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

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.