Non-linear wave propagation, with applications to physics and magnetohydrodynamics

Cover......Page 1
Series......Page 2
Title page......Page 3
Copyright page......Page 4
Preface......Page 5
Contents......Page 7
PART I — GENERAL THEORY......Page 11
1.1 The Wave and Hyperbolic Equations......Page 13
1.2 The Cauchy Problem and Characteristics......Page 19
1.3 Mixed Boundary and Initial Value Problems......Page 25
1.4 Non-Linear Equations, Quasi-Linear Systems, and Lipschitz Continuous Solutions......Page 34
1.5 Systems of Quasi-Linear Equations with Many Variables......Page 39
1.6 Quasi - Linear Hyperbolic First Order Systems and Characteristics......Page 43
1.7 Rays and Wave Fronts......Page 50
1.8 Illustrative Examples: The Maxwell Equations......Page 58
1.9 Hydrodynamics......Page 67
2.1 Riemann Invariants—Systems with Two Dependent Variables......Page 75
2.2 Generalised Riemann Invariants—System with $n$ Dependent Variables......Page 95
2.3 Mixed Boundary and Initial Value Problems......Page 104
2.4 Propagation of Discontinuities along Wave Fronts......Page 110
3.1 Conservation Laws......Page 121
3.2 Weak Solutions......Page 126
3.3 Evolutionary Conditions on Discontinuities in Conservation Laws of Hyperbolic Type......Page 134
3.4 Evolutionary Conditions on a General System......Page 135
3.5 General Shock Relations......Page 140
3.6 Hydrodynamic Discontinuities......Page 145
3.7 Numerical Solution of Non-Linear Hyperbolic Systems......Page 151
3.8 The Propagation of Weak Discontinuities along Rays......Page 159
3.9 Geometrical Acoustics—The Theory of Weak Shock Waves......Page 169
PART II — THE APPLICATION TO MAGNETOHYDRODYNAMICS......Page 175
4.1 Basic Equations and Assumptions......Page 177
4.2 The Adiabatic Reversible System and the Lundquist Equations......Page 180
4.3 The Characteristic Equations......Page 181
4.4 Wave Front Diagram......Page 187
4.5 Propagation of Weak Hydromagnetic Discontinuities......Page 199
5.1 Properties of Magnetic Lines of Force......Page 205
5.2 Simple Waves in One-Dimensional Propagation......Page 212
6.1 The Conservation Laws......Page 224
6.2 Fast and Slow Shocks......Page 229
6.3 Limit Shocks......Page 248
6.4 Transverse Shocks and Contact Discontinuities......Page 259
7.1 Further Considerations on the Evolutionary Condition......Page 266
7.2 The Piston Problem......Page 269
7.3 Riemann's Problem......Page 286
8.1 Weak Discontinuities in Steady Flows......Page 303
8.2 The Reducible Form of Plane Aligned-Field Flow......Page 309
8.3 Oblique Shock Waves......Page 321
8.4 The Discontinuity in the Static Case......Page 327
A. Basic Theorems in Matrix Theory......Page 331
B. The Rankine-Hugoniot Relation......Page 337
C. The Behaviour of $X^\pm_f$......Page 341
D. Shock Relations......Page 344
E. The Representation of the Hydromagnetic Equations......Page 349
F. Contact Transformations and the Legendre Transformation......Page 357
References to Part I......Page 367
References to Part II......Page 369
Index......Page 373
Instead of back cover......Page 380