Wavefronts And Rays As Characteristics And Asymptotics

by Michael A Slawinski, Andrej Bona

Overview

This textbook — incorporated with many illuminating examples and exercises — is aimed at graduate students of physical sciences and engineering. The purpose is to provide a background of physics and underlying mathematics for the concept of rays, filling the gap between mathematics and physics textbooks for a coherent treatment of all topics. The authors' emphasis and extremely good presentation of the theory of characteristics, which defines the rays, accentuate the beauty and versatility of this theory. To this end, the rigour of the formulation — by a pure mathematician's standards — is downplayed to highlight the physical meaning and to make the subject accessible to a wider audience. The authors describe in detail the theory of characteristics for different types of differential equations, the applications to wave propagation in different types of media, and the phenomena such as caustics.

Product Details

ISBN-13:	9789814295512
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	05/13/2011
Pages:	296
Product dimensions:	6.20(w) x 9.10(h) x 0.80(d)

Table of Contents

Preface vii

List of Figures xv

Acknowledgments xvii

1 Characteristic equations of first-order linear partial differential equations 1

Preliminary remarks 1

1.1 Motivational example 2

1.1.1 General and particular solutions 2

1.1.2 Characteristics 3

1.2 Directional derivatives 4

1.3 Nonlinear digression: Inviscid Burgers's equation 9

1.4 Taylor series of solutions 10

1.5 Incompatibility of side conditions 16

1.6 Semilinear equations 18

1.7 Systems of equations 23

Closing remarks 28

1.8 Exercises 30

2 Characteristic equations of second-order linear partial differential equations 47

Preliminary remarks 47

2.1 Motivational examples 48

2.1.1 Equation with directional derivative 48

2.1.2 Wave equation in one spatial dimension 53

2.1.3 Heat equation in one spatial dimension 58

2.1.4 Laplace equation in two spatial dimensions 61

2.2 Hyperbolic, parabolic and elliptic equations 62

2.3 Characteristics 64

2.3.1 Semilinear equations 64

2.3.2 Wave, heat and Laplace equations 70

2.3.3 Solution of wave equation 72

2.3.4 Systems of semilinear equations 73

2.3.5 Elastodynamic and Maxwell equations 76

2.3.6 Quasilinear equations 78

Closing remarks 81

2.4 Exercises 82

3 Characteristic equations of first-order nonlinear partial differential equations 97

Preliminary remarks 97

3.1 Motivational example 98

3.2 Characteristics 99

3.3 Side conditions 105

3.4 Physical applications 105

3.4.1 Elastodynamic equations 105

3.4.2 Maxwell equations 116

Closing remarks 117

3.5 Exercises 118

4 Propagation of discontinuities for linear partial differential equations 125

Preliminary remarks 125

4.1 Motivational example 126

4.2 Discontinuities and frequency content 128

4.3 Asymptotic series 133

4.3.1 General formulation 134

4.3.2 Choice of asymptotic sequence 140

4.4 Eikonal equation 142

4.4.1 Derivation 142

4.4.2 Solution 144

4.5 Transport equation 145

4.5.1 Derivation 145

4.5.2 Solution 148

4.6 Higher-order transport equations 155

4.7 Physical applications 158

4.7.1 Elastodynamic equations 158

4.7.2 Maxwell equations 160

Closing remarks 162

4.8 Exercises 164

5 Caustics 177

Preliminary remarks 177

5.1 Singularities of transport equations 178

5.2 Caustics as envelopes of characteristics 178

5.3 Phase change on caustics 180

5.3.1 Waves in isotropic homogeneous media 181

5.3.2 Method of stationary phase 183

5.3.3 Phase change 186

Closing remarks 193

5.4 Exercises 194

Afterword 203

Appendix A Integral theorems 207

Preliminary remarks 207

A.1 Divergence Theorem 208

A.1.1 Statement 208

A.1.2 Plausibility argument 208

A.2 Curl Theorem 212

A.2.1 Statement 212

A.2.2 Plausibility argument 213

Closing remarks 216

Appendix B Elastodynamic equations 217

Preliminary remarks 217

B.1 Cauchy's equations of motion 218

B.2 Stress-strain equations 222

B.3 Elastodynamic equations 223

B.4 Scalar and vector potentials 224

B.4.1 Isotropy and homogeneity 224

B.4.2 Equations of motion 225

B.4.3 Scalar and vector potentials 226

B.4.4 Wave equations 227

B.5 Equations of motion versus wave equations 229

Closing remarks 229

Appendix C Maxwell equations in vacuo 231

Preliminary remarks 231

C.1 Formulation 232

C.1.1 Fundamental equations 232

C.1.2 Coulomb's law 232

C.1.3 No-monopole law 233

C.1.4 Faraday's law 233

C.1.5 Ampère's law 234

C.1.6 Speed of light 235

C.1.7 Maxwell equations 235

C.2 Scalar and vector potentials 237

Closing remarks 240

Appendix D Fourier series and transforms 243

Preliminary remarks 243

D.1 Similarity of functions 244

D.2 Fourier series 245

D.3 Fourier transform 247

Closing remarks 250

Appendix E Distributions 251

Preliminary remarks 251

E.1 Definition of distributions 252

E.2 Operations on distributions 255

E.3 Symbol 258

E.4 Principal symbol 259

Closing remarks 259

List of symbols 261

Bibliography 263

Index 267