Derive the wave equation governing the velocity potential for linearised sound in a perfect gas. How is the pressure disturbance related to the velocity potential? Write down the spherically symmetric solution to the wave equation with time dependence $e^{i \omega t}$, which is regular at the origin.

A high pressure gas is contained, at density $\rho_{0}$, within a thin metal spherical shell which makes small amplitude spherically symmetric vibrations. Ignore the low pressure gas outside. Let the metal shell have radius $a$, mass $m$ per unit surface area, and elastic stiffness which tries to restore the radius to its equilibrium value $a_{0}$ with a force $-\kappa\left(a-a_{0}\right)$ per unit surface area. Show that the frequency of these vibrations is given by

$$\omega^{2}\left(m+\frac{\rho_{0} a_{0}}{\theta \cot \theta-1}\right)=\kappa \quad \text { where } \theta=\omega a_{0} / c_{0}$$