An asymptotic decoupling method for waves in layered media

Abstract

This paper presents a technique, asymptotic decoup- ling, for analysing wave propagation in multi-layered media. The technique leads to a hierarchy of approximations to the exact dispersion relation, obtained from finite-product approximations to low-order dispersion relations appearing as factors in the asymptotically decoupled limits. Levels of refinement may be added or removed according to the frequency range of interest, the degree of accuracy required, and the material and geometrical parameters of the different layers. This is shown to be particularly useful in stiff problems, because unlimited accuracy is obtainable without redundancy even when Young's moduli and the thicknesses of the layers differ by many orders of magnitude, for example in a stiff sandwich plate with a very soft core. Full details are presented for a non-trivial example, that of antisymmetric waves in a three-layered planar elastic waveguide. Comparisons are made with two widely used approximations, Tiersten's thin-skin approximation and the composite Timoshenko approximation. The mathematical basis of the paper is the asymptotic decoupling of the wave motion in different layers in the limit of indefinitely large or small density ratio.