Dynamic And Static Green's Functions In Transversely Isotropic Elastic Media

Abstract

Concise and numerically feasible dynamic and static Green's functions are obtained in dyadic form by solving the wave equation and the equilibrium equation with general source distribution in transversely isotropic (TI) media. The wave and equilibrium equations are solved by using an extended version of the Kupradze method originally developed for isotropic media. The dynamic Green's function is expressed through three scalar quantities characterizing the propagation of SH and P-SV waves in a transversely isotropic medium. The 2-D inverse Laplacian operator contained in previous Green's function expressions is eliminated without limiting to special cases and geometries. The final dyadic form is similar to that of the isotropic dyadic Green's function, and therefore lends itself to easy analytical and numerical manipulations. The static Green's function has the same dyadic form as the dynamic function except that the three scalars must be redefined. From the dynamic Green's function, displacements due to vertical, horizontal, and explosive sources are explicitly given. The displacements of the explosive source show that an explosive source in a TI medium excites not only the quasi-P wave, but also the quasi-SV wave. The singular properties of the Green's functions are also addressed through their surface integrals in the limit of coinciding receiver and source. The singular contribution is shown to be -1/2 when the static stress Green's function is integrated over a half elliptical surface.