Contents

Preface xi

§P.1. How this book came to be, and its peculiarities xi

§P.2. A bird’s eye view of hyperbolic equations xiv

Chapter 1. Simple Examples of Propagation 1

§1.1. The method of characteristics 2

§1.2. Examples of propagation of singularities using progressing

waves 12

§1.3. Group velocity and the method of nonstationary phase 16

§1.4. Fourier synthesis and rectilinear propagation 20

§1.5. A cautionary example in geometric optics 27

§1.6. The law of reflection 28

1.6.1. The method of images 30

1.6.2. The plane wave derivation 33

1.6.3. Reflected high frequency wave packets 34

§1.7. Snell’s law of refraction 36

Chapter 2. The Linear Cauchy Problem 43

§2.1. Energy estimates for symmetric hyperbolic systems 44

§2.2. Existence theorems for symmetric hyperbolic systems 52

§2.3. Finite speed of propagation 58

2.3.1. The method of characteristics. 58

2.3.2. Speed estimates uniform in space 59

2.3.3. Time-like and propagation cones 64

§2.4. Plane waves, group velocity, and phase velocities 71

vii