Editing Wave power (section)

=== Energy and energy flux ===
In a [[sea state]], the [[arithmetic mean|mean]] [[energy density]] per unit area of [[gravity wave]]s on the water surface is proportional to the wave height squared, according to linear wave theory:<ref name="Phillips" /><ref name="Goda">{{cite book | first=Y. | last=Goda | title=Random Seas and Design of Maritime Structures | year=2000 | publisher=World Scientific | isbn=978-981-02-3256-6 }}</ref>

:<math>E=\frac{1}{16}\rho g H_{m0}^2,</math>{{efn|Here, the factor for random waves is {{frac|1|16}}, as opposed to {{frac|1|8}} for periodic waves – as explained hereafter. For a small-amplitude sinusoidal wave <math display="inline"> \eta = a \cos 2\pi\left(\frac{x}{\lambda}-\frac{t}{T}\right)</math> with wave amplitude <math> a,</math> the wave energy density per unit horizontal area is <math display="inline"> E=\frac{1}{2}\rho g a^2,</math> or <math display="inline"> E=\frac{1}{8}\rho g H^2</math> using the wave height <math display="inline"> H = 2a</math> for sinusoidal waves. In terms of the variance of the surface elevation <math display="inline"> m_0 = \sigma_\eta^2 = \overline{(\eta-\bar\eta)^2} = \frac{1}{2}a^2,</math> the energy density is <math display="inline"> E=\rho g m_0</math>. Turning to random waves, the last formulation of the wave energy equation in terms of <math display="inline"> m_0</math> is also valid (Holthuijsen, 2007, p. 40), due to [[Parseval's theorem]]. Further, the [[significant wave height]] is ''defined'' as <math display="inline"> H_{m0} = 4\sqrt{m_0}</math>, leading to the factor {{frac|1|16}} in the wave energy density per unit horizontal area.}}<ref>{{Cite book
 | last = Holthuijsen
 | first = Leo H.
 | year = 2007
 | title = Waves in oceanic and coastal waters
 | publisher = Cambridge University Press
 | isbn = 978-0-521-86028-4
 | location = Cambridge
}}</ref>

where ''E'' is the mean wave energy density per unit horizontal area (J/m<sup>2</sup>), the sum of [[kinetic energy|kinetic]] and [[potential energy]] density per unit horizontal area. The potential energy density is equal to the kinetic energy,<ref name="Phillips" /> both contributing half to the wave energy density ''E'', as can be expected from the [[Equipartition theorem#Potential energy and harmonic oscillators|equipartition theorem]].

The waves propagate on the surface, where crests travel with the phase velocity while the energy is transported horizontally with the [[group velocity]]. The mean transport rate of the wave energy through a vertical [[plane (mathematics)|plane]] of unit width, parallel to a wave crest, is the energy [[flux]] (or wave power, not to be confused with the output produced by a device), and is equal to:<ref>{{cite journal | last=Reynolds |first=O. | author-link=Osborne Reynolds | year=1877 |title=On the rate of progression of groups of waves and the rate at which energy is transmitted by waves | journal=Nature | volume=16 |issue=408 | pages=343–44 | doi = 10.1038/016341c0 |bibcode = 1877Natur..16R.341. | doi-access=free }}<br>{{cite journal | title=On progressive waves | author=Lord Rayleigh (J. W. Strutt) | author-link=Lord Rayleigh | year=1877 | journal=Proceedings of the London Mathematical Society | volume=9 | issue=1 | pages=21–26 | doi=10.1112/plms/s1-9.1.21 | url=https://zenodo.org/record/1447762 }} Reprinted as Appendix in: ''Theory of Sound'' '''1''', MacMillan, 2nd revised edition, 1894.</ref><ref name="Phillips" />

:<math>P = E\, c_g, </math> with ''c<sub>g</sub>'' the group velocity (m/s).

Due to the [[dispersion (water waves)|dispersion relation]] for waves under gravity, the group velocity depends on the wavelength ''λ'', or equivalently, on the wave [[period (physics)|period]] ''T''.

[[Wave height]] is determined by wind speed, the length of time the wind has been blowing, fetch (the distance over which the wind excites the waves) and by the [[bathymetry]] (which can focus or disperse the energy of the waves). A given wind speed has a matching practical limit over which time or distance do not increase wave size. At this limit the waves are said to be "fully developed". In general, larger waves are more powerful but wave power is also determined by [[wavelength]], water [[density]], water depth and acceleration of gravity.