Equations of Mathematical Diffraction Theory

Mezhlum A. Sumbatyan, Antonio Scalia0415308496, 9780415308496, 9780203643488

Equations of Mathematical Diffraction Theory focuses on the comparative analysis and development of efficient analytical methods for solving equations of mathematical diffraction theory. Following an overview of some general properties of integral and differential operators in the context of the linear theory of diffraction processes, the authors provide estimates of the operator norms for various ranges of the wave number variation, and then examine the spectral properties of these operators. They also present a new analytical method for constructing asymptotic solutions of boundary integral equations in mathematical diffraction theory for the high-frequency case. Clearly demonstrating the close connection between heuristic and rigorous methods in mathematical diffraction theory, this valuable book provides you with the differential and integral equations that can easily be used in practical applications.

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Table of contents :
EQUATIONS OF MATHEMATICAL DIFFRACTION THEORY……Page 3
Differential and Integral Equations and Their Applications……Page 2
PREFACE……Page 5
AUTHORS……Page 8
Table of Contents……Page 9
Table of Contents……Page 0
1.1. Fourier Transform, Line Integrals of Complex­Valued Integrands, and Series in Residues……Page 12
Helpful remarks……Page 16
1.2. Convolution Integral Equations and the Wiener-Hopf Method……Page 17
Helpful remarks……Page 18
1.3. Summation of Divergent Series and Integrals……Page 20
1.4. Asymptotic Estimates of Integrals……Page 23
Helpful remarks……Page 28
1.5. Fredholm Theory for Integral Equations of the Second Kind……Page 32
Helpful remarks……Page 34
1.6. Fredholm Integral Equations of the First Kind……Page 35
Helpful remarks……Page 39
1.7. Singular Integral Equations with a Cauchy-Type Singularity in the Kernel……Page 40
Helpful remarks……Page 45
1.8. Hyper-Singular Integrals and Integral Equations……Page 46
Helpful remarks……Page 48
Linear Hydroaeroacoustics……Page 49
Electromagnetic wave theory……Page 51
Linear dynamic elasticity……Page 52
Helpful remarks……Page 53
2.1. Properties of the Potentials of Single and Double Layers……Page 55
2.2. Basic Integral Equations of the Diffraction Theory……Page 62
Formulation of diffraction problem……Page 65
2.3. Properties of Integral Operators of Diffraction Theory: General Case and Low Frequencies……Page 67
Low-frequency diffraction problem……Page 68
Arbitrary 3D obstacle: Acoustically hard boundary……Page 70
Acoustically hard obstacle……Page 71
Acoustically soft obstacle……Page 73
Helpful remarks……Page 74
Low-frequency scattering from a hard round disk……Page 75
Low-frequency diagram for acoustically hard sphere……Page 76
Scattering by acoustically soft sphere……Page 77
2.6. Asymptotic Character of the Kirchhoff Physical Diffraction Theory……Page 78
Helpful remarks……Page 82
3.1. Wave Operator in Acoustic Layer: Mode Expansion, Homogeneous and Inhomogeneous Waves……Page 83
Sommerfeld’s radiation condition……Page 86
Principle of extremely low absorption (Ignatowsky’s principle)……Page 87
Energy radiation condition (Mandelshtam’s principle)……Page 88
Principle of amplitude for extremely large time (Tikhonov-Samarsky principle)……Page 89
Helpful remarks……Page 90
3.3. Waves in Elastic Layer……Page 92
Helpful remarks……Page 96
3.4. Generalized Riemann’s Zeta Function and Summation of Some Oscillating Series……Page 97
Helpful remarks……Page 100
3.5. Application: Efficient Calculation of Wave Fields in a Layer of Constant Thickness……Page 101
3.6. Waves in the Stratified Half-Plane……Page 104
Helpful remarks……Page 109
4.1. General Spectral Properties of the Interior Problem for Laplacian……Page 110
Helpful remarks……Page 115
4.2. Explicit Formulas for Eigenfrequencies of Round Disc……Page 116
4.3. Some Variational Principles for Eigenvalues……Page 119
