José M. Carcione received the degree 'Licenciado in Ciencias Físicas' from Buenos Aires University in 1978, the degree 'Dottore in Fisica' from Milan University in 1984 and the PhD in Geophysics from Tel-Aviv University in 1987. He was awarded the Alexander von Humboldt scholarship for a position at the Geophysical Institute of Hamburg University, where he stayed from 1987 to 1989. Dr. Carcione received the 2007 Anstey award at the EAGE in London and the 2017 EAGE Conrad Schlumberger award in Paris. He has authored several books and has published more than 360 peer-reviewed articles.
Authored by the internationally renowned Jose M. Carcione, 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 stress-strain relations. The combination of this relation and the equations of momentum conservation lead to the equation of motion. The differential formulation is written in terms of memory variables, and Biot's theory is used to describe wave propagation in porous media. For each rheology, a plane-wave analysis is performed in order to understand the physics of wave propagation. 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. New to this edition: This new edition presents the fundamentals of wave propagation in Anisotropic, Anelastic, Porous Media while also incorporating the latest research from the past 7 years, including that of the author. The author presents all the equations and concepts necessary to understand the physics of wave propagation. These equations form the basis for modeling and inversion of seismic and electromagnetic data. Additionally, demonstrations are given, so the book can be used to teach post-graduate courses. Addition of new and revised content is approximately 30%. - Examines the fundamentals of wave propagation in anisotropic, anelastic and porous media- Presents all equations and concepts necessary to understand the physics of wave propagation, with examples- Emphasizes geophysics, particularly, seismic exploration for hydrocarbon reservoirs, which is essential for exploration and production of oil
Front Cover 1
Wave Fields in Real Media: Wave Propagation in Anisotropic, Anelastic, Porous and Electromagnetic Media 4
Copyright 5
Contents 6
Dedication 13
Preface 14
About the Author 22
Basic Notation 24
Glossary of Main Symbols 26
Chapter 1: Anisotropic Elastic Media 28
1.1 Strain-Energy Density and Stress–Strain Relation 29
1.2 Dynamical Equations 32
1.2.1 Symmetries and Transformation Properties 34
Symmetry Plane of a Monoclinic Medium 36
Transformation of the Stiffness Matrix 38
1.3 Kelvin–Christoffel Equation, Phase Velocity and Slowness 39
1.3.1 Transversely Isotropic Media 41
1.3.2 Symmetry Planes of an Orthorhombic Medium 43
1.3.3 Orthogonality of Polarizations 45
1.4 Energy Balance and Energy Velocity 45
1.4.1 Group Velocity 48
1.4.2 Equivalence Between the Group and Energy Velocities 50
1.4.3 Envelope Velocity 51
1.4.4 Example: Transversely Isotropic Media 52
1.4.5 Elasticity Constants from Phase and Group Velocities 53
1.4.6 Relationship Between the Slowness and Wave Surfaces 56
SH-Wave Propagation 56
1.5 Finely Layered Media 57
1.5.1 The Schoenberg–Muir Averaging Theory 62
Examples 63
1.6 Anomalous Polarizations 66
1.6.1 Conditions for the Existence of Anomalous Polarization 67
1.6.2 Stability Constraints 69
1.6.3 Anomalous Polarization in Orthorhombic Media 71
