The propagation of elastic waves in unconsolidated unsaturated granular mediums

Abstract

The theory makes use of three distinct idealized models. The general equations of motion are derived for a continuum model. This continuum consists of three components—a solid, a gas, and a liquid; the liquid adheres to the solid and has surface tension. The elastic coefficients occurring in the equations of motion are determined for a model consisting of randomly stacked spheres of different sizes; the interstices of this model contain both gas and liquid. The dissipation coefficients occurring in the equations of motion are determined for a model of short capillary tubes. The nine equations of motion are transformed into rotational and irrotational equations to yield plane, progressive, sinusoidal waves. The wave velocity and the attenuation are derived for the asymptotic cases of very low and very high frequencies. It is proved that there are three types of compressional waves and one type of shear wave. At very low frequencies only one type of compressional wave prevails, and at very high frequencies the dissipation for all four types of waves is proportional to the square root of the frequency.