Wind wave

In fluid dynamics, wind waves, or wind-generated winds, are surface waves that occur on the free surface of bodies of water (like oceans, seas, lakes, rivers, canals, puddles or ponds). 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 ripples, to waves over 100 ft (30 m) high.[1]

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 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.[2] Wind waves in the ocean are called ocean surface waves.

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 water surface movements, flow velocities and water pressure. The key statistics of wind waves (both seas and swells) in evolving sea states can be predicted with wind wave models.

Although waves are usually considered in the water seas of Earth, the hydrocarbon seas of Titan may also have wind-driven waves.[3]

Mavericks Surf Contest 2010b
Large wave
Video of large waves from Hurricane Marie along the coast of Newport Beach, California
Waves in pacifica 1
Ocean waves


Water wave diagram
Aspects of a water wave
Sjyang waveGeneration
Wave formation
Deep water wave
Water particle motion of a deep water wave
Ocean wave phases numbered
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.
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:[4]

  1. Wind speed or strength relative to wave speed—the wind must be moving faster than the wave crest for energy transfer
  2. The uninterrupted distance of open water over which the wind blows without significant change in direction (called the fetch)
  3. Width of area affected by fetch (at right angle to the distance)
  4. Wind duration — the time for which the wind has blown over the water.
  5. 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:

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.[5]

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:[6]

  1. The water is originally at rest.
  2. The water is not viscous.
  3. The water is irrotational.
  4. There is a random distribution of normal pressure to the water surface from the turbulent wind.
  5. 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.


  1. two-dimensional parallel shear flow
  2. incompressible, inviscid water and wind
  3. irrotational water
  4. slope of the displacement of the water surface is small[7]

Generally these wave formation mechanisms occur together on the water surface and eventually produce fully developed waves.

For example,[8] 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:

  1. 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[6])
  2. 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[6])
  3. The interactions between the waves on the surface generate longer waves[9] 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[10]).
Conditions necessary for a fully developed sea at given wind speeds, and the parameters of the resulting waves
Wind conditions Wave size
Wind speed in one direction Fetch Wind duration Average height Average wavelength Average period and speed
19 km/h (12 mph) 19 km (12 mi) 2 hr 0.27 m (0.89 ft) 8.5 m (28 ft) 3.0 sec, 10.2 km/h (9.3 ft/sec)
37 km/h (23 mph) 139 km (86 mi) 10 hr 1.5 m (4.9 ft) 33.8 m (111 ft) 5.7 sec, 21.4 km/h (19.5 ft/sec)
56 km/h (35 mph) 518 km (322 mi) 23 hr 4.1 m (13 ft) 76.5 m (251 ft) 8.6 sec, 32.0 km/h (29.2 ft/sec)
74 km/h (46 mph) 1,313 km (816 mi) 42 hr 8.5 m (28 ft) 136 m (446 ft) 11.4 sec, 42.9 km/h (39.1 ft/sec)
92 km/h (57 mph) 2,627 km (1,632 mi) 69 hr 14.8 m (49 ft) 212.2 m (696 ft) 14.3 sec, 53.4 km/h (48.7 ft/sec)
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 ft long wave must be either in shallow water or intermediate depth.


Munk ICCE 1950 Fig1
Classification of the spectrum of ocean waves according to wave period[11]
Porto Covo pano April 2009-4
Surf on a rocky irregular bottom. Porto Covo, west coast of Portugal

Three different types of wind waves develop over time:

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 waves" (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 25 m (82 ft) height was 2.2 times the significant wave height. Such waves are distinct from tides, caused by the Moon and Sun's gravitational pull, tsunamis that are caused by underwater earthquakes or landslides, and waves generated by underwater explosions or the fall of meteorites—all having far longer wavelengths than wind waves.

