direct effect on these forces. Hence, due to larger values of C L and C D , a higher N0 (as in the case of high-lofted clubs) leads to higher wind-induced deviations. Moreover, higher lofts of golf clubs also lead to higher launch angles of the ball (θ ). For equivalent ranges (R f ), a higher θ increases the upward component of the ball’s launch speed (u 0 ). Thus, the ball stays airborne for a longer duration of time, and hence experiences a greater influence of wind. Therefore, higher N0 and higher θ together lead to higher relative wind-induced deviations. Finally, all clubs show a similar trend in wind-induced deviations with respect to relative wind heading βr . As discussed in Sect. 4.2.1, the effect of wind becomes more profound as βr is increased. Figure 4.17 shows the relative deviation in range (δRf ) due to a u w = 5 m/s for four clubs: (1) Driver (blue), (2) 3-Iron (red), (3) 7-Iron (yellow) and (4) Pitching wedge (green). As expected, the deviations become higher as the loft of the club increases.

4.2.3 Aerodynamics and Wind Bearman and Harvey [9] reported the experiment data of C L and C D for two types of balls: (1) having conventional circular-shaped dimples and (2) having hexagonalshaped dimples. So far, in our analysis, we used the data of the hexagonal-dimpled ball

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to compute the trajectories. In this section, we compare the wind-induced deviations in the flight trajectories of both the balls and discuss the underlying cause of the difference. Figure 4.18 shows the flight trajectories of the two balls for four clubs’ shots: (1) Driver, (2) 3-Iron, (3) 7-Iron and (4) Pitching wedge. For all four clubs, the hexagonal-dimpled ball’s flight trajectory (blue curves) had higher R f and h M as compared to the trajectory of a conventional ball (red curves). However, for both the balls, the trends of club-wise trajectory characteristics (R f , h M and flight time T f ) are similar (see Fig. 4.19). Thus, we can say that the hexagonal-dimpled ball proves to be superior compared to the conventional ball for all clubs. Cause This is because the hexagonal dimples reduced the C D and enhanced C L of the ball over a majority of the operation range of golf balls. As a result, this ball flew both higher and further as compared to the conventional ball.

When these trajectories are subjected to a wind of 5 m/s along various directions, they are deflected as shown in Fig. 4.20. For both the balls, the trends of relative wind-induced deviations in range (δRf ) are similar. Figure 4.21 shows the trend over βr = 0◦ tp 180◦ for drive shots (solid curves) and 7-Iron shots (dashed curves). As discussed before in sec. 4.2.2, δRf increases as βr increases from 0◦ (tailwind) to 180◦ (headwind). Also, as expected, the relative deviation in a 7-Iron shot is more than that in a drive shot.

Fig. 4.17 Relative wind-induced deviations in Rf a wind of 5 m/s for β = 0◦ (tailwind) to 180◦ (headwind) for clubs: (1) Driver (blue), (2) 3-Iron (red), (3) 7-Iron (yellow) and (4) Pitching wedge (green)

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Fig. 4.18 Trajectories of a conventional golf ball (red) and a hexagonally-dimpled golf ball (blue) for the clubs: a Drive, b 3-Iron, c 7-Iron and d Pitching wedge

Fig. 4.19 Club-wise trajectory characteristics a R f , b h M and c T f for a conventional golf ball (red) and a hexagonally-dimpled golf ball (blue)

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Fig. 4.20 Deflections in the flight trajectories of a 7-Iron shot due to a 5 m/s wind with respect to βr , for a conventional golf ball (red) and a hexagonally-dimplied golf ball (blue) Fig. 4.21 Variation in δRf of a drive (solid curves) and a 7-Iron shot (dashed curves) due to a 5 m/s wind with respect to βr , for a conventional golf ball (red) and a hexagonally-dimplied golf ball (blue)

However, it is very clear from Figs. 4.20 and 4.21, that the conventional ball is affected less by the wind as compared to the hexagonal-dimpled ball. This is the observation both in terms of absolute deviations (Rf ), as well as relative deviations δRf (see Fig. 4.20).

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Cause The hexagonal-dimpled ball has a lower C D , which helps it to cover a longer horizontal range (R f ). This is further enhanced by its higher C L values which increase the net upward acting on the ball throughout its flight. As a result, it flies higher and stays airborne for a longer duration of time as compared to the conventional ball (see T f trends in Fig. 4.19), which, in turn, helps in increasing R f . However, as discussed in Sects. 4.2.1 and 4.2.2, a higher flight time also gives wind more time to influence the flight trajectory of the hexagonaldimpled ball as compared to the that of a conventional ball. Hence, the windinduced deviations are higher in case of the hexagonal-dimpled ball.

Therefore, despite its inferior performance in terms of R f and h M due to lower C L , a conventional ball might be a safer choice because of the reduced wind-induced deviations.

4.3 Summary The flight trajectory of a golf ball is significantly influenced by the aerodynamic forces that act on the ball. These forces, in turn, depend on the relative airspeed of the ball. The relative airspeed is the resultant of the ball’s velocity and the local wind velocity. Therefore, wind plays an integral part in the governing the ball’s flight. In fact, the trajectory is much more sensitive to the wind conditions as compared to other factors like launch speed, launch spin rate and launch angle. Thus, it is important to study how the wind impacts gameplay and how its effects vary as the physical setting changes. The physical setting comprises of three parameters: wind conditions (direction and speed), launch conditions (ball speed, spin rate and launch angle) and the aerodynamic characteristics of the ball. In the presence of a tailwind, the trajectory of the golf ball becomes longer and flatter. Also, it lands at a shallower angle with respect to the ground, which, in turn, increases the distance over which the ball bounces and rolls on the ground. As the wind direction changes from a tailwind to a headwind, the trajectory becomes shorter and more lobbed. The ball lands at a steeper angle with respect to the ground, which decreases the ground run distance. Moreover, the effect of a headwind is more profound than that of the tailwind, since ball stays airborne for a longer duration of time in the presence of a headwind. As a result, the wind gets more time to influence the trajectory. The wind-induced deviations also vary with respect to the launch conditions. Launch conditions are imparted to a golf ball when it is struck by a golf club to execute a shot. There are three major types of shots in the game of golf: drive, approach and putt. Out of these, drive and approach are aerial shots, whereas putt is a grounded shot wherein the ball only rolls on the surface of the ground. Drive

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and approach shot differ significantly based on their objectives, launch conditions and consequently, the trajectory shapes. In the case of a drive shot, the objective is to maximise the range, and thus the flight trajectory is long and flat, and the ground run is longer. On the other hand, the objective of an approach shot is to deliver the ball onto the putting green. Therefore, the key requirement is precision in launch conditions for accuracy in the final landing location. Therefore, its trajectory is short and lobbed so that the ground is shorter. Each type of shot is executed using a standard set of golf clubs, which comprise of a driver, woods, irons and a pitching wedge. Every club imparts a distinct set of launch conditions to the ball. As the loft of the club increases, the resultant trajectory becomes more lobbed. The more a trajectory is lobbed, the higher are the wind-induced deviations it experiences. Finally, aerodynamic characteristics of the golf ball also influence how the wind affects the ball’s trajectory. These characteristics are determined by the shape and sizes of the dimple on the ball’s surface. A hexagonal-dimpled golf ball has better flight performance (range and peak height) than a convention golf ball with circular dimples. This is because hexagonal dimples help in reducing the drag and enhancing the Magnus lift, which makes the ball fly further and higher. However, this comes at a cost. The enhanced lift keeps the ball airborne for a longer duration of time. As a result, the flight trajectory experience higher deviation in the presence of wind. These trends of wind-induced deviation with respect to the physical setting are crucial in understanding how wind impact gameplay in golf. A golfer must properly consider these factors while playing in windy conditions.

