Finding wave equations for sediment acoustic theories

Abstract

The two main theories for wave propagation in sediments, the Extended Biot theory and the Viscous Grain Shearing (VGS) theory, have been formulated in terms of dispersion equations only. Some have considered the lack of a wave equation to be a weakness that may indicate lack of a physical basis for these theories. In the Biot theory, it is the frequency-dependent viscodynamic operator which models viscous boundary effects in pores that poses the problem. It can be formulated as a ratio of Bessel functions for circular pores and as a ratio of tanh-functions for a 2D parallel plane duct. There is no simple equivalent time domain operator for use in the wave equation. In both geometries, the factor may, however, be approximated well with (1 + icω)1/2, where c is a constant. This factor also appears in the Cole-Davidson dielectric theory for complex media. The equivalent time domain operator, the fractional pseudo-differential operator, enables the formulation of wave equations for all three wave modes. The same operator turns out to be central for transforming the dispersion relations of the VGS theory to time-space wave equations. These wave equations enable a closer study of properties of the medium models than that obtained from dispersion relations alone.