File:Stokes3 nonlin celerity.svg

English: Nonlinear enhancement of the phase speed c as compared to the linear-theory phase speed c₀, according to Stokes' third-order theory for surface gravity waves:

${\displaystyle {\begin{aligned}c&=c_{_{0}}\,\left\{1+{\frac {9-10\,\sigma ^{2}+9\,\sigma ^{4}}{16\,\sigma ^{4}}}\,(ka)^{2}\right\}+{\mathcal {O}}\left((ka)^{4}\right),\quad {\text{with}}\\c_{_{0}}&={\sqrt {{\frac {g}{k}}\,\sigma }}\qquad {\text{and}}\qquad \sigma =\tanh \,kh.\end{aligned}}}$

On the horizontal axis is the relative water depth h / λ, with h the mean depth and λ the wavelength, while the vertical axis is the nonlinear phase-speed enhancement (c − c₀) / c₀ divided by the wave steepness ka squared (with k = 2π / λ the wavenumber and a the wave amplitude).
The blue line is valid for arbitrary water depth, while the dashed red line is the shallow-water limit (water depth small compared to the wavelength), and the dash-dot green line is the asymptotic limit for deep water waves.

The equation for the phase speed c (celerity) is based on Stokes' first definition of celerity, i.e. the mean horizontal flow velocity is zero below trough level.