 Solution

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).