Reflection and transmission of inhomogeneous waves with particular application to Rayleigh waves

Abstract

abstract The concept of inhomogeneous waves, which is quite common in the electromagnetic literature but not in the seismic, is reviewed. It is shown that the Rayleigh wave results from the constructive interference of a P inhomogeneous wave with an SV inhomogeneous wave. Of some interest is the fact that while the two inhomogeneous waves have prograde elliptical particle motions, they combine to yield the familiar retrograde motion near the surface. These two inhomogeneous waves are the only “allowed” inhomogeneous waves in a homogeneous elastic half-space. More can occur in a layered half-space but they are always discrete in number. However, “non-allowed” inhomogeneous waves often give a very good representation of the wave motion over a restricted area—a fact which is demonstrated for leaky modes and propagation of a Rayleigh wave on a three-quarter space. With this in mind, a formalism is developed for the transmission and reflection of the two allowed inhomogeneous waves making up the Rayleigh wave into nonallowed inhomogeneous waves. These are nonallowed in the sense that they do not fit the boundary conditions at a discrete number of points. Here the ray picture breaks down and diffractive effects occur. Our assumption is that for many interesting cases, these diffractive effects are small and can be ignored. The components of the resultant displacements and stresses from the nonallowed inhomogeneous waves on Rayleigh waves can be obtained by use of a relation due to Herrera, thus yielding the reflection and transmission coefficients. The agreement with previously published values is good. While only normal incidence is considered in this paper, the extension to non-normal incidence is straightforward. The required calculations are well within the capabilities of a small computer such as an IBM 1130.