Past exam of the waves and compressible flow course of the University of Oxford / 2023 / 2 / b / i

Substitute $W=f(z)e^{i(kx-\omega t)}$ into the transformed equation. It yieldsThe nonzero eigenfunctions of this Dirichlet boundary value problem are $f=A\sin (n\pi z/h)$, $n\ge 1$. Hence, with $N=\sqrt{\alpha g}$ and $b_n^2=(n\pi /h{)}^2+{\alpha }^2/4$,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.