The Causal Properties of the Compressional Wave in an Unconsolidated Marine Sediment

Abstract

The Viscous Grain Shearing (VGS) theory predicts the existence of a compressional wave and a shear wave in an unconsolidated marine sediment. Although it is known that, subject to certain constraints, the shear wave satisfies causality, the causal nature of the compressional wave is less well understood. In this paper, the VGS compressional-wave speed and attenuation are examined in three frequency regimes, where it is shown that they follow approximately frequency power laws. It is then proved that the VGS propagation factor, which is a combination of the phase speed and attenuation, is a causal transform: its inverse Fourier transform is zero for all times prior to the onset of the source. The derivation of this result, which is a necessary condition if the VGS compressional wave equation is to satisfy causality, includes the development of a technique for evaluating a class of previously unknown integrals. This integration procedure relies on a limiting argument combined with certain Fourier transforms, the latter taking the form of “improper” integrals, which, it is shown, can be expressed explicitly based on the properties of generalized functions. An expression for the impulse response of the VGS compressional wave is also developed and shown to satisfy causality, although the transition from zero to the peak level is abrupt, quite unlike the perfectly smooth behavior exhibited by the impulse response of the VGS shear wave, which is maximally flat everywhere in the medium at the instant the source is activated.