1.6.4 Anomalous Polarization in Monoclinic Media 71
1.6.5 The Polarization 72
1.6.6 Example 73
1.7 The Best Isotropic Approximation 76
1.8 Analytical Solutions 79
1.8.1 2D Green Function 79
1.8.2 3D Green Function 80
1.9 Reflection and Transmission of Plane Waves 81
1.9.1 Cross-Plane Shear Waves 84
Chapter 2: Viscoelasticity and Wave Propagation 90
2.1 Energy Densities and Stress–Strain Relations 92
2.1.1 Fading Memory and Symmetries of the Relaxation Tensor 94
2.2 Stress–Strain Relation for 1D Viscoelastic Media 95
2.2.1 Complex Modulus and Storage and Loss Moduli 95
2.2.2 Energy and Significance of the Storage and Loss Moduli 97
2.2.3 Non-negative Work Requirements and Other Conditions 98
2.2.4 Consequences of Reality and Causality 99
2.2.5 Summary of the Main Properties 101
Relaxation Function 101
Complex Modulus 101
2.3 Wave Propagation in 1D Viscoelastic Media 101
2.3.1 Wave Propagation for Complex Frequencies 106
2.4 Mechanical Models and Wave Propagation 108
2.4.1 Maxwell Model 110
2.4.2 Kelvin–Voigt Model 112
2.4.3 Zener or Standard Linear Solid Model 116
2.4.4 Burgers Model 119
2.4.5 Generalized Zener Model 121
Nearly Constant Q 123
2.4.6 Nearly Constant-Q Model with a ContinuousSpectrum 124
2.5 Constant-Q Model and Wave Equation 126
2.5.1 Phase Velocity and Attenuation Factor 127
2.5.2 Wave Equation in Differential Form: Fractional Derivatives 128
Propagation in Pierre Shale 129
2.6 Equivalence Between Source and Initial Conditions 130
2.7 Hysteresis Cycles and Fatigue 132
2.8 Distributed-Order Fractional Time Derivatives 136
2.8.1 The n Case 137
2.8.2 The Generalized Dirac CombFunction 137
2.9 The Concept ofCentrovelocity 138
2.9.1 1D Green Function and Transient Solution 139
2.9.2 Numerical Evaluation of the Velocities 140
2.9.3 Example 142
2.10 Memory Variables and Equation of Motion 144
2.10.1 Maxwell Model 144
2.10.2 Kelvin–VoigtModel 146
2.10.3 Zener Model 147
2.10.4 Generalized Zener Model 148
Chapter 3: Isotropic Anelastic Media 150
3.1 Stress–Strain Relation 151
3.2 Equations of Motion and Dispersion Relations 152
3.3 Vector Plane Waves 155
3.3.1 Slowness, Phase Velocity and Attenuation Factor 155
3.3.2 Particle Motion of the P-Wave 157
3.3.3 Particle Motion of the S-Waves 159
3.3.4 Polarization and Orthogonality 161
3.4 Energy Balance, Velocity and Quality Factor 162
3.4.1 P-Wave 164
3.4.2 S-Waves 170
3.5 Boundary Conditions and Snell Law 171
3.6 The Correspondence Principle 172
3.7 Rayleigh Waves 173
3.7.1 Dispersion Relation 174
3.7.2 Displacement Field 175
3.7.3 Phase Velocity and Attenuation Factor 176
3.7.4 Special Viscoelastic Solids 177
Incompressible Solid 177
Poisson Solid 177
Hardtwig Solid 177
3.7.5 Two Rayleigh Waves 177
3.8 Reflection and Transmission of SH Waves 178
3.9 Memory Variables and Equation of Motion 182
3.10 Analytical Solutions 184
3.10.1 Viscoacoustic Media 184
3.10.2 Constant-Q Viscoacoustic Media 185
3.10.3 Viscoelastic Media 186
3.10.4 Pekeris Solution for Lamb Problem 188
3.11 Constant-Q P- and S-Waves 189
3.11.1 Time Fractional Derivatives 189
3.11.2 Spatial Fractional Derivatives 191
3.12 Wave Equations Based on the Burgers Model 191
3.12.1 Propagation of P–SV Waves 192
3.12.2 Propagation of SH Waves 193
3.13 The Elastodynamic of a Non-Ideal Interface 193
3.13.1 The Interface Model 194
Boundary Conditions in Differential Form 195
3.13.2 Reflection and Transmission Coefficients of SH Waves 196
