Editing Wind wave (section)

==Physics of waves==
{{see also|Airy wave theory}}
[[File:Shallow water wave.png|thumb|upright=1.4|Stokes drift in shallow water waves ([[:File:Shallow water wave.gif|Animation]])]]

Wind waves are mechanical [[wave]]s that propagate along the interface between [[water]] and [[air]]; the restoring force is provided by gravity, and so they are often referred to as [[gravity wave|surface gravity waves]]. As the [[wind]] blows, pressure and friction perturb the equilibrium of the water surface and transfer energy from the air to the water, forming waves. The initial formation of waves by the wind is described in the theory of Phillips from 1957, and the subsequent growth of the small waves has been modeled by [[John W. Miles|Miles]], also in 1957.<ref>{{cite journal | first=O. M. | last=Phillips | year=1957 | title=On the generation of waves by turbulent wind | journal=Journal of Fluid Mechanics | volume=2 | issue=5 | pages=417–445 | doi=10.1017/S0022112057000233 | doi-broken-date=1 November 2024 |bibcode = 1957JFM.....2..417P | s2cid=116675962 }}</ref><ref>{{cite journal | first=J. W. | last=Miles | author-link=John W. Miles | year=1957 | title=On the generation of surface waves by shear flows | journal=Journal of Fluid Mechanics | volume=3 | issue=2 | pages=185–204 | doi=10.1017/S0022112057000567 | doi-broken-date=1 November 2024 |bibcode = 1957JFM.....3..185M | s2cid=119795395 }}</ref>

[[File:Deep water wave.png|thumb|upright=1.4|Stokes drift in a deeper water wave ([[:File:Deep water wave.gif|Animation]])]]
[[File:Orbital wave motion-Wiegel Johnson ICCE 1950 Fig 6.png|thumb|upright=1.3|Photograph of the water particle orbits under a—progressive and periodic—[[surface gravity wave]] in a [[wave flume]]. The wave conditions are: mean water depth ''d''&nbsp;=&nbsp;{{convert|2.50|ft|m|abbr=on}}, [[wave height]] ''H''&nbsp;=&nbsp;{{convert|0.339|ft|m|abbr=on}}, wavelength λ&nbsp;=&nbsp;{{convert|6.42|ft|m|abbr=on}}, [[period (physics)|period]] ''T''&nbsp;=&nbsp;1.12&nbsp;s.<ref>Figure 6 from: {{cite journal |first1=R. L. |last1=Wiegel |first2=J. W. |last2=Johnson |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=5–21 |doi=10.9753/icce.v1.2 |url=http://journals.tdl.org/ICCE/article/view/905|doi-access=free }}</ref>]]

In linear plane waves of one wavelength in deep water, [[fluid parcel|parcels]] near the surface move not plainly up and down but in circular orbits: forward above and backward below (compared to the wave propagation direction). As a result, the surface of the water forms not an exact [[sine wave]], but more a [[trochoid]] with the sharper curves upwards—as modeled in [[trochoidal wave]] theory. Wind waves are thus a combination of [[transversal wave|transversal]] and [[longitudinal wave|longitudinal]] waves.

When waves propagate in [[Waves and shallow water|shallow water]], (where the depth is less than half the wavelength) the particle trajectories are compressed into [[ellipse]]s.<ref>For the particle trajectories within the framework of linear wave theory, see for instance:
<br>[[#Phillips1977|Phillips (1977)]], page 44.<br>{{cite book | first=H. | last=Lamb | author-link=Horace Lamb | year=1994 | title=Hydrodynamics | publisher=Cambridge University Press | edition=6th| isbn=978-0-521-45868-9 }} Originally published in 1879, the 6th extended edition appeared first in 1932. See §229, page 367.<br>{{cite book | title=Fluid mechanics | author=L. D. Landau and E. M. Lifshitz | year=1986 | publisher=Pergamon Press | series=Course of Theoretical Physics | volume=6 | edition=Second revised | isbn=978-0-08-033932-0 }} See page 33.</ref><ref>A good illustration of the wave motion according to linear theory is given by [http://www.coastal.udel.edu/faculty/rad/linearplot.html Prof. Robert Dalrymple's Java applet] {{Webarchive|url=https://web.archive.org/web/20171114103146/http://www.coastal.udel.edu/faculty/rad/linearplot.html |date=2017-11-14 }}.</ref>

