Nonlinear Acoustic Waves in Micro-inhomogeneous Solids

Veniamin E. Nazarov from the Institute of Applied Physics, Russian Academy of Science, began resonance experiments in solids with microstructure as early as in the mid-1980s when the concept of contact nonlinearity in acoustic was completely new. Andrey V. Radostin is a senior scientific associate at the Institute of Applied Physics, Russian Academy of Science.

Preface ix

I. Introduction 1

I.1 Nonlinearity of Gases and Liquids 2

I.2 Nonlinearity of Homogeneous Solids 2

I.3 Micro-inhomogeneous Solids. General Considerations 6

References 9

1. Nonlinear Wave Processes in Homogeneous Media 11

1.1 A Progressive Approximation Approach: The Second Harmonic Generation in an Unbounded Medium 14

1.2 Generation of the Wave at a Difference Frequency: Parametric Array, Degenerate Interaction of Waves 15

1.3 Generation of the Wave at Second Harmonic Frequency in Resonators 17

1.4 Simple Waves and Formation of Discontinuities 19

1.5 Exact Solutions of the Burgers Equation 25

1.6 Nonlinear Wave Processes in Relaxing Media 27

1.7 Spherical and Cylindrical Waves 32

References 34

2. Physical Models and Mechanisms of the Structure Nonlinearity of Micro-inhomogeneous Media with Cracks and Cavities 37

2.1 Contact Mechanism of Nonlinearity for Cracks with Rough and Smooth Surfaces 38

2.2 Capillary and Viscous Mechanisms of Nonlinearity for Cracks Partially Filled with Liquid 43

2.2.1 The Equilibrium State of a Crack Partially Filled with Liquid 43

2.2.2 The Equation of State for a Crack Partially Filled with a Perfect Liquid 45

2.2.3 The Equation of State for a Crack Partially Filled with a Viscous Liquid 46

2.3 Acoustic Nonlinearity of Porous Water-like Materials with Cylindrical Cavities Partially Filled with Viscous Liquid 50

2.4 An Adhesion Mechanism of Hysteretic Nonlinearity for Cracks 58

References 63

3. ElasticWaves in Media with Strong Acoustic Nonlinearity 65

3.1 Nonlinear Propagation and Interaction of Acoustic Waves in Media with Cracks Possessing Adhesion 65

3.2 Acoustic Waves in Media with Bimodular Nonlinearity 68

3.3 Acoustic Waves in Media with Bimodular Nonlinearity and Linear Dissipation 70

3.4 Doubling of Acoustic Wave Frequency by a Nonlinear Layer 76

3.5 Propagation of High-Frequency Pulses in Media with Dissipative Nonlinearity 79

3.6 Interaction of Counter-Propagating Acoustic Waves in Media with Dissipative Nonlinearity 80

References 81

4. Wave Processes in Micro-inhomogeneous Solids with Hysteretic Nonlinearity 83

4.1 Hysteretic Equations of State for Solids: Phenomenological Models 84

4.2 Nonlinear Waves in an Unbounded Medium 88

4.2.1 Propagation of Unipolar Acoustic Pulses 95

4.2.2 Self-Demodulation of High-Frequency Acoustic Pulses 96

4.2.3 Nonlinear Travelling Waves in a Ring Resonator 97

4.3 Vibrations of Rods Composed of Materials with Hysteretic Nonlinearity and Linear Dissipation 99

4.4 Propagation of Unipolar Strain Pulses in Media with Hysteretic Nonlinearity of an Arbitrary Power-Law 103

References 107

5. Wave Processes in Nonlinear Micro-inhomogeneous Media with Relaxation 111

5.1 Rheological Model and Dynamic Equation of State of Nonlinear Micro-inhomogeneous Media with Relaxation 112

5.1.1 Second Harmonic Generation 116

5.1.2 Generation of the Difference Frequency Wave 120

5.1.3 Self-Demodulation of Pulses with a High-Frequency Carrier 124

5.2 Evolution of Acoustic Waves in Micro-inhomogeneous Media with Quadratic Elastic Nonlinearity and Relaxation 127

5.2.1 Stationary Symmetric Shock Waves 127

5.2.2 Evolution Equations for Low-Frequency and High-Frequency Acoustic Waves 129

5.2.3 Evolution of Initially Harmonic Waves 130

5.3 Wave Processes in Micro-inhomogeneous Elastic Media with Hysteretic Nonlinearity and Relaxation 132

5.3.1 Equation of State of Micro-inhomogeneous Medium with Hysteretic Nonlinearity and Relaxation 133

5.3.2 Nonlinear Propagation of a Quasi-Harmonic Wave 133

5.3.3 Frequency Dependencies of Nonlinear Coefficients for a Medium with Defects having Different Distribution over Relaxation Frequencies 135

