SCATTERING OF WAVES BY A SEMICYLINDRICAL GROOVE IN THE SURFACE OF AN ELASTIC HALF-SPACE

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Abstract
An elastic half-space, composed of homogeneous, isotropic material, contains a semicylindrical groove in its free surface; a time-harmonic elastic wave (a Rayleigh wave, plane PV-wave or plane SV-wave) is incident upon the groove, and is scattered by it. What is the scattered elastic field and how is the scattered energy apportioned in the far field? This problem is solved, in the context of the linear theory of elasticity, by a series-expansion method. Multipole expansion potentials are used which are singular along the axis of the semicylinder, satisfy the free-surface boundary conditions (except on the groove) and consist of outgoing waves at infinity; these potentials are proved to be complete for the expansion of all time-harmonic elastic wave fields which satisfy the free-surface conditions on the flat surface of the half-space, and which satisfy ‘outgoing-radiation’ conditions at infinity. However, although this natural set of expansion potentials is complete, it is shown that it is not linearly independent; indeed the potentials satisfy infinitely many finite linear dependencies. We show how to reduce this linearly dependent expansion set to a linearly independent set which is still complete. If the scattered field is now expanded in terms of this reduced set, The expansion coefficients are then unique, a necessary prerequisite for efficient numerical computation. The expansion coefficients are then chosen so as to satisfy (to any desired accuracy) the boundary data on the groove and the far scattered field is then evaluated. Numerical results are presented, showing the effect of the scattering for the various kinds of incident wave.

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