《Non-Linear Wave Propagation with Applications to Physicsand Magnetohydrodynamics》

PARTⅠ—GENERAL THEORY1

Chapter 1General Hyperbolic Equations3

1.1 The Wave and Hyperbolic Equations3

1.2 The Cauchy Problem and Characteristics9

1.3 Mixed Boundary and Initial Value Problems15

1.4 Non-Linear Equations,Quasi-Linear Systems,and Lipschitz Continuous Solutions24

1.5 Systems of Quasi-Linear Equations with Many Variables29

1.6 Quasi-Linear Hyperbolic First Order Systems and Charactcristics33

1.7 Rays and Wave Fronts40

1.8 Illustratiye Examples:The Maxwell Equations48

1.9 Hydrodynamics57

Chapter 2The Method of Characteristics65

2.1 Riemann Invariants—Systems with Two Dependent Variables65

2.2 Generalised Riemann Invariants—System with n Dependent Variables85

2.3 Mixed Boundary and Initial Value Problems94

2.4 Propagation of Discontinuities along Waye Fronts100

Chapter 3Conservation Laws and Weak Solutions111

3.1 Conservation Laws111

3.2 Weak Solutions116

3.3 Evolutionary Conditions on Discontinuities in Conservation Laws of Hyperbolic Type124

3.4 Evolutionary Conditions on a General System125

3.5 General Shock Relations130

3.6 Hydrodynamic Discontinuities135

3.7 Numerical Solution of Non-Linear Hyperbolic Systems141

3.8 The Propagation of Weak Discontinuities along Rays149

3.9 Geometrical Acoustics—The Theory of Weak Shock Waves159

PART Ⅱ—THE APPLICATION TO MAGNETOHYDRODYNAMICS165

Chapter 4The Fundamental Equations and Characteristics167

4.1 Basic Equations and Assumptions167

4.2 The Adiabatic Reversible System and the Lundquist Equations170

4.3 The Characteristic Equations171

4.4 Wave Front Diagram177

4.5 Propagation of Weak Hydromagnetic Discontinuities189

Chapter 5Simple Waves195

5.1 Properties of Magnetic Lines of Force195

5.2 Simple Waves in One-Dimensional Propagation202

Chapter 6Magnetohydrodynamic Shocks214

6.1 The Conservation Laws214

6.2 Fast and Slow Shocks219

6.3 Limit Shocks238

6.4 Transverse Shocks and Contact Discontinuities249

Chapter 7Interaction of Hydromagnetic Waves256

7.1 Further Considerations on the Evolutionary Condition256

7.2 The Piston Problem259

7.3 Riemann’s Problem276

Chapter 8Spatial Discontinuities293

8.1 Weak Discontinuities in Steady Flows293

8.2 The Reducible Form of Plane Aligned-Field Flow299

8.3 Oblique Shock Waves311

8.4 The Discontinuity in the Static Case317

Appendices321

A.Basic Theorems in Matrix Theory321

B.The Rankine-Hugoniot Relation327

C.The Behaviour of X？331

D.Shock Relations334

E.The Representation of the Hydromagnetic Equations339

F.Contact Transformations and the Legendre Transformation347

REFERENCES TO PARTⅠ357

REFERENCES TO PARTⅡ359

INDEX363