References 1. Robertson, S. J., Burnett, A. F., Newton, R. U., & Knight, P. W. (2012). Development of the Nine-Ball Skills Test to discriminate elite and high-level amateur golfers. Journal of sports sciences, 30, 431–437. https://doi.org/10.1080/02640414.2012.654398. 2. Erlichson, H. (1983). Maximum projectile range with drag and lift, with particular application to golf. American Journal of Physics, 51, 357–362. https://doi.org/10.1119/1.13248. 3. McPhee, J. J., & Andrews, G. C. (1988). Effect of sidespin and wind on projectile trajectory, with particular application to golf. American Journal of Physics, 56, 933–939. https://doi.org/ 10.1119/1.15363. 4. Stengel, R. F. (1992). On the flight of a golf ball in the vertical plane. Dynamics and Control, 2, 147–159. https://doi.org/10.1007/BF02169495. 5. Sajima, T., Yamaguchi, T., Yabu, M., & Tsunoda, M. (2006). The aerodynamic influence of dimple design on flying golf ball. The Engineering of Sport, 6(143–148), 26. https://doi.org/ 10.1007/978-0-387-46050-5_. 6. Naruo, T., & Mizota, T. (2006). Experimental verification of trajectory analysis of golf ball under atmospheric boundary layer. The Engineering of Sport, 6(149–154), 27. https://doi.org/ 10.1007/978-0-387-46050-5_. 7. Naruo, T., & Mizota, T. (2014). The influence of golf ball dimples on aerodynamic characteristics. Procedia Engineering, 72, 780–785. https://doi.org/10.1016/j.proeng.2014.06.132. 8. Yaghoobian, N., & Mittal, R. (2018). Experimental determination of baseball spin and lift. Sports Engineering, 21, 1–10. https://doi.org/10.1007/s12283-017-0239-9. 9. Bearman, P., & Harvey, J. (1976). Golf Ball Aerodynamics. Aeronautical Quarterly, 27(2), 112–122. https://doi.org/10.1017/S0001925900007617.

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10. My Online Golf Club, Golf shots explained https://www.myonlinegolfclub.com/Information/ TypesOfGolfShot 11. James, N., & Rees, G. D. (2008). Approach shot accuracy as a performance indicator for US PGA Tour golf professionals. International Journal of Sports Science & Coaching, 3, 145–160. https://doi.org/10.1260/174795408785024225. 12. World Meteorological Organization (2008). Guide to Meteorological Instruments and Methods of Observation https://www.weather.gov/media/epz/mesonet/CWOP-WMO8.pdf 13. TrackManGolf, TrackMan Average Tour Stats https://blog.trackmangolf.com/trackmanaverage-tour-stats/ 14. TrackManGolf, How to utilize the inconsistency number https://blog.trackmangolf.com/howto-utilize-the-consistency-number/

Chapter 5

Gameplay, the Course and Wind

The game has such a hold on golfers because they compete not only against an opponent, but also against the course, against par, and most surely-against themselves. —Arnold Palmer

Overview In Chap. 4, we presented a quantitative analysis of the effect of wind on a golf ball’s trajectory. This helped us understand how the wind-induced deviations vary as the physical setting (wind heading, launch conditions and aerodynamic characteristics) changes. We concluded that wind significantly affected the flight trajectories, and therefore, must be accounted for during a game. In this chapter, we take the discussion further while focusing more on the aspect of gameplay. With the help of our inferences from the quantitative analysis, we explain how golfers tackle the problem of wind under various circumstances during a game. A golfer generally accounts for headwind (or tailwind) by using a longer club (or shorter club) than he usually would [1]. In addition, he makes adjustments in his stance which, in turn, affects the swing and hence the launch conditions. Understanding how exactly to make such tweaks, however, demands good judgement, which comes with experience. In the case of crosswind, the situation becomes more challenging. The golfer not only needs to minimise the sideways deviation from the target, but also must judge which direction the ball runs in after hitting the ground (see Sect. 4.2.1). Players generally adjust their launch heading (ψ) when playing in the crosswind. However, experts may add some side-spin as well to introduce a side force against wind’s direction. In reality, the wind also changes direction with height which further increases the challenge. We discussed in Sect. 2.1.1, that all golfers practice consistency in their swings so that they always get predictable flight trajectories. Nonetheless, the wind really compels them to creatively tweak their well-practised deliveries and hence helps to differentiate between an amateur and an expert. Add to this carefully placed golf course elements (see Sect. 2.2), and the challenge gets compounded. In a well © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. Malik and S. Saha, Golf and Wind, https://doi.org/10.1007/978-981-15-9720-6_5

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designed golf course, which complements the local wind conditions, a misjudged shot can significantly increase the difficulty of the subsequent shots. For instance, in a par-4 course, if the tee shot lands too short of the farewell due to headwind, it could become quite difficult to deliver the ball onto the green in one shot. As a result, the player risks the chance of finishing over par score. As a result, the golfer not only needs to judge wind’s effect on each individual shot, but also requires an effective strategy to complete the game with minimal setback due to wind. In order to understand this aspect of wind’s effect, we raise the following new questions, which we aim to answer in this chapter: • How can the launch conditions be adapted to counter the effect of wind? How effective are these adjustments? • In which situations does the effect of wind become more critical? • How does the presence of wind affect a player’s strategy taking account of the golf course? We begin the discussion with Sect. 5.1, wherein we explain various strategies used by players to account for deflection in trajectory due to wind, and their limitations. Thereafter, in Sect. 5.2, we present case studies of three real golf courses, in order to examine how the effect of wind becomes critical in different types of golf courses (penal, heroic and strategic).

5.1 Playing Golf in Windy Conditions We concluded in Chap. 4, that the effect of wind, if not accounted for, can significantly change the outcome of a game for a golfer. In this section, we discuss some corrective measures that golfers use to compensate the effect of wind in various situations, i.e. headwind, tailwind and crosswind. We also discuss some challenges that come along with these practises, especially for amateur golfers.

