ISBN: 3-540-65592-1
TITLE: Acoustics of Layered Media II
AUTHOR: Brekhovskikh, Leonid M.; Godin, Oleg A.
TOC:

1. Reflection and Refraction of Spherical Waves 1
1.1 Integral Representation of the Sound Field 1
1.2 Reflected Wave 5
1.3 Refracted Wave 16
1.4 Very Large or Very Small Ratio of Media Densities. Reflection from an Impedance Boundary 21
1.5 Weak Boundaries 26
1.6 Reflection from a Moving Medium 33
2. Reflection of Bounded Wave Beams 41
2.1 Displacement of a Reflected Beam 42
2.1.1 Classical Expression for Displacement 42
2.1.2 Examples of Beam Displacement 44
2.2 Incidence Angle Close to Angle of Total Reflection 47
2.2.1 Displacement of the Maximum of the Beam Envelope 47
2.2.2 The Role of Absorption 52
2.2.3 Displacement of the "Centroid" of a Beam 53
2.3 Approach to Beam Displacement Using Energy Considerations 58
2.4 Incidence Angle Close to pi/2 63
2.5 Reflection from a Boundary with Refraction Index Close to Unity 65
2.6 The Goos-Hnchen Effect 71
2.7 "Nonspecular Effects" Accompanying Beam Reflection 72
2.7.1 Longitudinal Displacement of a Beam 73
2.7.2 Deviation of the Beam Reflection Angle from the Angle of Incidence 76
2.8 Some Remarks About Beam Reflection at a FluidSolid Interface 79
2.9 Concluding Remarks 79
3. The Lateral Wave 81
3.1 Physical Interpretation and Significance 81
3.2 The Ray Approach 85
3.2.1 Ray Displacement upon Reflection 85
3.2.2 Caustics of Usual and Diffracted Rays 86
3.2.3 Lateral Rays in a Moving Medium 89
3.3 Region of Observation of a Lateral Wave 91
3.3.1 Two Lossy Homogeneous Halfspaces in Contact 91
3.3.2 Physical Interpretation 93
3.3.3 The General Case 95
3.4 Lateral Waves in Layered Media 95
3.4.1 Very Large Horizontal SourceReceiver Separations 95
3.4.2 Review of Other Problems 99
3.5 Lateral Wave Generation by a Directional Source 102
3.5.1 Lateral Waves in Sound Beam Reflection 102
3.5.2 Distributed Sound Source 107
3.6 Weakly Uneven Boundaries 108
3.6.1 The Mean Field 109
3.6.2 Random Lateral Wave from a Plane Incident Wave 112
3.6.3 Random Lateral Wave from a Point Source 114
4. Exact Theory of the Sound Field in Inhomogeneous Moving Media 121
4.1 Wave Equation for Nonstationary (Nonsteady-State) Moving Media 121
4.1.1 Linearization of Hydrodynamics Equations 121
4.1.2 Exact Wave Equations 124
4.1.3 Sound Wave Equation for a Medium with Slow Currents 126
4.2 Reciprocity Relations 130
4.2.1 Reciprocity Principle for a Medium at Rest 130
4.2.2 Layered Moving Media. Flow Reversal Theorem 131
4.2.3 Flow Reversal Theorem and the Reciprocity Principle for Homogeneous Media and Homogeneous Flow 133
4.3 Exact Solutions of the Wave Equations for a Point Source 135
4.3.1 The Point Source in Homogeneous Moving Media 135
4.3.2 Integral Representation of the Field in a Layered Medium 142
4.3.3 Sound Field in a Medium Where Sound Velocity Is a Linear Function of z 144
4.3.4 Sound Field in a Medium Where the Squared Refraction Index Is a Linear or Quadratic Function of Coordinates 146
4.4 Discrete Spectrum of a Field. Normal Modes 150
4.4.1 Discrete Spectrum in a Medium at Rest 151
4.4.2 A Linear Source in a Waveguide 152
4.4.3 Discrete Spectrum of a Field of a Point Source in a Moving Medium 155
