Derive the wave equation governing the velocity potential for linearised sound waves in a perfect gas. How is the pressure disturbance related to the velocity potential?

A high pressure gas with unperturbed density $\rho_{0}$ is contained within a thin metal spherical shell which makes small amplitude spherically symmetric vibrations. Let the metal shell have radius $a$, mass $m$ per unit surface area, and an 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. Assume that there is a vacuum outside the spherical shell. Show that the frequencies $\omega$ of vibration satisfy

$$\theta^{2}\left(1+\frac{\alpha}{\theta \cot \theta-1}\right)=\frac{\kappa a_{0}^{2}}{m c_{0}^{2}}$$

where $\theta=\omega a_{0} / c_{0}, \alpha=\rho_{0} a_{0} / m$, and $c_{0}$ is the speed of sound in the undisturbed gas. Briefly comment on the existence of solutions.

[Hint: In terms of spherical polar coordinates you may assume that for a function $\psi \equiv \psi(r)$,

$$\nabla^{2} \psi=\frac{1}{r} \frac{\partial^{2}}{\partial r^{2}}(r \psi)$$