An analytic description of converging shock waves

Abstract

The similarity solution describing the motion of converging spherical and cylindrical shocks is governed by a set of three ordinary differential equations. Previous descriptions of the shock motion have been based on numerical solutions of these differential equations. In the present paper a study of the singular points of the differential equations leads to an analytic description of the flow and a determination of the similarity exponent which is in excellent agreement with the earlier numerical values. Limiting values of the ratio of specific heats are considered. It is shown that as the ratio tends to unity the shock becomes ‘freely propagating’ and the first terms in a power series for the similarity exponent are obtained. Large values of the ratio of specific heats are briefly considered and provide a further check on the analytic description of this paper. Finally in the Appendix the condition for the pressure to have a maximum is clarified and the location of the maximum provides further strong evidence of the high accuracy of the analytic approach of this paper.