Comparison of continuous and discontinuous models of heterogeneity in the propagation of transient planar stress waves

Abstract

In the characteristic differential equations governing propagation of linear one-dimensional waves through heterogeneous media, the only properties of significance are the sound speed c and the acoustic impedance ρc. The former occurs in the differential equations governing the (curved) characteristics, while the latter appears in the differential equations governing the evolution of particle velocity and strain along the characteristics. One might attempt to approximate the spatial variation of these material properties in a periodic array of homogeneous layers by smooth interpolating functions. An implicitly stable finite difference representation of the characteristic equations is used in the present work to consider propagation of a transient wave through a bi-material sequence of layers of thickness L. Waveforms are compared to those obtained from a model in which the layer properties are taken to vary sinusoidally, such that ρ = ρ0[1 + ε sin(πx/L)] and c = c0[1 + δ sin(πx/L)]. Of particular interest is the case where the wavelength is L, which, according to Floquet theory, represents a condition of uncertain stability for steady-state waves, but which leads to maximum interference for a harmonic transient wave in the sinusoidally varying medium.