Solution

Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 78 / 3 / i / Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 40D / c / i / Solution

For a layer $0<z<h$ with rigid lower boundary and traction-free upper boundary, an SH mode
$u_y=Y(z)e^{i(\kappa x-\omega t)}$ has
$$
Y(z)=\sin(q_nz),\qquad
q_n=\frac{(2n+1)\pi}{2h},
$$
and dispersion
$$
\omega^2=c_s^2(\kappa^2+q_n^2).
$$
Its cutoff is $\omega_n=c_sq_n$, its phase speed is $\omega/\kappa$, and its group speed is $c_s^2\kappa/\omega$.

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 Incoming links: Guided SH modes between rigid and free planes