Based on Biot wave theory of porous media

a model was established for non-homogeneous saturated half-space

in which porosity

density

shear modulus and coefficient of permeability were coupled with each other and varied along the depth at the same time. A three-dimensional (3D) dynamic control equation with soil skeleton displacement and pore pressure as the fundamental unknowns is constructed in the cylindrical coordinate system. Operator operation and Hankel integral transformation are used to solving the control equation

and the product decomposition of the vibration response of the half-space under the action of simple harmonic concentrated forces is obtained. The results obtained in this paper are reduced to homogeneous saturated half-space and elastic half-space

respectively

and compared with the classical Lamb solution

the accuracy of results is also verified. Based on the existing results

the coupling relations among porosity

density

shear modulus and coefficient of permeability are given and substituted into the derived results for numerical calculation. The dynamic responses of water-saturated foundation and gas saturated foundation (dry soil) are analyzed respectively. The effect of non-uniform gradient on the calculated results is also investigated. The results show that the coupling of four parameters along the depth of the foundation has a certain influence on the dynamic response of the foundation

and the attenuation rate of vibration displacement and pore pressure in the formation is accelerated. Since the viscosity of water is much higher than that of gas

the vibration attenuation in the water-saturated ground is much faster. The higher the degree of non-uniformity

the more obvious the influence of coupling effect.