Wave Fields in Real Media: Wave Propagation in Anisotropic, Anelastic, Porous and Electromagnetic Media examines the differences between an ideal and a real description of wave propagation, starting with the introduction of relevant constitutive relations. The differential formulation can be written in terms of memory variables, and Biot theory is used to describe wave propagation in porous media. For each constitutive relation, a plane-wave analysis is performed to illustrate the physics of wave propagation. New topics are the S-wave amplification function, Fermat principle and its relation to Snell law, bounds and averages of seismic Q, seismic attenuation in partially molten rocks, and more. This book contains a review of the main direct numerical methods for solving the equation of motion in the time and space domains. The emphasis is on geophysical applications for seismic exploration, but researchers in the fields of earthquake seismology, rock acoustics and material science - including many branches of acoustics of fluids and solids - may also find this text useful.
1. Wave Fields in Real Media - p. i 2. Cover - p. i 3. Title Page - p. iii 4. Copyright Page - p. iv 5. Quotes - p. v 6. Table of Contents - p. vi 7. About the Author - p. xvii 8. Preface - p. xix 9. Basic Notation - p. xxvii 10. Glossary of main symbols - p. xxix 11. Chapter 1 Anisotropic elastic media - p. 1 12. 1.1: Strain-energy density and stress-strain relations - p. 1 13. 1.2: Dynamical equations - p. 4 14. 1.3: Kelvin-Christoffel equation, phase velocity and slowness - p. 12 15. 1.4: Energy balance and energy velocity - p. 18 16. 1.5: Finely layered media - p. 29 17. 1.6: Anomalous polarizations - p. 39 18. 1.7: The best isotropic approximation - p. 49 19. 1.8: Analytical solutions - p. 52 20. 1.9: Reflection and transmission of plane waves - p. 55 21. Chapter 2 Viscoelasticity and wave propagation - p. 63 22. 2.1: Energy densities and stress-strain relations - p. 64 23. 2.2: Stress-strain relation for 1-D viscoelastic media - p. 67 24. 2.3: Wave propagation in 1-D viscoelastic media - p. 75 25. 2.4: Mechanical models and wave propagation - p. 84 26. 2.5: Constant-Q model and wave equation - p. 102 27. 2.6: The Cole-Cole model - p. 106 28. 2.7: Equivalence between source and initial conditions - p. 107 29. 2.8: Hysteresis cycles and fatigue - p. 110 30. 2.9: Distributed-order fractional time derivatives - p. 113 31. 2.10: The concept of centrovelocity - p. 115 32. 2.11: Memory variables and equation of motion - p. 121 33. 2.12: Instantaneous frequency and quality factor - p. 127 34. Chapter 3 Isotropic anelastic media - p. 135 35. 3.1: Stress-strain relations - p. 136 36. 3.2: Equations of motion and dispersion relations - p. 136 37. 3.3: Vector plane waves - p. 139 38. 3.4: Energy balance, velocity and quality factor - p. 146 39. 3.5: Boundary conditions and Snell law - p. 155 40. 3.6: The correspondence principle - p. 157 41. 3.7: Rayleigh waves - p. 158 42. 3.8: Reflection and transmission of SH waves - p. 164 43. 3.9: Memory variables and equation of motion - p. 167 44. 3.10: Analytical solutions - p. 169 45. 3.11: Constant-Q P- and S-waves - p. 175 46. 3.12: Wave equations based on the Burgers model - p. 177 47. 3.13: P-SV wave equation based on the Cole-Cole model - p. 179 48. 3.14: The elastodynamic of a non-ideal interface - p. 181 49. 3.15: SH-wave transfer function - p. 190 50. Chapter 4 Anisotropic anelastic media - p. 195 51. 4.1: Stress-strain relations - p. 196 52. 