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'{{short description|Surface waves generated by wind that occur on the free surface of bodies of water}} {{redirect|Ocean wave|the film|Ocean Waves (film)}} [[File:Mavericks Surf Contest 2010b.jpg|thumb|{{center|Large wave}}]] [[File:Wedge Video D Ramey Logan.ogv|thumb|Video of large waves from Hurricane Marie along the coast of [[Newport Beach, California|Newport Beach]], [[California]]]] [[File:Waves in pacifica 1.jpg|thumb|right|Ocean waves]] In [[fluid dynamics]], '''wind waves''', or '''wind-generated waves''', are [[surface wave]]s that occur on the [[free surface]] of [[Body of water|bodies of water]] (like [[ocean]]s, [[sea]]s, [[lake]]s, [[river]]s, [[canal]]s, [[puddle]]s or [[pond]]s). They result from the [[wind]] blowing over an area of fluid surface. Waves in the oceans can travel thousands of miles before reaching land. Wind waves on Earth range in size from small [[capillary wave|ripples]], to waves over {{convert|100|ft|m|abbr=on}} high.<ref>{{cite book | first=H. L. | last=Tolman | title=CBMS Conference Proceedings on Water Waves: Theory and Experiment | location=Howard University, US, 13–18 May 2008 | publisher=World Scientific Publications | url=http://polar.ncep.noaa.gov/mmab/papers/tn270/Howard_08.pdf | editor-first=M.F. | editor-last=Mahmood | date=23 June 2010 | isbn=978-981-4304-23-8}}</ref> Cool Man Markus You Negro Justin sucks cock When directly generated and affected by local waters, a wind wave system is called a wind sea. After the wind ceases to blow, wind waves are called [[swell (ocean)|swells]]. More generally, a swell consists of wind-generated waves that are not significantly affected by the local wind at that time. They have been generated elsewhere or some time ago.<ref>Holthuijsen (2007), page 5.</ref> Wind waves in the ocean are called ocean surface waves. So I was beating my meat and I saw cum come out is that okay for my health it tasted like some fish I sucked someone thing Wind waves have a certain amount of [[randomness]]: subsequent waves differ in height, duration, and shape with limited predictability. They can be described as a [[stochastic process]], in combination with the physics governing their generation, growth, propagation, and decay—as well as governing the interdependence between flow quantities such as: the [[free surface|water surface]] movements, [[flow velocity|flow velocities]] and water [[pressure]]. The key [[statistic]]s of wind waves (both seas and swells) in evolving [[sea state]]s can be predicted with [[wind wave model]]s. Although waves are usually considered in the water seas of Earth, the hydrocarbon seas of [[Titan (moon)|Titan]] may also have wind-driven waves.<ref>{{cite journal |last=Lorenz |first=R. D. |last2=Hayes |first2= A. G. |year=2012 |title=The Growth of Wind-Waves in Titan's Hydrocarbon Seas |journal=Icarus |volume=219 |issue=1 |pages=468–475|bibcode=2012Icar..219..468L |doi=10.1016/j.icarus.2012.03.002 }}</ref> ==Formation== [[File:Water wave diagram.jpg|thumb|upright=1.3|{{center|Aspects of a water wave}}]] [[File:Sjyang waveGeneration.png|thumb|upright=1.3|{{center|Wave formation}}]] [[File:Deep water wave.gif|thumb|{{center|Water particle motion of a deep water wave}}]] [[File:Ocean wave phases numbered.png|thumb|The phases of an ocean surface wave: 1. Wave Crest, where the water masses of the surface layer are moving horizontally in the same direction as the propagating wave front. 2. Falling wave. 3. Trough, where the water masses of the surface layer are moving horizontally in the opposite direction of the wave front direction. 