Off-center blast in a shocked medium

When multiple blasts occur at different times, the situation arises in which a blast wave is propagating into a medium that has already been shocked. Determining the evolution in the shape of the second shock is not trivial, as it is propagating into air that is not only non-uniform, but also non-stationary. To accomplish this task, we employ the method of Kompaneets to determine the shape of a shock in a non-uniform media. We also draw from the work of Korycansky (Astrophys J 398:184Ð"Ñ'Ð'ІÐ"Ñ'вÐ'Ñ™"189, 1992) on an off-center explosion in a medium with radially varying density. Extending this to treat non-stationary flow, and making use of approximations to the Sedov solution for the point blast problem, we are able to determine an analytic expression for the evolving shape of the second shock. In particular, we consider the case of a shock in air at standard ambient temperature and pressure, with the second shock occurring shortly after the original blast wave reaches it, as in a sympathetic detonation.