Summary. In this paper we study Biot’s full, time-dependent equations of dynamic poroelasticity with a view to understanding the effect of pore fluid on seismic wave propagation. Typical values of the constants appearing in the equations which are relevant to the rock surrounding earthquake sources are estimated from values appearing in the recent literature. We investigate the disturbance due to an instantaneous point body force acting in a uniform whole space. In fact we calculate the tensor fundamental solution since this has spherical symmetry, which is strongly exploited in our method of solution. The introduction of four scalar potentials enables us to reduce the problem to two decoupled second-order systems, each consisting of two coupled wave equations with friction in one space and one time dimension. By a further transformation these systems are expressed as symmetric hyperbolic systems of the first order, which are then solved by Laplace transforms. Because the dispersion equations are of higher than second degree only the large time saddle-point contributions are calculated. From these several phenomena emerge. (a) A P wave propagating with the P-wave speed appropriate to the ‘solid’ obtained by constraining the fluid to move with the solid matrix. However, instead of a 6 pulse shape familiar in elastodynamics this P wave has the shape of a Gaussian which appears to diffuse in a frame of reference moving with the P-wave speed. (b) An S wave with similar shape to P. (c) A long-term diffusion which is what one obtains from the equations reduced by setting the inertial terms to zero as in consolidation theory. We also investigate in an appendix a special case of dynamical compatibility in which the P wave remains sharp (i.e. a 6 pulse) and one of our two systems can be solved explicitly. The pulse diffusion amounts to a dissipation of the high frequency content qf seismic waves at a rate proportional to the square of the frequency.

The fundamental solution in dynamic poroelasticity

Time domain fundamental solution to Biot's complete equations of dynamic poroelasticity. Part I: Two-dimensional solution†

Employing Biot's theory for wave propagation in a porous solid, the frequency equation for radial vibrations of a poroelastic cylinder is obtained. The frequency equation has been derived in the form of a determinant involving Bessel functions. The roots of the frequency equation give the values of the characteristic circular frequency parameters of the first four modes for various geometries. These roots, which correspond to various modes, are numerically calculated and presented graphically. The results indicate that the effects of porosity are very pronounced.

Natural frequencies of a poroelastic hollow cylinder

On the application of Biot's theory to acoustic wave propagation in snow

Reflection and transmission of elastic waves at an elastic/porous solid saturated by two immiscible fluids

In this investigation, Biot’s theory was employed in the study of wave motions in fluid‐saturated porous rocks. Consistent with the described experimental arrangement, Biot’s equations were solved using a Laplace transformation. The theory predicts two dilatational waves: a slightly dispersed fast wave propagating ahead of a heavily dispersed and attenuated slow wave. By comparing these results with experimental results, it becomes evident that the measured waves are in fact the fast waves.Subject Classification: [43]20.15, [43]20.40.

P. N. Jogi

Papers

Study of wave motions in fluid‐saturated porous rocks