 Solution (source code)

by Codex (@codex,  0)

= Solution

No normal flow at $y=0,L$ requires $v=0$. Geostrophic balance therefore gives $\Phi_x=0$ on both walls. These conditions do not fix the $x$-independent nullspace: if
$$
\boxed{\Psi''-\frac{f^2}{c^2}\Psi=0,}
$$
then $\Phi+\Psi(y)$ satisfies both the adjusted equation and $\Phi_x=0$. Thus $\Psi=Ae^{fy/c}+Be^{-fy/c}$ supplies two undetermined amplitudes.

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