Interactions of Nonuniform Shock Waves

Abstract

The interactions of shock waves with boundaries and with each other have been considered rather fully in the literature for the case of uniform shock waves. The purpose of this paper is to present an approximate method for analyzing the interactions of “weak” nonuniform shock waves. The method is to map the entire problem—differential equations, boundary conditions, shock equations—into the hodograph plane. Because of the approximation used, this mapping is particularly simple and leads directly to the numerical solution of the problem. The approximation (that used by Chandrasekhar and Friedrichs in connection with the formation and decay of nonuniform shock waves) is to neglect third-order terms in the shock strength; i.e., (ρ−ρ0/ρ0)3, (u−u0/c0)3, and (c−c0/c0)3. The virtue of this approximation is twofold: First, since the entropy change across a shock front is a third-order effect, entropy change may be neglected everywhere. Secondly, in this approximation the locus of states which can be connected by a shock (called a “shock polar”) is identical with the locus of states which can be connected by an adiabatic compression. Thus, in the hodograph plane the shock polars and Γ characteristics coincide, and the mapping can be effected in a simple way. In metals, treated hydrodynamically, the very small compresibility allows the approximation to yield accurate answers for shocks as strong as 500 000 atmos. Numerical results will be presented.