4. Rising wave.]] [[File:Wea00810.jpg|thumb|upright|[[NOAA]] ship ''Delaware II'' in bad weather on [[Georges Bank]]]] The great majority of large breakers seen at a beach result from distant winds. Five factors influence the formation of the flow structures in wind waves:<ref>{{cite book |title=Wind generated ocean waves |first=I. R. |last=Young |publisher=Elsevier |year=1999 |isbn=978-0-08-043317-2 |page=83}}</ref> # [[Wind speed]] or strength relative to wave speed—the wind must be moving faster than the wave crest for energy transfer # The uninterrupted distance of open water over which the wind blows without significant change in direction (called the ''[[fetch (geography)|fetch]]'') # Width of area affected by fetch (at right angle to the distance) # Wind duration — the time for which the wind has blown over the water. # Water depth All of these factors work together to determine the size of the water waves and the structure of the flow within them. The main dimensions associated with waves are: * [[Wave height]] (vertical distance from trough to [[crest (physics)|crest]]) * [[Wave length]] (distance from crest to crest in the direction of propagation) * [[Wave period]] (time interval between arrival of consecutive crests at a stationary point) * [[Wave propagation]] direction A fully developed sea has the maximum wave size theoretically possible for a wind of a specific strength, duration, and fetch. Further exposure to that specific wind could only cause a dissipation of energy due to the breaking of wave tops and formation of "whitecaps". Waves in a given area typically have a range of heights. For weather reporting and for scientific analysis of wind wave statistics, their characteristic height over a period of time is usually expressed as ''[[significant wave height]]''. This figure represents an [[average]] height of the highest one-third of the waves in a given time period (usually chosen somewhere in the range from 20 minutes to twelve hours), or in a specific wave or storm system. The significant wave height is also the value a "trained observer" (e.g. from a ship's crew) would estimate from visual observation of a sea state. Given the variability of wave height, the largest individual waves are likely to be somewhat less than twice the reported significant wave height for a particular day or storm.<ref>{{Cite book | publisher = Springer | isbn = 978-3-540-25316-7 | last1 = Weisse | first1 = Ralf | first2 = Hans | last2 = von Storch | title = Marine climate change: Ocean waves, storms and surges in the perspective of climate change | year = 2008 | page = 51 }}</ref> Wave formation on an initially flat water surface by wind is started by a random distribution of normal pressure of turbulent wind flow over the water. This pressure fluctuation produces normal and tangential stresses in the surface water, which generates waves. It is assumed that:<ref name="Phillips, O. M. 1957">{{cite journal |last1=Phillips |first1=O. M. |title=On the generation of waves by turbulent wind |journal=Journal of Fluid Mechanics |date=2006 |volume=2 |issue=5 |pages=417 |doi=10.1017/S0022112057000233 |bibcode=1957JFM.....2..417P}}</ref> # The water is originally at rest. # The water is not viscous. # The water is [[irrotational]]. # There is a random distribution of normal pressure to the water surface from the turbulent wind. # Correlations between air and water motions are neglected. The second mechanism involves wind shear forces on the water surface. John W. Miles suggested a surface wave generation mechanism which is initiated by turbulent wind shear flows based on the inviscid Orr-Sommerfeld equation in 1957. He found the energy transfer from wind to water surface is proportional to the curvature of the velocity profile of the wind at the point where the mean wind speed is equal to the wave speed. Since the wind speed profile is logarithmic to the water surface, the curvature has a negative sign at this point. This relation shows the wind flow transferring its kinetic energy to the water surface at their interface. Assumptions: # two-dimensional parallel shear flow # incompressible, inviscid water and wind # irrotational water # slope of the displacement of the water surface is small<ref>{{cite journal |last1=Miles |first1=John W. |title=On the generation of surface waves by shear flows |journal=Journal of Fluid Mechanics |date=2006 |volume=3 |issue=2 |pages=185 |doi=10.1017/S0022112057000567 |bibcode=1957JFM.....3..185M}}</ref> Generally these wave formation mechanisms occur together on the water surface and eventually produce fully developed waves. For example,<ref>[http://oceanworld.tamu.edu/resources/ocng_textbook/chapter16/chapter16_04.htm