Compressible fluid of equilibrium density $\rho_{0}$, pressure $p_{0}$ and sound speed $c_{0}$ is contained in the region between an inner rigid sphere of radius $R$ and an outer elastic sphere of equilibrium radius $2 R$. The elastic sphere is made to oscillate radially in such a way that it exerts a spherically symmetric, perturbation pressure $\tilde{p}=\epsilon p_{0} \cos \omega t$ on the fluid at $r=2 R$, where $\epsilon \ll 1$ and the frequency $\omega$ is sufficiently small that

$$\alpha \equiv \frac{\omega R}{c_{0}} \leqslant \frac{\pi}{2}$$

You may assume that the acoustic velocity potential satisfies the wave equation

$$\frac{\partial^{2} \phi}{\partial t^{2}}=c_{0}^{2} \nabla^{2} \phi$$

(a) Derive an expression for $\phi(r, t)$.

(b) Hence show that the net radial component of the acoustic intensity (wave-energy flux) $\mathbf{I}=\tilde{p} \mathbf{u}$ is zero when averaged appropriately in a way you should define. Interpret this result physically.

(c) Briefly discuss the possible behaviour of the system if the forcing frequency $\omega$ is allowed to increase to larger values.

$\left[\right.$ For a spherically symmetric variable $\left.\psi(r, t), \nabla^{2} \psi=\frac{1}{r} \frac{\partial^{2}}{\partial r^{2}}(r \psi) .\right]$