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+# Airy linear wave animation
+
+Visualize the fundamental plane-wave solutions to Airy's linear wave equation. The mathematical details are very nicely summarized [in Wikipedia](https://en.wikipedia.org/wiki/Airy_wave_theory).
+
+The movement of water that takes place under the wave's surface explains why water depth is such an important factor in determining wave velocity.
+
+## [Click here for the interactive animation](https://maresb.github.io/airy-wave/)
+
+## Mathematical details
+
+In short, the velocity $v$ of a water wave is given by the dispersion relation
+
+$$
+v^2 = \frac{g\lambda}{2\pi}\tanh\frac{2\pi h}{\lambda}
+$$
+
+where $g$ is gravitational acceleration, $h$ is water depth, and $\lambda$ is the wavelength. This dispersion relation determines all dynamics of water waves in the small-ampltude limit.
+
+Special cases of this formula are the deep and shallow limits:
+
+$$v\approx\sqrt{\frac{g\lambda}{2\pi}} \textrm{ as } h\to\infty,$$
+$$v\approx\sqrt{gh} \textrm{ as } h\to0.$$
+
+
+## Why are the dynamics determined by this dispersion relation?
+
+More mathematically-amenable quantities are the angular wavenumber $k = 2\pi/\lambda$, and angular frequency $\omega = k v$. In terms of these quantities the relation becomes
+
+$$
+\omega^2 = g k \tanh(k h).
+$$
+
+In particular, given any initial conditions for the surface $z=\eta(x,0)$ and $\partial\eta/\partial t(x,0)$ in the small-amplitude limit, then the time evolution is determined by evolving the individual Fourier modes according to
+
+$$\exp(i (kx-\omega t)).$$
+
+For any $k\neq 0$ there is a pair of values $\pm \omega$ and hence a pair of Fourier coefficients $C_{k,+\omega}$, $C_{k,-\omega}$ that can be matched to the pair of $k$-Fourier coefficients coming from the initial conditions.
+
+## What are the assumptions?
+
+The primary assumption is that the amplitude $a$ of the waves is small so that second-order effects are negligible. (If the envelope of the surface wave is distorted from a pure sinusoid, then the amplitude $a$ has been set too large.)
+
+The dispersion relation arises from exactly solving the PDE of the fluid motion under the surface. That PDE is [Bernoulli's equation](https://en.wikipedia.org/wiki/Bernoulli%27s_principle#Unsteady_potential_flow) and assumes that water is incompressible and irrotational. While water is certainly not irrotational, it is still a convenient assumption since rotational effects don't play a very significant role in small waves.