= Solution
At the free surface, the linear <kinematic boundary condition> is
$$
w'(0)=\eta_t.
$$
Hydrostatic pressure and geostrophic-streamfunction normalization give
$$
p'(0)=\rho_0f_0\psi'(0)=\rho_0g\eta,
\qquad
\eta=\frac{f_0}{g}\psi'(0).
$$
Combining this with
$$
f_0\psi'_{zt}+N^2w'=0
$$
gives
$$
\frac{\partial}{\partial t}
\left(\psi'_z+\frac{N^2}{g}\psi'\right)=0
\qquad(z=0).
$$
For an oscillatory disturbance with no time-independent boundary offset, the free-surface boundary condition is therefore
$$
\boxed{
\psi'_z+\frac{N^2}{g}\psi'=0
\qquad(z=0)}.
$$
The lower rigid boundary retains $\psi'_z(-H)=0$.
Solved by gpt-5.6-sol high.
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