5.1.1 Compensating for Headwind and Tailwind We discussed in Sect. 2.1.1, controlling the shape of a ball’s flight trajectory is crucial in the game of golf. In the presence of wind, this skill becomes all-themore important. The trajectory is determined by the launch conditions of the ball (see Sect. 2.1, which are predominantly given by the club in use (see Sect. 4.2.2). Depending on the requirement, a golfer chooses the appropriate club. In the absence of wind, all golfers have a good idea about the range a golf ball covers when hit with each of their clubs. However, in the presence of wind, the trajectories are deflected and thus the range differs.

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Fig. 5.1 Compensating for headwind; trajectories of a regular 7 Iron shot [2], without wind (red solid), 7 Iron shot in presence of a 5 m/s headwind (red dashed) and a 5 Iron shot [2], in presence of a 5 m/s headwind (black solid)

Fig. 5.2 Compensating for tailwind; trajectories of a regular 7 Iron shot [2], without wind (red solid), 7 Iron shot in presence of a 5 m/s tailwind (red dashed) and an 8 Iron shot [2], in presence of a 5 m/s tailwind (black solid)

In practice, when playing a certain shot in the headwind, golfers use lowernumbered club (i.e. a longer club) than what they usually would use in the absence of wind. Since the launch conditions imparted by the longer club result in a trajectory having greater range, this helps to compensate the loss of carry due to headwind. Figure 5.1 shows the trajectories of a regular 7 Iron shot [2], without wind (red solid), 7 Iron shot in presence of a 5 m/s headwind (red dashed) and a 5 Iron shot [2], in presence of a 5 m/s headwind (black solid). Using the longer 5 Iron club compensates for the loss in range due to the headwind and brings the trajectory closer to the 7 Iron shot without any wind. Similarly, while playing the same shot in the tailwind, the golfer uses highernumbered club (i.e. a shorter club) than he would usually use. Figure 5.2 shows the trajectories of a regular 7 Iron shot [2], without wind (red solid), 7 Iron shot in presence of a 5 m/s tailwind (red dashed) and an 8 Iron shot [2], in presence of a 5 m/s tailwind (black solid). Here, the shorter 8 Iron club compensates for the additional range due to the tailwind and yields a trajectory closer to the 7 Iron shot without any wind.

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Fig. 5.3 Compensating for crosswind; trajectories of a regular 7 Iron shot [2] without wind (red solid), in presence of a 5 m/s crosswind (red dashed), having ψ = −7.5◦ in presence of a 5 m/s crosswind (black dashed) and having a sidespin of 1250 R P M (black solid)

5.1.2 Compensating for Crosswind Crosswinds pose a greater challenge compared to headwinds and tailwinds [1]. This is because the players need to account for deviation in range, drift from the target line, as well as the change in direction of the ground run. Therefore, golfers try to plan their shots in such a way that they can avoid crosswind, and play against headwind or tailwind instead. However, this is not always possible. Figure 5.3 shows the top view of trajectories of a golf ball launched in the presence of a 5 m/s crosswind directed along the y-axis. The solid red line is the trajectory of a typical 7 Iron shot hit in the absence of wind while the red dashed curve is the same shot played in the presence of the crosswind. A common corrective strategy used by all players is to change the launch heading of the ball [3]. A well judged adjustment helps to compensate the sideways deviation of the golf ball during its flight (see black dashed curve in Fig. 5.3). Another strategy used by experienced players is to draw or fade the ball into the wind [4]. That is, they intentionally impart some sidespin to the ball which generates a side force and keeps the ball nearly on track, despite the action of crosswind (see black solid curve in Fig. 5.3). However, this is can be very tricky, and a misjudgement can prove to be disastrous. Hence, this strategy is generally avoided by amateurs [3]).

5.1.3 Challenges and Limitations While changing the club or the heading angle helps in compensating for winds effect, this is not always sufficient. In certain environmental conditions, wind by itself brings along a degree uncertainty due to its spatial and temporal variation. This uncertainty further increases as the ball flies higher above the ground and enters turbulent wind flow. Moreover, if the flight time is longer, then the ball is more likely to be subjected to temporal variation of the wind-velocity profile. In such situations, it becomes

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very difficult to predict how the ball will behave while it is airborne. For instance, Yaghoobian and Mittal [5], studied the uncertainty in landing position due to spatially and temporally varying wind flow. They showed that the local plant canopy in the course caused a swirling wind flow. This poses a challenge which is nearly impossible for golfers to account for. Experienced players tackle this problem by keeping their launch angle and launch spin rate low. This makes the trajectory flatter and reduces the flight time. They are able to achieve this by modifying their stance and hence the swing. However, such practices are quite challenging and demand good judgement and skill. Since golf swings involve very high speeds, a bold attempt to tweak the launch conditions by an amateur could result in a mishit or inconsistent shots, causing more detriment than a benefit.

5.2 Case Studies In this section, we analyse how the local wind conditions affect the gameplay strategy at three real golf courses. We conduct this analysis by simulating the golf ball’s motion in virtual models of the golf courses, as described in Chap. 3. Visualising the ball’s trajectories in the context of real golf courses provides a useful insight into how a player thinks during a game, and hence helps us understand the impact of wind better. We conduct our analysis on golf courses having three distinctly different design styles (see Sect. 2.2.4): (1) hole 17 of Tournament Players Club (penal design), (2) hole 8 of Muirfield Golf Club (strategic design) and (3) hole 18 of Pebble Beach Golf Links (heroic design). This helps us understand how the course and wind together challenge the golfer to strategize the gameplay.

5.2.1 Penal: Hole 17, Tournament Players Club, Florida Hole 17 of Tournament Players Club Stadium Course in Ponte Vedra, Florida, USA (popularly known as TPC Sawgrass) is a very iconic golf course. It is considered to be one of the most difficult holes due to its notorious layout (see Fig. 5.4). This is a par-3 course having a large pond surrounding a unique island green. The layout is such that the only way to reach the putting green is by hitting a single approach shot from the teeing box. Since the water hazard surrounds the green from all directions, any misplayed shots would certainly result in a lost ball [7]. Hence, this course is said to have a penal design style (see Sect. 2.2.4). Even in the absence of wind, the design layout of this course makes it quite difficult to finish the game at par (i.e. one aerial shot and two putt shots), not only for amateurs, but also for experts. This is because the thought dropping the ball into water plays a mental game with the players. The nervousness causes the players to

