An elastic solid with density $\rho$ has Lamé moduli $\lambda$ and $\mu$. Write down equations satisfied by the dilational and shear potentials $\phi$ and $\boldsymbol{\psi}$.

For a two-dimensional disturbance give expressions for the displacement field $\mathbf{u}=\left(u_{x}, u_{y}, 0\right)$ in terms of $\phi(x, y ; t)$ and $\psi=(0,0, \psi(x, y ; t))$.

Suppose the solid occupies the region $y<0$ and that the surface $y=0$ is free of traction. Find a combination of solutions for $\phi$ and $\psi$ that represent a propagating surface wave (a Rayleigh wave) near $y=0$. Show that the wave is non-dispersive and obtain an equation for the speed $c$. [You may assume without proof that this equation has a unique positive root.]