= Solution
Substitute $W=f(z)e^{i(kx-\omega t)}$ into the transformed equation. It yields
$$
f''+\left(\frac{\alpha g k^2}{\omega^2}-k^2-\frac{\alpha^2}{4}\right)f=0,\qquad f(0)=f(h)=0.
$$
The nonzero <eigenfunctions> of this <Dirichlet boundary value problem> are $f=A\sin(n\pi z/h)$, $n\geq1$. Hence, with $N=\sqrt{\alpha g}$ and $b_n^2=(n\pi/h)^2+\alpha^2/4$,
$$
\boxed{\omega^2=\frac{N^2k^2}{k^2+b_n^2}=\frac{4\alpha gh^2k^2}{4n^2\pi^2+h^2(\alpha^2+4k^2)}}.
$$
The physical vertical <velocity> includes the factor $e^{\alpha z/2}\sin(n\pi z/h)$. The zero horizontal wavenumber is the zero-frequency limiting case, not a propagating oscillation.
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