Analytical Solution of Biot's Equations Based on Potential Functions Method

In this paper, a new analytical solution for Biot's equations is presented based on potential functions method. The primary coupled Biot's equations have been considered based on fluid and solid displacements in three-dimensional (3D) space. By defining some potential functions, the governing equations have been improved to a simpler form. Then the coupled Biot's equations have been replaced with four-decoupled equations, by doing some mathematical manipulations. For a case study, it is assumed that the incident wave is in xy-plane and for specific boundary conditions; the partial differential equations are converted to ordinary differential equations and solved analytically. Then two foams with different properties have been considered, and acoustical properties of these foams due to the new developed method have been compared with the corresponding results presented by transfer-matrix method. Good agreement between results verifies the new presented solution. Based on the potential function method, not only the acoustical properties of porous materials are calculated, but also the analytical values of all basic field variables, such as pressure, fluid, and solid displacements, are obtained for all points in the porous media. Furthermore, fundamental features, such as damped and undamped natural frequencies, and damping coefficient of porous materials are calculated by considering presented results. The obtained results show that maximum values of field variables, such as pressure, fluid, and solid displacements, happen at the damped natural frequencies of the porous media, as expected. By increasing material thickness, the effect of damping of porous material on damped natural frequency decreases. Damping decreases the first natural frequency of the foam up to 8.5%.