On Green’s functions for elastic waves in anisotropic media

The Green’s function, as a temporal Fourier transform of the point-source solution of the wave equation in an infinite medium, is obtained for the homogeneous, transversely isotropic solids, such as fiber composites or directionally solidificated steels of the columnar-grained polycrystalline structure. The surface integrals, representing the exact point-source solutions for the quasilongitudinal, quasitransverse, and purely transverse horizontally polarized shear waves, are approximated in terms of elementary functions, given that the elastic parameters characterizing the deviation of the solid from the isotropic medium are relatively small (weak anisotropy approximation). A brief discussion of the main features of the solution and a comparison to well-known eikonal (ray) approximations are presented.