On the Biot slow S-wave

Abstract

It is accepted widely that the Biot theory predicts only one shear wave representing the in-phase/unison shear motions of the solid and fluid constituent phases (fast S-wave). The Biot theory also contains a shear mode wherein the two constituent phases essentially undergo out-of-phase shear motions (slow S-wave). From the outset of the development of the Biot framework, the existence of this mode has remained unnoticed because of an oversight in decoupling its system of two coupled equations governing shear processes. Moreover, in the absence of the fluid strain-rate term in the Biot constitutive relation, the velocity of this mode is zero. Once the Biot constitutive relation is corrected for the missing fluid strain-rate term (i.e., fluid viscosity), this mode turns out to be, in the inertial regime, a diffusive process akin to a viscous wave in a Newtonian fluid. In the viscous regime, it degenerates to a process governed by a diffusion equation with a damping term. Although this mode is damped so heavily that it dies off rapidly near its source, overlooking its existence ignores a mechanism to draw energy from seismic waves (fast P- and S-waves) via mode conversion at interfaces and at other material discontinuities and inhomogeneities. To illustrate the consequence of generating this mode at an interface, I examine the case of a horizontally polarized fast S-wave normal incident upon a planar air-water interface in a porous medium. Contrary to the classical Biot framework, which suggests that the incident wave should be transmitted practically unchanged through such an interface, the viscosity-corrected Biot framework predicts a strong, fast S-wave reflection because of the slow S-wave generated at the interface.