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Fig. 5.4 Hole 17, Tournament Players Club in Florida. Image courtesy Google Earth, Data SIO, NOAA, US Navy, NGA, GEBCO Landsat/Copernicus [6]

become too careful which, in turn, hampers with their muscle memory and converts an otherwise simple approach shot into a daunting mission! In such a situation, the added uncertainty due to swirling winds in the course can prove to be a major challenge for the players, making it nearly impossible for amateurs to even finish the game at all. Through this case study, we aim to understand the extent of this challenge by simulating the golf ball’s motion when subjected to wind incident along all directions. However, before we proceed towards the analysis, we briefly discuss the parameters we use to model this course in our simulation programme using the process described in Sect. 3.2. Figure 5.5a shows the tops view of the golf course in Google Earth, annotated with the course features used to create its model in our simulation programme. We chose a domain of 160 m by 65 m wherein the tee is positioned at (10, 42) and hole at (133, 43). We captured the layout of the course using the pond, an island green, a bunker attached to the green and another island containing a tree. Table 5.1 summaries the parameters (positions, dimensions and orientations) of these features in the format as described in Sect. 2.2.2. Finally, we model the uncertainty due to swirling winds by subjecting the ball’s trajectory to winds incident along all the directions and hence obtain an uncertainty region within which the ball can land (see

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(a)

(b)

(c) Fig. 5.5 a Identifying the features to be including in the model and b–c compiled model of Tournament Players Club in MATLAB in top view and 3D view respectively

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Table 5.1 Parameters of the course features in hole 17 of TPC Sawgrass Feature x (m) y (m) a (m) b (m) Pond Island (green) Island (tree) Bunker

90 133 94 124

45 43 9 37

71 13 7 8

65 13 7 3

α (◦ ) −9 0 0 −60

Sect. 4.2.1). Although we neglect the spatial and temporal variation of wind, this scheme provides an upper bound of the deflection in the ball’s trajectory, considering a hypothetical case when a constant wind speed acts on the ball at all locations and along all possible directions. Figure 5.5b–c show the resultant golf course model in MATLAB in top view and 3D view, respectively. Case 1: No wind When there is no wind, a regular 9 Iron shot is suitable to deliver the ball safely onto the putting green. Nevertheless, this differs from golfer to golfer due to variation in the balls’ aerodynamic characteristics, clubs’ characteristics and the players’ swing. Using the aerodynamic characteristics of the hexagonal dimpled ball (Sect. 4.2.3), and the average launch conditions (u 0 , N0 and θ ) of a 9 Iron club (Sect. 4.2.2, a shot headed parallel to the x-axis (i.e. ψ = 0◦ ) lands the ball close to the centre of the island green (see Fig. 5.6a). Case 2: 2 m/s wind Seldom is the case when there is no wind at TPC Sawgrass [8]. As a result, it becomes important to consider the case in the presence of a mild 2 m/s wind incident along all directions. Figure 5.6b shows the original 9 Iron trajectory (black solid curve) and the trajectories deflected (dashed curves) by wind having various heading directions (β). Here, the black dashed curves are valid trajectories that land and halt on the island, whereas the red dashed curves and invalid trajectories that do land on the island but thereafter roll into the water in presence of tailwinds. Nevertheless, it is worth noting that for this value of wind speed, all shots still land on the island, and minor adjustments in the swing could prevent the excess rolling in presence of tailwinds. Therefore, even if the player does not account for the wind and delivers the 9 Iron shots as he normally would, he would be in a safe situation. This proves to be a boon for amateur players who may lack the skill required to compensate for the effect of wind, as discussed in Sect. 5.1. Case 3: 5 m/s wind The situation completely changes when we consider a moderately high wind speed of 5 m/s. Unlike case 2, all the trajectories of a 9 Iron shot under the effect of 5 m/s winds land the ball in the water (see Fig. 5.6c). In such a case, it becomes impossible for an amateur player to finish the game without taking corrective measures to account for wind. Even for experts, the challenge immensely increases. This is because of the swirling nature of the local wind profile, which makes it nearly impossible to judge how exactly the ball will be deflected. Therefore, an under-correction or over-correction can prove to be disastrous.

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(a)

(b)

(c) Fig. 5.6 Playing a regular 9 Iron shot [2]: a in the absence of wind, b in the presence of a 2 m/s wind and c in the presence of a 5 m/s wind along various directions; black solid curve: original trajectory, black dashed curve: deflected and valid trajectory, red dashed curve: deflected and invalid trajectory

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Fig. 5.7 Mean deviation in first bounce location due to winds incident along various directions v/s magnitude of wind speeds (m/s)

Discussion: As discussed before, the par-3 rating of TPC Sawgrass implies that it is possible to hit the ball into the hole in regulation through a total of three shots. These include one aerial approach shot from the tee, followed by two putt shots when the ball reaches the green. Moreover, owing to the layout of this course, a well delivered approach shot lands the ball onto the green while a misplayed shot lands it into water. In such a case, the player needs to retake the tee shot. This is unlike regular golf courses, wherein every shot generally brings the player closer to the green, even if the ball lands in a hazard like sand bunkers. As a result, in TPC Sawgrass, players have nearly no opportunity to compensate for the loss incurred by misplayed shots, which on the other hand may be possible in regular courses by re-planning their subsequent shots. Furthermore, the uncertainty introduced due to higher wind speeds can potentially raise the challenge to the extent that delivering a successful tee shot becomes impossible even for experienced players. Figure 5.7 shows the variation in the mean wind-induced deviation in the location of the ball’s first bounce w.r.t. the magnitude of wind speeds (red curve). The black dashed curve indicates the approximate radius of the island green (13 m). Hence, for wind speeds below 3 m/s, the uncertainty region due to wind-induced deviations is smaller than the island itself, which implies that the ball lands on the island irrespective of the direction of the wind (case 2). Whereas, for higher wind speeds, the uncertainty region is larger than the island, which increases the chances of the ball falling into water. Since the exact distribution of wind remains unknown to the golfers, it is nearly impossible to counter its effect. This sometimes forces the golfers to play the shot blindly, and hence introduces an element of luck in the game. While some degree of luck is necessary for the game to be thrilling, a player’s

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skill must remain decisive. As the magnitude of wind speed increases, the probability of the ball falling into water, and hence the luck factor, also increase and eventually supersede the player’s skill as the most decisive factor. At this point, the true essence of the game is lost and the distinction between amateurs and experts becomes cloudy. Therefore, it becomes important to understand when the game must be paused until the environment becomes more suitable. A further investigation of this case study can facilitate this understanding.

