On the eigenvalue problem of imploding shock waves

Abstract.

The motion of an adiabatic converging spherical or cylindrical shock wave in perfect gas can be described by a self-similar solution governed by only one single ordinary differential equation. The solving of this differential equation is a nonlinear eigenvalue problem complicated by the occurrence of a singular point in the phase plane. In this paper a new approximation method of determining the eigenvalue is presented. It is based on a quasi-removal of the singularity by some simple algebraic rearrangements of the differential equation. For various adiabatic coefficients in either cylindrical or spherical geometry this method guarantees eigenvalues which are almost as accurate as the eigenvalues achieved by the most sophisticated series expansion methods so far available.