Write down the linearised equations governing motion of an inviscid compressible fluid at uniform entropy. Assuming that the velocity is irrotational, show that it may be derived from a velocity potential $\phi(\mathbf{x}, t)$ satisfying the wave equation

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

and identify the wave speed $c_{0}$. Obtain from these linearised equations the energyconservation 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 $\mathbf{I}$ in terms of $\phi$.

Such a fluid occupies a semi-infinite waveguide $x>0$ of square cross-section $0<y<a$, $0<z<a$ bounded by rigid walls. An impenetrable membrane closing the end $x=0$ makes prescribed small displacements to

$$x=X(y, z, t) \equiv \operatorname{Re}\left[e^{-i \omega t} A(y, z)\right]$$

where $\omega>0$ and $|A| \ll a, c_{0} / \omega$. Show that the velocity potential is given by

$$\phi=\operatorname{Re}\left[e^{-i \omega t} \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \cos \left(\frac{m \pi y}{a}\right) \cos \left(\frac{n \pi z}{a}\right) f_{m n}(x)\right]$$

where the functions $f_{m n}(x)$, including their amplitudes, are to be determined, with the sign of any square roots specified clearly.

If $0<\omega<\pi c_{0} / a$, what is the asymptotic behaviour of $\phi$ as $x \rightarrow+\infty$ ? Using this behaviour and the energy-conservation equation averaged over both time and the crosssection, or otherwise, determine the double-averaged energy flux along the waveguide,

$$\left\langle\overline{I_{x}}\right\rangle(x) \equiv \frac{\omega}{2 \pi a^{2}} \int_{0}^{2 \pi / \omega} \int_{0}^{a} \int_{0}^{a} I_{x}(x, y, z, t) \mathrm{d} y \mathrm{~d} z \mathrm{~d} t$$

explaining why this is independent of $x$.