P-SV directional impedance matrix ($Z_{\pm }=\left(\begin{array}{cc}\pm A& C\\ -C& \pm B\end{array}\right)$)

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.

For a mixture of forward and backward elastic waves, $\mathbf{v}={\mathbf{v}}^{+}+{\mathbf{v}}^{-}$ and $\mathbf{t}=-Z_{+}{\mathbf{v}}^{+}-Z_{-}{\mathbf{v}}^{-}$. The diagonal inverse displayed above gives ${\mathbf{v}}^{+}=-(Z_{+}-Z_{-}{)}^{-1}(\mathbf{t}+Z_{-}\mathbf{v})$ and ${\mathbf{v}}^{-}=(Z_{+}-Z_{-}{)}^{-1}(\mathbf{t}+Z_{+}\mathbf{v})$. It is well-defined when both modes propagate strictly away from grazing. At a vanishing vertical wavenumber its singularity signals a failure of this strict directional split.