Springer Online Journal Archives 1860-2000
Abstract Reflection and transmission of elastic wave motion by a layer of compact inhomogeneities has been analyzed. For identical inhomogeneities whose geometrical centers are periodically spaced, the problem has been formulated and solved rigorously. The reflected and transmitted longitudinal and transverse wave motions have been expressed as superpositions of wavemodes, where each wavemode has its own cut-off frequency. At its cut-off frequency a mode converts from a standing into a propagating wavemode. The standing wavemodes decay exponentially with distance to the plane of the centers of the inhomogeneities. At small frequencies only the lowest order modes of longitudinal and transverse wave motion are propagating. Reflection and transmission coefficients have been defined in terms of the coefficients of the zeroth-order scattered wavemodes. These coefficients have been computed by a novel application of the Betti-Rayleigh reciprocal theorem. They are expressed as integrals over the surface of a single inhomogeneity, in terms of the displacements and tractions on the surface of the inhomogeneity. The system of singular integral equations for the surface fields has been solved numerically by the boundary integral equation method. Curves show the reflection and transmission coefficients for the reflected and transmitted longitudinal and transverse waves as functions of the frequency. Some results are also presented for planar distributions of cracks whose spacing and size are random variables. Finally, dispersion relations are discussed for solids which are completely filled with periodically spaced inhomogeneities.
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