Show that, for a plane acoustic wave, the acoustic intensity $\tilde{p} \mathbf{u}$ may be written as $\rho_{0} c_{0}|\mathbf{u}|^{2} \hat{\mathbf{k}}$ in the standard notation.

Derive the general spherically-symmetric solution of the wave equation. Use it to find the velocity potential $\phi(r, t)$ for waves radiated into an unbounded fluid by a pulsating sphere of radius

$$a\left(1+\varepsilon e^{i \omega t}\right) \quad(\varepsilon \ll 1)$$

By considering the far field, or otherwise, find the time-average rate at which energy is radiated by the sphere.

$\left[\right.$ You may assume that $\nabla^{2} \phi=\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial \phi}{\partial r}\right)$.]