= Solution
Use complex amplitudes $(U,W,\Sigma,P)(z)e^{i(kx-\omega t)}$, with $k\ne0$ and $\omega\ne0$, and take physical real parts. The two-dimensional linearized <Boussinesq equations> give
$$
ikU+W'=0,\quad -i\omega\Sigma=-N^2W,\quad
-i\omega U=-ikP/\rho_0,\quad -i\omega W=-P'/\rho_0+\Sigma.
$$
The first and third equations imply $U=iW'/k$ and $P=i\rho_0\omega W'/k^2$; the buoyancy equation gives $\Sigma=-iN^2W/\omega$. Insert these into the vertical momentum equation. The amplitude of the <internal gravity wave> satisfies
$$
\boxed{W''+k^2\left(\frac{N^2(z)}{\omega^2}-1\right)W=0.}
$$
Oscillatory vertical propagation requires $N^2>\omega^2$; in a well-mixed region $N=0$, the bounded solution is <evanescent> rather than a vertically propagating internal wave.
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