Similarity Solution for Low Mach Number Spherical Shocks

摘要

A nonlinear progressive wave equation (NPE) describes the evolution of a low Mach number shock wave. The NPE is the nonlinear time domain counterpart of the frequency domain linear parabolic wave equation (PE) for small angle propagation. The NPE in spherical symmetry admits a similarity solution that specifies both the shape of the pulse and the shock strength as a function of range. For finite amplitude spherical waves whether self-similar or not, the theory predicts constancy of an impulse integral corresponding to that of linear theory. For the self-similar waves, theory and available data are in qualitative agreement in the following areas: (1) The shock strength decreases with range as an approximate power law; (2) the temporal behavior of the solution at fixed range is a shock discontinuity followed by a roughly exponential decay; and (3) the effective relaxation time behind the shock is in reasonable agreement with data for slightly more than the first decade in range. Reprints. (JHD)