English: Several integral properties of Stokes waves on deep water as a function of wave steepness. The wave steepness is defined as the ratio of wave height H to the wavelength λ. These properties of surface gravity waves are made dimensionless using the wavenumber k = 2π / λ, gravitational acceleration g and the fluid density ρ.

Shown are the kinetic energy density T , the potential energy density V , the total energy density E = T + V , the horizontal wave momentum density I , and the relative enhancement of the phase speed c . Wave energy densities T , V and E are integrated over depth and averaged over one wavelength, so they are energies per unit of horizontal area; the wave momentum density I is similar. The dashed black lines show 1/16 ( kH ) ² and 1/8 ( kH ) ² , being the values of the integral properties as derived from (linear) Airy wave theory . The maximum wave height occurs for a wave steepness H / λ of 0.1412, above which no periodic surface gravity waves exist.

Note that the shown wave properties have a maximum for a wave height less than the maximum wave height (see e.g. E.D. Cokelet (1977) "Steep gravity waves in water of arbitrary uniform depth", Philosophical Transactions of the Royal Society of London , A 286 (1335), pp. 183–230, doi:10.1098/rsta.1977.0113 ).

This figure is a remake and adaptation of Figure 1 in: L.W. Schwartz and J.D. Fenton (1982) "Strongly nonlinear waves", Annual Review of Fluid Mechanics 14 , pp 39–60, doi:10.1146/annurev.fl.14.010182.000351 .