Starting from the equations governing sound waves linearized about a state with density $\rho_{0}$ and sound speed $c_{0}$, derive the acoustic energy equation, giving expressions for the local energy density $E$ and energy flux $\mathbf{I}$.

A sphere executes small-amplitude vibrations, with its radius varying according to

$$r(t)=a+\operatorname{Re}\left(\epsilon e^{i \omega t}\right)$$

with $0<\epsilon \ll a$. Find an expression for the velocity potential of the sound, $\tilde{\phi}(r, t)$. Show that the time-averaged rate of working by the surface of the sphere is

$$2 \pi a^{2} \rho_{0} \omega^{2} \epsilon^{2} c_{0} \frac{\omega^{2} a^{2}}{c_{0}^{2}+\omega^{2} a^{2}}$$

Calculate the value at $r=a$ of the dimensionless ratio $c_{0} \bar{E} /|\overline{\mathbf{I}}|$, where the overbars denote time-averaged values, and comment briefly on the limits $c_{0} \ll \omega a$ and $c_{0} \gg \omega a$.