Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 40C (Waves)

Substitution gives the displayed relation with $c_P^2=(\lambda +2\mu )/\rho$ and $c_S^2=\mu /\rho$. Resolving $A$ parallel and perpendicular to $k$ gives a longitudinal P mode of speed $c_P$ and two transverse polarizations of speed $c_S$, conventionally SV in and SH normal to the propagation plane.

In the lower solid $l^2+m^2={\omega }^2/c_S^2$. Put $m^{\prime 2}={\omega }^2/c_S^{\prime 2}-l^2$ and $Z={\mu }^{\prime }{m}^{\prime }\cot (m^{\prime }h)$. Matching displacement and shear stress at $y=0$, with zero displacement at $h$, givesFor real propagating parameters $|R|=1$: the rigidly backed lossless layer transmits no mean energy, so all incident energy is reflected (though with a phase shift).