To propagate plane waves across a horizontal interface between two transversely isotropic homogeneous media, we must proceed somewhat different than previously. Because we do not know what the vertical wavenumber component will be across a horizontal interface (Snell's law only guarantees the continuity of the horizontal slownesses), we must solve the eigensystem in equation (1) in terms of pz (which gives simpler expressions than solving in terms of kz) for the upper and lower media, and impose the necessary boundary conditions at the interface. Rearranging equation (1) and expressing it in terms of slownesses, we obtain
Computing this system's eigenvalues requires the solution of a sixth- order polynomial in px, py, and pz. The solution expresses the vertical slowness pz as a sixth-order polynomial of a pair of horizontal slownesses px and py. If the system has monoclinic symmetry or higher, as is the case of transverse isotropy, we only need to solve a cubic polynomial in pz2.
Each of the six eigenvalues corresponds to a particle-motion direction . Wave-types are separated into three up-going and three down-going by computing the z-component of their Poynting vector. Its sign gives the sense of direction of the vertical energy flow.