Preface ��P.1.How this book came to be, and its peculiarities ��P.2.A bird's eye view of hyperbolic equations Chapter 1.Simple Examples of Propagation ��1.1.The method of characteristics ��1.2.Examples of propagation of singularities using progressing waves ��1.3.Group velocity and the method of nonstationary phase ��1.4.Fourier synthesis and rectilinear propagation ��1.5.A cautionary example in geometric optics ��1.6.The law of reflection 1.6.1.The method of images 1.6.2.The plane wave derivation 1.6.3.Reflected high frequency wave packets ��1.7.Snell's law of refraction Chapter 2.The Linear Cauchy Problem ��2.1.Energy estimates for symmetric hyperbolic systems ��2.2.Existence theorems for symmetric hyperbolic systems 62.3.Finite speed of propagation 2.3.1.The method of characteristics 2.3.2.Speed estimates uniform in space 2.3.3.Time-like and propagation cones ��2.4.Plane waves, group velocity, and phase velocities ��2.5.Precise speed estimate ��2.6.Local Cauchy problems Appendix 2.I.Constant coefficient hyperbolic systems Appendix 2.II.Functional analytic proof of existence Chapter 3.Dispersive Behavior ��3.1.Orientation ��3.2.Spectral decomposition of solutions ��3.3.Large time asymptotics ��3.4.Maximally dispersive systems 3.4.1.The L1 �� Lo decay estimate 3.4.2.Fixed time dispersive Sobolev estimates 3.4.3.Strichartz estimates Appendix 3.I.Perturbation theory for semisimple eigenvalues Appendix 3.I��.The stationary phase inequality Chapter 4.Linear Elliptic Geometric Optics ��4.1.Euler's method and elliptic geometric optics with constant coefficients ��4.2.Iterative improvement for variable coefficients and nonlinear phases ��4.3.Formal asymptotics approach ��4.4.Perturbation approach ��4.5.Elliptic regularity ��4.6.The Microlocal Elliptic Regularity Theorem Chapter 5.Linear Hyperbolic Geometric Optics ��5.1.Introduction ��5.2.Second order scalar constant coefficient principal part 5.2.1.Hyperbolic problems 5.2.2.The quasiclassical limit of quantum mechanics ��5.3.Symmetric hyperbolic systems ��5.4.Rays and transport