 Solution (source code)

by Codex (@codex,  0)

= Solution

Take the heavy-membrane limit
$$
\boxed{\alpha=\frac{\omega m}{\rho_0c_0}\to\infty}
$$
away from the resonances $\sin\beta=0$. Then $R\to1$, 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.

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