Dynamic Green's functions of spherical P1, P2 and SV sources embedded in a water-saturated half-space

The dynamic Green's functions of spherical dilatational source P1, P2 and SV-waves embedded in a water saturated half-space in the frequency domain are presented based on Biot's theory. Firstly, the spherical source in the space domain is expressed as the summation of cylindrical waves in the wave number domain using the Hankel transformation, and the image of the spherical source is introduced. For the dilatational source, the shear stress becomes zero with non-zero normal stress and pore pressure at the surface of the half-space, which are defined as the residual normal stress and pore pressure;while for the shear source, the normal stress and pore pressure become zero with none-zero shear stress, which is defined as the residual shear stress. Finally, the total responses in the wave number domain are obtained by adding the responses of the reversed residual stress, the residual pore pressure, the spherical source and the virtual spherical source, while the Green's functions in the space domain can be obtained by inverse Hankel transformation. The dynamic Green's functions presented in this paper can be used as the fundamental solutions for the indirect boundary element method, which is expected to contribute to solve symmetric three dimensional wave scattering problems in water saturated half-space.