Yet, the largest ever recorded wind waves are common — not rogue — waves in extreme sea states. For example: 29.1 m (95 ft) high waves have been recorded on the RRS Discovery in a sea with 18.5 m (61 ft) significant wave height, so the highest wave is only 1.6 times the significant wave height.[12] The biggest recorded by a buoy (as of 2011) was 32.3 m (106 ft) high during the 2007 typhoon Krosa near Taiwan.[13]

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 760 km/h (470 mph). Wind waves (deep-water waves) have a period of about 20 seconds.

Wave type Typical wavelength Disturbing force Restoring force
Capillary wave < 2 cm Wind Surface tension
Wind wave 60–150 m (200–490 ft) Wind over ocean Gravity
Seiche Large, variable; a function of basin size Change in atmospheric pressure, storm surge Gravity
Seismic sea wave (tsunami) 200 km (120 mi) 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:

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 of the wavelength on the period (the dispersion relation).

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

where g is the acceleration due to gravity, 9.8 meters (32 feet) per second squared. Because g and π (3.14) are constants, the equation can be reduced to:

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:

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.

Shoaling and 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 shoaling.

Wave refraction is the process that occurs when waves interact with the sea bed as the wave crests 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 speeds, with those parts of the wave in deeper water moving faster than those in shallow water. This process continues until the crests become (nearly) parallel to the depth contours. Rays—lines 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.[15]


Big wave breaking in Santa Cruz
Big wave breaking

Some waves undergo a phenomenon called "breaking".[16] A breaking wave is one whose base can no longer support its top, causing it to collapse. A wave breaks when it runs into 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 > 0.17 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 > 0.8 h.[17] 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 surfers or surf lifesavers. Their varying characteristics make them more or less suitable for surfing, and present different dangers.

  1. 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.
  2. 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.
  3. 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 clapotic waves)

Physics of waves

Shallow water wave
Stokes drift in shallow water waves (Animation)

Wind waves are mechanical waves that propagate along the interface between water and air; the restoring force is provided by gravity, and so they are often referred to as 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 Miles, also in 1957.[18][19]

Deep water wave
Stokes drift in a deeper water wave (Animation)
Orbital wave motion-Wiegel Johnson ICCE 1950 Fig 6
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 = 2.50 ft (0.76 m), wave height H = 0.339 ft (0.103 m), wavelength λ = 6.42 ft (1.96 m), period T = 1.12 s.[20]

In linear plane waves of one wavelength in deep water, 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 and longitudinal waves.

When waves propagate in shallow water, (where the depth is less than half the wavelength) the particle trajectories are compressed into ellipses.[21][22]

In reality, for 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.[23][24]

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 wave motion of small-amplitude waves – well approximated by


c = phase speed;
λ = wavelength;
d = water depth;
g = acceleration due to gravity at the Earth's surface.

In deep water, where , so and the hyperbolic tangent approaches , the speed approximates

In SI units, with in m/s, , when 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 equations are applicable, combining 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 c when the wavelength approaches infinity) can be approximated by

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


S = surface tension of the air-water interface;
= density of the water.[25]

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.[26] 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 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 of the wave moves faster than the trough. This causes surf, a breaking of the waves.

The movement of wind waves can be captured by wave energy devices. The energy density (per unit area) of regular sinusoidal waves depends on the water density , gravity acceleration and the wave height (which, for regular waves, is equal to twice the amplitude, ):

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


Global Wave Height Speed
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 (33.6 mph; 54.1 km/h) 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.

Surfers are very interested in the 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 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 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.[27]

Seismic signals

Ocean water waves generate land seismic waves that propagate hundreds of kilometers into the land.[28] These seismic signals usually have the period of 6 ± 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 Longuet-Higgins in 1950, after in 1941 Pierre Bernard suggested this relation with standing waves on the basis of observations.[29][30]

