Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 40A / a / Solution

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.