= Solution
Let $\phi_+$ and $p'_+$ denote the fields in $y>0$, and $\phi_-$ and $p'_-$ those in $y<0$. The linearized <kinematic boundary condition> equates the normal fluid velocity on both faces to the membrane velocity:
$$
\boxed{
\partial_y\phi_+(x,0,t)=\eta_t(x,t)
=\partial_y\phi_-(x,0,t).
}
$$
The linear acoustic pressure relation is
$$
p'_\pm=-\rho_0\partial_t\phi_\pm.
$$
Pressure below pushes upward and pressure above pushes downward. The transverse force balance for a membrane with mass per area $m$ and tension $T$ is consequently
$$
\boxed{
m\eta_{tt}=T\eta_{xx}+p'_- -p'_+,
}
$$
or, in terms of the velocity potentials,
$$
m\eta_{tt}-T\eta_{xx}
=\rho_0(\phi_{+,t}-\phi_{-,t})
\qquad(y=0).
$$
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