Propagation of a curved weak shock

Abstract

Propagation of a curved shock is governed by a system of shock ray equations which is coupled to an infinite system of transport equations along these rays. For a two-dimensional weak shock, it has been suggested that this system can be approximated by a hyperbolic system of four partial differential equations in a ray coordinate system, which consists of two independent variables (ζ, t) where the curves t = constant give successive positions of the shock and ζ = constant give rays. The equations show that shock rays not only stretch longitudinally due to finite amplitude on a shock front but also turn due to a non-uniform distribution of the shock strength on it. These changes finally lead to a modification of the amplitude of the shock strength. Since discontinuities in the form of kinks appear on the shock, it is necessary to study the problem by using the correct conservation form of these equations. We use such a system of equations in conservation form to construct a total-variation-bounded finite difference scheme. The numerical solution captures converging shock fronts with a pair of kinks on them – the shock front emerges without the usual folds in the caustic region. The shock strength, even when the shock passes through the caustic region, remains so small that the small-amplitude theory remains valid. The shock strength ultimately decays with a well-defined geometrical shape of the shock front – a pair of kinks which separate a central disc from a pair of wings on the two sides. We also study the ultimate shape and decay of shocks of initially periodic shapes and plane shocks with a dent and a bulge.