Plane longitudinal waves in a liquid saturated porous geometrically nonlinear medium

This paper considers the propagation of plane longitudinal waves in a liquid-saturated porous medium with allowance for the nonlinear relationship between deformations and displacements of the solid phase. This porous liquid-saturated medium is examined herein within the framework of the classical Biot’s theory. It is shown that a mathematical model allowing for a geometric nonlinearity may be reduced to a system of evolutionary equations with respect to displacements of the medium skeleton and liquid in pores. The system of evolutionary equations, in its turn, depending on the availability of viscosity, is reduced to asimple wave equation or the generalized Burgers equation. The solution of the Riemann equation is obtained for a bell-shaped initial profile. The solution for the generalized Burgers equation has been found in the form of a stationary shock wave. The relationship between the amplitude and width of the shock wave front is established.