5.2.2 Strategic: Hole 8, Muirfield Golf Club, Scotland Hole 8 of Muirfield Golf Club, Scotland, is a classic example of a course having a strategic design style. This is a par-4 course, which means that it requires two aerial shots to reach the green in regulation. The distinct dogleg fairway and numerous strategically placed sand bunkers (see Fig. 5.8), offers the players two different strategies to finish the game in regulation, that is, to reach the putting green in two aerial shots. In the first strategy, the golfer needs to cover a greater total distance but can place the ball safely away from any sand bunkers. This reduces the risk of landing the ball in the bunker due to inconsistency in launch conditions. On the other hand, the second strategy allows the golfer to reach the green by covering a lower

Fig. 5.8 Hole 8, Muirfield Golf Clubs, Scotland. Image courtesy Google Earth, CNES/Airbus, Data SIO, NOAA, US Navy, NGA, GEBCO, IBCAO, Landsat/Copernicus, Mexam Technologies [9]

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(a)

(b)

(c) Fig. 5.9 a Identifying the features to be included in the model and b–c compiled model of Muirfield Golf Club in MATLAB in top view and 3D view, respectively

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Table 5.2 Parameters of the course features in hole 8 of Muirfield Golf Club Feature x (m) y (m) a (m) b (m) Green Bunker 1 Bunker 2 Bunker 3 Bunker 4 Bunker 5 Bunker 6 Bunker 7 Bunker 8 Bunker 9 Bunker 10 Bunker 11 Bunker 12 Bunker 13

358 186 193 200 205 222 262 299 318 312 315 332 364 380

191 145 137 143 158 175 177 203 199 204 183 171 176 200

21 4 3 3.5 3.5 5 3 3 7.5 7 2.5 5.5 3.5 3.5

15 2.5 2.5 2.75 2.75 2.5 2 2 1.75 1.75 2.5 1.5 2.5 2

α (◦ ) 0 −60 90 90 90 90 90 −45 90 10 0 80 0 −60

total range, while incurring a risk of landing the ball in one of the bunkers. Hence, accuracy in launch conditions becomes important. Unlike in the case of the penal golf course, TCP Sawgrass (18th), the Muirfield course (8th) is a fairly simple golf course wherein the stakes are not very high. None of these strategies poses any heavy penalty on the players, and both let the player finish the game in regulation. Nevertheless, the player is free to adopt any one of the strategies depending on his strengths, weaknesses and confidence level at that time (distance v/s accuracy). Through this case study, we aim to explore the ways in which wind along different directions brings about a change in the otherwise simple playing strategy. As in the case of the penal course, we carry out this analysis by simulating golf shots in a model golf course, as described below. Figure 5.9a shows the tops view of the golf course in Google Earth, annotated with the course features used to create its model in our simulation programme. We chose a domain of 400 by 230 m wherein the tee is positioned at (20, 30) and the hole at (358, 191). We captured the layout and shape of the course by carving out the out-of-bound regions as indicated in black, and hence placing sand bunkers and the putting green at appropriate locations. Table 5.2 summarises the parameters (positions, dimensions and orientations) of these features in the format as described in Sect. 2.2.2. Figure 5.9b–c show the resultant golf course model in MATLAB in top view and 3D view, respectively. Case 1: No wind As shown in Fig. 5.10a–b, this course offers routes two reach the putting green, both involving two aerial shots. The first strategy (red trajectories) allows the player to place the first shot safely away from any of the sand bunkers in the course. This way, any inadvertent inconsistency or inaccuracy in the launch does

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(a)

(b) Fig. 5.10 Playing in the absence of wind: a 3D view and b top view; red trajectories: greater total distance covered but less accuracy required (1st strategy); blue trajectories: shorter total distance but more accuracy required (2nd strategy)

not pose the risk of landing the ball into a hazard. So to say, the player has more room for error. The first shot is a standard 5-Wood shot [2], with ψ = 42◦ . This is followed by a 5-Iron shot [2], with ψ = 5◦ . The total distance covered by the two shots in this route is 217 m + 178 m = 395 m. However, the players who are confident about their long-distance shots can cut the dogleg corner and reduce this total distance by opting for the second route (blue trajectories). However, this requires the players to place to shot quite close to the bunkers, due to which, accuracy becomes important. Here, tees off with a drive shot [2], along ψ = 33◦ , followed by a lofted 9-Iron shot [2], along ψ = 11◦ , thus reducing the total distance to 246 m + 135.5 m = 381.5 m. Depending on the comfort

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(a)

(b) Fig. 5.11 Playing in the presence of: a a 5 m/s crosswind inwards (β = −60◦ ) and b a 5 m/s crosswind outwards (β = 120◦ ); red trajectories: route 1, blue trajectories: route 2, purple trajectory: alternative drive shot; solid curves: wind absent, dashed curves: wind present

zone of the player with his clubs, he can opt for any of the strategies and still finish the game in regulation. Case 2: Crosswind (inward) Figure 5.11a shows the effect of a 5 m/s crosswind on the tee shots when it blows into the corner (β = −60◦ ). Note that in case of the blue route, the ball goes out of bounds due to wind-induced deviation. On the other hand,

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the deflection in the tee shot of the red route results in lowering the total distance to the green, since the ball tends to cut the corner while still staying clear of the sand bunkers. Therefore, the red route becomes a safe choice for an amateur who cannot account for the wind’s effect. Case 3: Crosswind (outward) When the same wind blows in the opposite direction (β = 120◦ ), deflection in the tee shot of the red route lands the ball out of bounds. Whereas, that in the blue route deflects clear of the sand bunkers and lands the ball on safe ground. That said, the presence of wind always increases the risk associated with placing close to the bunkers. Thus, opting for either of these routes (red and blue) is not very suitable for an amateur golfer. Instead, an alternative drive shot aimed towards the centre of the course (purple curves) proves to be a safe choice when the wind deflects the ball outwards. Although this does give a distance advantage as in the case of the blue route, it helps in avoiding the hazards. Discussion: Hole 8 of Muirfield golf club is a simple golf course that does not pose any heavy penalty for the golfer for misdeliveries, as in the case of TPC Sawgrass. It’s par-4 rated strategic design style offers two possible routes for the golfer to choose from (see Fig. 5.10a, b). While one route (red trajectories) gives a safe room to play away from hazards, the other (blue trajectories) offers a lower total distance as a trade for more accuracy. The total distance can be reduced by adopting a long-range tee shot which cuts the corner of the distinct dogleg-shaped fairway and lands closer to the putting green. Nevertheless, the strategically placed sand bunkers around the probable landing site of such a shot calls for high accuracy. That said, even if the ball lands in a bunker, it is not a heavy penalty like in the case of TPC Sawgrass. This is because the player does not need to start the game from the beginning, with a tee shot. Thus, when there is no wind in the picture, the golfers choose any one of the strategies based on the strengths and comfort zone (distance v/s accuracy). However, the presence of wind introduces a bias, and depending on the wind’s direction, tends to make one strategy more favourable than the other. This is especially true for inexperienced golfers who lack the skill of judging wind’s effect and thus countering it, the best strategy is to play safe. In this case study, we analysed the effects of 5 m/s crosswinds which blow (1) into the corner (case 1) and (2) out of the corner (case 2), as shown in Fig. 5.11a–b. In case 1, we found that the wind itself aids in reducing the total distance covered by deflecting the ball such that it cuts the corner. In the process, the 5-wood shot (red route) lands safely onto the fairway, clear of the bunkers. However, the drive shot (blue route) lands the ball out of bounds. Therefore, the red route becomes more favourable. In case 2, however, the wind tends to increase the total distance to the green by deflecting the ball away from line-of-sight. Here, the 5-wood shot (red route) lands the ball out

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of bounds while the drive shot (blue route) lands on the fairway, quite close to the bunkers. Therefore, a third strategy (purple trajectory in Fig. 5.11b) in which a drive shot is placed towards the centre of the fairway proves suitable. This provides the best of both worlds, keeping the ball safe from any hazards, at the same time landing it closer to the green. Thus, we see how the presence of wind compels a golfer to replan his gameplay strategy with regards to wind-induced deviations.

