The behavior of induced discontinuities behind curved shocks in isotropic linear elastic materials

In this paper we examine the behavior of the induced discontinuities behind curved longitudinal and transverse shock waves in isotropic linear elastic materials. It is shown that in either case the governing differential equation of the induced discontinuity differs from that of the shock amplitude. The latter depends linearly on the second fundamental form of the shock surface and exhibits purely geometrical effects. The former, however, depends non-linearly on the second fundamental form of the shock surface, and on the shock amplitude. These terms are dominant for a strong shock and their effects diminish as the shock weakens. In particular, the governing differential equation for an acceleration wave is obtained in the limit as the shock amplitude vanishes. The results obtained are quite unexpected, and they demonstrate the complex evolutionary behavior of mechanical waves due to geometrical considerations alone.