4.4.4 More About the Structure of the Discrete Spectrum of a Point Source in a Moving Medium 160
4.4.5 Formulas of More Convenience 166
4.5 Phase and Group Velocities of Modes 169
4.5.1 Generalized Orthogonality of Modes 170
4.5.2 Mode Phase and Group Velocities in a Medium at Rest 174
4.5.3 Phase and Group Velocities in a Moving Medium 177
4.6 The Epstein Waveguide 182
4.6.1 Waveguide with a Free Boundary 182
4.6.2 Waveguide with an Absolutely Rigid Boundary 186
4.6.3 Comparison with Results Obtained in the WKB Approximation 188
5. High Frequency Sound Fields 193
5.1 Geometrical Acoustics Approximation for a Localized Source 193
5.1.1 Ray Series. Eikonal Function 193
5.1.2 Ray Equations. Ray Tube. Power Density Flow 194
5.1.3 A Three-Dimensionally Inhomogeneous Moving Medium 196
5.1.4 Layered Media and Horizontal Flow 200
5.2 Ray Acoustics as a Limiting Case of Wave Theory 202
5.2.1 The Case of a Moving Medium 202
5.2.2 Waveguide Sound Field in the Ray Approximation 205
6. The Field at and near a Caustic 209
6.1 Simple Caustics 210
6.1.1 Definition 210
6.1.2 Caustics in Waveguides. Qualitative Results 211
6.1.3 The Sound Field near an Ordinary Point of a Caustic 211
6.1.4 Field near a Caustic in Terms of Ray Quantities 215
6.1.5 Limits of Validity 216
6.2 Reference Functions Method 217
6.2.1 Caustics in Media at Rest 217
6.2.2 The Reference Functions Method for Solving One-Dimensional Wave Equations 221
6.2.3 Moving Media 223
6.3 A Cusp of a Caustic and Other Peculiarities of Ray Structures 229
6.3.1 Uniform Asymptotics 229
6.3.2 Local Asymptotics 236
6.3.3 General Aspects of Field Singularities 239
7. Wave Propagation in a Range Dependent Waveguide 243
7.1 Reference Waveguide Method 244
7.1.1 Interaction of Modes 244
7.1.2 Coupling Coefficients of Modes 247
7.1.3 Solution of Coupling Equations by the Method of Successive Approximations 254
7.2 Propagation of Waves in Three Dimensions in a Range Dependent Waveguide 263
7.2.1 Horizontal Rays 263
7.2.2 Calculation of Mode Amplitude at the Ray 265
7.2.3 The Field in the Region of a Simple Caustic of Horizontal Rays 272
7.2.4 Applicability Conditions of the Adiabatic Approximation. Smooth Perturbations Method 275
7.3 Waveguide Propagation in a Three-Dimensional Inhomogeneous Moving Medium 282
7.3.1 Equations for the Sound Field in Compressed Coordinates 283
7.3.2 Boundary Conditions for a Sound Field in a Moving Medium 285
7.3.3 Horizontal (Modal) Rays in a Moving Medium 288
7.3.4 Adiabatic Invariants 293
7.3.5 Sound Field of a Point Source 295
7.3.6 Flow Reversal Theorem for Modes 297
7.4 The Sound Field in the Vicinity of the Cutoff Section of a Waveguide 300
7.4.1 Waveguide with Impedance Boundaries 300
7.4.2 Modes and Continuous Spectrum 304
7.4.3 Penetration of a Mode Through a Cutoff Section in a Pekeris Waveguide 312
7.5 Rays in Irregular Waveguides 320
7.5.1 The Ray Invariant in a Medium at Rest 320
7.5.2 Accuracy of Conservation of the Ray Invariant 325
7.5.3 The Ray Invariant in a Moving Medium 326
7.6 Parabolic Equation Method 329
7.6.1 Standard Parabolic Equation 330
7.6.2 Relations Between Solutions to PE and the Helmholtz Equation 334
7.6.3 One-Way Wave Equations 342
7.6.4 Effect of Density Inhomogeneities and of Shear Waves 350
