= P-SV directional impedance matrix
{c}
{title2=$Z_\pm=\begin{pmatrix}\pm A&C\\-C&\pm B\end{pmatrix}$}
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 $Z_-=-Z_+^\dagger$. The <Hermitian part> of $Z_+$ is positive diagonal; thus forward/backward fluxes have opposite signs and mixed directional flux cross terms cancel.
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