On converging shock waves of spherical and polyhedral form

The behaviour of converging spherical shock waves is considered using Whitham's theory of geometrical shock dynamics. An analysis of converging shocks whose initial shape takes the form of regular polyhedra is presented. The analysis of this problem is motivated by the earlier work on converging cylindrical shocks discussed in Schwendeman & Whitham (1987). In that paper, exact solutions were reported for converging polygonal shocks in which the initial shape re-forms repeatedly as the shock contracts. For the polyhedral case, the analysis is performed both analytically and numerically for an equivalent problem involving shock propagation in a converging channel with triangular cross-section. It is found that a repeating sequence of shock surfaces composed of nearly planar pieces develops, although the initial planar surface does not re-form, and that the increase in strength of the shock at each iterate in the sequence follows the same behaviour as for a converging spherical shock independent of the convergence angle of the channel. In this sense, the shocks are stable and the result is analogous to that found in the two-dimensional case. A numerical study of converging spherical shocks subject to smooth initial perturbations in strength shows a strong tendency to form surfaces composed of nearly planar pieces suggesting that the stability result is fairly general.