For Stokes waves under the action of gravity, the third-order dispersion relation is – according to Stokes's first definition of celerity:
This third-order dispersion relation is a direct consequence of avoiding secular terms, when inserting the second-order Stokes solution into the third-order equations (of the perturbation series for the periodic wave problem).
In deep water (short wavelength compared to the depth):
and in shallow water (long wavelengths compared to the depth):
As shown above, the long-wave Stokes expansion for the dispersion relation will only be valid for small enough values of the Ursell parameter: .