4.2: Fracture-induced anisotropic attenuation - p. 203 53. 4.3: Stiffness tensor from oscillatory experiments - p. 210 54. 4.4: Wave velocities, slowness and attenuation vector - p. 213 55. 4.5: Energy balance and fundamental relations - p. 215 56. 4.6: Propagation of SH waves - p. 224 57. 4.7: Wave propagation in symmetry planes - p. 232 58. 4.8: Summary of plane-wave equations - p. 241 59. 4.9: Memory variables and equation of motion - p. 245 60. 4.10: Fermat principle and its relation to Snell law - p. 251 61. 4.11: Analytical transient solution for SH waves - p. 255 62. Chapter 5 The reciprocity principle - p. 259 63. Chapter 6 Reflection and transmission coefficients - p. 275 64. 6.1: Reflection and transmission of SH waves - p. 276 65. 6.1.1: Symmetry plane of a homogeneous monoclinic medium - p. 276 66. 6.1.2: Complex stiffnesses - p. 278 67. 6.1.3: Reflection and transmission coefficients - p. 280 68. 6.1.4: Propagation, attenuation and energy directions - p. 282 69. 6.1.5: Brewster and critical angles - p. 289 70. 6.1.6: Phase velocities and attenuations - p. 294 71. 6.1.7: Energy-flux balance - p. 296 72. 6.1.8: Energy velocities and quality factors - p. 297 73. 6.1.9: The physical solution in ray-tracing algorithms - p. 300 74. 6.2: Reflection and transmission of qP-qSV waves - p. 302 75. 6.3: Interfaces separating a solid and a fluid - p. 319 76. 6.4: Scattering coefficients of a set of layers - p. 323 77. Chapter 7 Biot theory for porous media - p. 329 78. 7.1: Isotropic media. Stress-strain relations - p. 331 79. 7.2: Gassmann equation and effective stress - p. 334 80. 7.3: Pore-pressure build-up in source rocks - p. 346 81. 7.4: The asperity-deformation model - p. 348 82. 7.5: Anisotropic media. Stress-strain relations - p. 353 83. 7.6: Kinetic energy - p. 361 84. 7.7: Dissipation potential - p. 367 85. 7.8: Lagrange equations and equation of motion - p. 369 86. 7.9: Plane-wave analysis - p. 378 87. 7.10: Strain energy for inhomogeneous porosity - p. 387 88. 7.11: Boundary conditions - p. 393 89. 7.12: Reflection and transmission coefficients - p. 400 90. 7.13: Extension of Biot theory to a composite frame - p. 403 91. 7.14: Extension of Biot theory to partial saturation - p. 413 92. 7.15: Squirt-flow dissipation - p. 416 93. 7.16: The mesoscopic-loss mechanism. White model - p. 419 94. 7.17: The Biot-Gardner effect - p. 427 95. 7.18: Green function for poro-viscoacoustic media - p. 430 96. 7.19: Green function for a fluid/solid interface - p. 436 97. 7.20: Poro-viscoelasticity - p. 440 98. 7.21: Bounds and averages on velocity and quality factor - p. 444 99. 7.22: Anelasticity models based on ellipsoidal pores - p. 449 100. 7.23: Effect of melt on wave propagation - p. 452 101. 7.24: Anisotropy and poro-viscoelasticity - p. 458 102. 7.25: Gassmann equation for a solid pore infill - p. 473 103. 7.26: Mesoscopic loss in layered and fractured media - p. 476 104. 7.27: Fluid-pressure diffusion - p. 480 105. 7.28: Thermo-poroelasticity - p. 499 106. Chapter 8 The acoustic-electromagnetic analogy - p. 513 107. 8.1: Maxwell equations - p. 516 108. 8.2: The acoustic-electromagnetic analogy - p. 517 109. 8.3: A viscoelastic form of the electromagnetic energy - p. 525 110. 8.4: The analogy for reflection and transmission - p. 537 111. 8.5: The layer problem - p. 554 112. 8.6: 3-D electromagnetic theory and the analogy - p. 567 113. 8.7: Plane-wave theory - p. 571 114. 8.8: Electromagnetic diffusion in anisotropic media - p. 579 115. 