Editing Wind wave (section)

==Spectrum==
[[File:Munk ICCE 1950 Fig1.svg|thumb|upright=1.75|Classification of the [[spectrum]] of ocean waves according to wave [[period (physics)|period]]<ref>{{Cite journal | last=Munk |first=Walter H. |author-link=Walter Munk |year=1950 |title=Proceedings 1st International Conference on Coastal Engineering |journal=Coastal Engineering Proceedings |issue=1 |location=Long Beach, California |publisher=[[American Society of Civil Engineers|ASCE]] |pages=1–4 |doi=10.9753/icce.v1.1 |url=http://journals.tdl.org/ICCE/article/view/904 |doi-access=free }}</ref> ]]

Ocean waves can be classified based on: the disturbing force that creates them; the extent to which the disturbing force continues to influence them after formation; the extent to which the restoring force weakens or flattens them; and their wavelength or period. Seismic sea waves have a period of about 20 minutes, and speeds of {{convert|760|km/h|mph|abbr=on}}. Wind waves (deep-water waves) have a period up to about 20 seconds.

{| class="wikitable"
|+<ref name="Garrison">{{cite book|author=Tom Garrison|title =Oceanography: An Invitation to Marine Science|publisher =Yolanda Cossio|year =2009|isbn =978-0495391937|edition =7th}}</ref>
|-
! Wave type!! Typical wavelength !! Disturbing force !! Restoring force
|-
| Capillary wave|| < 2&nbsp;cm || Wind || Surface tension
|-
| Wind wave || {{convert|60|–|150|m|ft|abbr=on}} || Wind over ocean || Gravity
|-
| [[Seiche]] || Large, variable; a function of basin size || Change in atmospheric pressure, storm surge || Gravity
|-
| Seismic sea wave (tsunami) || {{convert|200|km|mi|abbr=on}} || Faulting of sea floor, volcanic eruption, landslide || Gravity
|-
| Tide || Half the circumference of Earth || Gravitational attraction, rotation of Earth || Gravity
|}

The speed of all ocean waves is controlled by gravity, wavelength, and water depth. Most characteristics of ocean waves depend on the relationship between their wavelength and water depth. Wavelength determines the size of the orbits of water molecules within a wave, but water depth determines the shape of the orbits. The paths of water molecules in a wind wave are circular only when the wave is traveling in deep water. A wave cannot "feel" the bottom when it moves through water deeper than half its wavelength because too little wave energy is contained in the water movement below that depth. Waves moving through water deeper than half their wavelength are known as deep-water waves. On the other hand, the orbits of water molecules in waves moving through shallow water are flattened by the proximity of the sea bottom surface. Waves in water shallower than 1/20 their original wavelength are known as shallow-water waves. Transitional waves travel through water deeper than 1/20 their original wavelength but shallower than half their original wavelength.

In general, the longer the wavelength, the faster the wave energy will move through the water. The relationship between the wavelength, period and velocity of any wave is:

:::<math> C = {L}/{T} </math>

where C is speed (celerity), L is the wavelength, and T is the period (in seconds). Thus the speed of the wave derives from the functional dependence <math> L(T) </math> of the wavelength on the period (the [[dispersion relation]]).

The speed of a deep-water wave may also be approximated by:

:::<math> C = \sqrt{{gL}/{2\pi}} </math>

where g is the acceleration due to gravity, {{convert|9.8|m|ft|abbr=off|sp=us}} per second squared. Because g and π (3.14) are constants, the equation can be reduced to:

:::<math> C = 1.251\sqrt{L} </math>

when C is measured in meters per second and L in meters. In both formulas the wave speed is proportional to the square root of the wavelength.

The speed of shallow-water waves is described by a different equation that may be written as:

:::<math> C = \sqrt{gd} = 3.1\sqrt{d} </math>

where C is speed (in meters per second), g is the acceleration due to gravity, and d is the depth of the water (in meters). The period of a wave remains unchanged regardless of the depth of water through which it is moving. As deep-water waves enter the shallows and feel the bottom, however, their speed is reduced, and their crests "bunch up", so their wavelength shortens.
{{clear}}

===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>