5.2.3 Heroic: Hole 18, Pebble Beach Golf Links, California Hole 18 of Pebble Beach Golf Links, California, USA, is a famous example of a golf course having a heroic design style. It’s a beautiful 18-hole course in which the final 18th hole is located along the coast of Pebble Beach (see Fig. 5.12a). This is a par-5 course that offers two possible paths to reach the putting green (see Fig. 5.12b). The par-5 rating of the course implies that players need 5 shots, including 3 aerial shots, to finish the game in regulation (black curves). Interestingly, the unique layout of this course opens the door to another strategy by which the player can reach the putting green in just 2 aerial shots (red curves). Herein, the players must execute two long-distance shots, in both of which the ball travels over the ocean and lands close to the seashore. This allows the players to take advantage of a more favourable heading angle, and thus reduce the effective distance to the green. However, this reward of a lower score comes at the cost of a much higher risk of losing the ball in the ocean, hence facing a heavy penalty. Hence, golfers who are willing to gamble and play the heroic choice get a clear advantage over those who play the game in regulation. In this case study, we aim to understand how the local wind conditions change the stakes of the heroic strategy as the wind heading (β) varies. We first consider the case when there is no wind in order to understand the necessary conditions to reach the putting green in just two aerial shots. Next, we add a moderately high wind of 5 m/s [8], to the setup and analyse how a headwind, tailwind and crosswind affect these conditions, and hence the stakes. We carry out this analysis using the model golf course described in Sect. 3.2. Case 1: No wind In this golf course, it is possible to reach the putting green in just two aerial shots, which is one shot less than when the game is played in regulation. In the absence of wind, a drive shot followed by a 3-wood shot can land the ball onto the green. These shots must be played very close to the seashore, such that the ball may fly over the ocean. Figure 5.13a–b show the narrow window which the players have so that this strategy yields success. Here, the red trajectories are the possible drive shots launched from the tee within a range of heading angles (ψ) such that the ball does not fall in water or hit any other obstacle (like trees, bunkers, etc.). The magenta band on the ground denotes the locus of final halt locations of the ball (see

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(a)

(b) Fig. 5.12 a Hole 18, Pebble Beach Golf Links, California. Image courtesy Google Earth, Data MVARI, Data SIO, NOAA, US Navy, NGA, GEBCO Landsat/Copernicus [10]; b Regular playing strategy of three shots (black curves) and heroic playing strategy of two shots (red curves)

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(a)

(b) Fig. 5.13 Playing in the absence of wind: a 3D view and b top view; red trajectories: driver shots played from tee, black trajectories: 3 Wood shots played to the putting green, magenta band: landing region of driver shots, cyan band: launching region of 3 Wood shots

Sect. 4.2.1, after the drive shots. Similarly, the cyan band on the ground denotes all the possible locations from where the ball may be launched using a 3-wood club, so that it lands and halts within the putting green. Therefore, the intersection of the magenta and cyan regions denotes the region where the ball must land after the drive shot, so that the second 3-wood shot lands the ball onto the putting green. The black trajectories denote the possible 3-wood shots launched from this intersection region at the appropriate heading angles. Hereafter, we refer to this intersection region of the magenta and cyan bands as the “window” for executing the heroic strategy. It is worth noting that such a window is only obtained when a driver and a 3-wood shot is played in combination. Moreover, a driver club can only be used in the tee shot, since its low loft angle makes it impossible to deliver high range shots when the

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(a)

(b)

(c) Fig. 5.14 Playing in the presence of: a a 5 m/s tailwind (β = −10◦ ), b a 5 m/s headwind (β = 170◦ ) and c a 5 m/s sea breeze (β = −100◦ ); magenta band: landing region of possible first shots, cyan band: launching region of possible second shots

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ball is placed on the ground without a tee. As a result, the only possible arrangement is one when the first shot is played with a driver and the second with a 3-wood club. Even in this case, the window is very small, which indicates that opting for the heroic strategy demands confidence, skill and good judgement on part of the player. Case 2: Tailwind Naturally, the heroic window is modified when we introduce tailwind of magnitude 5 m/s and directed approximately from the tee towards the hole having β = −10◦ (see Fig. 5.14a). Since the tailwind aids the flight of the ball, helping it cover larger distances, the heroic strategy is no longer limited to just using the driver and 3-wood combination. The golfer now gets a wide range of options including the driver, 3-wood, 5-wood or a hybrid. As a result, the window becomes considerably larger, and allows the golfers to execute shots that are not very close to the seashore, while still reaching the green in two aerial shots. Even though such a strategy remains more challenging as compared to playing in regulation, the gambling stake is greatly reduced since the players no longer need to play dangerously close to the sea. This, in turn, could encourage even amateur players to attempt the 2-shot strategy. Hence, the course loses its characteristic heroic nature due to low stakes. Case 3: Headwind On the other hand, a headwind of magnitude 5 m/s and heading β = 170◦ diminishes the window to null (see Fig. 5.14b). In this situation, the headwind shortens the range of the balls, thus making it impossible to reach the green in two aerial shots. This way, even experts do not attempt the heroic strategy since the risk of losing the ball in water does not out-weigh the reward of finishing with a lower score. They might as well play the game in regulation, wherein they get ample room to safely avoid hazards, and still reach the green in 3 aerial shots. Hence, the course loses its heroic nature due to unrealistically high stakes. Case 4: Crosswind The situation is different, and more interesting when there is a crosswind in the picture. Figure 5.14c illustrates what happens when we introduce a sea breeze of magnitude 5 m/s and heading β = −100◦ in the simulation. If the golfer plays the drive shots exactly the same way as in case 1 (as if there was no wind in the picture), then the ball tends to deflect away from the seashore. Consequently, most of the shots which yielded success in case 1 now stand invalid (red curves), since the ball hits the tree in the vicinity. Nevertheless, some of the shots can still yield success, provided the player makes an appropriate modification in the second shot’s launch heading to account for the crosswind’s effect (black curves). However, this margin is very small and thus cannot be relied upon (see Fig. 5.14c). On the other hand, the original window still remains open for more experienced golfers who choose to account for the crosswind’s effect. This requires them to confidently play their shots by aiming for the ocean at just the right angle, so that the ball deflects along the crosswind and safely lands on the ground while avoiding all hazards (white dashed curves). Naturally, this situation is much more challenging than the case without any wind. The golfers not only need to be confident about delivering consistent and errorfree shots, but also must possess excellent judgement of the ball behaviour in wind. In addition, as discussed in Sect. 5.2.1, there always exists the factor of luck due to uncertainty in wind conditions. Nonetheless, unlike case 3, it is very well possible to

