Snell's law in transversely isotropic media
P. F. Daley
The problem of reflection and transmission of waves at a plane boundary separating two transversely isotropic media is considered subject to the general condition that the axes of anisotropy in both media need not align with the intervening interface. Slowness vectors and surfaces are employed in the treatment presented. The characteristics or rays, defining the direction of energy propagation, are normal to the slowness surface, which makes slowness space ideally suited for treatment of anisotropic problems, as both Snell's Law and ray properties are formulated in terms of slowness. Ray-vector magnitudes are not dealt with here as this topic warrants a separate treatment, and the inclusion of which would create needless complexity.
The exact eikonals (Hamiltonians) of the coupled quasi-compressional (qP) and quasi-shear (qSV) wave propagation are used in the derivations and are homogeneous of order 2 in powers of slowness, which is a requisite for the use of the theory of characteristics. Once this condition is violated through simplifications of an eikonal, the theory of characteristics is not applicable. As it is an extremely useful mathematical tool for dealing with problems related to wave propagation, negating it seems counterproductive, unless it is done for a specific rather than general purpose. The possibility of components of the slowness vector becoming complex is briefly considered. This is motivated by the fact that, for post-critical regions of quantities such as the PP reflection coefficient at an interface between two transversely isotropic media, this becomes a factor in the proper computation of certain quantities.