Mathematical methods for wave phenomena

Preface......Page 6
Contents......Page 3
1.1 First-Order Quasi-Linear Differential Equations......Page 11
1.2 An Illustrative Example......Page 17
1.3 First-Order Nonlinear Differential Equations......Page 22
1.4 Examples-The Eikonal Equation-and More Theory......Page 28
1.5 Propagation of Wave Fronts......Page 37
1.6 Variable Index of Refraction......Page 48
1.7 Higher DimensionsFirst-Order Quasi-Linear Differential Equations References......Page 52
References......Page 54
2.1 The Dirac Delta Function and Related Distributions......Page 55
2.2 Fourier Transforms......Page 62
2.3 Fourier Transforms of Distributions......Page 68
2.4 Multidimensional Fourier Transforms......Page 71
2.5 Asymptotic Expansions......Page 76
2.6 Asymptotic Expansions of Fourier Integrals with Monotonic Phase......Page 83
2.7 The Method of Stationary Phase......Page 87
2.8 Multidimensional Fourier Integrals......Page 92
References......Page 99
3.1 Prototype Second-Order Equations......Page 102
3.2 Some Simple Examples......Page 104
References......Page 109
4 THE WAVE EQUATION IN ONE SPACE DIMENSION......Page 0
4.1 Characteristics for the Wave Equation in One Space Dimension......Page 110
4.2 The Initial Boundary Value Problem......Page 115
4.3 The Initial Boundary Value Problem Continued......Page 119
4.4 The Adjoint Equation and the Riemann Function......Page 130
4.5 The Green?ˉs Function......Page 140
4.6 Asymptotic Solution of the Klein-Gordon Equation......Page 143
4.7 More on Asymptotic Solutions......Page 146
References......Page 154
5.1 Characteristics and 111-Posed Cauchy Problems......Page 155
5.2 The Energy Integral, Domain of Dependence, and Uniqueness......Page 160
5.3 The Green?ˉs Function......Page 163
5.4 Scattering Problems......Page 168
References......Page 173
6.1 Green?ˉs Identities and Uniqueness Results......Page 175
6.2 Some Special Features of Laplace?ˉs Equation......Page 180
6.3 Green?ˉs Functions......Page 184
6.4 Problems in Unbounded Domains and the Sommerfeld Radiation Condition......Page 190
6.5 Some Exact Solutions......Page 202
References......Page 212
7.1 Watson?ˉs Lemma......Page 214
7.2 The Method of Steepest Descents: Preliminary Results......Page 221
7.3 Formulas for the Method of Steepest Descents......Page 235
7.4 The Method of Steepest Descents: Implementation......Page 239
References......Page 250
8.1 Scattering by a Half-Space: Analysis by Steepest Descents......Page 251
8.2 Introduction to Ray Methods......Page 267
8.3 Determination of Ray Data......Page 278
8.4 The Kirchhoff Approximation......Page 291
References......Page 309
9.1 The Singular Function and the Characteristic Function......Page 312
9.2 Physical Optics Far-Field Inverse Scattering (POFFIS)......Page 322
9.3 The Seismic Inverse Problem......Page 331
References......Page 347
Index......Page 348