Elastic Waves in Solids, Volume 1

Preface ix

List of Main Symbols xiii

Chapter 1 Propagation in an Unbounded Solid 1

1.1 Reviewing the mechanics of continuous media 2

1.1.1 Conservation equations 2

1.1.2 Kinematics of continuous media 9

1.1.3 Poynting’s theorem: energy balance 10

1.1.4 Stress–strain relationship: Maxwell relations 12

1.2 Isotropic solid 14

1.2.1 Constitutive equations 14

1.2.2 Equations of propagation, wave decoupling 16

1.2.3 Traveling, plane, sinusoidal waves 21

1.2.4 Polarization 25

1.2.5 Acoustic intensity 26

1.2.6 Cylindrical and spherical waves 27

1.3 Anisotropic solid 32

1.3.1 Symmetry and elasticity tensor 32

1.3.2 Propagation equation, phase velocity, polarization 41

1.3.3 Propagation in an orthotropic material 43

1.3.4 Group velocity and energy velocity 45

1.3.5 Slowness surface and wave surface 48

1.4 Piezoelectric solid 54

1.4.1 Constitutive equations 54

1.4.2 Reduction in the number of independent piezoelectric constants 59

1.4.3 Plane waves in a piezoelectric crystal 61

1.5 Viscoelastic media 70

1.5.1 Constitutive equation of linear viscoelasticity 71

1.5.2 Simple rheological models 72

1.5.3 Velocity and attenuation in a viscoelastic medium 74

1.5.4 Time–temperature superposition principle 77

1.5.5 Newtonian fluid 78

Chapter 2 Reflection and Transmission at an Interface 81

2.1 Boundary conditions 82

2.2 Direction and polarization of reflected and transmitted waves 85

2.2.1 Graphical construction 86

2.2.2 Wave decoupling 87

2.2.3 Critical angle, evanescent wave and total reflection 89

2.2.4 Conservation of energy 91

2.3 Isotropic solid: transverse horizontal wave 93

2.3.1 Reflection and transmission between two solids 93

2.3.2 Plate between two solids, impedance matching 96

2.4 Isotropic media: longitudinal and transverse vertical waves 100

2.4.1 Reflection on a free surface 100

2.4.2 Solid–fluid interface 105

2.5 Anisotropic medium: diffraction matrix 116

2.5.1 Analytical resolution 117

2.5.2 Expression for the stresses 119

2.5.3 Sorting the solutions 120

2.5.4 Considerations of symmetry 121

2.5.5 Reflection and transmission coefficients, interface waves 124

2.5.6 Interface between an orthotropic solid and an isotropic solid 127

Chapter 3 Surface Waves and Interface Waves 131

3.1 Surface waves 132

3.1.1 Isotropic solid: Rayleigh wave 132

3.1.2 Anisotropic solid 141

3.1.3 Piezoelectric crystal 151

3.2 Interface waves 164

3.2.1 Isotropic solid-perfect fluid interface 164

3.2.2 Interface between two isotropic solids 169

3.3 Bleustein–Gulyaev wave 173

Chapter 4 Guided Elastic Waves 179

4.1 Waveguide, group velocity 180

4.1.1 Elementary planar waveguide 181

4.1.2 Velocity of a wave packet 184

4.1.3 Propagation of a Gaussian pulse 187

4.2 Transverse horizontal waves 189

4.2.1 Guided TH modes 190

4.2.2 Love wave 190

4.2.3 Love wave in an inhomogeneous medium 192

4.3 Lamb waves 196

4.3.1 Free isotropic plate 196

4.3.2 Isotropic plate immersed in a fluid 221

4.3.3 Free anisotropic plate 226

4.4 Cylindrical guides 235

4.4.1 Compressional modes 239

4.4.2 Flexural modes 243

4.4.3 Torsional modes 244

4.4.4 Tubular waveguide 246

Appendix 1 Differential Operators in Cylindrical and Spherical Coordinates 247

Appendix 2 Symmetry and Tensors 253

Appendix 3 Transport of Energy 279

References 287

Index 295