Validation of an algorithm for wave propagations in graded materials with an analytical solution

Abstract

When a mechanical stress pulse, which is propagating in an elastic medium, encounters a material- or phase interface, which generally represents a change of the acoustic impedance, it is split up into a part, which propagates further into the new material and another part, which is reflected. The amplitude ratio of the reflected and the transmitted part is governed by the normalized difference of the acoustic impedance only, provided that the impedance change is a pure step function in space. If the acoustic impedance change is broadened spatially, the ratio of the transmitted and reflected part becomes frequency dependent and the effect can therefore be used for filter-, damping-, acoustic isolation-, and/or spectrum analysis purpose or for quantitative analysis of interface. The effect is of growing importance for micro- and nanostructures since the relative size of the interface layers is generally larger than in macroscopic structures. In this work, a pulse propagating in a linear elastic graded material is described with analytical solutions and one dimensional simulations. The numerical scheme is based on the Finite-Difference Time-Domain method (FDTD). The validation of the numerical model occurs by comparing the simulated pulse propagation-history with an analytical solution based on.¹ On-coming research is also given at the end of this study.