(a) Derive the wave equation for perturbation pressure for linearised sound waves in a compressible gas.

(b) For a single plane wave show that the perturbation pressure and the velocity are linearly proportional and find the constant of proportionality, i.e. the acoustic impedance.

(c) Gas occupies a tube lying parallel to the $x$-axis. In the regions $x<0$ and $x>L$ the gas has uniform density $\rho_{0}$ and sound speed $c_{0}$. For $0<x<L$ the temperature of the gas has been adjusted so that it has uniform density $\rho_{1}$ and sound speed $c_{1}$. A harmonic plane wave with frequency $\omega$ and unit amplitude is incident from $x=-\infty$. If $T$ is the (in general complex) amplitude of the wave transmitted into $x>L$, show that

$$|T|=\left(\cos ^{2} k_{1} L+\frac{1}{4}\left(\lambda+\lambda^{-1}\right)^{2} \sin ^{2} k_{1} L\right)^{-\frac{1}{2}}$$

where $\lambda=\rho_{1} c_{1} / \rho_{0} c_{0}$ and $k_{1}=\omega / c_{1}$. Discuss both of the limits $\lambda \ll 1$ and $\lambda \gg 1$.