New Directions in Linear Acoustics and Vibration: Quantum Chaos, Random Matrix Theory and Complexity

Title......Page 5
Copyright......Page 6
Contents......Page 7
Foreword......Page 9
Reference......Page 11
Introduction......Page 13
1.1 Introduction......Page 17
1.2.1 Modes in a Rectangular Enclosure......Page 18
1.2.2 The Length Spectrum of a Circle......Page 21
1.3 The Quantum–Acoustic Analogy......Page 22
1.3.1 The Semiclassical Limit......Page 23
1.3.2 How to Read the Quantum Literature......Page 24
1.4 The Semiclassical Trace Formula......Page 25
1.5 The Nature of the Approximation......Page 27
1.6.1 Billiard Dynamics......Page 28
1.6.2 Stability Matrix......Page 29
1.6.4 Families of Orbits......Page 30
1.6.5 Finding Periodic Orbits......Page 31
1.7.1 Acoustics......Page 32
1.7.2 Plates......Page 34
2.1 Introduction......Page 36
2.2.1 Ray Chaos in Cavities – Periodic Orbits......Page 37
2.2.2 The Semiclassical Approach for Chaotic Systems – The Trace Formula......Page 39
2.2.3 Speckle-Like and Scarred Wavefunctions......Page 42
2.3.1.1 Form Factor......Page 45
2.3.1.2 Length Spectrum......Page 48
2.3.2 From Ballistic to Scattering Systems: Point-Like Scatterers and Diffractive Orbits......Page 49
2.4 Conclusion......Page 53
3.1 Introduction......Page 54
3.2 The Gaussian Orthogonal Ensemble......Page 56
3.2.1 Other Ensembles......Page 58
3.2.2 Eigenvalue Statistics......Page 59
3.3 Numerical Examples......Page 62
3.4 Relevance for Acoustics......Page 63
3.4.1 Applications......Page 68
3.5 Summary......Page 70
4.1 Introduction......Page 71
4.2 Definition and Properties of Gaussian Random Functions......Page 74
4.3 Gaussian Random Waves from Scattered Speckle Patterns......Page 79
4.4 The Gaussian Random Wave Hypothesis for Modes......Page 81
4.5 Ergodic Wavefunctions and Sabine’s Formula for Reverberation Time......Page 85
5.1 Introduction......Page 89
5.2 Description of Trajectories in the Vicinity of a Periodic Orbit......Page 90
5.3 Short-Wavelength Construction of Resonances of Periodic Orbits......Page 94
5.4 Resonances with Minimum Dispersion: The Scar Function......Page 98
5.5 Eigenfunctions Calculation......Page 103
5.7 Appendix......Page 106
6.1 Introduction......Page 108
6.2 Closed Systems......Page 109
6.3 Open Systems: Scattering......Page 112
6.4 Open Systems: Resonances......Page 116
6.5 Summary......Page 121
7.1 Introduction......Page 122
7.2.1 Periodic Orbits: Classical Quantities......Page 123
7.2.2.1 Interpretations and Properties of the Transfer Operator......Page 124
7.2.3 Trace Formula from the Transfer Operator......Page 127
7.3.2 Ray Dynamics......Page 130
7.3.4 Trace Formula......Page 131
7.4.1 Smooth Term: Results......Page 133
7.5 Discussion......Page 134
8 Mesoscopics in Acoustics......Page 135
8.1 Enhanced Backscatter......Page 136
8.2 Localization......Page 138
8.3 Power-Law Dissipation......Page 140
8.4 Other Mesoscopic Phenomena......Page 141
8.6 Summary......Page 142
9.1 Introduction......Page 143
9.2 Wave Equations with Spatially Varying Coefficients......Page 144
9.3 Statistics of the Response......Page 146
9.3.1 Mean Response......Page 147
9.3.2 Mean Response within the First-Order Smoothing Approximation......Page 149
9.3.3 Covariance of the Response......Page 150
9.4 Scattering from Discrete Scatterers......Page 152
9.5 Summary......Page 157
10.1 Introduction......Page 158
10.2 Time Reversal of Acoustic Waves: Basic Principles......Page 159
10.3 Time-Reversal Mirrors......Page 160
10.3.1 An Ideal Time-Reversal Experiment......Page 162
10.4 Time-Reversal Mirror in Complex Media......Page 164
10.4.1 Time Reversal in an Acoustic Waveguide......Page 165
10.4.1.1 Time Reversal: Matched Filter or Inverse Filter......Page 168
10.4.2 Time Reversal in Chaotic Cavities......Page 171
10.4.2.1 Phase Conjugation and the Effect of Bandwidth......Page 174
10.4.3 Time Reversal in Open Systems: Random Medium......Page 175
10.5 Focusing below the Diffraction Limit......Page 179
10.6 Conclusion......Page 180
11.1 Introduction......Page 181
11.2 Waves and Rays......Page 182
11.2.1 The Paraxial Optical Approximation......Page 184
11.2.2 Eikonal Approximations......Page 185
11.3.1 Rays and Action-Angle Variables......Page 187
11.3.2 Modes and Action Quantization......Page 189
11.4.1 Stability Analysis in Range-Dependent Systems: Numerical Approach......Page 190
11.4.2 Chaos: Resonances, KAM Theory, and Extreme Sensitivity......Page 193
11.4.3 Stochastic Approximations: Action Diffusion......Page 195
11.5 Discussion......Page 198
12.1 Introduction: Seismic Waves and Data......Page 200
12.2 Equipartition of Seismic Waves......Page 203
12.3 Weak Localization......Page 207
12.4 Field Correlations of Seismic Waves and Green’s Function......Page 211
12.5 Tomography and Temporal Changes from Seismic Noise......Page 213
12.6 Conclusion......Page 217
13 Random Matrices in Structural Acoustics......Page 218
13.1.1 Mean Boundary Value Problem......Page 222
13.1.3 Reduced Mean Computational structural–acoustic Model......Page 224
13.2 Parametric Probabilistic Approach of the System Parameter Uncertainties......Page 225
13.3.1 Concept of the Nonparametric Probabilistic Approach......Page 227
13.3.2 Nonparametric Stochastic Reduced Model......Page 228
13.3.3.2 Positive and Semi-Positive Definite Random Matrices......Page 229
13.4 Random Matrix Theory......Page 230
13.4.2 Why the Gaussian Orthogonal Ensemble Cannot Be Used in the Low-and Medium-Frequency Ranges......Page 231
13.4.3.1 Definition of the Ensemble SG+......Page 232
13.4.3.2 Effective Construction by Using the Maximum Entropy Principle......Page 233
13.4.3.5 Invariance of Ensemble SG+ under Real Orthogonal Transformations......Page 234
13.4.3.7 Invertibility and Convergence Property When Dimension Goes to Infinity......Page 235
13.4.3.8 Probability Density Functions of the Random Eigenvalues......Page 236
13.4.4.1 Dispersion Parameters Controlling the Level of Uncertainties......Page 237
13.4.4.3 Confidence Regions of the Random Responses......Page 238
13.6 Experimental Validation......Page 239
13.7 Bibliographical Comments......Page 240
14.1 Introduction......Page 243
14.2 Natural Frequency Statistics......Page 245
14.3 The Response of a Single Component......Page 248
14.4 The Statistics of a Random Dynamic Stiffness Matrix......Page 254
14.5 The Mean Response of a Built-Up System......Page 256
14.6 The Variance of the Response of a Built-Up System......Page 259
14.7 Concluding Remarks......Page 261
References......Page 263
Index......Page 283