4.4. Weyl-Carleman Theory of Asymptotic Distribution of Large Eigenvalues……Page 124
Helpful remarks……Page 127
4.5. Exact Explicit Results for Some Polygons……Page 128
Helpful remarks……Page 133
4.6. Explicit Analytical Results for Some Polyhedra……Page 134
Helpful remarks……Page 140
5.1. Integral Operators in Diffraction by Linear Screen and by a Gap in the Screen……Page 141
Helpful remarks……Page 145
5.2. Operator Equation in Diffraction Problem on a Crack in Unbounded Elastic Medium……Page 146
5.3. High-Frequency Asymptotics in Diffraction by Linear Obstacles in Unbounded Medium……Page 150
5.4. High-Frequency Asymptotics for Diffraction by Linear Obstacles in Open Waveguides……Page 153
Helpful remarks……Page 158
5.5. High-Frequency Diffraction by a Linear Discontinuity in the Waveguide……Page 159
Helpful remarks……Page 164
5.6. Waves in Elastic Half-Space. Factorization of the Rayleigh Function……Page 165
5.7. Integral Equation of the Mixed Boundary Value Problem for Elastic Layer……Page 168
Helpful remarks……Page 171
6.1. Schoch’s Method: Exact Representation of 3D Wave Fields by One-Dimensional Quadratures……Page 172
Helpful remarks……Page 175
6.2. High-Frequency Wave Fields in Elastic Half-Space……Page 176
6.3. Asymptotic Nature of the Geometrical Diffraction Theory……Page 178
Helpful remarks……Page 181
6.4. High-Frequency Diffraction with Re-Reflections……Page 182
Helpful remarks……Page 185
6.5. Application: Examples of High-Frequency Multiple Diffraction……Page 187
6.6. Application: Physical Diffraction Theory for Nonconvex Obstacles……Page 191
6.7. Short-Wave Integral Operator in Diffraction by a Flaw in Elastic Medium……Page 193
Helpful remarks……Page 196
6.8. High-Frequency Asymptotics of Integral Operator in a Three-Dimensional Diffraction Theory……Page 197
Helpful remarks……Page 199
7.1. Some Basic Results in a Local Differential Geometry of Smooth Convex Surfaces……Page 201
Helpful remarks……Page 204
7.2. Reducing Inverse Problem of the Short-Wave Diffraction to Minkowski Problem……Page 205
Helpful remarks……Page 206
7.3. Explicit Results for a Differential Operator of the 2D Inverse Problem……Page 207
Helpful remarks……Page 208
7.4. Exact Explicit Inversion of the Basic Operator in the Case of Axial Symmetry……Page 209
7.5. Nonlinear Differential Operator of the Three-Dimensional Inverse Problem……Page 211
7.6. Reconstruction of Nonconvex Obstacles in the High-Frequency Range: 2D Case……Page 214
Helpful remarks……Page 218
7.7. Reconstruction of Nonconvex Obstacles in the High-Frequency Range: 3D Case……Page 219
Helpful remarks……Page 224
8.1. Ill-Posed Problems for Operator Equations of the First Kind: General Properties……Page 225
Helpful remarks……Page 227
8.2. Regularization of Ill-Posed Problems with the Help of Smoothing Functional……Page 228
Helpful remarks……Page 231
8.3. Iterative Methods for Operator Equations of the First Kind……Page 232
8.4. Comparison of Various Methods for Reconstruction of the Scatterer Geometry……Page 237
8.5. General Inverse Diffraction Problem: Combination of Iterations and Smoothing……Page 241
8.6. A Correct Treatment of Ill-Posed Boundary Equations in Acoustics of Closed Regions……Page 248
Helpful remarks……Page 253
8.7. Ill-Posed Method of Auxiliary Sources in Diffraction Theory……Page 254
8.8. A Method of Global Random Search in Inverse Problems……Page 257
8.9. Ill-Posed Problem on Reconstruction of Convex Hull of the Obstacle in Acoustic Medium……Page 259
Helpful remarks……Page 263
9.1. Steepest Descent Method: Stability and Improvement of the Convergence……Page 264
9.2. Galerkin Methods for Integral Equations of the First Kind with Weakly Singular Kernels……Page 268
Helpful remarks……Page 272
9.3. Integral Equations of the Physical Diffraction Theory in the Case of Nonconvex Obstacles……Page 273
Helpful remarks……Page 276
9.4. Numerical Methods in Singular Integral Equations with the Cauchy-Type Kernel……Page 277
9.5. Numerical Methods for Hyper-Singular Integral Equations……Page 281
Helpful remarks……Page 285
REFERENCES……Page 286