Table of Contents
Preface vii
Changes from, First Edition xi
List of Figures xix
Acknowledgments xxi
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 19
1.7 Systems of equations 23
Closing remarks 29
1.8 Exercises 29
2 Characteristic equations of second-order linear partial differential equations 49
Preliminary remarks 49
2.1 Motivational examples 50
2.1.1 Equation with directional derivative 50
2.1.2 Wave equation in one spatial dimension 55
2.1.3 Heat equation in one spatial dimension 60
2.1.4 Laplace equation in two spatial dimensions 63
2.2 Hyperbolic, parabolic and elliptic equations 64
2.3 Characteristics 66
2.3.1 Semilinear equations 66
2.3.2 Wave, heat and Laplace equations 72
2.3.3 Solution of wave equation 74
2.3.4 Systems of semilinear equations 75
2.3.5 Elastodynamic and Maxwell equations 78
2.3.6 Quasilinear equations 80
Closing remarks 83
2.4 Exercises 83
3 Characteristic equations of first-order nonlinear partial differential equations 99
Preliminary remarks 99
3.1 Motivational example 100
3.2 Characteristics 101
3.3 Side conditions 107
3.4 Physical applications 107
3.4.1 Elastodynamic equations 107
3.4.2 Maxwell equations 118
Closing remarks 119
3.5 Exercises 119
4 Propagation of discontinuities for linear partial differential equations 127
Preliminary remarks 127
4.1 Motivational example 128
4.2 Discontinuities and frequency content 130
4.3 Asymptotic series 135
4.3.1 General formulation 136
4.3.2 Choice of asymptotic sequence 142
4.4 Eikonal equation 144
4.4.1 Derivation 144
4.4.2 Solution 147
4.5 Transport equation 147
4.5.1 Derivation 147
4.5.2 Solution 150
4.6 Higher-order transport equations 158
4.7 Physical applications 160
4.7.1 Elastodynamic equations 160
4.7.2 Maxwell equations 162
Closing remarks 164
4.8 Exercises 165
5 Caustics 179
Preliminary remarks 179
5.1 Singularities of transport equations 180
5.2 Caustics as envelopes of characteristics 180
5.3 Phase change on caustics 182
5.3.1 Formulation 182
5.3.2 Waves in isotropic homogeneous media 183
5.3.3 Method of stationary phase 185
5.3.4 Phase change 189
Closing remarks 195
5.4 Exercises 196
Afterword 205
Appendix A Integral theorems 209
Preliminary remarks 209
A.1 Divergence Theorem 210
A.1.1 Statement 210
A.1.2 Plausibility argument 211
A.1.3 Corollary 216
A.2 Curl Theorem 217
A.2.1 Statement 217
A.2.2 Plausibility argument 218
Closing remarks 221
Appendix B Elastodynamic equations 223
Preliminary remarks 223
B.1 Cauchy's equations of motion 224
B.2 Stress-strain equations: Hookcan solids 232
B.3 Elastodynamic equations: anisotropy, inhomogeneity 233
B.4 Elastodynamic equations: isotropy, homogeneity 235
B.4.1 Equations of motion 235
B.4.2 Scalar and vector potentials 237
B.4.3 Wave equations 237
B.5 Equations of motion versus wave equations 239
Closing remarks 240
Appendix C Maxwell equations in vacuo 241
Preliminary remarks 241
C.1 Formulation 242
C.1.1 Fundamental equations 242
C.1.2 Coulomb's law 242
C.1.3 No-monopole law 243
C.1.4 Faraday's law 243
C.1.5 Ampère's law 244
C.1.6 Speed of light 245
C.1.7 Maxwell equations 245
C.2 Scalar and vector potentials 247
Closing remarks 250
Appendix D Fourier series and transforms 253
Preliminary remarks 253
D.1 Similarity of functions 254
D.2 Fourier series 256
D.3 Fourier transform 260
D.3.1 Formulation 260
D.3.2 Application to differential equations 262
Closing remarks 265
Appendix E Distributions 267
Preliminary remarks 267
E.1 Definition of distributions 268
E.2 Operations on distributions 272
E.3 Symbol 277
E.4 Principal symbol 278
Closing remarks 279
Appendix F Green's functions 281
Preliminary remarks 281
F.1 Electrostatic equation 282
F.2 Wave equations 285
F.2.1 Three spatial dimensions 285
F.2.2 Two spatial dimensions 289
F.2.3 One spatial dimension 290
F.2.4 Green's function and initial value problem 292
F.3 Elastodynamic equations 298
Closing remarks 301
List of symbols 303
Bibliography 305
Index 309
