The transition from acceleration wave to shock wave

In certain circumstances the amplitude of an acceleration wave is predicted by singular surface theory to become infinite after only a finite distance of propagation and it is widely believed that this corresponds to the formation of a shock wave, though no proof appears to have been given. We prove that this is so for one-dimensional motions of an elastic half-space by utilising an exact solution obtained from simple wave theory. This result is extended to dilatational cylindrical and spherical wave propagation in an elastic material.