Method of the Degenerate Hodograph

Abstract

The vast majority of exact solutions in continuum mechanics have been obtained by the method of the degenerate hodograph. This method deals with solutions which are distinguished by finite relations between the dependent variables. Solutions with degenerate hodograph form a class of solutions called multiple waves. Riemann waves and Prandtl-Meyer flows are the simplest solutions of this class1. The main problem with the theory of multiple waves is obtaining a compatible system of equations in the space of the dependent and independent variables. The chapter starts by giving the main definitions and basic facts of the theory. Simple waves of systems with two independent variables are closely related to the Riemann invariants. Attempts to generalize the notion of Riemann invari-ants to equations with more than two independent variables are discussed. One of these approaches deals with simple integral elements. The simplest case of multiple waves is the case of simple waves. The first application of simple waves for multi-dimensional flows was made for isen-tropic flows of an ideal gas. From a group analysis point of view a multiple wave is a partially invariant solution. For example, a simple wave is a partially invariant solution with the defect one; the defect of a double wave is equal to two. In the theory of partially invariant solutions there is the problem of re-ducibility to a smaller defect. solutions, irreducible to invariant, take a special place among partially invariant solutions. This is related to the fact that the problem of compatibility for an invariant multiple wave is easier than for a partially invariant multiple wave. Hence, the problem of reducibility arises. There are few theorems which state sufficient conditions of reducibility. One of them ' ~ ~ ~ l i c a t i o n s of the method of degenerate hodograph to the gas dynamic equations can be found in [I601 and the references therein.