A homogenisation approach to elastic waves in directional media

Abstract

We present a new approach to the problem of modelling elastic wave propagation in a heterogeneous medium, where the material properties vary substantially on a fine length scale in the vertical direction. In contrast lateral heterogeneity occurs on a much longer length scale. We employ the elastic wave equations in a form which is similar to those used in the propagator matrix method. We exploit the smallness of the ratio between the vertical and lateral length scales and apply homogenisation techniques to derive a hierarchy of equations which govern successive approximations to the full wave field. This system of equations is solved successively, using averaging with respect to the fine scale variable. The first order solution may be derived from an effective medium theory in which our formulae for the effective material parameters are the same as the Backus average formulae, although different averaging techniques are employed. Furthermore, beyond the upscaled effective medium solution, we obtain an averaged equation for the second order correction. Note that our theory is also applicable to wave propagation in irregular structures with both layering and quite general spatial variation of the material parameters. In this sense our approach provides a generalisation of the familiar effective medium theories for elastic wave propagation. The implications of the second order theory are illustrated numerically and discussed in detail.