7.6.5 Parabolic Approximation for a Sound Field in a Moving Fluid 353
7.6.6 The Acoustic Energy Conservation Law and Reciprocity Relations in the Narrow-Angle Parabolic Approximation 357
8. Energy Conservation and Reciprocity for Waves in Three-Dimensionally Inhomogeneous Moving Media 361
8.1 Oscillatory Displacement of Fluid Particles 361
8.2 Linearized Equations of Hydrodynamics 363
8.2.1 Euler and Continuity Equations 363
8.2.2 Equation of State 365
8.2.3 Equations Governing the Linear Waves 367
8.2.4 An Alternative Version of the Mixed EulerianLagrangian Representation of the Equations of Motion 369
8.3 Linearized Boundary Conditions 370
8.4 Flow Reversal Theorem 373
8.4.1 Solenoidal Flow 374
8.4.2 General Steady Ambient Flow 377
8.4.3 Reciprocity Principle for Acoustic-Gravity Waves 379
8.4.4 Comparison to Other Versions of FRT 381
8.5 Conservation of Wave Action 384
8.6 Wave Energy Conservation Law 387
8.7 Applications to Sound Waves in a Plane-Stratified Moving Fluid 392
8.7.1 Quasi-Plane Waves 392
8.7.2 Reciprocity Relations for Plane-Wave Transmission Coefficients 393
8.7.3 Violations of FRT and the Wave Energy Conservation 395
8.7.4 Output of Sound Sources in a Flow 396
8.8 Summary 398
Appendix A. The Reference Integrals Method 401
A.1 The Method of Steepest Descent 401
A.1.1 Integrals over an Infinite Contour 401
A.1.2 Integrals over Semi-infinite Contours 405
A.1.3 Integrals with Finite Limits 406
A.1.4 The Contribution of Branch Points 406
A.1.5 Integrals with Saddle Points of Higher Orders 407
A.1.6 Several Saddle Points 409
A.1.7 Concluding Remarks 409
A.2 Integrals over a Real Variable 410
A.2.1 Asymptotics of Laplace Integrals 410
A.2.2 Stationary Phase Method. Asymptotics of Fourier Integrals 411
A.2.3 Asymptotics of Multiple Fourier Integrals 412
A.2.4 Asymptotics of Multiple Laplace Integrals 414
A.2.5 Contributions of Critical Points on a Boundary 414
A.3 Uniform Asymptotics of Integrals 414
A.3.1 The Concept of Uniform Asymptotics 414
A.3.2 A Pole and a Simple Stationary Point 415
A.3.3 A Single Simple Stationary Point and a Branch Point 418
A.3.4 Semi-infinite Contours 421
A.3.5 Other Cases 423
A.3.6 Concluding Remarks 425
Appendix B. Differential Equations of Coupled-Mode Propagation in Fluids with Sloping Boundaries and Interfaces 429
B.1 Derivation of the Differential Equations for Mode Coupling 430
B.2 Mode-Coupling Coefficients in Terms of Environmental Gradients 435
B.3 Energy Conservation and Symmetry of the Mode Coupling Coefficients 438
B.4 Convergence of Normal Mode Expansions and its Implications on the Mode-Coupling Equations: Two Examples 440
Appendix C. Reciprocity and Energy Conservation Within the Parabolic Approximation 447
C.1 Definitions and Basic Relationships 448
C.1.1 Range-Independent One-Way Wave Equations 449
C.1.2 Equivalence of Reciprocity and Energy Conservation 455
C.2 Energy Conserving and Reciprocal One-Way Wave Equation 456
C.3 Generalized Claerbout PE (GCPE) 461
C.3.1 GCPE Derivation 461
C.3.2 Local Reciprocity and Energy Balance Relations 462
C.3.3 Media with Interfaces 464
C.4 Comparison of Different One-Way Approximations 468
C.5 Conclusion 472
References 475
Subject Index 517
END
