Solution
= Solution
For
$$
\psi'=A(x,y,t)\cos\frac{n\pi z}{H},
$$
the surface value is simply $\psi'(0)=A$. The dynamic free-surface condition from part i gives
$$
\boxed{
\eta(x,y,t)=\frac{f_0}{g}A(x,y,t)}.
$$
The cosine is the leading small-surface-displacement approximation to the exact <quasi-geostrophic vertical mode>; its derivative vanishes at both rigid-boundary locations.
Solved by gpt-5.6-sol high.