An elastic solid of density $\rho$ has Lamé moduli $\lambda$ and $\mu$. From the dynamic equation for the displacement vector $\mathbf{u}$, derive equations satisfied by the dilatational and shear potentials $\phi$ and $\psi$. Show that two types of plane harmonic wave can propagate in the solid, and explain the relationship between the displacement vector and the propagation direction in each case.

A semi-infinite solid occupies the half-space $y<0$ and is bounded by a traction-free surface at $y=0$. A plane $P$-wave is incident on the plane $y=0$ with angle of incidence $\theta$. Describe the system of reflected waves, calculate the angles at which they propagate, and show that there is no reflected $P$-wave if

$$4 \sigma(1-\sigma)^{1 / 2}(\beta-\sigma)^{1 / 2}=(1-2 \sigma)^{2}$$

where

$$\sigma=\beta \sin ^{2} \theta \quad \text { and } \quad \beta=\frac{\mu}{\lambda+2 \mu}$$