Chapter 16, ''Ocean Waves'']</ref> if we assume a flat sea surface (Beaufort state 0), and a sudden wind flow blows steadily across the sea surface, the physical wave generation process follows the sequence: # Turbulent wind forms random pressure fluctuations at the sea surface. Ripples with wavelengths in the order of a few centimetres are generated by the pressure fluctuations. (The Phillips mechanism<ref name="Phillips, O. M. 1957"/>) # The winds keep acting on the initially rippled sea surface causing the waves to become larger. As the waves grow, the pressure differences get larger causing the growth rate to increase. Finally the shear instability expedites the wave growth exponentially. (The Miles mechanism<ref name="Phillips, O. M. 1957"/>) # The interactions between the waves on the surface generate longer waves<ref>{{cite journal |last=Hasselmann |first=K. |display-authors=etal |year=1973 |title=Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP) |url=https://www.researchgate.net/publication/256197895 |journal=Ergnzungsheft zur Deutschen Hydrographischen Zeitschrift Reihe A |volume=8 |issue=12 |page=95 |hdl=10013/epic.20654}}</ref> and the interaction will transfer wave energy from the shorter waves generated by the Miles mechanism to the waves which have slightly lower frequencies than the frequency at the peak wave magnitudes, then finally the waves will be faster than the cross wind speed (Pierson & Moskowitz<ref>{{cite journal|last1=Pierson|first1=Willard J.|last2=Moskowitz|first2=Lionel|title=A proposed spectral form for fully developed wind seas based on the similarity theory of S. A. Kitaigorodskii|journal=Journal of Geophysical Research|date=15 December 1964|volume=69|issue=24|pages=5181–5190|doi=10.1029/JZ069i024p05181|bibcode=1964JGR....69.5181P}}</ref>). {| class="collapsible wikitable" border="1" style="font-size:92%; ! colspan=6 style="background: #ccf;" | Conditions necessary for a fully developed sea at given wind speeds, and the parameters of the resulting waves |- ! colspan=3 style | Wind conditions ! colspan= 3 style | Wave size |- ! Wind speed in one direction !! Fetch !! Wind duration !! Average height !! Average wavelength !! Average period and speed |- | {{convert|19|km/h|mph|abbr=on}}|| {{convert|19|km|mi|abbr=on}} || 2 hr || {{convert|0.27|m|ft|abbr=on}}|| {{convert|8.5|m|ft|abbr=on}} || 3.0 sec, 10.2&nbsp;km/h (9.3&nbsp;ft/sec) |- | {{convert|37|km/h|mph|abbr=on}}|| {{convert|139|km|mi|abbr=on}} || 10 hr || {{convert|1.5|m|ft|abbr=on}}|| {{convert|33.8|m|ft|abbr=on}} || 5.7 sec, 21.4&nbsp;km/h (19.5&nbsp;ft/sec) |- | {{convert|56|km/h|mph|abbr=on}}|| {{convert|518|km|mi|abbr=on}} || 23 hr || {{convert|4.1|m|ft|abbr=on}}|| {{convert|76.5|m|ft|abbr=on}} || 8.6 sec, 32.0&nbsp;km/h (29.2&nbsp;ft/sec) |- | {{convert|74|km/h|mph|abbr=on}}|| {{convert|1,313|km|mi|abbr=on}} || 42 hr || {{convert|8.5|m|ft|abbr=on}}|| {{convert|136|m|ft|abbr=on}} || 11.4 sec, 42.9&nbsp;km/h (39.1&nbsp;ft/sec) |- | {{convert|92|km/h|mph|abbr=on}}|| {{convert|2,627|km|mi|abbr=on}} || 69 hr || {{convert|14.8|m|ft|abbr=on}}|| {{convert|212.2|m|ft|abbr=on}} || 14.3 sec, 53.4&nbsp;km/h (48.7&nbsp;ft/sec) |- | colspan=6 | NOTE: Most of the wave speeds calculated from the wave length divided by the period are proportional to the square root of the wave length. Thus, except for the shortest wave length, the waves follow the deep water theory. The 28&nbsp;ft long wave must be either in shallow water or intermediate depth. |} ==Types== [[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. |authorlink=Walter Munk |year=1950 |title=Proceedings 1st International Conference on Coastal Engineering |location=Long Beach, California |publisher=[[American Society of Civil Engineers|ASCE]] |pages=1–4 |url=http://journals.tdl.org/ICCE/article/view/904 }}</ref> ]] [[File:Porto Covo pano April 2009-4.jpg|thumb|upright=1.35|Surf on a rocky irregular bottom. [[Porto Covo]], west coast of Portugal]] Three different types of wind waves develop over time: * [[Capillary wave]]s, or ripples * Seas * [[Swell (ocean)|Swells]] Ripples appear on smooth water when