Applied Wave Mathematics (eBook)

Preface 6
Contents 7
List of Contributors 9
CENS, CMA and the CENS-CMA Project 11
CENS 1999-2009 11
CMA 14
The CENS-CMA Project 15
Part I Waves in Solids 17
Overview 18
Deformation Waves in Solids 21
Introduction 21
General ideas 21
Notes from history 23
Description of what follows 23
Basic theory 23
Advanced theories 26
General ideas 26
Separation of macro- and microstructure 27
Balance of pseudomomentum 29
Internal variables 30
Model governing equations 31
Basic linear theory 31
Wave hierarchy 32
Nonlinearities 33
One-wave models 34
Final remarks 36
References 37
The Perturbation Technique for Wave Interaction in Prestressed Material 39
Introduction 39
Prelude 40
Basic relations in continuum mechanics 42
Coordinate systems 43
Conservation laws 43
The constitutive equation 44
Initial and boundary conditions 46
Compatibility conditions 46
Governing equations 47
The perturbation technique 49
The prestressed state 50
Counterpropagating waves 51
Harmonic waves 54
Nondestructive characterization of plane strain 56
Conclusions 60
References 60
Waves in Inhomogeneous Solids 62
Introduction 62
Governing equations 63
The wave-propagation algorithm 64
Averaged quantities 65
Numerical fluxes 65
Second-order corrections 67
The conservative wave propagation algorithm 67
Excess quantities and numerical fluxes 68
Excess quantities at the boundaries between cells 69
One-dimensional waves in periodic media 70
One-dimensional weakly nonlinear waves in periodic media 72
One-dimensional linear waves in laminates 74
Nonlinear elastic waves in laminates under impact loading 76
Comparison with experimental data 79
Waves in functionally graded materials 83
Concluding remarks 85
References 86
Part II Mesoscopic Theory 89
Overview 90
References 93
Dynamics of Internal Variables from the Mesoscopic Background for the Example of Liquid Crystals and Ferrofluids 94
Introduction to liquid crystals 94
Some properties of liquid crystals 94
Mesoscopic theory of complex materials 98
Complex materials 98
Examples of internal structure 99
The mesoscopic concept 102
Mesoscopic balance equations 104
Mesoscopic theory of uniaxial liquid crystals 104
Mesoscopic balance equations 106
Macroscopic balance equations 107
Macroscopic constitutive quantities 108
Order parameters 109
Differential equation for the distribution function and for the alignment tensors 110
Example of a closed differential equation for the second order alignment tensor 111
Landau theory of phase transitions as a special case 114
A remark on constitutive theory and the Second Law of Thermodynamics 116
A set of differential equations for the moments and a second order differential equation for the alignment tensor 117
Application of the mesoscopic theory to dipolar media 121
Orientation distribution function and alignment tensors 121
Exploitation of the balance of spin 122
Equation of motion for the magnetization 123
Summary 126
Summary of the mesoscopic theory 126
References 127
Towards a Description of Twist Waves in Mesoscopic Continuum Physics 131
Introduction 131
Mesoscopic Continuum Physics 133
Generalization of vector fields to the mesoscopic space 133
Mesoscopic balances and a transport theorem 135
Orientation waves 137
Twist waves in classical macroscopic theory 137
Twist waves in mesoscopic theory 140
Comparison 143
Mesoscopic mass density, orientation distribution function and macroscopic director 144
Macroscopic balance equations 144
Conclusions and outlook 147
References 148
Part III Exploiting the Dissipation Inequality 150
Overview 151
References 153
Weakly Nonlocal Non-equilibrium Thermodynamics -- Variational Principles and Second Law 154
Introduction 154
Second law and weakly nonlocal constitutive spaces 156
Thermodynamic evolution of internal variables 157
First order nonlocality -- relaxation 157
Second order nonlocality -- the Ginzburg-Landau equation 159
Dual internal variables -- Hamiltonian structure 161
Classical Irreversible Thermodynamics 169
One component fluids -- second order nonlocal in the density 171
Fluid mechanics in general 171
Schrödinger-Madelung fluid 176
Summary and outlook 178
Appendix -- Farkas's lemma and some of its consequences 180
Affine Farkas's lemma 181
Liu's theorem 181
References 183
Part IV Waves in Fluids 188
Overview 189
Surface waves at the cutting edge of research and applications 189
References 192
Long Ship Waves in Shallow Water Bodies 193
Introduction 193
Linear ship wakes 196
Kelvin wedge 196
Navigational speeds 201
Distribution of wave heights and periods 203
Patterns of wakes from fast ferries 205
Changes in the ship wave pattern 205
Realistic spatial patterns of ship waves 209
Ship waves at a fixed point 211
Contribution of ship wakes to local hydrodynamics 215
Parameters of wind waves in semi-enclosed seas 215
Ship wakes versus wind waves 216
Ship wakes and the coast 220
Excessive near-bottom velocities and impulse loads 222
Conclusions 224
References 225
Modelling of Ship Waves from High-speed Vessels 229
Waves generated by ships 230
Governing equations and results from linear wave theory 230
Steady ship wakes and ship wave parameters 233
Wave making resistance to steady ship motion 236
Ship wake patterns in deep and shallow water 239
Why is wake wash from high-speed vessels a problem? 245
Properties of long wave model equations 246
Approximations for shallow water flow 246
Comparison between different model equations 248
Numerical modelling of ship waves 252
Numerical models based on Boussinesq-type equations 253
Ship representation by a moving pressure disturbance 254
The influence of the dispersive and nonlinear components 256
Concluding Remarks 260
References 260
New Trends in the Analytical Theory of Long Sea Wave Runup 264
Introduction 264
Basic equations and parameters 266
Method of solution: hodograph transformation 268
Linear approximation of nonlinear long wave runup 271
The relation between linear and nonlinear runup properties 275
Runup of solitary waves 280
Runup of periodic waves 288
Conclusion 293
References 294
Part V Mathematical Methods 296
Overview 297
The Pseudospectral Method and Discrete Spectral Analysis 299
Introduction 299
The model equations 301
The pseudospectral method 303
Approximation of space derivatives 303
The discrete Fourier transform 304
The essence of the pseudospectral method 305
The pseudospectral method and different equation types 307
Filtering and other practical tips 310
Discrete spectral analysis 313
Spectral amplitudes and spectral densities 313
Cumulative spectrum and time averaged normalised spectral densities 315
Applications 316
Conclusions 324
References 327
Foundations of Finite Element Methods for Wave Equations of Maxwell Type 332
Introduction 332
Analysis of the finite element method for waves 334
Linear wave equations 334
Convergence theory for linear equations 337
Consistency 342
Eigenvalue approximation 346
Construction of finite element spaces 351
Algebra 351
Differential geometry 361
Finite elements on cellular complexes 372
Conclusion 387
References 388
An Introduction to the Theory of Scalar Conservation Laws with Spatially Discontinuous Flux Functions 391
Introduction 391
The Riemann problem 394
Existence of a solution 395
Vanishing viscosity and smoothing 413
The Cauchy problem 419
A model equation 420
Uniqueness of entropy solutions 447
References 459
Index 461