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reach the putting green via the heroic route. Moreover, the added challenge further enhances the heroic nature of the golf course, reserving the opportunity only for the best of the best. Discussion: The par-5 hole 18 at Pebble Beach Golf Link has a heroic design style, which is an amalgamation of penal and strategic styles. Just as in Muirfield (8th), the course has two different routes that the golfer can opt for depending on his skill. However, the two strategies differ significantly in the rewards and risks involved. The regular strategy lets the golfer finish the game in regulation (3 aerial shots), while the heroic strategy enables the player to finish 1 below par! However, this comes at a significant risk of losing the ball in the ocean, which is similar to the case of the penal course, TPC Sawgrass (17th). Therefore, a significant risk can offer a significant reward, which encourages experts to gamble for the heroic strategy, as opposed to amateurs. This, in turn, adds to the thrill of the game. In the absence of wind, the course layout makes the stakes of the heroic sufficiently high to encourage the experts and discourage the amateurs from opting for the heroic strategy. This helps in clearly distinguishing between the two. However, in the presence of wind along various directions, this situation changes. While a 5 m/s headwind makes the heroic strategy impossible to execute even for experts, a tailwind of the same magnitude makes it easy for even the amateurs to attempt. In both these cases, the course loses its signature heroic character. On the other hand, a crosswind due to a sea breeze further enhances this characteristic. This is because the crosswind amplifies the challenge without altering its feasibility. In this case, the heroic route is attempted only by the most experienced golfers who are confident about their long-distance shots, as well as their judgement of how wind affects the flight trajectory. Knowledge of the wind conditions during the game can help the player understand his stance in the game, and hence enable him to plan his gameplay strategy to finish the game to the best of his abilities. This is a good example of when the wind conditions couple with the course layout and greatly affect the gameplay.

5.3 Summary Wind has a significant impact on the flight trajectories of golf balls, and hence the gameplay strategies. Therefore, it must be taken into account during a game. While the quantitative analysis presented in Chap. 4 helps the golfers understand how wind affects the trajectories, the discussion in this chapter helps to link that knowledge to the game itself.

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An experienced golfer uses various techniques to counter the effect of wind by suitably adapting his launch conditions such that the deflection caused by wind is nullified, or rather minimised. While playing in a headwind, the player switches to a longer club than he would use in the absence of wind, which, in turn, increases the range to help recover the loss of carry due to headwind. This way, when the headwind makes the trajectory shorter and more lobbed, it comes closer to the shape desired by the golfer originally. Similarly, while playing in tailwind, a golfer uses a shorter club so that the ball does not overshoot the target when pushed ahead by the wind. Apart from this, golfer often make tweaks and adjustments to their stance, which, in turn, reflects in their swing and hence the launch conditions. For instance, keeping the ball closer to the rear leg and bending forward helps in reducing the launch spin rate and hence keeps the trajectory flatter. Such adjustments further help in nullifying the effect of wind. However, it must be noted that changing the club is a fairly simple technique that even amateur golfers can adopt. On the other hand, making tweaks in the stance and swing poses an overhead risk of inducing inconsistency in the shots, which is something very undesirable in golf. As a result, such advanced techniques demand a lot of experience and practice and can thus only be adopted by experts. Playing in crosswinds is even more challenging since the in addition to deviation in range, the players also need to account for drift from the target line and change in the direction of the ground run. The most commonly used strategy for playing in crosswinds is to aim the shot into the wind. This way, the wind deflects the ball along its flow, and hence it lands closer to its target line. However, this is not as straightforward as changing the club while playing in headwind or tailwind. It takes an ample amount of experience to develop a good judgement about the extent of wind’s effect and more importantly the required modification in the launch heading. Hence, players try to avoid crosswind as much as possible. However, more experienced players also use sidespin to tackle crosswinds. The side force generated as a result of the sidespin nullifies the deflective force due to wind and keeps the ball quite close to the original path. However, similar to modifying the stance while playing in headwind or tailwind, a mishit caused by an amateur’s bold attempt at such an advanced technique, can prove to be disastrous. Finally, the uncertainty that exists due to spatial and temporal variation of the wind-velocity profile due to the golf course layout greatly adds to the challenge. This is because such variation is beyond the knowledge and judgement of the golfers, and hence nearly impossible to account for. That said, maintaining a flatter trajectory (lower launch angle and launch spin rate) minimises the wind-induced deviation (see Sect. 4.2.2), and is, therefore, a thumb rule followed by experienced players while playing in any kind of wind. In order to provide a realistic perspective to our analysis of wind-induced deviations in Chap. 4, we studied how the wind plays a direct impact on the gameplay strategies in three distinct kinds of golf courses: penal, strategic and heroic (see Sect. 2.2.4). In the penal course, there exists a mandatory shot which, if misdelivered, can land a grave penalty for the player. In the case of TPC Sawgrass (17th), this critical delivery is a 9-Iron tee shot which is also an approach shot. The player needs to deliver the ball onto an island green, and a mishit can land the ball in water, compelling the player to re-attempt the shot. Even in the absence of wind this iconic

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course is already quite challenging for amateur golfers. However, the local wind conditions in certain cases (wind speed greater than 3 m/s) make it nearly impossible for even experts to finish the game with a par score since a lot of balls land in water. This is especially due to the spatial and temporal variation of wind which leads to uncertainty. Therefore, the local wind conditions enhance the penal nature of this course. On the other hand, a strategic course offers multiple routes to the green, none of which pose a heavy risk. All these routes are similar in terms of overall difficulty and reward. For instance, in Muirfield (8th), golfers can adopt any one of the two routes that the course offers. While one lets the player place his shots safely away from any hazards, the other cuts down the total distance by playing a bit closer to sand bunkers. Nevertheless, no heavy penalty is posed on the player in either of the routes and both consist of two aerial shots that let the player finish the game in regulation rating of par-4. Therefore, in the absence of wind, the golfer can adopt any of the routes based on his strengths comfort zones, like distance v/s accuracy. However, when the wind comes in the picture, it makes one route more favourable than the other depending on the direction in which it is blowing, thus introducing a bias. The choice of playing strategy is now influenced more by wind than by the strengths of the player. Finally, a heroic golf course is an amalgamation of the penal and strategic design styles. For instance, Pebble Beach (18th) offers two routes to finish the game. However, unlike in a strategic design, here the two routes greatly differ in terms of the rewards and the risks associated. While one route lets the player safely finish the game in regulation rating of par-5 (three aerial shots to the green), the other route lets the player reach the putting green in just two aerial shots. However, this significant reward requires the player to take a significant risk of placing his shots dangerously close to the seashore. This way, a bad shot can land the ball into the water, much like in the case of a penal design. Such a layout allows the player to gamble in the game by taking a significant risk to earn a significant reward of a lower score. However, due to the risks involved, this is only attempted by experts, not amateurs. Hence, such a design style helps to clearly distinguish between amateurs and experts. Here, the wind blowing along different directions have a different effect on the gameplay strategy. While a headwind makes the heroic strategy impossible to execute, a tailwind makes it much simpler so that even amateurs can attempt it. The course loses its iconic heroic nature in both these cases. On the other hand, a sea breeze raises risks associated with the heroic route due to the additional challenge of correctly accounting for the deflection caused to the wind. However, since the delivery is still possible, it helps in picking out the best of the best players, hence further enhancing the heroic nature of the course. In summary, we find that the gameplay strategy is significantly impacted by the wind in all the design styles: penal, strategic and heroic. However, the nature of the impact is distinctly different for each of the three styles. In the penal course, the uncertainty introduced by wind intensifies the penal characteristic of the course, thus making the situation more grave than it already was. Whereas in the strategic course, the direction of wind compels the player to choose his route based on the implication

