= Solution
At leading order the two wall conditions may be evaluated on the undeformed line $y=b$: shifting the evaluation point multiplies small perturbations and creates only second-order errors. The dynamic relation and the linear Bernoulli relation give $\eta=p'/(\rho_0k)=-\phi_t/k$. Differentiate this in time and use $\phi_y=\eta_t$:
$$
\boxed{k\phi_y+\phi_{tt}=0\quad\text{at }y=b.}
$$
The sign follows from the convention that positive <pressure> displaces the upper wall upward.
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