1 Introduction ................................................. 1
2 The Equations of Inviscid Compressible Flow .................. 5
2.1 The Field Equations ..................................... 5
2.2 Initial and Boundary Conditions ........................ 13
2.3 Vorticity and Irrotationality .......................... 14
2.3.1 Homentropic Flow ................................ 14
2.3.2 Incompressible Flow ............................. 17
Exercises ................................................... 19
3 Models for Linear Wave Propagation .......................... 23
3.1 Acoustics .............................................. 23
3.2 Surface Gravity Waves in Incompressible Flow ........... 26
3.3 Internal Waves ......................................... 28
3.4 Acoustic Waves in a Stratified Fluid ................... 31
3.5 Waves in Rotating Incompressible Flows ................. 32
3.6 Waves in Dissociating Gases ............................ 33
3.7 Waves in Viscous Incompressible Flow ................... 34
3.8 Other Linear Wave Models in Mechanics and
Electromagnetism ....................................... 36
3.8.1 Electromagnetic Waves ........................... 36
3.8.2 Waves in Plasmas ................................ 37
3.8.3 Elastic Waves ................................... 42
3.8.4 Plastic Waves ................................... 44
Exercises ................................................... 49
4 Theories for Linear Waves ................................... 59
4.1 Wave equations and Hyperbolicity ....................... 59
4.2 Fourier Series, Eigehvalues and Resonance .............. 61
4.3 Fourier Integrals and the Method of Stationary Phase ... 65
4.4 Dispersion and Group Velocity .......................... 70
4.4.1 Dispersion Relations ............................ 71
4.4.2 Stability and Dissipation ....................... 76
4.4.3 Other Approaches to Group Velocity .............. 78
4.5 The Frequency Domain ................................... 80
4.5.1 Homogeneous Media ............................... 80
4.5.2 Scattering Problems in Homogeneous Media ........ 82
4.6 Inhomogeneous Media .................................... 85
4.6.1 The Frequency Domain ............................ 85
4.6.2 The Time Domain ................................. 88
4.7 Stationary Waves ....................................... 90
4.7.1 Stationary Surface Waves on a Running Stream .... 91
4.7.2 Steady Flow in Slender Nozzles .................. 93
4.7.3 Compressible Flow past Thin Wings ............... 94
4.7.4 Compressible Flow past Slender Bodies ........... 99
4.8 *High-frequency Waves ................................. 102
4.8.1 The Eikonal Equation ........................... 102
4.8.2 Ray Theory ..................................... 103
4.8.3 Paraxial Approximations ........................ 107
4.9 Dimensionality and the Wave Equation .................. 108
Exercises .................................................. 113
5 Nonlinear Waves in Fluids .................................. 131
5.1 Introduction .......................................... 131
5.2 Models for Nonlinear Waves ............................ 133
5.2.1 One-dimensional Unsteady Gas Dynamics .......... 133
5.2.2 Two-dimensional Steady Homentropic Gas
Dynamics ....................................... 134
5.2.3 Shallow Water Theory ........................... 136
5.3 *Nonlinearity and Dispersion .......................... 139
5.3.1 The Korteweg-de Vries Equation ................. 139
5.3.2 The Nonlinear Schrodinger Equation ............. 142
5.3.3 Resonance ...................................... 147
5.4 Smooth Solutions for Hyperbolic Models ................ 150
5.4.1 The Piston Problem for One-dimensional
Unsteady Gas Dynamics .......................... 151
5.4.2 Prandtl-Meyer Flow ............................. 154
5.4.3 The Dam Break Problem .......................... 156
5.5 The Hodograph Transformation .......................... 158
Exercises .................................................. 160
6 Shock Waves ................................................ 175
6.1 Discontinuous Solutions ............................... 175
6.1.1 Introduction to Weak Solutions ................. 176
6.1.2 Rankine-Hugoniot Shock Conditions .............. 182
6.1.3 Shocks in Two-dimensional Steady Flow .......... 184
6.1.4 Jump Conditions in Shallow Water Flow .......... 190
6.1.5 Delta Shocks in Shallow Water .................. 193
6.2 Other Flows involving Shock Waves ..................... 194
6.2.1 Shock Tubes .................................... 194
6.2.2 Oblique Shock Interactions ..................... 195
6.2.3 Steady Quasi-one-dimensional Gas Flow .......... 199
6.2.4 Shock Waves with Chemical Reactions ............ 201
6.2.5 Open Channel Flow .............................. 203
6.3 Further Limitations of Linearised Gas Dynamics ........ 204
6.3.1 Transonic Flow ................................. 204
6.3.2 The Far Field for Flow past a Thin Wing ........ 206
6.3.3 Hypersonic Flow ................................ 208
6.4 Shocks in Solids and Plasmas .......................... 211
6.4.1 The Piston Problem in Elastoplasticity ......... 212
6.4.2 Shock Tube Problems in Cold Plasmas ............ 213
Exercises .................................................. 217
7 Epilogue ................................................... 231
Appendix ...................................................... 233
A.1 Heat Engines and Entropy ............................... 233
References ................................................. 237
Index ......................................................... 239
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