Singular perturbation analysis of the scattering of Rayleigh waves by thin surface layers

Abstract

Introduction.Surface waves progagating in elastic or piezoelectric half-spaces will be scattered upon interaction with thinly plated, finite or semi-infinite surface regions. At large distances from the scatterer along the surface, reflected and transmitted surface waves will appear. The remainder of the field after subtraction of all the surface waves is known as the radiated wave. In plane strain, the surface waves decay exponentially with depth but do not decay with distance along the surface, while the radiated wave decays as r~x/2 along any ray, and as r~3'2 along the surface. The power flux associated with the incident wave will be partitioned among the reflected, transmitted, and radiated waves.Numerous analyses of such interactions have appeared [1][2][3][4][5][6][7]. A common feature among all of these is the representation of the surface layer by a set of boundary conditions [8] to be applied to fields in the substrate. (These boundary conditions are accurate to within terms of order (layer thickness)/(wavelength).)The resulting boundary value problems have been solved by various methods. The Wiener-Hopf technique is useful in some very special cases with semi-infinite layers [1-3] but even here usually leads to integrals which can be evaluated only numerically. A variational approach [4][5] and various related modal expansion techniques [6] are more widely applicable and have proven successful for many applications, but have not given any insight into the radiated wave, as they incorporate an assumed field consisting only of surface waves. A direct, approximate, numerical solution has been obtained for a single strip [7], but the numerical analysis is exceedingly complex. Nevertheless, that study suggests that the power flux in the radiated wave may in some instances exceed that in the reflected surface wave.Thus, in spite of this proliferation of existing approaches, it would still seem useful to devise a method which is capable of treating finite and semi-infinite layers, elastic and piezoelectric substrates, inertial and stiffening effects of the layer, plane and non-plane problems; which accounts for body waves as well as surface waves, and which leads to simple expressions for the scattered field. The purpose of this paper is to present, by way of example, a singular perturbation approach to this class of problems-an approach which appears to possess the attributes listed above. The problem treated in detail is the scattering of a Rayleigh wave by a finite, thin strip. In order not to obscure the fundamental concepts, the density of the layer is taken to vanish, but this restriction is not essential to the success of the method. The problem is formulated in terms of a singular integral equation governing the shearing traction between the strip and the substrate. A parameter