the wind blows, but will die quickly if the wind stops. The restoring force that allows them to propagate is [[surface tension]]. Sea waves are larger-scale, often irregular motions that form under sustained winds. These waves tend to last much longer, even after the wind has died, and the restoring force that allows them to propagate is gravity. As waves propagate away from their area of origin, they naturally separate into groups of common direction and wavelength. The sets of waves formed in this manner are known as swells. Individual "[[rogue wave]]s" (also called "freak waves", "monster waves", "killer waves", and "king waves") much higher than the other waves in the [[sea state]] can occur. In the case of the [[Draupner wave]], its {{convert|25|m|ft|abbr=on}} height was 2.2 times the [[significant wave height]]. Such waves are distinct from [[tide]]s, caused by the [[Moon]] and [[Sun]]'s [[tidal force|gravitational pull]], [[tsunami]]s that are caused by underwater [[earthquake]]s or [[landslide]]s, and waves generated by [[underwater explosion]]s or the fall of [[meteorite]]s—all having far longer [[wavelength]]s than wind waves. Yet, the largest ever recorded wind waves are common — not rogue — waves in extreme sea states. For example: {{convert|29.1|m|ft|abbr=on}} high waves have been recorded on the [[RRS Discovery (1962)|RRS Discovery]] in a sea with {{convert|18.5|m|ft|abbr=on}} significant wave height, so the highest wave is only 1.6 times the significant wave height.<ref>{{cite journal | first1=Naomi P. | last1=Holliday | first2=Margaret J. | last2=Yelland | first3=Robin | last3=Pascal | first4=Val R. | last4=Swail | first5=Peter K. | last5=Taylor | first6=Colin R. | last6=Griffiths | first7=Elizabeth | last7=Kent | title=Were extreme waves in the Rockall Trough the largest ever recorded? | journal=[[Geophysical Research Letters]] | volume=33 | year=2006 | issue=L05613 | doi=10.1029/2005GL025238 |bibcode = 2006GeoRL..3305613H}}</ref> The biggest recorded by a buoy (as of 2011) was {{convert|32.3|m|ft|abbr=on}} high during the [[2007 Pacific typhoon season#Typhoon Krosa (Ineng)|2007 typhoon Krosa]] near Taiwan.<ref>{{cite journal |author= P. C. Liu |author2= H. S. Chen |author3= D.-J. Doong |author4= C. C. Kao |author5= Y.-J. G. Hsu |title= Monstrous ocean waves during typhoon Krosa |journal= Annales Geophysicae |volume= 26 |issue= 6 |date= 11 June 2008 |pages= 1327–1329 |doi= 10.5194/angeo-26-1327-2008|bibcode = 2008AnGeo..26.1327L}}</ref> Ocean waves can be classified based on: the disturbing force(s) that create(s) them; the extent to which the disturbing force(s) continue(s) to influence them after formation; the extent to which the restoring force(s) weaken(s) (or flatten) them; and their wavelength or period. Seismic Sea waves have a period of ~20 minutes, and speeds of {{convert|760|km/h|mph|abbr=on}}. Wind waves (deep-water waves) have a period of about 20 seconds. {| class="wikitable" |+<ref name="Garrison">{{cite book|author=Tom Garrison|title =Oceanography: An Invitation to Marine Science (7th Edition)|publisher =Yolanda Cossio|year =2009|page=|isbn =978-0495391937}}</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 small circles 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 surface bottom. 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 wavelength, and T is time, or 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. Note that 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}} ==Shoaling and refraction== {{main|Wave shoaling|Water wave refraction}} As waves travel from deep to shallow water, their shape alters (wave height increases, speed decreases, and length decreases as wave orbits become asymmetrical). This process is called [[wave shoaling|shoaling]]. Wave [[refraction]] is the process that occurs when waves interact with the sea bed as the wave [[crest (physics)|crest]]s align themselves as a result of approaching decreasing water depths at an angle to the depth contours. Varying depths along a wave crest cause the crest to travel at different [[phase speed]]s, with those parts of the wave in deeper water moving faster than