P-SV displacement potentials ($u_x={\varphi }_x-{\psi }_z,u_z={\varphi }_z+{\psi }_x$)

In a uniform isotropic material, these scalar P-SV displacement potentials separate longitudinal P waves from vertically polarized SV-waves in an $x,z$ plane. They satisfy ${\varphi }_{tt}={\alpha }^2\Delta \varphi$ and ${\psi }_{tt}={\beta }^2\Delta \psi$, with compressional and shear speeds $\alpha ,\beta$. Reversing the sign convention for $\psi$ changes its polarization column but not the physical P-SV directional impedance matrix or the separate modal energy fluxes.