Wave propagation in micro-heterogeneous porous media: a model based on an integro-differential wave equation

A model hyperbolic partial differential equation with singular convolution operators and infinitely smooth solutions is studied. It is shown that short pulses, including finite-bandwidth pulses, propagate with a delay with respect to the wavefront. For a two-parameter family of such equations Green's functions are obtained in a simple self-similar form. As an application, it is demonstrated that the Gurevich-Lopatnikov dispersion law for a thin-layered porous medium can be approximated by a hyperbolic equation with singular memory.