(a) Write down the linearised equations governing motion of an inviscid compressible fluid at uniform entropy. Assuming that the velocity is irrotational, show that the velocity potential $\phi(\mathbf{x}, t)$ satisfies the wave equation and identify the wave speed $c_{0}$. Obtain from these linearised equations the energy-conservation equation

$$\frac{\partial E}{\partial t}+\nabla \cdot \mathbf{I}=0$$

and give expressions for the acoustic-energy density $E$ and the acoustic-energy flux, or intensity, I.

(b) Inviscid compressible fluid with density $\rho_{0}$ and sound speed $c_{0}$ occupies the regions $y<0$ and $y>0$, which are separated by a thin elastic membrane at an undisturbed position $y=0$. The membrane has mass per unit area $m$ and is under a constant tension $T$. Small displacements of the membrane to $y=\eta(x, t)$ are coupled to small acoustic disturbances in the fluid with velocity potential $\phi(x, y, t)$.

(i) Write down the (linearised) kinematic and dynamic boundary conditions at the membrane. [Hint: The elastic restoring force on the membrane is like that on a stretched string.]

(ii) Show that the dispersion relation for waves proportional to $\cos (k x-\omega t)$ propagating along the membrane with $|\phi| \rightarrow 0$ as $y \rightarrow \pm \infty$ is given by

$$\left\{m+\frac{2 \rho_{0}}{\left(k^{2}-\omega^{2} / c_{0}^{2}\right)^{1 / 2}}\right\} \omega^{2}=T k^{2}$$

Interpret this equation by explaining physically why all disturbances propagate with phase speed $c$ less than $(T / m)^{1 / 2}$ and why $c(k) \rightarrow 0$ as $k \rightarrow 0$.

(iii) Show that in such a wave the component $\left\langle I_{y}\right\rangle$ of mean acoustic intensity perpendicular to the membrane is zero.