The maximum wave steepness, for periodic and propagating deep-water waves, is , so the wave height is about one-seventh () of the wavelength λ. And surface gravity waves of this maximum height have a sharp wave crest – with an angle of 120° (in the fluid domain)  – also for finite depth, as shown by Stokes in 1880.
An accurate estimate of the highest wave steepness in deep water () was already made in 1893, by John Henry Michell, using a numerical method. A more detailed study of the behaviour of the highest wave near the sharp-cornered crest has been published by Malcolm A. Grant, in 1973. The existence of the highest wave on deep water with a sharp-angled crest of 120° was proved by John Toland in 1978. The convexity of η(x) between the successive maxima with a sharp-angled crest of 120° was independently proven by C.J. Amick et al. and Pavel I. Plotnikov in 1982
The highest Stokes wave – under the action of gravity – can be approximated with the following simple and accurate representation of the free surface elevation η(x,t):
with 
for 
and shifted horizontally over an integer number of wavelengths to represent the other waves in the regular wave train. This approximation is accurate to within 0.7% everywhere, as compared with the "exact" solution for the highest wave.
Another accurate approximation – however less accurate than the previous one – of the fluid motion on the surface of the steepest wave is by analogy with the swing of a pendulum in a grandfather clock.
Large library of Stokes waves computed with high precision for the case of infinite depth, represented with high accuracy (at least 27 digits after decimal point) as a Padé approximant can be found at StokesWave.org