Viscoelastic Waves in Layered Media Book

Preface xi

1 One-Dimensional Viscoelasticity 1

1.1 Constitutive Law 2

1.2 Stored and Dissipated Energy 5

1.3 Physical Models 7

1.4 Equation of Motion 15

1.5 Problems 17

2 Three-Dimensional Viscoelasticity 19

2.1 Constitutive Law 19

2.2 Stress-Strain Notation 20

2.3 Equation of Motion 23

2.4 Correspondence Principle 25

2.5 Energy Balance 26

2.6 Problems 30

3 Viscoelastic P, SI, and SII Waves 32

3.1 Solutions of Equation of Motion 32

3.2 Particle Motion for P Waves 37

3.3 Particle Motion for Elliptical and Linear S Waves 40

3.3.1 Type-I or Elliptical S (SI) Wave 42

3.3.2 Type-II or Linear S (SII) Wave 45

3.4 Energy Characteristics of P, SI, and SII Waves 46

3.4.1 Mean Energy Flux (Mean Intensity) 46

3.4.2 Mean Energy Densities 50

3.4.3 Energy Velocity 53

3.4.4 Mean Rate of Energy Dissipation 54

3.4.5 Reciprocal Quality Factor, Q-1 55

3.5 Viscoelasticity Characterized by Parameters for Homogeneous P and S Waves 57

3.6 Characteristics of Inhomogeneous Waves in Terms of Characteristics of Homogeneous Waves 59

3.6.1 Wave Speed and Maximum Attenuation 60

3.6.2 Particle Motion for P and SI Waves 64

3.6.3 Energy Characteristics for P, SI, and SII Waves 67

3.7 P, SI, and SII Waves in Low-Loss Viscoelastic Media 75

3.8 P, SI, and SII Waves in Media with Equal Complex Lamé Parameters 82

3.9 P, SI, and SII Waves in a Standard Linear Solid 84

3.10 Displacement and Volumetric Strain 86

3.10.1 Displacement for General P and SI Waves 86

3.10.2 Volumetric Strain for a General P Wave 92

3.10.3 Simultaneous Measurement of Volumetric Strain and Displacement 93

3.11 Problems 96

4 Framework for Single-BoundaryReflection-Refraction and Surface-Wave Problems 98

4.1 Specification of Boundary 98

4.2 Specification of Waves 99

4.3 Problems 106

5 General P, SI, and SII Waves Incident on a Viscoelastic Boundary 107

5.1 Boundary-Condition Equations for General Waves 107

5.2 Incident General SI Wave 109

5.2.1 Specification of Incident General SI Wave 109

5.2.2 Propagation and Attenuation Vectors; Generalized Snell's Law 111

5.2.3 Amplitude and Phase 114

5.2.4 Conditions for Homogeneity and Inhomogeneity 115

5.2.5 Conditions for Critical Angles 120

5.3 Incident General P Wave 123

5.3.1 Specification of Incident General P Wave 123

5.3.2 Propagation and Attenuation Vectors; Generalized Snell's Law 125

5.3.3 Amplitude and Phase 126

5.3.4 Conditions for Homogeneity and Inhomogeneity 127

5.3.5 Conditions for Critical Angles 129

5.4 Incident General SII Wave 130

5.4.1 Specification of Incident General SII Wave 130

5.4.2 Propagation and Attenuation Vectors; Generalized Snell's Law 131

5.4.3 Amplitude and Phase 133

5.4.4 Conditions for Homogeneity and Inhomogeneity 134

5.4.5 Conditions for Critical Angles 134

5.4.6 Energy Flux and Energy Flow Due to Wave Field Interactions 135

5.5 Problems 141

6 Numerical Models for General Waves Reflected and Refracted at Viscoelastic Boundaries 143

6.1 General SII Wave Incident on a Moderate-Loss Viscoelastic Boundary (Sediments) 144

6.1.1 Incident Homogeneous SII Wave 145

6.1.2 Incident Inhomogeneous SII Wave 151

6.2 P Wave Incident on a Low-Loss Viscoelastic Boundary (Water, Stainless-Steel) 155

6.2.1 Reflected and Refracted Waves 156

6.2.2 Experimental Evidence in Confirmation of Theory of Viscoelastic Waves 163

6.2.3 Viscoelastic Reflection Coefficients for Ocean, Solid-Earth Boundary 165

6.3 Problems 169

7 General SI, P, and SII Waves Incident on a Viscoelastic Free Surface 170

7.1 Boundary-Condition Equations 170

7.2 Incident General SI Wave 172

7.2.1 Reflected General P and SI Waves 172

7.2.2 Displacement and Volumetric Strain 176

7.2.3 Numerical Model for Low-Loss Media (Weathered Granite) 181

7.3 Incident General P Wave 192

7.3.1 Reflected General P and SI Waves 192

7.3.2 Numerical Model for Low-Loss Media (Pierre Shale) 196

7.4 Incident General SII Wave 203

7.5 Problems 204

8 Rayleigh-Type Surface Wave on a Viscoelastic Half Space 206

8.1 Analytic Solution 206

8.2 Physical Characteristics 210

8.2.1 Velocity and Absorption Coefficient 210

8.2.2 Propagation and Attenuation Vectors for Component Solutions 211

8.2.3 Displacement and Particle Motion 212

8.2.4 Volumetric Strain 217

8.2.5 Media with Equal Complex Lamé Parameters (A = M) 219

8.3 Numerical Characteristics of Rayleigh-Type Surface Waves 225

8.3.1 Characteristics at the Free Surface 227

8.3.2 Characteristics Versus Depth 232

8.4 Problems 241

9 General SII Waves Incident on Multiple Layers of Viscoelastic Media 246

9.1 Analytic Solution (Multiple Layers) 247

9.2 Analytic Solution (One Layer) 254

9.3 Numerical Response of Viscoelastic Layers (Elastic, Earth's Crust, Rock, Soil) 255

9.4 Problems 261

10 Love-Type Surface Waves in Multilayered Viscoelastic Media 262

10.1 Analytic Solution (Multiple Layers) 262

10.2 Displacement (Multiple Layers) 265

10.3 Analytic Solution and Displacement (One Layer) 267

10.4 Numerical Characteristics of Love-Type Surface Waves 270

10.5 Problems 278

11 Appendices 279

11.1 Appendix 1 - Properties of Riemann-Stieltjes Convolution Integral 279

11.2 Appendix 2 - Vector and Displacement-Potential Identities 279

11.2.1 Vector Identities 279

11.2.2 Displacement-Potential Identities 280

11.3 Appendix 3 - Solution of the Helmholtz Equation 280

11.4 Appendix 4 - Roots of Squared Complex Rayleigh Equation 284

11.5 Appendix 5 - Complex Root for a Rayleigh-Type Surface Wave 286

11.6 Appendix 6 - Particle-Motion Characteristics for a Rayleigh-Type Surface Wave 288

References 292

Additional Reading 295

Index 296