The Reverse Blast Wave

The Sedov-Taylor-von Neumann point explosion or blast wave problem is perhaps the most famous example of a self-similar solution of the one-dimensional (1D) compressible Euler equations. The mathematical structure of this problem has been exhaustively studied since the 1940s, with a key result being identification of the underlying Lie point symmetry group that gives rise to it. The structure of this symmetry group has several important consequences: (1) It encodes all self-similar behaviors featured in the blast wave solution, (2) It categorizes the blast wave solution among broader classes of self-similar solutions admitted by the underlying equations of motion, and (3) It enables the construction of new solutions from existing results. In view of these outcomes, this work explores the intersection of the classical blast wave solution with additional Lie point symmetry structures admitted by the 1D compressible Euler equations. Emphasis is placed on scaling transformations in the time variable and their symmetry-preserving consequences for the remaining independent and dependent variables. In doing so a new class of closed-form solutions is derived using the blast wave solution as a kernel; for certain values of the symmetry-preserving scaling parameter these solutions are found to correspond to implosions driven by 1D cylindrical or spherical pistons.