5 - Relativistic simple waves

Summary

Introduction

Simple waves are exact solutions of quasi-linear equations representing traveling waves. They are the most natural nonlinear analog of the plane traveling waves of the linear theory. Although they correspond to special initial conditions, they are still sufficiently general to be of physical interest (as witnessed by Friedrichs' theorem, which, loosely speaking, states that any one dimensional smooth solution neighboring a constant state must be a simple wave). Simple waves are exact analytical solutions which show clearly some of the main features of nonlinear wave propagation in general, such as steepening, breaking, and shock formation. They are also sufficiently complex to be useful as benchmarks against which to test numerical codes.

In classical fluid dynamics simple waves are of paramount importance for several reasons. Simple waves are the basic ingredients for constructing analytical solutions to the Riemann problem (or shock-tube problem: the evolution of an initial state corresponding to two adjacent fluids at different pressures). These solutions are among the standard tests for numerical hydrodynamical codes (Sod, 1978) because they comprise some of the key features of general hydrodynamical behavior (nonlinear steepening and the occurrence of discontinuities). Also, analytical solutions to the Riemann problem can be used in order to construct sophisticated numerical codes able to deal very accurately with shock front tracking (Plohr, Glimm, and McBryan, 1983). Simple waves are also used in order to construct approximate analytical solutions to the problem of weak shock decay (Landau and Lifshitz, 1959a; Whitham, 1974; Courant and Friedrichs, 1976) and in Whitham's theory of geometric shock dynamics (Whitham, 1974).