Wave Fields in Real Media - Wave Propagation in Anisotropic, Anelastic, Porous and Electromagnetic Media

Front Cover

Wave Fields in Real Media

Copyright Page

Table of Contents

Preface

About the author

Basic notation

Glossary of main symbols

Chapter 1 Anisotropic elastic media

1.1 Strain-energy density and stress-strain relation

1.2 Dynamical equations

1.2.1 Symmetries and transformation properties

Symmetry plane of a monoclinic medium

Transformation of the stiffness matrix

1.3 Kelvin-Christoffel equation, phase velocity and slowness

1.3.1 Transversely isotropic media

1.3.2 Symmetry planes of an orthorhombic medium

1.3.3 Orthogonality of polarizations

1.4 Energy balance and energy velocity

1.4.1 Group velocity

1.4.2 Equivalence between the group and energy velocities

1.4.3 Envelope velocity

1.4.4 Example: Transversely isotropic media

1.4.5 Elasticity constants from phase and group velocities

1.4.6 Relationship between the slowness and wave surfaces

SH-wave propagation

1.5 Finely layered media

1.6 Anomalous polarizations

1.6.1 Conditions for the existence of anomalous polarization

1.6.2 Stability constraints

1.6.3 Anomalous polarization in orthorhombic media

1.6.4 Anomalous polarization in monoclinic media

1.6.5 The polarization

1.6.6 Example

1.7 The best isotropic approximation

1.8 Analytical solutions for transversely isotropic media

1.8.1 2-D Green's function

1.8.2 3-D Green's function

1.9 Reflection and transmission of plane waves

1.9.1 Cross-plane shear waves

Chapter 2 Viscoelasticity and wave propagation

2.1 Energy densities and stress-strain relations

2.1.1 Fading memory and symmetries of the relaxation tensor

2.2 Stress-strain relation for 1-D viscoelastic media

2.2.1 Complex modulus and storage and loss moduli

2.2.2 Energy and significance of the storage and loss moduli

2.2.3 Non-negative work requirements and other conditions

2.2.4 Consequences of reality and causality

2.2.5 Summary of the main properties

Relaxation function

Complex modulus

2.3 Wave propagation concepts for 1-D viscoelastic media

2.3.1 Wave propagation for complex frequencies

2.4 Mechanical models and wave propagation

2.4.1 Maxwell model

2.4.2 Kelvin-Voigt model

2.4.3 Zener or standard linear solid model

2.4.4 Burgers model

2.4.5 Generalized Zener model

Nearly constant Q

2.4.6 Nearly constant-Q model with a continuous spectrum

2.5 Constant-Q model and wave equation

2.5.1 Phase velocity and attenuation factor

2.5.2 Wave equation in differential form. Fractional derivatives.

Propagation in Pierre shale

2.6 The concept of centrovelocity

2.6.1 1-D Green's function and transient solution

2.6.2 Numerical evaluation of the velocities

2.6.3 Example

2.7 Memory variables and equation of motion

2.7.1 Maxwell model

2.7.2 Kelvin-Voigt model

2.7.3 Zener model

2.7.4 Generalized Zener model

Chapter 3 Isotropic anelastic media

3.1 Stress-strain relation

3.2 Equations of motion and dispersion relations

3.3 Vector plane waves

3.3.1 Slowness, phase velocity and attenuation factor

3.3.2 Particle motion of the P wave

3.3.3 Particle motion of the S waves

3.3.4 Polarization and orthogonality

3.4 Energy balance, energy velocity and quality factor

3.4.1 P wave

3.4.2 S waves

3.5 Boundary conditions and Snell's law

3.6 The correspondence principle

3.7 Rayleigh waves

3.7.1 Dispersion relation

3.7.2 Displacement field

3.7.3 Phase velocity and attenuation factor

3.7.4 Special viscoelastic solids

Incompressible solid

Poisson solid

Hardtwig solid

3.7.5 Two Rayleigh waves

3.8 Reflection and transmission of cross-plane shear waves

3.9 Memory variables and equation of motion

3.10 Analytical solutions

3.10.1 Viscoacoustic media

3.10.2 Constant-Q viscoacoustic media

3.10.3 Viscoelastic media

3.11 The elastodynamic of a non-ideal interface

3.11.1 The interface model

Boundary conditions in differential form

3.11.2 Reflection and transmission coefficients of SH waves

Energy loss

3.11.3 Reflection and transmission coefficients of P-SV waves

Energy loss

Examples

Chapter 4 Anisotropic anelastic media

4.1 Stress-strain relations

4.1.1 Model 1: Effective anisotropy

4.1.2 Model 2: Attenuation via eigenstrains

4.1.3 Model 3: Attenuation via mean and deviatoric stresses

4.2 Wave velocities, slowness and attenuation vector

4.3 Energy balance and fundamental relations

4.3.1 Plane waves. Energy velocity and quality factor

4.3.2 Polarizations

4.4 The physics of wave propagation for viscoelastic SH waves

4.4.1 Energy velocity

4.4.2 Group velocity

4.4.3 Envelope velocity

4.4.4 Perpendicularity properties

4.4.5 Numerical evaluation of the energy velocity

4.4.6 Forbidden directions of propagation

4.5 Memory variables and equation of motion in the time domain

4.5.1 Strain memory variables

4.5.2 Memory-variable equations

4.5.3 SH equation of motion

4.5.4 qP-qSV equation of motion

4.6 Analytical solution for SH waves in monoclinic media

Chapter 5 The reciprocity principle

5.1 Sources, receivers and reciprocity

5.2 The reciprocity principle

5.3 Reciprocity of particle velocity. Monopoles

5.4 Reciprocity of strain

5.4.1 Single couples

Single couples without moment

Single couples with moment

5.4.2 Double couples

Double couple without moment. Dilatation.