8.9: Analytical solution for anisotropic media - p. 584 116. 8.10: Elastic medium with Fresnel wave surface - p. 588 117. 8.11: Finely layered media - p. 591 118. 8.12: The time-average and CRIM equations - p. 594 119. 8.13: Kramers-Kronig relations - p. 595 120. 8.14: The reciprocity principle - p. 601 121. 8.15: Babinet principle - p. 602 122. 8.16: Alford rotation - p. 603 123. 8.17: Cross-property relations - p. 606 124. 8.18: Poroacoustic and electromagnetic diffusion - p. 607 125. 8.19: Electro-seismic wave theory - p. 612 126. 8.20: Gravitational waves - p. 614 127. Chapter 9 Numerical methods - p. 617 128. 9.1: Equation of motion - p. 617 129. 9.2: Time integration - p. 619 130. 9.3: Calculation of spatial derivatives - p. 626 131. 9.4: Source implementation - p. 632 132. 9.5: Boundary conditions - p. 634 133. 9.6: Absorbing boundaries - p. 635 134. 9.7: Model and modeling design. Seismic modeling - p. 637 135. 9.8: Concluding remarks - p. 640 136. 9.9: Appendix - p. 641 137. 9.9.1: Fractional calculus - p. 641 138. 9.9.2: Electromagnetic-diffusion code - p. 653 139. 9.9.3: Finite-differences code for SH waves - p. 658 140. 9.9.4: Finite-difference code for SH (TM) waves - p. 663 141. 9.9.5: Pseudospectral Fourier method - p. 671 142. 9.9.6: Pseudospectral Chebyshev method - p. 675 143. 9.9.7: Pseudospectral sine/cosine method - p. 678 144. 9.9.8: Earthquake sources. The moment-tensor - p. 679 145. 9.9.9: 3-D anisotropic media. Free-surface boundary treatment - p. 682 146. 9.9.10: Modeling in cylindrical coordinates - p. 685 147. Examinations - p. 689 148. Chronology of main discoveries - p. 695 149. Leonardo's manuscripts - p. 711 150. A list of scientists - p. 717 151. Bibliography - p. 729 152. Name index - p. 763 153. Subject index - p. 781
Wave Fields in Real Media: Wave Propagation in Anisotropic, Anelastic, Porous and Electromagnetic Media examines the differences between an ideal and a real description of wave propagation, starting with the introduction of relevant constitutive relations. The differential formulation can be written in terms of memory variables, and Biot theory is used to describe wave propagation in porous media. For each constitutive relation, a plane-wave analysis is performed to illustrate the physics of wave propagation. New topics are the S-wave amplification function, Fermat principle and its relation to Snell law, bounds and averages of seismic Q, seismic attenuation in partially molten rocks, and more. This book contains a review of the main direct numerical methods for solving the equation of motion in the time and space domains. The emphasis is on geophysical applications for seismic exploration, but researchers in the fields of earthquake seismology, rock acoustics and material science - including many branches of acoustics of fluids and solids - may also find this text useful.
1. Wave Fields in Real Media - p. i 2. Cover - p. i 3. Title Page - p. iii 4. Copyright Page - p. iv 5. Quotes - p. v 6. Table of Contents - p. vi 7. About the Author - p. xvii 8. Preface - p. xix 9. Basic Notation - p. xxvii 10. Glossary of main symbols - p. xxix 11. Chapter 1 Anisotropic elastic media - p. 1 12. 1.1: Strain-energy density and stress-strain relations - p. 1 13. 1.2: Dynamical equations - p. 4 14. 1.3: Kelvin-Christoffel equation, phase velocity and slowness - p. 12 15. 1.4: Energy balance and energy velocity - p. 18 16. 1.5: Finely layered media - p. 29 17. 1.6: Anomalous polarizations - p. 39 18. 1.7: The best isotropic approximation - p. 49 19. 