In reality, for [[:wikt:finite|finite]] values of the wave amplitude (height), the particle paths do not form closed orbits; rather, after the passage of each crest, particles are displaced slightly from their previous positions, a phenomenon known as [[Stokes drift]].<ref>For nonlinear waves, the particle paths are not closed, as found by [[George Gabriel Stokes]] in 1847, see [[#Stokes1847|the original paper by Stokes]]. Or in [[#Phillips1977|Phillips (1977)]], page 44: ''"To this order, it is evident that the particle paths are not exactly closed&nbsp;... pointed out by Stokes (1847) in his classical investigation"''.</ref><ref>Solutions of the particle trajectories in fully nonlinear periodic waves and the Lagrangian wave period they experience can for instance be found in:<br>{{cite journal| author=J. M. Williams| title=Limiting gravity waves in water of finite depth | journal=[[Philosophical Transactions of the Royal Society A]] | volume=302 | issue=1466 | pages=139–188 | year=1981| doi=10.1098/rsta.1981.0159 |bibcode = 1981RSPTA.302..139W | s2cid=122673867 }}<br>{{cite book| title=Tables of progressive gravity waves | author=J. M. Williams | year=1985 | publisher=Pitman | isbn=978-0-273-08733-5 }}</ref>

As the depth below the free surface increases, the radius of the circular motion decreases. At a depth equal to half the [[wavelength]] λ, the orbital movement has decayed to less than 5% of its value at the surface. The [[phase speed]] (also called the celerity) of a surface gravity wave is—for pure [[periodic function|periodic]] wave motion of small-[[amplitude]] waves—well approximated by

:<math>c=\sqrt{\frac{g \lambda}{2\pi} \tanh \left(\frac{2\pi d}{\lambda}\right)}</math>

where
:''c'' = [[phase speed]];
:''λ'' = [[wavelength]];
:''d'' = water depth;
:''g'' = [[standard gravity|acceleration due to gravity at the Earth's surface]].

In deep water, where <math>d \ge \frac{1}{2}\lambda</math>, so <math>\frac{2\pi d}{\lambda} \ge \pi</math> and the hyperbolic tangent approaches <math>1</math>, the speed <math>c</math> approximates

:<math>c_\text{deep}=\sqrt{\frac{g\lambda}{2\pi}}.</math>

In SI units, with <math>c_\text{deep}</math> in m/s, <math>c_\text{deep} \approx 1.25\sqrt\lambda</math>, when <math>\lambda</math> is measured in metres.
This expression tells us that waves of different wavelengths travel at different speeds. The fastest waves in a storm are the ones with the longest wavelength. As a result, after a storm, the first waves to arrive on the coast are the long-wavelength swells.

For intermediate and shallow water, the [[Boussinesq approximation (water waves)|Boussinesq equations]] are applicable, combining [[dispersion (water waves)|frequency dispersion]] and nonlinear effects. And in very shallow water, the [[shallow water equations]] can be used.

If the wavelength is very long compared to the water depth, the phase speed (by taking the [[Limit of a function|limit]] of {{var|c}} when the wavelength approaches infinity) can be approximated by

:<math>c_\text{shallow} = \lim_{\lambda\rightarrow\infty} c = \sqrt{gd}.</math>

On the other hand, for very short wavelengths, [[surface tension]] plays an important role and the phase speed of these [[gravity-capillary wave]]s can (in deep water) be approximated by

:<math>c_\text{gravity-capillary}=\sqrt{\frac{g \lambda}{2\pi} + \frac{2\pi S}{\rho\lambda}}</math>

where
:''S'' = [[surface tension]] of the air-water interface;
:<math>\rho</math> = [[density]] of the water.<ref name=physics_handbook>{{cite book|title=Physics Handbook for Science and Engineering|year=2006|publisher=Studentliteratur|page=263|author=Carl Nordling, Jonny Östermalm|edition=Eight|isbn=978-91-44-04453-8}}</ref>

When several wave trains are present, as is always the case in nature, the waves form groups. In deep water, the groups travel at a [[group velocity]] which is half of the [[phase speed]].<ref>In deep water, the [[group velocity]] is half the [[phase velocity]], as is shown [[Gravity wave#Quantitative description|here]]. Another reference is [http://musr.physics.ubc.ca/~jess/hr/skept/Waves/node12.html] {{Webarchive|url=https://web.archive.org/web/20000312125351/http://musr.physics.ubc.ca/~jess/hr/skept/Waves/node12.html|date=2000-03-12}}.</ref> Following a single wave in a group one can see the wave appearing at the back of the group, growing, and finally disappearing at the front of the group.

As the water depth <math>d</math> decreases towards the [[coast]], this will have an effect: wave height changes due to [[wave shoaling]] and [[refraction]]. As the wave height increases, the wave may become unstable when the [[crest (physics)|crest]] of the wave moves faster than the [[trough (physics)|trough]]. This causes ''surf'', a breaking of the waves.

The movement of wind waves can be captured by [[wave power|wave energy devices]]. The energy density (per unit area) of regular sinusoidal waves depends on the water [[density]] <math>\rho</math>, gravity acceleration <math>g</math> and the wave height <math>H</math> (which, for regular waves, is equal to twice the [[amplitude]], <math>a</math>):

:<math>E=\frac{1}{8}\rho g H^2=\frac{1}{2}\rho g a^2.</math>

The velocity of propagation of this energy is the [[group velocity]].