Double couple without moment and monopole force

Double couple without moment and single couple

5.5 Reciprocity of stress

Chapter 6 Reflection and transmission of plane waves

6.1 Reflection and transmission of SH waves

6.1.1 Symmetry plane of a homogeneous monoclinic medium

6.1.2 Complex stiffnesses of the incidence and transmission media

6.1.3 Reflection and transmission coefficients

6.1.4 Propagation, attenuation and energy directions

6.1.5 Brewster and critical angles

6.1.6 Phase velocities and attenuations

6.1.7 Energy-flux balance

6.1.8 Energy velocities and quality factors

6.2 Reflection and transmission of qP-qSV waves

6.2.1 Propagation characteristics

6.2.2 Properties of the homogeneous wave

6.2.3 Reflection and transmission coefficients

6.2.4 Propagation, attenuation and energy directions

6.2.5 Phase velocities and attenuations

6.2.6 Energy-flow balance

6.2.7 Umov-Poynting theorem, energy velocity and quality factor

6.2.8 Reflection of seismic waves

6.2.9 Incident inhomogeneous waves

Generation of inhomogeneous waves

Ocean bottom

6.3 Reflection and transmission at fluid/solid interfaces

6.3.1 Solid/fluid interface

6.3.2 Fluid/solid interface

6.3.3 The Rayleigh window

6.4 Reflection and transmission coefficients of a set of layers

Chapter 7 Biot's theory for porous media

7.1 Isotropic media. Strain energy and stress-strain relations

7.1.1 Jacketed compressibility test

7.1.2 Unjacketed compressibility test

7.2 The concept of effective stress

7.2.1 Effective stress in seismic exploration

Pore-volume balance

Acoustic properties

7.2.2 Analysis in terms of compressibilities

7.3 Anisotropic media. Strain energy and stress-strain relations

7.3.1 Effective-stress law for anisotropic media

7.3.2 Summary of equations

Pore pressure

Total stress

Effective stress

Skempton relation

Undrained-modulus matrix

7.3.3 Brown and Korringa's equations

Transversely isotropic medium

7.4 Kinetic energy

7.4.1 Anisotropic media

7.5 Dissipation potential

7.5.1 Anisotropic media

7.6 Lagrange's equations and equation of motion

7.6.1 The viscodynamic operator

7.6.2 Fluid flow in a plane slit

7.6.3 Anisotropic media

7.7 Plane-wave analysis

7.7.1 Compressional waves

Relation with Terzaghi's law

The diffusive slow mode

7.7.2 The shear wave

7.8 Strain energy for inhomogeneous porosity

7.8.1 Complementary energy theorem

7.8.2 Volume-averaging method

7.9 Boundary conditions

7.9.1 Interface between two porous media

Deresiewicz and Skalak's derivation

Gurevich and Schoenberg's derivation

7.9.2 Interface between a porous medium and a viscoelastic medium

7.9.3 Interface between a porous medium and a viscoacoustic medium

7.9.4 Free surface of a porous medium

7.10 The mesoscopic loss mechanism. White model

7.11 Green's function for poro-viscoacoustic media

7.11.1 Field equations