Energy Loss 196
3.13.3 Reflection and Transmission Coefficients of P–SV Waves 198
Energy Loss 199
Examples 200
Chapter 4: Anisotropic Anelastic Media 204
4.1 Stress–Strain Relations 206
4.1.1 Model 1: Effective Anisotropy 208
4.1.2 Model 2: Attenuation via Eigenstrains 209
4.1.3 Model 3: Attenuation via Mean and Deviatoric Stresses 211
4.2 Fracture-Induced Anisotropic Attenuation 212
4.2.1 The Equivalent Monoclinic Medium 214
4.2.2 The Orthorhombic Equivalent Medium 215
4.2.3 HTI Equivalent Media 217
4.3 Stiffness Tensorfrom OscillatoryExperiments 219
4.4 Wave Velocities, Slowness and Attenuation Vector 222
4.5 Energy Balance and Fundamental Relations 224
4.5.1 Plane Waves: Energy Velocity and Quality Factor 226
4.5.2 Polarizations 231
4.6 Propagation of SH Waves 232
4.6.1 Energy Velocity 232
4.6.2 Group Velocity 234
4.6.3 Envelope Velocity 235
4.6.4 Perpendicularity Properties 235
4.6.5 Numerical Evaluation of the Energy Velocity 237
4.6.6 Forbidden Directions of Propagation 239
4.7 Wave Propagation in Symmetry Planes 241
4.7.1 Properties of the Homogeneous Wave 243
4.7.2 Propagation, Attenuation and Energy Directions 243
4.7.3 Phase Velocities and Attenuations 244
4.7.4 Energy Balance, Velocity and Quality Factor 244
4.7.5 Explicit Equations in Symmetry Planes 245
4.8 Memory Variables and Equation of Motion 248
4.8.1 Strain Memory Variables 249
4.8.2 Memory-Variable Equations 251
4.8.3 SH Equation of Motion 253
4.8.4 qP–qSV Equation of Motion 253
4.9 Analytical Solution for SH Waves 255
Chapter 5: The Reciprocity Principle 258
5.1 Sources, Receivers and Reciprocity 259
5.2 The Reciprocity Principle 260
5.3 Reciprocity of Particle Velocity: Monopoles 261
5.4 Reciprocity of Strain 262
5.4.1 Single Couples 263
Single Couples Without Moment 264
Single Couples with Moment 264
5.4.2 Double Couples 265
Double Couple Without Moment: Dilatation 265
Double Couple Without Moment and Monopole Force 265
Double Couple Without Moment and SingleCouple 267
5.5 Reciprocity of Stress 267
5.6 Reciprocity Principle for Flexural Waves 269
5.6.1 Equation of Motion 270
5.6.2 Reciprocity of the Deflection 271
5.6.3 Reciprocity of the Bending Moment 272
Chapter 6: Reflection and Transmission of Plane Waves 274
6.1 Reflection and Transmission of SH Waves 275
6.1.1 Symmetry Plane of a Homogeneous Monoclinic Medium 276
6.1.2 Complex Stiffnesses 278
6.1.3 Reflection and TransmissionCoefficients 279
6.1.4 Propagation, Attenuation and Energy Directions 282
6.1.5 Brewster and Critical Angles 289
6.1.6 Phase Velocities and Attenuations 293
6.1.7 Energy-Flux Balance 295
6.1.8 Energy Velocities and Quality Factors 297
6.2 Reflection and Transmission of qP–qSV Waves 299
6.2.1 Phase Velocities and Attenuations 301
6.2.2 Energy-Flow Balance 302
6.2.3 Reflection of Seismic Waves 304
6.2.4 Incident Inhomogeneous Waves 312
Generation of Inhomogeneous Waves 315
Ocean Bottom 316
6.3 Interfaces Separating a Solid and a Fluid 318
6.3.1 Solid/Fluid Interface 318
6.3.2 Fluid/Solid Interface 319
6.3.3 The Rayleigh Window 320
6.4 Scattering Coefficients of a Set of Layers 322
Chapter 7: Biot Theory for Porous Media 326
7.1 Isotropic Media – Stress–Strain Relations 329
7.1.1 Jacketed Compressibility Test 330
7.1.2 Unjacketed Compressibility Test 331
7.2 The Concept of Effective Stress 332
7.2.1 Effective Stress in Seismic Exploration 335
Pore-Volume Balance 337
