File:Stokes3 nonlin celerity.svg

Summary

<a href="https://en.wikipedia.org/wiki/Nonlinear_system" class="extiw" title="en:Nonlinear system">Nonlinear</a> enhancement of the <a href="https://en.wikipedia.org/wiki/phase_velocity" class="extiw" title="en:phase velocity">phase speed</a> c as compared to the linear-theory phase speed c₀, according to <a href="https://en.wikipedia.org/wiki/George_Gabriel_Stokes" class="extiw" title="en:George Gabriel Stokes">Stokes</a>' third-order theory for <a href="https://en.wikipedia.org/wiki/surface_gravity_wave" class="extiw" title="en:surface gravity wave">surface gravity waves</a>:

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On the horizontal axis is the relative water depth h / λ, with h the mean depth and λ the <a href="https://en.wikipedia.org/wiki/wavelength" class="extiw" title="en:wavelength">wavelength</a>, 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 <a href="https://en.wikipedia.org/wiki/wavenumber" class="extiw" title="en:wavenumber">wavenumber</a> and a the wave <a href="https://en.wikipedia.org/wiki/amplitude" class="extiw" title="en:amplitude">amplitude</a>).
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 <a href="https://en.wikipedia.org/wiki/flow_velocity" class="extiw" title="en:flow velocity">flow velocity</a> is zero below <a href="https://en.wikipedia.org/wiki/trough_(physics)" class="extiw" title="en:trough (physics)">trough</a> level.

Licensing

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