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:

${\displaystyle \begin{array}{cc}c& =c_{{}_0}\{1+\frac{9-10{\sigma }^2+9{\sigma }^4}{16{\sigma }^4}(ka{)}^2\}+𝒪((ka{)}^4),\text{with}\\ c_{{}_0}& =\sqrt{\frac{g}{k}\sigma }\text{and}\sigma =\tanh kh.\end{array}}$

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.