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of wind rather than his strengths as was the case without wind. Thus, wind alters the decision-making strategy without posing any additional hazard. Finally, in the heroic course, wind acting along different directions bring about a direct change in the stakes involved in gambling for the heroic route. It either eliminates gambling factor by making the stakes too low (tailwind) or impossibly high (headwind), or it raises the stakes just enough (crosswind) to further enhance the iconic heroic characteristic of the course. Therefore, the coupling that occurs between the wind conditions and golf course layout has a decisive impact on how the game takes its course.

References 1. PGA, A Lesson Learned: Playing in the wind https://www.pga.com/archive/golf-instruction/ lesson-learned/fundamentals/playing-in-wind-lesson-learned 2. TrackManGolf, TrackMan Average Tour Stats https://blog.trackmangolf.com/trackmanaverage-tour-stats/ 3. Free Online Golf Tips, Tips for Glaying Golf in the Wind - Crosswind, etc. https://free-onlinegolf-tips.com/advanced-golf-tips/trouble-shot-tips/how-to-play-great-golf-in-the-wind/ 4. Meandmygolf, How to draw and fade your shots https://youtu.be/lwNeCfN5n8A 5. Yaghoobian, N., & Mittal, R. (2018). Experimental determination of baseball spin and lift. Sports Engineering, 21, 1–10. https://doi.org/10.1007/s12283-017-0239-9. 6. Google Earth, Data SIO, NOAA, US Navy, NGA, GEBCO Landsat/Copernicus. Tournament Players Club, Florida. http://earth.google.com 7. PGA Tour, Russell Knox implodes on No. 17 at THE PLAYERS https://youtu.be/BRJdDD0fj9c 8. Willy Weather, USA weather forecast. https://wind.willyweather.com/ 9. Google Earth, CNES/Airbus, Data SIO, NOAA, US Navy, NGA, GEBCO, IBCAO, Landsat/Copernicus, Mexam Technologies. Muirfield Golf Club. http://earth.google.com 10. Google Earth, Data MVARI, Data SIO, NOAA, US Navy, NGA, GEBCO Landsat/Copernicus. Pebble Beach Golf Links, California. http://earth.google.com

Chapter 6

Concluding Thoughts

Golf is a science, the study of a lifetime, in which you can exhaust yourself but never your subject. —David Forgan

The flight of a golf ball is majorly governed by aerodynamic forces acting on the ball. Since these forces depend on the relative airspeed of the ball, wind has a direct impact on the golf ball’s trajectory. Moreover, the impact of wind is much more significant than that of other parameters like ball speed, spin rate and launch angle. Therefore, the effect of wind becomes a vital subject in the game of golf. However, since the the impact of wind on the game as a whole, as opposed to just on individual flight trajectories. In Chap. 2, we discussed physical models that govern the motion of a golf ball: flight, bouncing and rolling. The local wind conditions were modelled using a logarithmic velocity profile, that captures the effect of the atmospheric boundary layer. Thereafter in Chap. 2, we discussed the algorithm and numerical techniques that we use to simulate a golf ball’s motion in a synthesised golf course environment. We also described the procedure we use to model a real or an arbitrary golf course and explain how we incorporate it in the simulation programme so that the ball interacts with it. In Chap. 4, we presented a detailed quantitative analysis of the variation in windinduced deviations with respect to three parameters that govern the ball’s trajectory: wind conditions, launch conditions and aerodynamic characteristics. We found that the effect of headwind is more profound than that of a tailwind of the same magnitude. Also, the wind-induced deviations are higher for more lobbed shots compared to flat shots. Furthermore, the trajectory of a golf ball having hexagonal dimples undergoes a higher deflection in the presence of wind, as compared to that of a conventional golf ball having circular dimples. Finally, in Chap. 5, we discussed some © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. Malik and S. Saha, Golf and Wind, https://doi.org/10.1007/978-981-15-9720-6_6

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strategies that players use to counter the effect of wind, and presented case studies of three real golf courses having distinct design styles: penal, strategic and heroic. This helped us examine which situation makes the effect of wind more critical. We inferred that penal courses tend to become prohibitive for all players in the presence of wind above a certain threshold. However, it is still possible to play on a strategic course when it is windy. Nevertheless, depending on the direction of wind, one route tends to become more favourable than the other. On the other hand, a heroic course increases the gap between experts and amateurs in the presence of wind, thus only letting the best of the best players seek advantage of the heroic route. In this book, we neglected the spatial and temporal variation of wind which occurs when the wind gusts and swirls around trees, buildings, hills and valleys. Instead, we approximated the wind-velocity profile using a logarithmic law, due to the formation of atmospheric boundary layer over a generic golf-course-like terrain. Similarly, we neglected the inconsistency of golf shots in the case studies and instead assumed that the golfer is able to deliver his swings perfectly every time. These assumptions along the way helped us keep our analysis simple to comprehend and get results that explained the broader picture. However, it would be interesting to see how the inclusion of more complexity in the simulation models leads to change in the prediction. It could also open the doors to new questions about wind’s impact on gameplay. For instance, what are the challenges while playing in gusty wind speeds? How does an uneven terrain bring about a change in the local wind-velocity profile? In such a situation, is there a notable difference in the way the flight trajectory is deflected? How do other course elements like trees, buildings, bunkers and water bodies contribute to swirling wind speeds? What are the new challenges introduced in the gameplay due to swirling winds? Such complex situations could be demystified by including the spatial and temporal variation of wind speeds on the golf course. Similarly, modelling the inconsistency of golf shots for the case studies could help us better understand the differences between the gameplay of an amateur and an expert. These new questions offer a promising direction to continue the study in future.