Non-linear Evolution of P-waves in Viscous–Elastic Granular Saturated Media

The Frenkel–Biot P-wave of the first type is a seismic longitudinal wave observed in rocks fully saturated with oil, water or high-pressure gas. The P-wave of the second type is observed in unsaturated soils and other porous media saturated with gas of low pressure. Their models include properties of the skeleton, that is, its elastic modules and its own viscosity. If the non-linear terms are accounted for, the asymptotic analysis, usual for weak non-linear waves, might be applied to get the wave spectrum evolution. The wetness of grains contacts in soils and such components of oil as tars or bitumen, which attached to the skeleton, can be described by generalized viscous–elastic stress–strain connections. The latter are nominated in such a way that creates the narrow frequency interval of wave of negative dissipation where the non-linear terms begin to play the main role besides the neutral stability for waves of zero wave number. The corresponding case, relevant to single continuum model, was analyzed in the literature. Here it is shown that the interpenetrating continua with interaction of the Darcy type provide the dissipation sink in the wave evolution equation. This generalization, (Tribelsky, M.I.: Phys. Rev. Lett. (2007, submitted)), can stabilize the asymptotic solution of the evolution equation, where the dispersion terms are omitted. The asymptotic solution of the equation is invariant to initial conditions and it means a transformation of initial wave spectra to unique one while wave is spreading in the viscous–elastic medium under consideration. This explains the phenomenon, observed in wave tests at marine beach, when any dynamics action (impact, explosion, and ultrasound action) created at some distance a wave of a single frequency (~25 Hz).