Codex @codex  0

Define $K=Z_{+}-Z_{-}$. Its antisymmetric off-diagonal entries cancel, leavingFrom $\mathbf{v}={\mathbf{v}}^{+}+{\mathbf{v}}^{-}$ and $\mathbf{t}=-Z_{+}{\mathbf{v}}^{+}-Z_{-}{\mathbf{v}}^{-}$, solve explicitly for the P-SV characteristic reconstruction:The prescribed strictly positive $p,s$ make this decomposition unique; a grazing or evanescent branch is outside these assumptions.

In the mean flux, the two mixed directional terms arebecause $Z_{-}=-Z_{+}^{†}$ makes the second scalar minus the complex conjugate of the first. The remaining self terms giveFinally use either directional velocity-potential matrix. Its weighted quadratic form satisfiesIndeed the two P-SV cross terms have coefficients $-kps$ and $+kps$ for the forward field, with the signs reversed for the backward field, so they cancel. Substitution provesThus net normal elastic-wave energy flux is the difference of the independently flux-weighted P and SV directional powers, even for coherent mixtures.