 P-SV directional impedance matrix

For real frequency $\omega >0$ and both vertical wavenumbers $p=k_{\alpha }>0$, $s=k_{\beta }>0$, let $D=k^2+ps$, $A=\rho \omega p/D$, $B=\rho \omega s/D$, and $C=k(2\mu D-\rho {\omega }^2)/(\omega D)$. With traction $\mathbf{t}=({\sigma }_{xz},{\sigma }_{zz})$ and particle velocity $\mathbf{v}=(v_x,v_z)$, each directional P wave/SV-wave field obeys $\mathbf{t}=-Z_{\pm }\mathbf{v}$. Direct displacement-potential substitution yields $Z_{-}=-Z_{+}^{†}$. The Hermitian part of $Z_{+}$ is positive diagonal; thus forward/backward fluxes have opposite signs and mixed directional flux cross terms cancel.