Show that in an acoustic plane wave the velocity and perturbation pressure are everywhere proportional and find the constant of proportionality.

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 gas is cooled so that it has uniform density $\rho_{1}$ and sound speed $c_{1}$. A harmonic plane wave with frequency $\omega$ is incident from $x=-\infty$. Show that the amplitude of the wave transmitted into $x>L$ relative to that of the incident wave is

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

where $\lambda=\rho_{1} c_{1} / \rho_{0} c_{0}$ and $k_{1}=\omega / c_{1}$.

What are the implications of this result if $\lambda \gg 1$ ?