1.8: Analytical solutions - p. 52 20. 1.9: Reflection and transmission of plane waves - p. 55 21. Chapter 2 Viscoelasticity and wave propagation - p. 63 22. 2.1: Energy densities and stress-strain relations - p. 64 23. 2.2: Stress-strain relation for 1-D viscoelastic media - p. 67 24. 2.3: Wave propagation in 1-D viscoelastic media - p. 75 25. 2.4: Mechanical models and wave propagation - p. 84 26. 2.5: Constant-Q model and wave equation - p. 102 27. 2.6: The Cole-Cole model - p. 106 28. 2.7: Equivalence between source and initial conditions - p. 107 29. 2.8: Hysteresis cycles and fatigue - p. 110 30. 2.9: Distributed-order fractional time derivatives - p. 113 31. 2.10: The concept of centrovelocity - p. 115 32. 2.11: Memory variables and equation of motion - p. 121 33. 2.12: Instantaneous frequency and quality factor - p. 127 34. Chapter 3 Isotropic anelastic media - p. 135 35. 3.1: Stress-strain relations - p. 136 36. 3.2: Equations of motion and dispersion relations - p. 136 37. 3.3: Vector plane waves - p. 139 38. 3.4: Energy balance, velocity and quality factor - p. 146 39. 3.5: Boundary conditions and Snell law - p. 155 40. 3.6: The correspondence principle - p. 157 41. 3.7: Rayleigh waves - p. 158 42. 3.8: Reflection and transmission of SH waves - p. 164 43. 3.9: Memory variables and equation of motion - p. 167 44. 3.10: Analytical solutions - p. 169 45. 3.11: Constant-Q P- and S-waves - p. 175 46. 3.12: Wave equations based on the Burgers model - p. 177 47. 3.13: P-SV wave equation based on the Cole-Cole model - p. 179 48. 3.14: The elastodynamic of a non-ideal interface - p. 181 49. 3.15: SH-wave transfer function - p. 190 50. Chapter 4 Anisotropic anelastic media - p. 195 51. 4.1: Stress-strain relations - p. 196 52. 4.2: Fracture-induced anisotropic attenuation - p. 203 53. 4.3: Stiffness tensor from oscillatory experiments - p. 210 54. 4.4: Wave velocities, slowness and attenuation vector - p. 213 55. 4.5: Energy balance and fundamental relations - p. 215 56. 4.6: Propagation of SH waves - p. 224 57. 4.7: Wave propagation in symmetry planes - p. 232 58. 4.8: Summary of plane-wave equations - p. 241 59. 4.9: Memory variables and equation of motion - p. 245 60. 4.10: Fermat principle and its relation to Snell law - p. 251 61. 4.11: Analytical transient solution for SH waves - p. 255 62. Chapter 5 The reciprocity principle - p. 259 63. Chapter 6 Reflection and transmission coefficients - p. 275 64. 6.1: Reflection and transmission of SH waves - p. 276 65. 6.1.1: Symmetry plane of a homogeneous monoclinic medium - p. 276 66. 6.1.2: Complex stiffnesses - p. 278 67. 6.1.3: Reflection and transmission coefficients - p. 280 68. 6.1.4: Propagation, attenuation and energy directions - p. 282 69. 6.1.5: Brewster and critical angles - p. 289 70. 6.1.6: Phase velocities and attenuations - p. 294 71. 6.1.7: Energy-flux balance - p. 296 72. 6.1.8: Energy velocities and quality factors - p. 297 73. 6.1.9: The physical solution in ray-tracing algorithms - p. 300 74. 6.2: Reflection and transmission of qP-qSV waves - p. 302 75. 6.3: Interfaces separating a solid and a fluid - p. 319 76. 6.4: Scattering coefficients of a set of layers - p. 323 77. Chapter 7 Biot theory for porous media - p. 329 78. 7.1: Isotropic media. Stress-strain relations - p. 331 79. 7.2: Gassmann equation and effective stress - p. 334 80. 7.3: Pore-pressure build-up in source rocks - p. 346 81. 7.4: The asperity-deformation model - p. 348 82. 7.5: Anisotropic media. Stress-strain relations - p. 353 83. 