Self‐similar solution for spherical weak shocks

Abstract

The nonlinear progressive wave equation (NPE), recently developed for investigation of weak shock refraction [B. E. McDonald and W. A. Kuperman, Proc. Ocean Acoust. Workshop, edited by D. Lee and M. Schultz (Yale Univ., 1–3 August 1983)], admits similarity solutions appropriate to spherically symmetric explosions. The NPE reduces the second‐order wave equation with lowest‐order nonlinear correction to a first‐order initial value problem in time (i.e., the forward propagating characteristic is automatically selected while the backward propagating one is eliminated). When refraction is absent, the NPE admits an analytic, closed form similarity solution with similarity variable x = r/ct. Results for peak pressure and relaxation time versus range closely resemble those of P. H. Rogers [J. Acoust. Soc. Am. 62, 1412–1419 (1977)] who assumed the pulse shape to be initially exponential. (Our solution predicts the actual form of the self‐similar pulse within a constant of integration.) Both our results and Rogers' show good agreement with pressure data, but only fair agreement with relaxation time data. This seems to call for some unspecified change in the theory or perhaps a reinterpretation of the experimental data. [Work supported by ONR.]