Wave equations of the two common sediment acoustic theories

Abstract

Two popular theories for wave propagation in sediments, the Biot poroelastic theory and the Viscous Grain Shearing (VGS) theory, have been formulated in terms of dispersion relations. The lack of explicit wave equations for these models could be considered to be a weakness. Hence, providing wave equations that are fully consistent with these theories might confirm that they have a physical basis. In the Biot theory, it is the frequency-dependent viscodynamic operator that poses the problem as there is no equivalent time-domain operator. Previous work has shown that this factor can be approximated well with a simplified square root operator. Drawing on the parallel to the Cole-Davidson dielectric theory for complex media where the same factor appears, and using the equivalent time-domain operator, which is a fractional pseudo-differential operator, enables the formulation of wave equations for all three wave modes of the Biot theory. The same operator turns out to be central for transforming the dispersion relations of the VGS theory to time-space wave equations. In this case, the relation between the dispersion equations and the wave equations is exact. The wave equations of these two theories will enable a closer study of the properties of the medium models.

Wave equations of the two common sediment acoustic theories