Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 333 / 4 / i / Solution

Let every perturbation be proportional to $\exp [i(kx+mz-\omega t)]$ and define the intrinsic frequencyEliminating pressure with incompressibility, and then eliminating buoyancy, gives the internal gravity wave dispersion relationThe vertical group velocity isThus energy propagates upward when $\hat{\omega }m<0$ and downward when $\hat{\omega }m>0$. Equivalently, on the positive-intrinsic-frequency branch upward propagation requires $m<0$, while on the negative branch it requires $m>0$.