Write down the equation governing linearized displacements $\mathbf{u}(\mathbf{x}, t)$ in a uniform elastic medium of density $\rho$ and Lamé constants $\lambda$ and $\mu$. Derive solutions for monochromatic plane $P$ and $S$ waves, and find the corresponding wave speeds $c_{P}$ and $c_{S}$.

Such an elastic solid occupies the half-space $z>0$, and the boundary $z=0$ is clamped rigidly so that $\mathbf{u}(x, y, 0, t)=\mathbf{0}$. A plane $S V$-wave with frequency $\omega$ and wavenumber $(k, 0,-m)$ is incident on the boundary. At some angles of incidence, there results both a reflected $S V$-wave with frequency $\omega^{\prime}$ and wavenumber $\left(k^{\prime}, 0, m^{\prime}\right)$ and a reflected $P$-wave with frequency $\omega^{\prime \prime}$ and wavenumber $\left(k^{\prime \prime}, 0, m^{\prime \prime}\right)$. Relate the frequencies and wavenumbers of the reflected waves to those of the incident wave. At what angles of incidence will there be a reflected $P$-wave?

Find the amplitudes of the reflected waves as multiples of the amplitude of the incident wave. Confirm that these amplitudes give the sum of the time-averaged vertical fluxes of energy of the reflected waves equal to the time-averaged vertical flux of energy of the incident wave.

[Results concerning the energy flux, energy density and kinetic energy density in a plane elastic wave may be quoted without proof.]