At the beginning of last century, Lamb obtained the displacement solutions of a wave field on the semi-infinite space subjected to a concentrated force. These solutions are widely applied in soil dynamics,seismic engineering and geophysics. Lamb's achievements are still being regarded as an important tool to solve the response and vibration problems in current research and engineering. However, it should be noted that Lamb's solutions only satisfy a homogeneous single medium. There are many dynamic problems of the multiphase or saturated porous medium in practice. Thus, scientists and technicians have been concerned with the displacement solutions to a semi-infinite space of a saturated porous medium subjected to a concentrated force.Philippacopoulos obtained the solutions on the semi-infinite space of a saturated porous medium subjected to a concentrated force in 1987. This step occurred for 73 years after Lamb finished the above-mentioned work, then 25 years after that Biot put forward a dynamics equation of a saturated porous medium. It is the mathematical difficulty that inhibits the development, and the most distinct difficulty is the coupling of fast and slow dilational waves in the dynamic equation of a saturated porous medium. Philippacopoulos have not solved the coupling problem. His solutions are gained with a corresponding determinant of Eigen equations,which equals to zero, so that the solutions are complex and the deduction is complicated. The loads applied to a saturated porous medium have been varied such as the moving, the torsions, the swing and so on. At this stage we should decouple the fast and slow dilational waves for regularization of solving the dynamics problem of a saturate porous medium. Based on the Homholtz's solution for the Biot dynamics equation of a saturated porous medium with complex-coefficient and Fourier transformation, especially, according to the reverse of the fast and slow dilational waves on vibration phase, the authors solved the coupling of the fast and slow dilational waves and obtained Green's function to the dynamics equation of a saturated porous medium in 1999. These results are consistent with Chen's results (1994) which were obtained by referring to the continuity of the solution to the dynamics equation with an inhomogeneous term δ function and the discontinuity of its first order derivative. Utilizing the Green functions and its transformation form in axisymmetric cylindrical coordinates, using Sommerfetd's integral and the influence of a free surface field, the authors obtained the dynamic displacement solutions of a saturated porous medium subjected to a concentrated force in the semiinfinite space. Naturally, the solutions are consistent with the Philippacopoulos's solutions. When a saturated porous medium decays a single-phase medium, the results are consistent with Lamb's results. The whole 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 fast and slow dilational waves of a saturated porous medium in this paper, it also provides reference to regularization and generalization for solving the dynamics problem in the semi-infinite space of a saturated porous medium.%利用作者根据饱和多孔介质动力学方程快、慢纵波解耦求得的集中力作用下饱和多孔介质Green函数解答,通过柱坐标变换,运用Sommerfeld积分,再根据自由表面应力为零的特征,添加自由表面影响场,从而求得半无限空间集中力作用下饱和多孔介质动力学问题的解答.其结果与Philippacopoulos解答结果一致;当饱和多孔蜕化为单相时与Lamb的方程一致.整个推导过程明了,物理意义也较为清晰;方法符合常规解法.因此,该方法为简化与规范饱和多孔介质动力学问题的解答提供了基础;并且能为一直未解决的半无限空间饱和多孔介质动力学问题的流相解答(诸如孔隙压、排水量)提供解决途径.
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