 Solution

Forthe normal derivative at $x=0,L$ contains $\sin (\pi x/L)$ and the normal derivative at $y=0,L$ contains $\sin (2\pi y/L)$. Both therefore vanish at the side walls.

At $z=0$, ${\varphi }_t$ has exactly the same horizontal dependence as the imposed pressure. The dynamic boundary condition is consequently consistent provided $\eta$ has that same horizontal mode. This is the rectangular standing surface-gravity mode selected by the forcing.