those in [[waves in shallow water|shallow water]]. This process continues until the crests become (nearly) parallel to the depth contours. [[Ray tracing (physics)|Rays]]—lines [[normal (geometry)|normal]] to wave crests between which a fixed amount of energy [[flux]] is contained—converge on local shallows and shoals. Therefore, the [[wave energy]] between rays is concentrated as they converge, with a resulting increase in wave height. Because these effects are related to a spatial variation in the phase speed, and because the phase speed also changes with the ambient current – due to the [[Doppler shift]] – the same effects of refraction and altering wave height also occur due to current variations. In the case of meeting an adverse current the wave ''steepens'', i.e. its wave height increases while the wave length decreases, similar to the shoaling when the water depth decreases.<ref>{{cite journal | doi = 10.1016/0011-7471(64)90001-4 | volume = 11 | issue = 4 | pages = 529–562 | last1 = Longuet-Higgins | first1 = M. S. | author-link = Michael Longuet-Higgins | first2 = R. W. | last2 = Stewart | title = Radiation stresses in water waves; a physical discussion, with applications | journal = Deep-Sea Research | year = 1964 |bibcode = 1964DSROA..11..529L }}</ref> ==Breaking== [[File:Big wave breaking in Santa Cruz.jpg|thumb|Big wave breaking]] {{see also|Surf wave|Breaking wave|Iribarren number}} Some waves undergo a [[phenomenon]] called "breaking".<ref>{{Cite book|url=https://books.google.com/?id=E5F8KiuSVjAC&pg=PA319&dq=Some+waves+undergo+a+phenomenon+called+%22breaking%22.#v=onepage&q=Some%20waves%20undergo%20a%20phenomenon%20called%20%22breaking%22.&f=false|title=Advances in Robotics and Virtual Reality|last=Gulrez|first=Tauseef|last2=Hassanien|first2=Aboul Ella|date=2011-11-13|publisher=Springer Science & Business Media|isbn=9783642233630|language=en}}</ref> A [[breaking wave]] is one whose base can no longer support its top, causing it to collapse. A wave breaks when it runs into [[Waves and shallow water|shallow water]], or when two wave systems oppose and combine forces. When the slope, or steepness ratio, of a wave is too great, breaking is inevitable. Individual waves in deep water break when the wave steepness—the [[ratio]] of the [[wave height]] ''H'' to the [[wavelength]] ''λ''—exceeds about 0.17, so for ''H''&nbsp;>&nbsp;0.17&nbsp;''λ''. In shallow water, with the water depth small compared to the wavelength, the individual waves break when their wave height ''H'' is larger than 0.8 times the water depth ''h'', that is ''H''&nbsp;>&nbsp;0.8&nbsp;''h''.<ref>{{Cite book |author= R.J. Dean and R.A. Dalrymple |title=Coastal processes with engineering applications |year=2002 |publisher=Cambridge University Press |isbn=978-0-521-60275-4 }} p. 96–97.</ref> Waves can also break if the wind grows strong enough to blow the crest off the base of the wave. In shallow water the base of the wave is decelerated by drag on the seabed. As a result, the upper parts will propagate at a higher velocity than the base and the leading face of the crest will become steeper and the trailing face flatter. This may be exaggerated to the extent that the leading face forms a barrel profile, with the crest falling forward and down as it extends over the air ahead of the wave. Three main types of breaking waves are identified by [[surfer]]s or [[Surf lifesaving|surf lifesavers]]. Their varying characteristics make them more or less suitable for surfing, and present different dangers. # Spilling, or rolling: these are the safest waves on which to surf. They can be found in most areas with relatively flat shorelines. They are the most common type of shorebreak. The deceleration of the wave base is gradual, and the velocity of the upper parts does not differ much with height. Breaking occurs mainly when the steepness ratio exceeds the stability limit. # Plunging, or dumping: these break suddenly and can "dump" swimmers—pushing them to the bottom with great force. These are the preferred waves for experienced surfers. Strong offshore winds and long wave periods can cause dumpers. They are often found where