Solution
= Solution
Substitution into $\nabla^2\phi=0$ gives
$$
Z''-\left[\left(\frac\pi L\right)^2
+\left(\frac{2\pi}{L}\right)^2\right]Z=0.
$$
Set
$$
k=\frac{\pi\sqrt5}{L}.
$$
Then $Z=Ae^{kz}+Be^{-kz}$. Since the fluid occupies $z<0$, decay as $z\to-\infty$ forces $B=0$. Absorbing $A$ into $F$ and choosing $Z(0)=1$ gives
$$
\boxed{Z(z)=e^{kz},
\qquad k=\frac{\pi\sqrt5}{L}}.
$$
This is the usual exponential depth dependence of a <deep-water gravity wave>.