The Green's function solution of Lamb's problem for a saturated porous medium

At the beginning of last century, Lamb obtained thedisplacement solutions of a wave field on the semi-infinite space subjectedto a concentrated force. These solutions are widely applied in soildynamics, seismic engineering and geophysics. Lamb's achievements are stillbeing regarded as an important tool to solve the response and vibrationproblems in current research and engineering. However, it should be notedthat Lamb's solutions only satisfy a homogeneous single medium. There aremany dynamic problems of the multi-phase or saturated porous medium inpractice. Thus, scientists and technicians have been concerned with thedisplacement solutions to a semi-infinite space of a saturated porous mediumsubjected to a concentrated force. Philippacopoulos obtained the solutions onthe semi-infinite space of a saturated porous medium subjected to aconcentrated force in 1987. This step occurred for 73 years after Lambfinished the above-mentioned work, then 25 years after that Biot put forwarda dynamics equation of a saturated porous medium. It is the mathematicaldifficulty that inhibits the development, and the most distinct difficultyis the coupling of fast and slow dilational waves in the dynamic equation ofa saturated porous medium. Philippacopoulos have not solved the couplingproblem. His solutions are gained with a corresponding determinant of Eigenequations, which equals to zero, so that the solutions are complex and thededuction is complicated. The loads applied to a saturated porous mediumhave been varied such as the moving, the torsions, the swing and so on. Atthis stage we should decouple the fast and slow dilational waves forregularization of solving the dynamics problem of a saturate porous medium.Based on the Homholtz's solution for the Biot dynamics equation of asaturated porous medium with complex-coefficient and Fourier transformation,especially, according to the reverse of the fast and slow dilational waveson vibration phase, the authors solved the coupling of the fast and slowdilational waves and obtained Green's function to the dynamics equation of asaturated porous medium in 1999. These results are consistent with Chen'sresults (1994) which were obtained by referring to the continuity of thesolution to the dynamics equation with an inhomogeneous term \delta function and the discontinuity of its first order derivative. Utilizing theGreen functions and its transformation form in axisymmetric cylindricalcoordinates, using Sommerfeld's integral and the influence of a free surfacefield, the authors obtained the dynamic displacement solutions of asaturated porous medium subjected to a concentrated force in thesemi-infinite space. Naturally, the solutions are consistent with thePhilippacopoulos's solutions. When a saturated porous medium decays asingle-phase medium, the results are consistent with Lamb's results. Thewhole process of deduction is common with a clear physical meaning. Hence,it not only justifies the method of solving (by means of) a coupling of fastand slow dilational waves of a saturated porous medium in this paper, italso provides reference to regularization and generalization for solving thedynamics problem in the semi-infinite space of a saturated porous medium.