Rigid elastic waveguide mode

In a homogeneous elastic layer $-H<z<H$ with rigid boundaries, in-plane propagating modes mix P- and SV-waves. For phase speed $c>c_P>c_S$, one even-horizontal, odd-vertical family obeys

A shear-horizontal wave mode has ${\omega }_n(k)=c_s\sqrt{k^2+(n\pi /h{)}^2}$. At $x=Vt$ with $0<|V|<c_s$, a nondegenerate stationary Fourier phase gives generic $t^{-1/2}$ decay. Faster rays have no stationary point; compactly supported data give eventual zero by finite propagation speed.