Transient Wave Propagation in a Harmonically Heterogeneous Elastic Solid

Transient propagation of a one-dimensional dilatational wave in a harmonically heterogeneous elastic solid is studied by several techniques. A regular perturbation analysis in terms of the characteristics of the differential equation shows that initiation of a temporally harmonic excitation that generates a signal whose wavelength is twice the periodicity of the heterogeneity leads to secularity in the first approximation. The frequency at which this situation occurs matches the frequency at which Floquet theory predicts that steady-state waves may be unstable. A finite difference algorithm based on integrating along the characteristics is developed and implemented to obtain a numerical solution. In the critical case, backscattering of the wave from the heterogeneity results in a mixture of propagating and standing wave features. However, rather than being unstable, the heterogeneity in this condition is shown to result in maximum interference with forward propagation. A comparable analysis for a step excitation on the boundary provides additional insight into the underlying propagation phenomena.