High order asymptotic approximations for surface Love waves of SH type in transversely isotropic media

Uniform asymptotics of the surface Love waves for the case of transverse isotropy of an elastic medium (being a special type of anisotropy) is obtained. The waves under investigation are similar to the SH Love waves well known from the isotropic theory of elasticity. The separate Love mode is represented as a sum of a space time (ST) caustic expansion (for a surface wave with a real eikonal), and two correction terms being the ST ray solutions with complex eikonals (inhomogeneous waves). Due to the specific structure of the elasticity tensor in a transversely isotropic medium, the boundary surface is necessarily a plane. The procedure for finding the coefficients of the ray series is different from that for an isotropic medium. The recursion process involved in the computation of higher terms of the asymptotic expansion allows one to see the transition of the relations obtained into the known ray solutions for isotropic media. Two inhomogeneous waves manifest itself in high-order terms of the asymptotics of the surface waves. The contribution of inhomogeneous waves to the wave field is shown to depend on the first terms (next the principal term) of the ray series. The validity of the asymptotic formulas obtained is justified by the relations found for the elasticity parameters of the transversely isotropic medium, which ensure the existence of SH Love waves in the medium.