= Reflection of a P-wave from a rigid plane
{title2=$A=(ab-\kappa^2)/(ab+\kappa^2)$}
For a stable isotropic solid in $y<0$, use $u_x=\phi_x+\psi_y$, $u_y=\phi_y-\psi_x$. An incident <P wave> with <frequency> $\omega$, tangential <wavenumber> $\kappa$ and upward normal <wavenumber> $a$ reflects into a P potential $Ae^{i(\kappa x-ay-\omega t)}$ and an SV potential $Be^{i(\kappa x-by-\omega t)}$, where $b^2=\omega^2/c_s^2-\kappa^2$. Vanishing of both displacements at the rigid plane gives $\kappa(1+A)-bB=0$, $a(1-A)-\kappa B=0$. Hence $A=(ab-\kappa^2)/(ab+\kappa^2)$ and $B=2a\kappa/(ab+\kappa^2)$. Because $c_p>c_s$ and $\kappa^2\leq\omega^2/c_p^2$, $b^2>0$ and the converted <S wave> propagates rather than being evanescent.
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