Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 40C (Waves)

Write $\rho ={\rho }_0+\tilde{\rho }$, $p=p_0+\tilde{p}$, and let $\tilde{\mathbf{u}}$ be small. Linearized mass and momentum conservation areThe linearized equation of state is
$\tilde{p}=c_0^2\tilde{\rho }$. Differentiate the first equation in time and eliminate the velocity with the second to obtain the wave equation

With the convention $\tilde{p}=\mathrm{Re}(\hat{p}{e}^{i\omega t})$, a wave travelling in the positive $x$-direction hasThe linearized momentum equation givesfor this right-moving wave; the sign is reversed for a left-moving wave.

In $x<0$, writeIn the cavity, the rigid condition $\hat{u}(L)=0$ gives
$\hat{p}=C\cos k(x-L)$. Continuity of fluid velocity at the membrane and the membrane equationgive, after eliminating $C$ and $\hat{\xi }$,The numerator is the complex conjugate of the denominator, hence $|R|=1$. No mean acoustic energy can pass through the rigid end, so all incident energy is ultimately reflected.

The membrane is stationary when the incident and reflected velocities cancel, so $R=1$. The formula gives this while allowing a nonzero cavity pressure whenThese are standing-wave resonances of the closed cavity: both the membrane at $x=0$ and the rigid wall at $x=L$ are velocity nodes, while the pressure field is nontrivial.

Take the heavy-membrane limitaway from the resonances $\sin \beta =0$. Then $R\to 1$, the membrane velocity tends to zero, and the cavity amplitude obtained from the matching equations is $O({\alpha }^{-1})$ relative to the pressure in $x<0$. The membrane therefore acts as an almost rigid reflecting wall and scarcely excites the cavity.