An Existence Theorem for the Point Source Blast Wave Equation

Abstract

We consider the existence of a solution for the point-source blast wave propagation caused by instantaneous explosion. No similarity solutions of the Euler equations satisfy the conservation law on the shock front, so far as the atmospheric pressure ahead of the shock is not negligible. To describe the initial state, it is assumed that the total amount of energy carried by the blast wave is constant. By a hodograph transform this free boundary problem is converted into an eigenvalue problem for a system defined on a bounded rectangle such that this initial state assumption is satisfied. The solution is prescribed in the form of a power series expansion in one of the variables y = C 2/U 2 for front shock speed U and sound velocity C. Its convergence is shown by applying the fixed point theory of contractive mapping defined through linearization of the system. Our solution is local in y and exact there.