Reflection and refraction of elastic waves at a corrugated interface

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Abstract
The effect of a corrugated interface on wave propagation is considered by using the method that was first applied to acoustical gratings by Rayleigh. The problem is what happens when a plane P wave is incident on a corrugated interface that separates two semi-infinite media. As is well known, there are irregular (scattered) waves as well as regular waves. By assuming both the amplitude and the slope of a corrugated interface to be small, quantities of the order of the square of corrugation amplitude are taken into account. In the case of normal incidence for three models considered, the effect of corrugation on reflection is larger than the effect of corrugation on refraction; the amplitude of the regularly reflected waves decreases, and that of the regularly refracted waves and of the irregular waves increases, as the corrugation amplitude becomes larger. Generally, the larger the velocity contrast, the larger the variation of wave amplitude with the wavelength and the amplitude of corrugation. The S wave component generally becomes larger as the wavelength of corrugation becomes smaller. Boundary waves exist, depending upon the ratio of wavelength of corrugation to that of the incident wave. For a specified interface, it is possible that there is a significant difference in wave amplitude as a function of the elastic constants. In the case of oblique incidence, computation was carried out for angles of incidence smaller than 15° for one model. For these small angles of incidence, almost all results for the case of normal incidence still hold. Furthermore, it can be concluded that the effect of the angle of incidence on reflected S waves is larger than for the other waves and that large differences in the amplitudes of waves at different angles of incidence may be expected for the irregular waves.