In fluid dynamics, the **wave height** of a surface wave is the difference between the elevations of a crest and a neighbouring trough. *Wave height* is a term used by mariners, as well as in coastal, ocean and naval engineering.

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.

## Definitions

Depending on context, wave height may be defined in different ways:

\({\displaystyle H=2a.\,}\)

- For a periodic wave, it is simply the difference between the maximum and minimum of the surface elevation
*z*=*η*(*x*–*c*p*t*):

\({\displaystyle H=\max \left\{\eta (x\,-\,c_{p}\,t)\right\}-\min \left\{\eta (x-c_{p}\,t)\right\},\,}\) with *c*p the phase speed (or propagation speed) of the wave. The sine wave is a specific case of a periodic wave.

- In random waves at sea, when the surface elevations are measured with a wave buoy, the individual wave height
*H*m 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. - Significant wave height
*H*1/3, or*Hs*or*Hsig*, as determined directly from the time series of the surface elevation, is defined as the average height of that one-third of the*N*measured waves having the greatest heights:

\({\displaystyle H_{1/3}={\frac {1}{{\frac {1}{3}}\,N}}\,\sum _{m=1}^{{\frac {1}{3}}\,N}\,H_{m}}\) where *H*m represents the individual wave heights, sorted into descending order of height as *m* increases from 1 to *N*. Only the highest one-third is used, since this corresponds best with visual observations of experienced mariners, whose vision apparently focuses on the higher waves.

- Significant wave height
*H*m0, defined in the frequency domain, is used both for measured and forecasted wave variance spectra. Most easily, it is defined in terms of the variance*m*0 or standard deviation*ση*of the surface elevation:

\({\displaystyle H_{m_{0}}=4{\sqrt {m_{0}}}=4\sigma _{\eta },\,}\) where *m*0, the zeroth-moment of the variance spectrum, is obtained by integration of the variance spectrum. In case of a measurement, the standard deviation *ση* is the easiest and most accurate statistic to be used.

- Another wave-height statistic in common usage is the root-mean-square (or RMS) wave height
*H*rms, defined as:

\({\displaystyle H_{\text{rms}}={\sqrt {{\frac {1}{N}}\sum _{m=1}^{N}H_{m}^{2}}},\,}\) with *H*m again denoting the individual wave heights in a certain time series.

## See also

## References

- Holthuijsen, Leo H. (2007),
*Waves in Oceanic and Coastal Waters*, Cambridge University Press, ISBN 0-521-86028-8, 387 pages. - Kinsman, Blair (1984),
*Wind waves: their generation and propagation on the ocean surface*, Dover Publications, ISBN 0-486-49511-6, 704 pages. - Phillips, Owen M. (1977),
*The dynamics of the upper ocean*(2nd ed.), Cambridge University Press, ISBN 0-521-29801-6, viii & 336 pages.