Acoustic Properties 339
7.2.2 Analysis in Terms of Compressibilities 340
7.3 Pore-Pressure Buildup in Source Rocks 344
7.4 The Asperity-Deformation Model 346
7.5 Anisotropic Media – Stress–Strain Relations 350
7.5.1 Effective-Stress Law for Anisotropic Media 355
7.5.2 Summary of Equations 357
7.5.3 Brown and Korringa Equations 358
Transversely Isotropic Medium 358
7.6 Kinetic Energy 359
7.6.1 Anisotropic Media 362
7.7 Dissipation Potential 364
7.7.1 Anisotropic Media 365
7.8 Lagrange Equations and Equation of Motion 366
7.8.1 The Viscodynamic Operator 367
7.8.2 Fluid Flow in aPlane Slit 368
7.8.3 Anisotropic Media 374
7.9 Plane-Wave Analysis 375
7.9.1 Compressional Waves 375
Relation With Terzaghi Law and the Second P-Wave 379
The Diffusive Slow Mode 380
7.9.2 The Shear Wave 381
7.10 Strain Energy for Inhomogeneous Porosity 383
7.10.1 Complementary Energy Theorem 383
7.10.2 Volume-Averaging Method 385
7.11 Boundary Conditions 389
7.11.1 Interface Between Two Porous Media 389
Deresiewicz and Skalak's Derivation 389
Gurevich and Schoenberg's Derivation 392
7.11.2 Interface Between a Porous Medium and a ViscoelasticMedium 394
7.11.3 Interface Between a Porous Medium and a Viscoacoustic Medium 394
7.11.4 Free Surface of a Porous Medium 395
7.12 Squirt-Flow Dissipation 395
7.13 The Mesoscopic Loss Mechanism – White Model 398
7.14 Mesoscopic Loss in Layered and Fractured Media 405
7.14.1 Effective Fractured Medium 407
7.15 Green Function for Poro-Viscoacoustic Media 409
7.15.1 Field Equations 409
7.15.2 The Solution 411
7.16 Green Function for a Fluid/Solid Interface 414
7.17 Poro-Viscoelasticity 419
7.18 Fluid-Pressure Diffusion in Anisotropic Media 423
7.18.1 Biot Classical Equation and Fractional-Derivative Version 423
7.18.2 Frequency-Wavenumber Domain Analysis 425
7.19 Anisotropy and Poro-Viscoelasticity 430
7.19.1 Stress–Strain Relations 431
7.19.2 Biot–Euler Equation 432
7.19.3 Time-HarmonicFields 432
7.19.4 Inhomogeneous Plane Waves 435
7.19.5 Homogeneous Plane Waves 438
7.19.6 Wave Propagation in Femoral Bone 440
7.20 Gassmann Equation for a Solid Pore Infill 444
Chapter 8: The Acoustic–Electromagnetic Analogy 448
8.1 Maxwell Equations 452
8.2 The Acoustic–Electromagnetic Analogy 453
8.2.1 Kinematics and Energy Considerations 458
8.3 A Viscoelastic Form of the Electromagnetic Energy 460
8.3.1 Umov–Poynting Theorem for Harmonic Fields 461
8.3.2 Umov–Poynting Theorem for Transient Fields 463
The Debye–Zener Analogy 466
The Cole–Cole Model 471
8.4 The Analogy for Reflection and Transmission 472
8.4.1 Reflection and Refraction Coefficients 473
Propagation, Attenuation and Ray Angles 473
Energy-Flux Balance 474
8.4.2 Application of the Analogy 474
Refraction Index and Fresnel Formulae 474
Brewster (Polarizing) Angle 475
Critical Angle: Total Reflection 477
Reflectivity and Transmissivity 478
Dual Fields 480
Sound Waves 481
8.4.3 The Analogy Between TM and TE Waves 483
Green Analogies 483
8.4.4 Brief HistoricalReview 487
8.5 The Single-Layer Problem 488
8.5.1 TM–SH–TE Analogy 492
8.5.2 Analogy with Quantum Mechanics: Tunnel Effect 492
8.6 3D Electromagnetic Theory and the Analogy 493
8.6.1 The Form of the Tensor Components 494
8.6.2 Electromagnetic Equations in Differential Form 495
8.7 Plane-Wave Theory 497
8.7.1 Slowness, Phase Velocity andAttenuation 500
8.7.2 Energy Velocity and Quality Factor 502
8.8 Electromagnetic Diffusion in Anisotropic Media 506
