REFLECTION AND TRANSMISSION COEFFICIENTS

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

$$\begin{eqnarray} ({\bf P C} {\bf P}^{T} & - & \rho {\bf I}) \, \omega^{2} \vec{\... ... 0 & 0 & p_{z} & p_{y} & p_{x} & 0 \end{array} \right) \nonumber\end{eqnarray}$$ (8)

where $\bf I$ is the $3\times 3$ identity matrix.

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_z².

Each of the six eigenvalues $p_z^{\lambda}$ corresponds to a particle-motion direction $\vec{\bf v}_{\lambda}$. 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.