Riemann and Shock Waves in a Porous Liquid-Saturated Geometrically Nonlinear Medium

Abstract

Within the framework of the classical Biot theory, the propagation of plane longitudinal waves in a porous liquidsaturated medium is considered with account for the nonlinear connection between deformations and displacements of solid phase. It is shown that the mathematical model accounting for the geometrical nonlinearity of the medium skeleton can be reduced to a system of evolution equations for the displacements of the skeleton of medium and of the liquid in pores. The system of evolution equations, in turn, depending on the presence of viscosity, is reduced to the equation of a simple wave or to the equation externally resembling the Burgers equation. The solution of the Riemann equation is obtained for a bell-shaped initial profile; the characteristic wave breaking is shown. In the second case, the solution is found in the form of a stationary shock wave having the profile of a nonsymmetric kink. The relationship between the amplitude and width of the shock wave front has been established. It is noted that the behavior of nonlinear waves in such media differs from the standard one typical of dissipative nondispersing media, in which the propagation of waves is described by the classical Burgers equation.