File:Stokes wave max height.svg

English: Stokes waves of maximum wave height on deep water, under the action of gravity. Shown is the simple and accurate approximation of the free surface elevation η(x,t):

${\displaystyle {\frac {\eta }{\lambda }}=A\,\cosh \,\left({\frac {x-ct}{\lambda }}\right),\qquad {\text{with}}\qquad A={\frac {1}{{\sqrt {3}}\,\sinh \left({\frac {1}{2}}\right)}},\qquad {\text{for }}|x-ct|\leq {\tfrac {1}{2}}\,\lambda ,}$

where λ is the wavelength, H is the wave height and c is the phase speed. This approximation – with a maximum error less than 0.7% in the free-surface elevation (relative to the wave height) – was discovered by:
R.C.T. Rainey and M.S. Longuet-Higgins (2006) "A close one-term approximation to the highest Stokes wave on deep water". Ocean Engineering 33(14–15), pp. 2012–2024.

The maximum wave steepness is H / λ ≈ 0.141. Surface gravity waves of maximum height have a sharp wave crest of 120°.