= Solution
Let every perturbation be proportional to $\exp[i(kx+mz-\omega t)]$ and define the <intrinsic frequency>
$$
\widehat\omega=\omega-kU.
$$
Eliminating pressure with <incompressibility>, and then eliminating buoyancy, gives the <internal gravity wave> dispersion relation
$$
\boxed{
\widehat\omega^2
=\frac{N^2k^2}{k^2+m^2}
},
\qquad
\omega=kU\mathbin{\pm}\frac{N|k|}{\sqrt{k^2+m^2}}.
$$
The vertical <group velocity> is
$$
c_{gz}=\frac{\partial\omega}{\partial m}
=-\frac{\widehat\omega m}{k^2+m^2}.
$$
Thus energy propagates upward when $\widehat\omega m<0$ and downward when $\widehat\omega m>0$. Equivalently, on the positive-intrinsic-frequency branch upward propagation requires $m<0$, while on the negative branch it requires $m>0$.
Back to article page
