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

Abstract

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 Padé 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 Padé 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 Padé 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; v; and the outgoing Rayleigh wave is determined for all values of v (< 0.5).