Stokes wave

Example

A Stokes wave is a periodic travelling wave of permanent form that solves the full nonlinear water wave problem. Stokes constructed it as a power series in the wave steepness $ka$, where $a$ is the amplitude and $k$ the wavenumber. In deep water, with $\theta = kx - \omega t$, the surface elevation to second order is $$\eta = a\cos\theta + \tfrac{1}{2}ka^2\cos(2\theta) + O(k^2a^3),$$ and the angular frequency to third order is $$\omega = \sqrt{gk}\left(1 + \tfrac{1}{2}k^2a^2\right) + O(k^3a^3).$$ The first term of $\eta$ is the linear wave of Airy wave theory. The second harmonic makes the crests sharper and the troughs flatter. The frequency now depends on the amplitude, so steeper waves travel faster than the linear dispersion relation predicts.