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Substitute the harmonic plane waveinto the elastic wave equation. The vector identitygivesIf $\mathbf{A}$ is parallel to $\mathbf{K}$, this becomesand gives a longitudinal P-wave. If $\mathbf{A}\cdot \mathbf{K}=0$, it becomesand gives either polarization of an S-wave.

For $\mathbf{K}=(k,0,m)$, convenient polarization vectors areThe SV-wave lies in the $xz$ propagation plane, whereas the SH-wave is perpendicular to that plane.

Taking the divergence of the Cauchy momentum equation and commuting constant-coefficient derivatives givesThus the dilatation is a wave of speedTaking the curl, and using $\nabla \times \nabla f=0$, givesso the rotation propagates atThese are respectively the speeds of P-waves and S-waves.

Let $k_S=\omega /c_S$ and $k_P=\omega /c_P$. Snell law for elastic waves fixes the reflected waves' common tangential wavenumber:Choose the reflected SV-wave and converted reflected P-wave asThe polarizations are respectively perpendicular and parallel to their wavevectors. Phase matching makes all three exponential factors equal at $z=0$. The rigid boundary condition for an elastic wave ${\mathbf{u}}_I+{\mathbf{u}}_{RS}+{\mathbf{u}}_{RP}=0$ there givesSolving this two-by-two linear system yieldsThese fields give the Reflection of an SV-wave from a rigid plane.