Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 333 / 1 / v / 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: ifthen $\Phi +\Psi (y)$ satisfies both the adjusted equation and ${\Phi }_x=0$. Thus $\Psi =A{e}^{fy/c}+B{e}^{-fy/c}$ supplies two undetermined amplitudes.