Radiation from a point source and scattering theory in a fluid-saturated porous solid

Abstract

The time harmonic Green function for a point load in an unbounded fluid‐saturated porous solid is derived in the context of Biot’s theory. The solution contains the two compressional waves and one transverse wave that are predicted by the theory and have been observed in experiments. At low frequency, the slow compressional wave is diffusive and only the fast compressional and transverse waves radiate energy. At high frequency, the slow wave radiates, but with a decay radius which is on the order of cm in rocks. The general problem of scattering by an obstacle is considered. The point load solution may be used to obtain scattered fields in terms of the fields on the obstacle. Explicit expressions are presented for the scattering amplitudes of the three waves. Simple reciprocity relations between the scattering amplitudes for plane‐wave incidence are also given. These hold under the interchange of incident and observation directions and are completely general results. Finally, the point source solution is Fourier transformed to get the solution for a load which is a delta function in time as well as space. We obtain a closed form expression when there is no damping. The three waves radiate from the source as distinct delta function pulses. With damping present, asymptotic approximations show the slow wave to be purely diffusive. The fast and transverse waves propagate as pulses. The pulses are Gaussian‐shaped, which broaden with increasing time or radial distance.