8.8.1 Differential Equations 506
8.8.2 Dispersion Relation 507
8.8.3 Slowness, Kinematic Velocities, Attenuation and Skin Depth 508
8.8.4 Umov–Poynting Theorem and Energy Velocity 509
8.8.5 Fundamental Relations 510
8.9 Analytical Solution for Anisotropic Media 511
8.9.1 The Solution 513
8.10 Elastic Medium with Fresnel Wave Surface 514
8.10.1 Fresnel Wave Surface 514
8.10.2 Equivalent Elastic Medium 516
8.11 Finely Layered Media 517
8.12 The Time-Average and CRIM Equations 520
8.13 The Kramers–Kronig Dispersion Relations 521
8.14 The ReciprocityPrinciple 523
8.15 Babinet Principle 524
8.16 Alford Rotation 525
8.17 Cross-Property Relations 527
8.18 Poro-Acoustic and Electromagnetic Diffusion 529
8.18.1 Poro-Acoustic Equations 530
8.18.2 Electromagnetic Equations 531
The TM and TEEquations 531
Phase Velocity, Attenuation Factor and Skin Depth 532
Analytical Solutions 533
8.19 Electro-Seismic Wave Theory 534
Chapter 9: Numerical Methods 536
9.1 Equation of Motion 537
9.2 Time Integration 538
9.2.1 Classical Finite Differences 540
9.2.2 Splitting Methods 542
9.2.3 Predictor–Corrector Methods 542
The Runge–Kutta Method 542
Fractional Calculus 543
9.2.4 Spectral Methods 543
9.2.5 Algorithms forFinite-ElementMethods 545
9.3 Calculation of Spatial Derivatives 546
9.3.1 Finite Differences 546
9.3.2 PseudospectralMethods 548
9.3.3 The Finite-Element Method 550
9.4 Source Implementation 552
9.5 Boundary Conditions 553
9.6 Absorbing Boundaries 554
9.7 Model and Modelling Design. Seismic Modelling 556
9.8 Concluding Remarks 559
9.9 Appendix 560
9.9.1 The FractionalDerivative 560
Grünwald–Letnikov and Central-Difference Approximations 561
9.9.2 Electromagnetic-Diffusion Code 562
9.9.3 Finite-differences code for SH waves 568
9.9.4 Finite-Difference Code for SH (TM)Waves 575
9.9.5 Pseudospectral Fourier Method 584
Calculation of Fractional Derivatives 587
9.9.6 Pseudospectral Chebyshev Method 587
9.9.7 Pseudospectral Sine/cosine Method 590
9.9.8 Earthquake Sources. The Moment–Tensor 592
9.9.9 3D Anisotropic Media. Free-Surface Boundary Treatment 595
9.9.10 Modelling in Cylindrical Coordinates 598
Equations for Axis-Symmetric Single-Phase Media 598
Equations for Poroelastic Media 599
Examinations 602
Chronology of MainDiscoveries 606
Leonardo's Manuscripts 618
A List of Scientists 624
Bibliography 634
Name Index 664
Subject Index 678
Preface
This book presents the fundamentals of wave propagation in anisotropic, anelastic and porous media, including electromagnetic waves. This new edition incorporates research work performed during the last seven years on several relevant topics, which have been distributed in the various chapters. The emphasis is on geophysical applications for hydrocarbon exploration, but researchers in the fields of earthquake seismology, rock acoustics and material science – including many branches of acoustics of fluids and solids (acoustics of materials, nondestructive testing, etc.) – may also find this text useful. This book can be considered, in part, a monograph, since much of the material represents my own original work on wave propagation in anisotropic, viscoelastic media. Although it is biased to my scientific interests and applications, I have, nevertheless, sought to retain the generality of the subject matter, in the hope that the book will be of interest and use to a wide readership.