7.6: Kinetic energy - p. 361 84. 7.7: Dissipation potential - p. 367 85. 7.8: Lagrange equations and equation of motion - p. 369 86. 7.9: Plane-wave analysis - p. 378 87. 7.10: Strain energy for inhomogeneous porosity - p. 387 88. 7.11: Boundary conditions - p. 393 89. 7.12: Reflection and transmission coefficients - p. 400 90. 7.13: Extension of Biot theory to a composite frame - p. 403 91. 7.14: Extension of Biot theory to partial saturation - p. 413 92. 7.15: Squirt-flow dissipation - p. 416 93. 7.16: The mesoscopic-loss mechanism. White model - p. 419 94. 7.17: The Biot-Gardner effect - p. 427 95. 7.18: Green function for poro-viscoacoustic media - p. 430 96. 7.19: Green function for a fluid/solid interface - p. 436 97. 7.20: Poro-viscoelasticity - p. 440 98. 7.21: Bounds and averages on velocity and quality factor - p. 444 99. 7.22: Anelasticity models based on ellipsoidal pores - p. 449 100. 7.23: Effect of melt on wave propagation - p. 452 101. 7.24: Anisotropy and poro-viscoelasticity - p. 458 102. 7.25: Gassmann equation for a solid pore infill - p. 473 103. 7.26: Mesoscopic loss in layered and fractured media - p. 476 104. 7.27: Fluid-pressure diffusion - p. 480 105. 7.28: Thermo-poroelasticity - p. 499 106. Chapter 8 The acoustic-electromagnetic analogy - p. 513 107. 8.1: Maxwell equations - p. 516 108. 8.2: The acoustic-electromagnetic analogy - p. 517 109. 8.3: A viscoelastic form of the electromagnetic energy - p. 525 110. 8.4: The analogy for reflection and transmission - p. 537 111. 8.5: The layer problem - p. 554 112. 8.6: 3-D electromagnetic theory and the analogy - p. 567 113. 8.7: Plane-wave theory - p. 571 114. 8.8: Electromagnetic diffusion in anisotropic media - p. 579 115. 8.9: Analytical solution for anisotropic media - p. 584 116. 8.10: Elastic medium with Fresnel wave surface - p. 588 117. 8.11: Finely layered media - p. 591 118. 8.12: The time-average and CRIM equations - p. 594 119. 8.13: Kramers-Kronig relations - p. 595 120. 8.14: The reciprocity principle - p. 601 121. 8.15: Babinet principle - p. 602 122. 8.16: Alford rotation - p. 603 123. 8.17: Cross-property relations - p. 606 124. 8.18: Poroacoustic and electromagnetic diffusion - p. 607 125. 8.19: Electro-seismic wave theory - p. 612 126. 8.20: Gravitational waves - p. 614 127. Chapter 9 Numerical methods - p. 617 128. 9.1: Equation of motion - p. 617 129. 9.2: Time integration - p. 619 130. 9.3: Calculation of spatial derivatives - p. 626 131. 9.4: Source implementation - p. 632 132. 9.5: Boundary conditions - p. 634 133. 9.6: Absorbing boundaries - p. 635 134. 9.7: Model and modeling design. Seismic modeling - p. 637 135. 9.8: Concluding remarks - p. 640 136. 9.9: Appendix - p. 641 137. 9.9.1: Fractional calculus - p. 641 138. 9.9.2: Electromagnetic-diffusion code - p. 653 139. 9.9.3: Finite-differences code for SH waves - p. 658 140. 9.9.4: Finite-difference code for SH (TM) waves - p. 663 141. 9.9.5: Pseudospectral Fourier method - p. 671 142. 9.9.6: Pseudospectral Chebyshev method - p. 675 143. 9.9.7: Pseudospectral sine/cosine method - p. 678 144. 9.9.8: Earthquake sources. The moment-tensor - p. 679 145. 9.9.9: 3-D anisotropic media. Free-surface boundary treatment - p. 682 146. 9.9.10: Modeling in cylindrical coordinates - p. 685 147. Examinations - p. 689 148. Chronology of main discoveries - p. 695 149. Leonardo's manuscripts - p. 711 150. A list of scientists - p. 717 151. Bibliography - p. 729 152. Name index - p. 763 153. Subject index - p. 781
