Discussion of the Dispersion Relation
A plot of the dispersion relation \[\omega^2 - f^2 = \frac{k^2}{m^2} \left( N^2 - \omega^2 \right)\] showing $\omega$ as a function of $k$ for various values of $m$. All curves pass through the point $\omega = f$, which represents inertial oscillations. Typically, $N \gg f$ in most of the atmosphere and the ocean
- High-frequency regime ($\omega \sim N$, but $\omega \le N$): In this range $f^2$ is negligible in comparison with $\omega^2$ in the denominator of the dispersion relation, which reduces to $$ m^2 \cong \frac{k^2}{\omega^2} \left( N^2 - \omega^2 \right) \quad\text{or}\quad \omega^2 \cong \frac{N^2 k^2}{m^2 + k^2}. $$ Using $\tan \theta = m/k$, this gives $\omega = N \cos \theta$. Thus high-frequency internal waves are the same as the non-rotating internal waves
- Low-frequency regime ($\omega \sim f$, but $\omega \ge f$): In this range $\omega^2$ can be neglected in comparison to $N^2$ in the dispersion relation, which becomes $$ m^2 \cong \frac{k^2 N^2}{\omega^2 - f^2} \quad\text{or}\quad \omega^2 \cong f^2 + \frac{k^2 N^2}{m^2}. $$ The low-frequency limit is obtained by making the hydrostatic assumption that neglecting $\partial w / \partial t$ in the vertical equation of motion
- Mid-frequency regime ($f \ll \omega \ll N$): In this range the dispersion relation with $l = 0$ simplifies to $$ m^2 \cong k^2 N^2 / \omega^2 $$ so that both the hydrostatic and the non-rotating assumptions are applicable