On the stability of plane shocks

Abstract

The stability of perturbed normal shock waves is considered. Shock perturbations depend directly upon the disturbances in the flow adjacent to the shock. In the present paper an initially stationary shock is assumed to be perturbed by acoustic waves reaching it from the downstream side. This case corresponds to the situation occurring in shock diffraction or reflexion. Two-dimensional problems of this type have been investigated previously, both analytically and experimentally. These previous analytic results have, in all cases, indicated that the perturbations of the shock decay with time as $t^{\frac{3}{2}}$, while experimentally both t^½ and $t^{\frac{3}{2}}$ decays have been observed. It is demonstrated in the present investigation that, when waves are continuously generated at a point or points behind the shock, a t^½ decay of the shock perturbations will occur, corresponding to the decay of the incident waves. However, when the source of waves is located only at the shock, as in a diffraction problem, $t^{-\frac{3}{2}}$ decay occurs owing to the cancellation, to lowest order, of the incident wave by its reflexion from the shock. These results explain the divergence between theory and experiment in this area, since the experiments giving the slower decay contained a source of waves behind the shock.

It is concluded that shock stability can only be considered in the context of the type of disturbances incident upon the shock.