The concepts of anisotropy, anelasticity1 and poroelasticity in physical media have gained much attention in recent years. The applications of these studies cover a variety of fields, including physics and geophysics, engineering and soil mechanics, underwater acoustics, etc. In particular, in the exploration of oil and gas reservoirs, it is important to predict the rock porosity, the presence of fluids (type and saturation), the preferential directions of fluid flow (anisotropy), the presence of abnormal pore-pressures (overpressure), etc. These microstructural properties and in situ rock conditions can be obtained, in principle, from seismic and electromagnetic properties, such as travel times, amplitude information and wave polarization. These measurable quantities are affected by the presence of anisotropy and attenuation mechanisms. For instance, shales are naturally bedded and possess intrinsic anisotropy at the microscopic level. Similarly, compaction and the presence of microcracks and fractures make the skeleton of porous rocks anisotropic. The presence of fluids implies relaxation phenomena, which causes wave dissipation. The use of modelling and inversion for the interpretation of the wave response of reservoir rocks requires an understanding of the relationship between the seismic and electromagnetic properties and the rock characteristics, such as permeability, porosity, tortuosity, fluid viscosity, stiffness, dielectric permittivity, electrical conductivity, etc.
Wave simulation is a theoretical field of research that began nearly four decades ago, in close relationship with the development of computer technology and numerical algorithms for solving differential and integral equations of several variables. In the field of research known as computational physics, algorithms for solving problems using computers are important tools that provide insight into wave propagation for a variety of applications.
In this book, I examine the differences between an ideal and a real description of wave propagation, where ideal means an elastic (lossless), isotropic and single-phase medium, and real means an anelastic, anisotropic and multi-phase medium. The first realization is, of course, a particular case of the second, but it must be noted that, in general, the real description is not a simple and straightforward extension of the ideal description.
The analysis starts by introducing the constitutive equation (strain–stress relation) appropriate for the particular rheology.2 This relation and the equations of conservation of linear momentum are combined to give the equation of motion, a second-order or a first-order matrix differential equation in time, depending on the formulation of the field variables. The differential formulation for lossy media is written in terms of memory (hidden) variables or alternatively, fractional derivatives. Biot theory is essential to describe wave propagation in multi-phase (porous) media from the seismic to the ultrasonic frequency range, representative of field and laboratory experiments, respectively. The acoustic–electromagnetic analogy reveals that the different physical phenomena have the same mathematical formulation. For each constitutive equation, a plane-wave analysis is performed in order to understand the physics of the wave propagation (i.e., calculation of phase, group and energy velocities, and quality and attenuation factors). For some cases, it is possible to obtain an analytical solution for transient wave fields in the space-frequency domain, which is then transformed to the time domain by a numerical Fourier transform. The book concludes with a review of the so-called direct numerical methods for solving the equations of motion in the time-space domain. The plane-wave theory and the analytical solutions serve to test the performance (accuracy and limitations) of the modelling codes.
A brief description of the main concepts discussed in this book follows.
Chapter 1: Anisotropic Elastic Media. In anisotropic lossless media, the directions of the wavevector and Umov–Poynting vector (ray or energy-flow vector) do not coincide. This implies that the phase and energy velocities differ. However, some ideal properties prevail: there is no dissipation, the group-velocity vector is equal to the energy-velocity vector, the wavevector is normal to the wave-front surface, the energy-velocity vector is normal to the slowness surface, plane waves are linearly polarized and the polarization of the different wave modes are mutually orthogonal. Methods used to calculate these quantities and provide the equation of motion for inhomogeneous media are shown. I also consider the seismic properties of finely stratified media composed of anisotropic layers, anomalously polarized media and the best isotropic approximation of anisotropic media. Finally, the analysis of a reflection–transmission problem and analytical solutions along the symmetry axis of a transversely isotropic medium are discussed.
