Transverse cylindrical simple waves and shock waves in elastic non-conductors

Abstract

A modulated simple wave theory is developed for transverse cylindrical motions of an unstrained incompressible isotropic elastic non-conductor with the aid of a modified version of Hunter and Keller's “Weakly Nonlinear Geometrical Optics” method. This theory is then used to construct shock wave solutions using the shock-fitting method. The evolution law thus derived shows that the effect of nonlinearity on the evolution of transverse cylindrical shock waves is cumulative, but that by the time it becomes most pronounced, geometrical spreading has already attenuated the shock amplitude until it is exponentially small. It follows that the linear theory gives satisfactory results for the propagation of transverse cylindrical shock waves. This is in sharp contrast to the situation for plane transverse shock waves whose amplitudes decay in the presence of material nonlinearities whilst the linear theory predicts constant amplitudes. Where it is present, geometrical spreading would appear to be a more potent decay mechanism than material nonlinearity.