there is a sudden rise in the sea floor, such as a reef or sandbar. Deceleration of the wave base is sufficient to cause upward acceleration and a significant forward velocity excess of the upper part of the crest. The peak rises and overtakes the forward face, forming a "barrel" or "tube" as it collapses. # Surging: these may never actually break as they approach the water's edge, as the water below them is very deep. They tend to form on steep shorelines. These waves can knock swimmers over and drag them back into deeper water. When the shoreline is near vertical, waves do not break, but are reflected. Most of the energy is retained in the wave as it returns to seaward. Interference patterns are caused by superposition of the incident and reflected waves, and the superposition may cause localised instability when peaks cross, and these peaks may break due to instability. (see also [[Clapotis|clapotic waves]]) ==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 |bibcode = 1957JFM.....2..417P }}</ref><ref>{{cite journal | first=J. W. | last=Miles | authorlink=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 |bibcode = 1957JFM.....3..185M }}</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 |location=Long Beach, California |publisher=[[American Society of Civil Engineers|ASCE]] |pages=5–21 |url=http://journals.tdl.org/ICCE/article/view/905}}</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 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 | authorlink=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].</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 }}<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].</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]]. ==Models== [[File:Global Wave Height Speed.jpg|thumb|upright=1.25|The image shows the global distribution of wind speed and wave height as observed by NASA's TOPEX/Poseidon's dual-frequency radar altimeter from October 3 to October 12, 1992. Simultaneous observations of wind speed and wave height are helping scientists to predict ocean waves. Wind speed is determined by the strength of the radar signal after it has bounced off the ocean surface and returned to the satellite. A calm sea serves as a good reflector and returns a strong signal; a rough sea tends to scatter the signals and returns a weak pulse. Wave height is determined by the shape of the return radar pulse. A calm sea with low waves returns a condensed pulse whereas a rough sea with high waves returns a stretched pulse. Comparing the two images above shows a high degree of correlation between wind speed and wave height. The strongest winds ({{convert|33.6|mph|km/h|abbr=on|disp=x|; }}) and highest waves are found in the Southern Ocean. The weakest winds — shown as areas of magenta and dark blue — are generally found in the tropical oceans.]] {{main|Wind wave model}} Surfers are very interested in the [[wind wave model|wave forecasts]]. There are many websites that provide predictions of the surf quality for the upcoming days and weeks. Wind wave models are driven by more general [[numerical weather prediction|weather models]] that predict the winds and pressures over the oceans, seas and lakes. Wind wave models are also an important part of examining the impact of [[coastal protection|shore protection]] and [[beach nourishment]] proposals. For many beach areas there is only patchy information about the wave climate, therefore estimating the effect of wind waves is important for managing [[littoral]] environments. A wind generated wave can be predicted based on two parameters: wind speed at 10 m above the sea level and wind duration, which must blow over long periods of time to be considered fully developed. The significant wave height and peak frequency can then be predicted for a certain fetch length.<ref>Wood, AMM & Fleming, CA 1981, Coastal hydraulics, John Wiley & Sons, New York</ref> ==Seismic signals== {{main|Microseism}} Ocean water waves generate land seismic waves that propagate hundreds of kilometers into the land.