A theory for the diffraction of SH waves by randomly rough surfaces in two dimensions

Abstract

In this paper we shall consider a mathematical model for the diffraction of SH waves by randomly rough surfaces. We use the crack Green function method to formulate the problem (see G. R. Wickham, Proc. R. Soc. A 378 (1981) 241-261, P. A. Lewis and G. R. Wickham, Phil. Trans. R. Soc. A 340 (1992) 503-529). This method gives rise to a Fredholm integral equation of the second kind for the jump in in the potential across the scatterer (also known as the crack opening displacement). To analyse the expectation of the crack opening displacement we use a smoothing method in which we perform an ensemble average over all realizations of a certain class of surfaces. A closure assumption is made which is valid for surfaces with small root mean square height. This gives rise to a coupled system of Fredholm integral equations for the average of the crack opening displacement and the average of the random components of the crack opening displacement and kernel. If we make the further assumption that the root mean square gradients are also small, then this coupled system reduces to a single Fredholm integral of the second kind for the expectation of the crack opening displacement. Numerical results are presented for the average crack opening displacement and the average scattered field.