Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 40D (Waves)

Taking the divergence of the elastic equation givesTaking its curl givesThus the Helmholtz separation of elastic waves produces longitudinal P-waves and transverse S-waves with

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.

Seek an SH displacementThe S-wave equation givesRigidity at $z=0$ gives $Y(0)=0$, while zero tangential traction at the free surface gives
$\mu Y^{\prime }(h)=0$. ThussoThese are the guided SH modes between rigid and free planes.

At zero horizontal wavenumber, the dispersion relation gives the vertical-mode cutoff

Sincethe phase and group velocities are

The preceding formulas giveTherefore

A horizontal wavelength $h$ means $\kappa =2\pi /h$. For $n=1$,Hence