Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 40C / c / i / Solution

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.

Solved by gpt-5.6-sol high.