5.4 Simulation of the Dynamic Hystereses in Micro-inhomogeneous Media Characterized by Imperfect Elasticity and Relaxation 138

5.5 Nonlinear Wave Processes in Media Containing Cracks Partially Filled with a Viscous Liquid 142

5.5.1 Equation of State for a Rod Containing Cracks Partially Filled with a Viscous Liquid 142

5.5.2 Nonlinear Propagation and Interaction of Elastic Waves in a Rod with Cracks 144

5.5.3 Analysis of the Wave Processes in a Rod with Identical Cracks 147

5.5.4 Analysis of the Wave Processes in a Rod with Cracks Distributed over the Radii 148

References 150

6. Wave Processes in Polycrystalline Solids with Dissipative and Reactive Nonlinearity Caused by Dislocations 153

6.1 Modification of the Linear Part of the Granato–Lücke Dislocation Theory of Absorption and Equation of State with Resonance Dissipative and Reactive Nonlinearity for Polycrystalline Solids 154

6.2 Attenuation of High-Frequency Pulses and the Phase Delay of its Carrier Frequency under the Action of a Low-Frequency Wave 158

6.3 Amplitude–Phase Self-Action Phenomena in the High-Frequency Wave of Finite Amplitude 160

References 163

7. Experimental Studies of Nonlinear Acoustic Phenomena in Polycrystalline Rocks and Metals 165

7.1 Experimental Setup 166

7.2 Nonlinear Shift in Resonance Frequency in a Glass Rod with Artificial Cracks 166

7.3 Low-Frequency Amplitude-Dependent Internal Friction and High-Frequency Dissipative Nonlinearity of Coarse-Grained Sandstone 167

7.3.1 Phenomena of LF Hysteretic Nonlinearity: Amplitude-Dependent Losses and Shifts of Resonant Frequencies 168

7.3.2 Nonlinear Attenuation of an Ultrasonic Pulse under the Action of a LF Wave 173

7.4 Effect of an Intense Sound Wave on the Acoustic Properties of a Fine-Grained Sandstone Rod Resonator 176

7.5 Nonlinear Acoustic Phenomena in Limestone 180

7.5.1 Phenomena of LF Hysteretic Nonlinearity 180

7.5.2 Manifestation of the HF Dissipative Nonlinearity 186

7.6 Oscillograms of the Free Boundary Nonlinear Oscillations of a Magnesite Resonator 187

7.7 High-Frequency Nonlinear Acoustic Phenomena in Marble 189

7.7.1 Attenuation and Phase Delay of the Carrier Frequency of Weak Ultrasonic Pulses under the Action of a LF Powerful Wave 189

7.7.2 Self-Action of Finite-Amplitude Ultrasonic Pulses 192

7.8 Sound by Sound Damping of Polycrystalline Zinc 195

7.9 Modulation of Sound by Sound in Copper Subjected to Different Degrees of Annealing 199

References 203

8. Experimental Studies of Nonlinear Acoustic Phenomena in Granular Media 205

8.1 Self-Demodulation of Acoustic Pulses in Partially Water-Saturated River Sand 206

8.1.1 Experimental Setup 206

8.1.2 Experimental Results 207

8.2 Self-Action of Acoustic Waves in Systems with Dissipative Nonlinearity 212

8.2.1 Description of the Experiment 213

8.2.2 Analytical Description of the Phenomenon of Self-Brightening 216

8.3 Amplification of Sound by Sound in Systems with Dissipative Nonlinearity 221

8.4 Self-Action of a Low-Frequency Acoustic Wave and Generation of the Second Harmonic in Dry and Water-Saturated River Sand 222

8.4.1 Experimental Scheme and Method of Measurement 222

8.4.2 Measurement Results 223

8.4.3 Analytical Description of the Phenomena of Self-Action and Generation of the Second Harmonic 226

8.5 Amplitude Modulation of Sound by Sound in Water-Saturated River Sand 228

References 231

9. Nonlinear Phenomena in Seismic Waves 233

9.1 Static Deformation of the Earth’s Surface Near the Harmonic Source of Seismic Vibrations 233

9.2 Amplitude Modulation of Sound by Sound in Sandy Soil 237

9.3 Self-Action of the Seismic Wave in Sandy Soil 237

9.4 Amplitude–Phase Modulation of a Seismo–Acoustic Wave under Diurnal Tides 242

References 245

Index 249