Internal waves in the atmosphere are frequently found in the lee (the downstream side) of mountains. In stably stratified conditions, the flow of air over a mountain causes a vertical displacement of fluid particles, which sets up internal waves as the air moves downstream of the mountain. If the amplitude is large and the air is moist, the upward motion causes condensation and cloud formation. Due to the effect of a mean flow, such lee waves are stationary with respect to the ground

The frequency of lee waves is much larger than $f$, so that rotational effects are negligible. The dispersion relation is therefore $$ \omega^2 = \frac{N^2 k^2}{m^2 + k^2} $$

We now have to introduce the effects of the mean flow. The dispersion relation \(\omega^2 = \frac{N^2 k^2}{m^2 + k^2}\) is still valid if $\omega$ is interpreted as the intrinsic frequency, the frequency measured in a frame of reference moving with the mean flow. In a medium moving with a velocity \(\mathbf{U}\), the observed frequency of waves at a fixed point is Doppler shifted to \[ \omega_0 = \underbrace{\omega}_{\text{intrinsic frequency}} + \underbrace{\mathbf{K} \cdot \mathbf{U}}_{\text{Doppler shift}} \]

A stationary wave $\omega_0 = 0$ requires the intrinsic frequency is $$ \omega = -\mathbf{K} \cdot \mathbf{U} = kU $$ (Here $-\mathbf{K} \cdot \mathbf{U}$ is positive because \(\mathbf{K}\) is westward and \(\mathbf{U}\) is eastward)

The dispersion relation then gives $$ U = \frac{N}{\sqrt{k^2 + m^2}} $$ If the flow speed $U$ is given, and the mountain introduces a typical horizontal wave number $k$, then the preceding equation determines the vertical wave number $m$ that generates stationary waves. Waves that do not satisfy this condition would radiate away