 Solution

Use complex amplitudes $(U,W,\Sigma ,P)(z)e^{i(kx-\omega t)}$, with $k\ne 0$ and $\omega \ne 0$, and take physical real parts. The two-dimensional linearized Boussinesq equations giveThe first and third equations imply $U=i{W}^{\prime }/k$ and $P=i{\rho }_0\omega W^{\prime }/k^2$; the buoyancy equation gives $\Sigma =-i{N}^{2}W/\omega$. Insert these into the vertical momentum equation. The amplitude of the internal gravity wave satisfiesOscillatory 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.