Radiation and scattering of waves on an elastic half-space; A non-commutative matrix Wiener-Hopf problem

Many problems in linear elastodynamics, or dynamic fracture mechanics, can be reduced to Wiener-Hopf functional equations defined in some region of a complex transform plane. Apart from a few simple cases, the inherent coupling between shear and compressional body motions gives rise to coupled systems of equations, and so the resulting Wiener-Hopf kernels are of matrix form. The key step in the solution of a Wiener-Hopf equation, which is to decompose the kernel into a product of two factors with particular analyticity properties, can be accomplished explicitly for scalar kernels. However, apart from special matrices which yield commutative factorizations, no procedure has yet been devised to exactly factorize general matrix kernels. It is the aim of this article to show that a new procedure for obtaining approximate factors of matrix kernels is applicable to the class of matrix kernels found in elasticity. This is performed, for ease of exposition, by way of a simple but non-trivial example: a linear elastic half-space, occupying the region y > 0, where (x,y,z) are cartesian coordinates, has a free boundary on x > 0, y = 0, and has imposed time-harmonic displacements on x < 0, y = 0. Via the substitution of a scalar function in the kernel by its Pade approximant, an approximate solution to this boundary value problem is obtained explicitly for three different forcing cases. The approximate factorization technique described herein is simple to apply, is shown to converge to the exact result as the Pade number increases, and a maximum error bound is easily obtained. Numerical evaluation of the explicit results reveals that convergence to the exact solution is extremely rapid (i.e. only a small number of poles and zeros are required in the Pade approximant), which is confirmed by a global energy balance calculation, and so numerical calculations are extremely rapid. The outgoing cylindrically spreading compressional and shear wave coefficients are evaluated for several values of Poisson's ratio, nu, and the outgoing Rayleigh wave is determined for all values of nu (<0.5). Copyright (C) 1996 Elsevier Science Ltd.