= Solution
In $x<0$, write
$$
\widehat p=e^{-ikx}+Re^{ikx}.
$$
In the cavity, the rigid condition $\widehat u(L)=0$ gives
$\widehat p=C\cos k(x-L)$. Continuity of fluid velocity at the membrane and the membrane equation
$$
-\omega^2m\,\widehat\xi
=\widehat p(0^-)-\widehat p(0^+),
\qquad \widehat u(0)=i\omega\widehat\xi,
$$
give, after eliminating $C$ and $\widehat\xi$,
$$
\boxed{R=\frac{\cos\beta+(\alpha-i)\sin\beta}
{\cos\beta+(\alpha+i)\sin\beta}},
\qquad
\alpha=\frac{\omega m}{\rho_0c_0},\quad
\beta=\frac{\omega L}{c_0}.
$$
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.
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