Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 40A (Waves)

Write the SH displacement as $u=U(z)e^{i(kx-\omega t)}$ with $c=\omega /k$. A trapped mode requiresIn the substrate $U=A{e}^{qz}$, whereand the stress-free condition at $z=h$ makes the layer solution $U=B\cos (m(z-h))$, whereContinuity of displacement and shear stress at $z=0$ eliminates $A,B$ and givesThe bars lost from the converted statement are required dimensionally and by the stated ordering of wave speeds.

Both sides of the dispersion relation are positive on an interval beginning at zero phase, so a zeroth branch exists for every $k>0$. Higher branches first appear when $c=c_s$, for which the right side vanishes and $mh=n\pi$. ThusandThe $n=1$ dispersion curve begins on the line $\omega =c_{s}k$ at $(k_c^{(1)},{\omega }_c^{(1)})$, bends below it, and approaches the layer shear-wave line $\omega ={\bar{c}}_{s}k$ from above as $k\to \infty$.