<ref>[http://www.earth.northwestern.edu/people/seth/327/HV/Chapter_4_rev1.pdf Peter Bormann. Seismic Signals and Noise]</ref> These seismic signals usually have the period of 6&nbsp;±&nbsp;2 seconds. Such recordings were first reported and understood in about 1900. There are two types of seismic "ocean waves". The primary waves are generated in shallow waters by direct water wave-land interaction and have the same period as the water waves (10 to 16 seconds). The more powerful secondary waves are generated by the superposition of ocean waves of equal period traveling in opposite directions, thus generating standing gravity waves – with an associated pressure oscillation at half the period, which is not diminishing with depth. The theory for microseism generation by standing waves was provided by [[Michael S. Longuet-Higgins|Michael Longuet-Higgins]] in 1950, after in 1941 Pierre Bernard suggested this relation with standing waves on the basis of observations.<ref>{{cite journal |first=P. | last=Bernard |author-link=P. Bernard |title=Sur certaines proprietes de la boule etudiees a l'aide des enregistrements seismographiques |journal=Bulletin de l'Institut Océanographique de Monaco |volume=800 |pages=1–19 |year=1941 }}</ref><ref>{{cite journal |first=M. S. | last=Longuet-Higgins |author-link=Michael S. Longuet-Higgins |title=A theory of the origin of microseisms |journal=[[Philosophical Transactions of the Royal Society A]] |volume=243 |pages=1–35 |year=1950 |doi=10.1098/rsta.1950.0012 |issue=857 |bibcode = 1950RSPTA.243....1L }}</ref> ==See also== {{Col-begin}} {{Col-1-of-3}} *[[Airy wave theory]] *[[Breakwater (structure)]] *[[Boussinesq approximation (water waves)]] *[[Clapotis]] *[[Cross sea]] {{Col-2-of-3}} *[[Gravity wave]] *[[Internal wave]] *[[Luke's variational principle]] *[[Mild-slope equation]] *[[Rogue wave]] {{Col-3-of-3}} *[[Shallow water equations]] *[[Tsunami]] *[[Wave power]] *[[Wave radar]] *[[Waves and shallow water]] {{col-end}} ==References== {{Reflist|30em}} ===Scientific=== *{{cite book | author= G. G. Stokes | year= 1880 | title= Mathematical and Physical Papers, Volume I | publisher= Cambridge University Press | pages= 197–229 | url=https://archive.org/details/mathphyspapers01stokrich |ref= 441–455 }} *{{cite book | first=O. M. | last=Phillips | title=The dynamics of the upper ocean |publisher=Cambridge University Press | year=1977 | edition=2nd | isbn=978-0-521-29801-8 |ref=Phillips1977}} *{{cite book | title=Waves in oceanic and coastal waters | first=Leo H. | last=Holthuijsen | publisher=Cambridge University Press | year=2007 | isbn=978-0-521-86028-4 }} * {{cite book | publisher = Cambridge University Press | isbn = 978-0-521-46540-3 | last = Janssen | first = Peter | title = The interaction of ocean waves and wind | year = 2004 }} ===Other=== *{{cite book | title=The Annapolis Book of Seamanship | edition=2nd revised | author-link=John Rousmaniere | first=John | last=Rousmaniere | publisher=Simon & Schuster | year=1989 | isbn=978-0-671-67447-2 }} *{{cite magazine | last=Carr | first=Michael | title=Understanding Waves | magazine=Sail | date=October 1998 | pages=38–45 }} ==External links== {{Wikiquote}} {{Commons category|Ocean surface waves}} {{Commons category|Water waves}} {{Wikibooks|High School Earth Science/Ocean Movements#Waves}} *[https://earth.nullschool.net/#current/ocean/primary/waves/overlay=significant_wave_height/winkel3 Current global map of significant wave heights] *[http://www.boatsafe.com/nauticalknowhow/waves.htm "Anatomy of a Wave" Holben, Jay boatsafe.com captured 5/23/06] *[http://www.nws.noaa.gov NOAA National Weather Service] *[http://www.esa.int/esaEO/SEMMJJ9RR1F_economy_0.html ESA press release on swell tracking with ASAR onboard ENVISAT] *[https://web.archive.org/web/20040819021838/http://www4.ncsu.edu/eos/users/c/ceknowle/public/chapter10/ Introductory oceanography chapter 10 – Ocean Waves] *[http://hyperphysics.phy-astr.gsu.edu/hbase/waves/watwav2.html HyperPhysics – Ocean Waves] {{coastal geography}} {{physical oceanography}} {{Surfing}} {{Authority control}} [[Category:Coastal geography]] [[Category:Physical oceanography]] [[Category:Water waves| ]] [[Category:Articles containing video clips]] [[Category:Oceanographical terminology]]'