7.11.2 The solution

7.12 Green's function at a fluid/porous medium interface

7.13 Poro-viscoelasticity

7.14 Anisotropic poro-viscoelasticity

7.14.1 Stress-strain relations

7.14.2 Biot-Euler's equation

7.14.3 Time-harmonic fields

7.14.4 Inhomogeneous plane waves

7.14.5 Homogeneous plane waves

7.14.6 Wave propagation in femoral bone

Chapter 8 The acoustic-electromagnetic analogy

8.1 Maxwell's equations

8.2 The acoustic-electromagnetic analogy

8.2.1 Kinematics and energy considerations

8.3 A viscoelastic form of the electromagnetic energy

8.3.1 Umov-Poynting's theorem for harmonic fields

8.3.2 Umov-Poynting's theorem for transient fields

The Debye-Zener analogy

The Cole-Cole model

8.4 The analogy for reflection and transmission

8.4.1 Reflection and refraction coefficients

Propagation, attenuation and ray angles

Energy-flux balance

8.4.2 Application of the analogy

Refraction index and Fresnel's formulae

Brewster (polarizing) angle

Critical angle. Total reflection

Reflectivity and transmissivity

Dual fields

Sound waves

8.4.3 The analogy between TM and TE waves

Green's analogies

8.4.4 Brief historical review

8.5 3-D electromagnetic theory and the analogy

8.5.1 The form of the tensor components

8.5.2 Electromagnetic equations in differential form

8.6 Plane-wave theory

8.6.1 Slowness, phase velocity and attenuation

8.6.2 Energy velocity and quality factor

8.7 Analytical solution for anisotropic media

8.7.1 The solution

8.8 Finely layered media

8.9 The time-average and CRIM equations

8.10 The Kramers-Kronig dispersion relations

8.11 The reciprocity principle

8.12 Babinet's principle

8.13 Alford rotation

8.14 Poro-acoustic and electromagnetic diffusion

8.14.1 Poro-acoustic equations

8.14.2 Electromagnetic equations

The TM and TE equations

Phase velocity, attenuation factor and skin depth

Analytical solutions

8.15 Electro-seismic wave theory

Chapter 9 Numerical methods

9.1 Equation of motion

9.2 Time integration

9.2.1 Classical finite differences

9.2.2 Splitting methods

9.2.3 Predictor-corrector methods

The Runge-Kutta method

9.2.4 Spectral methods

9.2.5 Algorithms for finite-element methods

9.3 Calculation of spatial derivatives

9.3.1 Finite differences

9.3.2 Pseudospectral methods

9.3.3 The finite-element method

9.4 Source implementation

9.5 Boundary conditions

9.6 Absorbing boundaries

9.7 Model and modeling design – Seismic modeling

9.8 Concluding remarks

9.9 Appendix

9.9.1 Electromagnetic-diffusion code

9.9.2 Finite-differences code for the SH-wave equation of motion

9.9.3 Finite-differences code for the SH-wave and Maxwell's equations

9.9.4 Pseudospectral Fourier Method

9.9.5 Pseudospectral Chebyshev Method

Examinations

Chronology of main discoveries

Leonardo's manuscripts

A list of scientists

Bibliography

Name index

Subject index