Chapter 2: Viscoelasticity and Wave Propagation. Attenuation is introduced in the form of Boltzmann superposition law, which implies a convolutional relation between the stress and strain tensors through the relaxation and creep matrices. The analysis is restricted to the one-dimensional case, where some of the consequences of anelasticity become evident. Although phase and energy velocities are the same, the group velocity loses its physical meaning. The concept of centrovelocity for non-harmonic waves is discussed. The uncertainty in defining the strain and rate of dissipated-energy densities is overcome by introducing relaxation functions based on mechanical models. The concepts of memory variable and fractional derivative are introduced to avoid time convolutions and obtain a time-domain differential formulation of the equation of motion.
Chapter 3: Isotropic Anelastic Media. The space dimension reveals other properties of anelastic (viscoelastic) wave fields. There is a distinct difference between the inhomogeneous waves of lossless media (interface waves) and those of viscoelastic media (body waves). In the former case, the direction of attenuation is normal to the direction of propagation, whereas for inhomogeneous viscoelastic waves, that angle must be less than π/2. Furthermore, for viscoelastic inhomogeneous waves, the energy does not propagate in the direction of the slowness vector and the particle motion is elliptical in general. The phase velocity is less than that of the corresponding homogeneous wave (for which planes of constant phase coincide with planes of constant amplitude); critical angles do not exist in general, and, unlike the case of lossless media, the phase velocity and the attenuation factor of the transmitted waves depend on the angle of incidence. There is one more degree of freedom, since the attenuation vector is playing a role at the same level as the wavenumber vector. Snell law, for instance, implies continuity of the tangential components of both vectors at the interface of discontinuity. For homogeneous plane waves, the energy-velocity vector is equal to the phase-velocity vector. The last part of the chapter analyzes the viscoelastic wave equation expressed in terms of fractional time derivatives, and provides expressions of the reflection and transmission coefficients corresponding to a partially welded interface.
Chapter 4: Anisotropic Anelastic Media. In isotropic media there are two well-defined relaxation functions, describing purely dilatational and shear deformations of the medium. The problem in anisotropic media is to obtain the time dependence of the relaxation components with a relatively reduced number of parameters. Fine layering has an “exact” description in the long-wavelength limit. The concept of eigenstrain allows us to reduce the number of relaxation functions to six; an alternative is to use four or two relaxation functions when the anisotropy is relatively weak. Fracture-induced anisotropic attenuation is studied, and harmonic quasi-static numerical experiments are designed to obtain the stiffness components of anisotropic anelastic media. The analysis of SH waves suffices to show that in anisotropic viscoelastic media, unlike the lossless case: the group-velocity vector is not equal to the energy-velocity vector, the wavevector is not normal to the energy-velocity surface, the energy-velocity vector is not normal to the slowness surface, etc. However, an energy analysis shows that some basic fundamental relations still hold: for instance, the projection of the energy velocity onto the propagation direction is...
| Erscheint lt. Verlag | 8.12.2014 |
|---|---|
| Sprache | englisch |
| Themenwelt | Naturwissenschaften ► Geowissenschaften ► Geologie |
| Naturwissenschaften ► Geowissenschaften ► Geophysik | |
| Naturwissenschaften ► Physik / Astronomie ► Elektrodynamik | |
| Naturwissenschaften ► Physik / Astronomie ► Mechanik | |
| Technik | |
| ISBN-10 | 0-08-100003-0 / 0081000030 |
| ISBN-13 | 978-0-08-100003-8 / 9780081000038 |
| Haben Sie eine Frage zum Produkt? |
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