Generation of shear and compression waves in an inhomogeneous elastic medium

Abstract

This paper considers an inhomogeneous solid sediment, whose properties vary continuously with depth, lying between two homogeneous media—an upper fluid layer representing the ocean and a semi-infinite, homogeneous, solid substrate. The problem considered is that of determining the reflection coefficient of a plane wave incident on the sediment from above. It is assumed that the shear modulus in the sediment is small compared with the bulk modulus. This results in a relatively simple pair of coupled equations governing the generation of shear and compression waves. The equations clearly demonstrate how a density gradient in the sediment results in the continuous generation of shear waves within the bulk of the medium, and not just at the interfaces between adjacent media. With a suitably chosen density variation in the sediment, the equations can be solved analytically. Exact solutions are derived for the isovelocity case, and are used to investigate the effect of a continuous density variation within the sediment, on the reflection of an incoming plane wave from the upper layer. These exact solutions are also compared with a simpler method for calculating reflection coefficient, in which density gradient terms are ignored and solutions for homogeneous media are matched simply by ensuring that the density is given its correct value at each interface. The shape of the density profile in the sediment has a significant effect on reflection coefficient if wavelength is comparable to the thickness of the sediment. The influence of the density profile decreases rapidly at higher frequencies, and the simple impedance matching technique is found to be adequate for wavelengths which are small compared with sediment thickness.