# Intuition behind Airy waves dispersion relation

Using Airy wave theory, one can derive the dispersion relation of water waves (under some physical assumptions): $$\omega^2 = gk\tanh{kh}$$ where $k$ is the wave number, $h$ the distance from the surface at rest and the bottom of the sea, and $g$ is Earth's gravitational pull. Under the approximation of $h>\frac{\pi}{k}$ ("deep water approximation"), one gets that $$\omega \approx \sqrt{gk}$$ and under the approximation of $h < \frac{\pi}{10k}$ ("shallow water approximation"), one gets $$\omega \approx \sqrt{gh}|k|$$ (results acquired from wikipedia).

How does this settle with intuition? I would expect water waves to act more strangely in shallow water, since we have more boundary effects from the bottom of the sea (reflections and stuff), while in deep water, these effect decay before they reach the surface.

If you were to simply guess the form of $v_{\text{ph}}$, the only parameters at your disposition are $k$, $h$ and $g$. There are two ways in which you can make something that has the dimension of a velocity, \begin{align} v_{\text{ph}} \sim \sqrt{g h}\, , && v_{\text{ph}} \sim \sqrt{\frac{g}{k}} \, . \end{align}
In the deep water limit, $h$ does not play a role and you get $v_{\text{ph}} \sim \sqrt{g/k}$ and $\omega \sim \sqrt{g k}$. Then you know that deep water waves will dispersion.
Note that $k>0$ so in shallow water $\omega \approx \sqrt{gh}k$. These are pure surface waves whose amplitudes are much smaller than the depth, and is also assumed that no other waves propagate, thus reflections from the bottom are completely ignored.