Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 77 / 1 / i / Solution

For $0<\omega <N$, put $\alpha =\omega /\sqrt{N^2-{\omega }^2}$. The dispersion relation permits $k=\pm \alpha m$. On the positive-frequency branch, for $k>0$,The radiation condition therefore selects $k=\alpha m$, since energy must travel into the fluid. Substitution into the internal-wave polarization relations gives the complex fieldsThe physical fields are their real parts; $v=0$. The vertical component of group velocity is negative, so this positive-$m$ plane internal gravity wave carries energy down and right, although its phase propagates in a different direction.