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| [[File:Wea00800,1.jpg|thumb|300px|Merchant ship labouring in heavy seas as a huge wave looms astern]]
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| In [[fluid dynamics]], the '''wave height''' of a [[ocean surface wave|surface wave]] is the difference between the [[elevation]]s of a [[crest (physics)|crest]] and a neighbouring [[trough (physics)|trough]].<ref name=Kinsman_38>{{harvtxt|Kinsman|1984|page=38}}</ref> Wave height is a term used by [[mariner]]s, as well as in [[coastal engineering|coastal]], [[offshore construction|ocean]] and [[naval engineering]].
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| At sea, the term [[significant wave height]] is used as a means to introduce a well-defined and standardized [[statistic]] to denote the characteristic height of the [[random]] waves in a [[sea state]]. It is defined in such a way that it more–or–less corresponds to what a [[mariner]] observes when estimating visually the average wave height.
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| == Several definitions for different situations ==
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| [[File:Sine wave amplitude.svg|right|thumb|Wave characteristics.]]
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| *For a ''[[sine wave]]'', the wave height ''H'' is twice the [[amplitude]]:<ref name=Kinsman_38/>
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| :: <math>H = 2a. \,</math>
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| *For a ''[[periodic function|periodic wave]]'' it is simply the difference between the [[maximum]] and [[minimum]] of the surface elevation ''z'' = ''η''(''x'' – ''c''<sub>p</sub> ''t''):<ref name=Kinsman_38/>
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| :: <math>H = \max\left\{ \eta(x\,-\,c_p\,t) \right\} - \min\left\{ \eta(x - c_p\,t) \right\}, \, </math>
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| :with ''c''<sub>p</sub> the [[phase speed]] (or propagation speed) of the wave. The sine wave is a specific case of a periodic wave.
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| *In random waves at sea, when the surface elevations are measured with a [[weather buoy|wave buoy]], the ''individual wave height'' ''H''<sub>m</sub> of each individual wave—with an [[integer]] label ''m'', running from 1 to ''N'', to denote its position in a sequence of ''N'' waves—is the difference in elevation between a wave crest and trough in that wave. For this to be possible, it is necessary to first split the measured [[time series]] of the surface elevation into individual waves. Commonly, an individual wave is denoted as the [[time interval]] between two successive downward-crossings through the [[average]] surface elevation (upward crossings might also be used). Then the individual wave height of each wave is again the difference between maximum and minimum elevation in the time interval of the wave under consideration.<ref name=Holt_24_28>{{harvtxt|Holthuijsen|2007|pages=24–28}}</ref>
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| *''[[Significant wave height]]'' ''H''<sub>1/3</sub>, or ''H''<sub>''s''</sub> or ''H''<sub>''sig''</sub>, in the ''[[time domain]]'', is defined as the average height of the one-third part of the measured waves—which are ''N'' in number—having the largest wave heights:<ref name=Holt_24_28/>
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| :: <math>H_{1/3} = \frac{1}{\frac13\,N}\, \sum_{m=1}^{\frac13\,N}\, H_m,</math>
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| :with ''H''<sub>m</sub> the individual wave heights, sorted in such a way that the highest wave has ''m''=1 and the lowest wave is for ''m''=''N''. Only the highest one-third is used, since this corresponds best with visual observations of experienced mariners: eyes and brain apparently focus on the higher waves seen.<ref name=Holt_24_28/>
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| *''Significant wave height'' ''H''<sub>m0</sub>, defined in the ''[[frequency domain]]'', is used both for measured and [[forecasting|forecast]]ed wave [[power spectrum|variance spectra]]. Most easily, it is defined in terms of the [[variance]] ''m''<sub>0</sub> or [[standard deviation]] ''σ''<sub>''η''</sub> of the surface elevation:<ref>{{harvtxt|Holthuijsen|2007|page=70}}</ref>
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| :: <math>H_{m_0} = 4 \sqrt{m_0} = 4 \sigma_\eta, \, </math>
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| :where ''m''<sub>0</sub>, the zeroth-[[moment (mathematics)|moment]] of the variance spectrum, is obtained by [[integration (mathematics)|integration]] of the variance spectrum. In case of a measurement, the standard deviation ''σ''<sub>''η''</sub> is the easiest and most accurate statistic to be used.
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| *Another wave-height statistic in common usage is the ''[[root-mean-square]] wave height'' (or ''RMS wave height'') ''H''<sub>rms</sub>, defined as:<ref name=Holt_24_28/>
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| :: <math>H_\text{rms} = \sqrt{ \frac{1}{N} \sum_{m=1}^N H_m^2}, \, </math>
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| :with ''H''<sub>m</sub> again denoting the individual wave heights in a certain [[time series]].
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| == See also ==
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| *[[Sea state]]
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| *[[Significant wave height]]
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| *[[Wind wave]]
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| == Notes ==
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| {{reflist}}
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| == References ==
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| {{ref begin}}
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| *{{Citation | title=Waves in Oceanic and Coastal Waters | first=Leo H. | last=Holthuijsen | publisher=Cambridge University Press | year=2007 | isbn=0-521-86028-8 }}, 387 pages.
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| *{{Citation | last=Kinsman | first=Blair | title=Wind waves: their generation and propagation on the ocean surface | year=1984 | publisher=Dover Publications | isbn=0-486-49511-6 }}, 704 pages.
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| *{{Citation | first=Owen M. | last=Phillips | title=The dynamics of the upper ocean | publisher=Cambridge University Press | year=1977 | edition=2nd | isbn=0-521-29801-6 }}, viii & 336 pages.
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| {{ref end}}
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| {{physical oceanography}}
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| [[Category:Water waves]]
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| [[Category:Physical oceanography]]
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