Editing Wind wave (section)

===Spectral models===
[[Sea state]] can be described by the '''sea wave spectrum''' or just '''wave spectrum''' <math>S(\omega, \Theta)</math>. It is composed of a '''wave height spectrum''' (WHS) <math>S(\omega)</math> and a '''wave direction spectrum''' (WDS) <math>f(\Theta)</math>. Many interesting properties about the sea state can be found from the wave spectra.

WHS describes the [[spectral density]] of [[wave height]] [[variance]] ("power") versus [[wave frequency]], with [[dimension (physics)|dimension]] <math>\{S(\omega)\} = \{{\text{length}}^2\cdot\text{time}\}</math>.
The relationship between the spectrum <math>S(\omega_j)</math> and the wave amplitude <math>A_j</math> for a wave component <math>j</math> is:
: <math>\frac{1}{2} A_j^2 = S(\omega_j)\, \Delta \omega</math>{{citation needed|date=February 2021}}{{Clarify|define the variables. What do the symbols stand for?|date=May 2023}}
Some WHS models are listed below. 
* The International Towing Tank Conference (ITTC) <ref>{{citation | title=International Towing Tank Conference (ITTC) | url=http://ittc.sname.org/ | access-date=11 November 2010 }}</ref> recommended spectrum model for fully developed sea (ISSC<ref>International Ship and Offshore Structures Congress</ref> spectrum/modified [[Pierson-Moskowitz spectrum]]):<ref>{{citation | doi=10.1029/JZ069i024p05181 | first1=W. J. | last1=Pierson | first2=L. | last2=Moscowitz | title=A proposed spectral form for fully developed wind seas based on the similarity theory of S A Kitaigorodskii | journal=Journal of Geophysical Research | volume=69 | issue=24 | pages=5181–5190 | year=1964 | bibcode=1964JGR....69.5181P}}</ref>
:: <math>
\frac{S(\omega)}{H_{1/3}^2 T_1} = \frac{0.11}{2\pi} \left(\frac{\omega T_1}{2\pi}\right)^{-5} \mathrm{exp} \left[-0.44 \left(\frac{\omega T_1}{2\pi}\right)^{-4} \right]
</math>
* ITTC recommended spectrum model for limited [[Fetch (geography)|fetch]] ([[JONSWAP spectrum]])
:: <math>
S(\omega) = 155 \frac{H_{1/3}^2}{T_1^4 \omega^5} \mathrm{exp} \left(\frac{-944}{T_1^4 \omega^4}\right)(3.3)^Y,
</math>
:where
:: <math>Y = \exp \left[-\left(\frac{0.191 \omega T_1 -1}{2^{1/2}\sigma}\right)^2\right]</math>
:: <math>\sigma =
\begin{cases}
0.07 & \text{if }\omega \le 5.24 / T_1, \\
0.09 & \text{if }\omega > 5.24 / T_1.
\end{cases}
</math>
:(The latter model has since its creation improved based on the work of Phillips and Kitaigorodskii to better model the wave height spectrum for high [[wavenumber]]s.<ref>{{cite journal|last1=Elfouhaily|first1=T.|last2=Chapron|first2=B.|last3=Katsaros|first3=K.|last4=Vandemark|first4=D.|title=A unified directional spectrum for long and short wind-driven waves|journal=[[Journal of Geophysical Research]]|date=July 15, 1997|volume=102|issue=C7|pages=15781–15796|url=http://archimer.ifremer.fr/doc/00091/20226/17877.pdf|doi=10.1029/97jc00467|bibcode = 1997JGR...10215781E |doi-access=free}}</ref>)

As for WDS, an example model of <math>f(\Theta)</math> might be:
: <math>f(\Theta) = \frac{2}{\pi}\cos^2\Theta, \qquad -\pi/2 \le \Theta \le \pi/2</math>
Thus the sea state is fully determined and can be recreated by the following function where <math>\zeta</math> is the wave elevation, <math>\epsilon_{j}</math> is uniformly distributed between 0 and <math>2\pi</math>, and <math>\Theta_j</math> is randomly drawn from the directional distribution function <math>\sqrt{f(\Theta)}:</math><ref>{{citation | first=E. R. | last=Jefferys | title=Directional seas should be ergodic | journal=Applied Ocean Research | volume=9 | issue=4 | year=1987 | pages=186–191 | doi=10.1016/0141-1187(87)90001-0 | bibcode=1987AppOR...9..186J }}</ref>
: <math>\zeta = \sum_{j=1}^N \sqrt{2 S(\omega_j) \Delta \omega_j}\; \sin(\omega_j t - k_j x \cos \Theta_j - k_j y \sin \Theta_j + \epsilon_{j}).</math>