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
*p*_{z} (which gives simpler expressions than solving in terms of *k*_{z})
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

(8) | ||

Computing this system's eigenvalues requires the solution of a sixth-
order polynomial in *p*_{x}, *p*_{y}, and *p*_{z}. The solution expresses
the vertical slowness *p*_{z} as a sixth-order polynomial of a pair of
horizontal slownesses *p*_{x} and *p*_{y}. If the system has monoclinic
symmetry or higher, as is the case of transverse isotropy, we only
need to solve a cubic polynomial in *p*_{z}^{2}.

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.

11/12/1997