An acoustic plane wave (not necessarily harmonic) travels at speed $c_{0}$ in the direction $\hat{\mathbf{k}}$, where $|\hat{\mathbf{k}}|=1$, through an inviscid, compressible fluid of unperturbed density $\rho_{0}$. Show that the velocity $\tilde{\mathbf{u}}$ is proportional to the perturbation pressure $\tilde{p}$, and find $\tilde{\mathbf{u}} / \tilde{p}$. Define the acoustic intensity $\mathbf{I}$.

A harmonic acoustic plane wave with wavevector $\mathbf{k}=k(\cos \theta, \sin \theta, 0)$ and unitamplitude perturbation pressure is incident from $x<0$ on a thin elastic membrane at unperturbed position $x=0$. The regions $x<0$ and $x>0$ are both occupied by gas with density $\rho_{0}$ and sound speed $c_{0}$. The kinematic boundary conditions at the membrane are those appropriate for an inviscid fluid, and the (linearized) dynamic boundary condition

$$m \frac{\partial^{2} X}{\partial t^{2}}-T \frac{\partial^{2} X}{\partial y^{2}}+[\tilde{p}(0, y, t)]_{-}^{+}=0$$

where $T$ and $m$ are the tension and mass per unit area of the membrane, and $x=X(y, t)$ (with $|k X| \ll 1$ ) is its perturbed position. Find the amplitudes of the reflected and transmitted pressure perturbations, expressing your answers in terms of the dimensionless parameter

$$\alpha=\frac{\rho_{0} c_{0}^{2}}{k \cos \theta\left(m c_{0}^{2}-T \sin ^{2} \theta\right)} .$$

Hence show that the time-averaged energy flux in the $x$-direction is conserved across the membrane.