Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 40D / b

A P-wave is longitudinal. For
$\hat{k}=(\sin \theta ,0,\cos \theta )$, its amplitude isparallel to the wavevector. Substitution gives $\omega =c_p\kappa$.

An SV-wave is transverse but polarized in the vertical plane containing the wavevector:It is perpendicular to $\hat{k}$ and has $\omega =c_s\kappa$.

An SH-wave is polarized horizontally, normal to the vertical propagation plane:It is also perpendicular to $\hat{k}$ and has $\omega =c_s\kappa$. A horizontal boundary distinguishes these two otherwise degenerate S polarizations.