See also


  1. ^ Tolman, H. L. (23 June 2010). Mahmood, M.F., ed. "CBMS Conference Proceedings on Water Waves: Theory and Experiment" (PDF). Howard University, US, 13–18 May 2008: World Scientific Publications. ISBN 978-981-4304-23-8.
  2. ^ Holthuijsen (2007), page 5.
  3. ^ Lorenz, R. D.; Hayes, A. G. (2012). "The Growth of Wind-Waves in Titan's Hydrocarbon Seas". Icarus. 219: 468–475. Bibcode:2012Icar..219..468L. doi:10.1016/j.icarus.2012.03.002.
  4. ^ Young, I. R. (1999). Wind generated ocean waves. Elsevier. p. 83. ISBN 0-08-043317-0.
  5. ^ Weisse, Ralf; von Storch, Hans (2008). Marine climate change: Ocean waves, storms and surges in the perspective of climate change. Springer. p. 51. ISBN 978-3-540-25316-7.
  6. ^ a b c Phillips, O. M. (2006). "On the generation of waves by turbulent wind". Journal of Fluid Mechanics. 2 (5): 417. Bibcode:1957JFM.....2..417P. doi:10.1017/S0022112057000233.
  7. ^ Miles, John W. (2006). "On the generation of surface waves by shear flows". Journal of Fluid Mechanics. 3 (2): 185. Bibcode:1957JFM.....3..185M. doi:10.1017/S0022112057000567.
  8. ^ Chapter 16, Ocean Waves
  9. ^ Hasselmann, K.; et al. (1973). "Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP)". Ergnzungsheft zur Deutschen Hydrographischen Zeitschrift Reihe A. 8 (12): 95. hdl:10013/epic.20654.
  10. ^ Pierson, Willard J.; Moskowitz, Lionel (15 December 1964). "A proposed spectral form for fully developed wind seas based on the similarity theory of S. A. Kitaigorodskii". Journal of Geophysical Research. 69 (24): 5181–5190. Bibcode:1964JGR....69.5181P. doi:10.1029/JZ069i024p05181.
  11. ^ Munk, Walter H. (1950). "Proceedings 1st International Conference on Coastal Engineering". Long Beach, California: ASCE: 1–4.
  12. ^ Holliday, Naomi P.; Yelland, Margaret J.; Pascal, Robin; Swail, Val R.; Taylor, Peter K.; Griffiths, Colin R.; Kent, Elizabeth (2006). "Were extreme waves in the Rockall Trough the largest ever recorded?". Geophysical Research Letters. 33 (L05613). Bibcode:2006GeoRL..3305613H. doi:10.1029/2005GL025238.
  13. ^ P. C. Liu; H. S. Chen; D.-J. Doong; C. C. Kao; Y.-J. G. Hsu (11 June 2008). "Monstrous ocean waves during typhoon Krosa" (PDF). Annales Geophysicae. European Geosciences Union. 26 (6): 1327–1329. Bibcode:2008AnGeo..26.1327L. doi:10.5194/angeo-26-1327-2008.
  14. ^ Tom Garrison (2009). Oceanography: An Invitation to Marine Science (7th Edition). Yolanda Cossio. ISBN 978-0495391937.
  15. ^ Longuet-Higgins, M. S.; Stewart, R. W. (1964). "Radiation stresses in water waves; a physical discussion, with applications". Deep-Sea Research. 11 (4): 529–562. Bibcode:1964DSROA..11..529L. doi:10.1016/0011-7471(64)90001-4.
  16. ^ Gulrez, Tauseef; Hassanien, Aboul Ella (2011-11-13). Advances in Robotics and Virtual Reality. Springer Science & Business Media. ISBN 9783642233630.
  17. ^ R.J. Dean and R.A. Dalrymple (2002). Coastal processes with engineering applications. Cambridge University Press. ISBN 0-521-60275-0. p. 96–97.
  18. ^ Phillips, O. M. (1957). "On the generation of waves by turbulent wind". Journal of Fluid Mechanics. 2 (5): 417–445. Bibcode:1957JFM.....2..417P. doi:10.1017/S0022112057000233.
  19. ^ Miles, J. W. (1957). "On the generation of surface waves by shear flows". Journal of Fluid Mechanics. 3 (2): 185–204. Bibcode:1957JFM.....3..185M. doi:10.1017/S0022112057000567.
  20. ^ Figure 6 from: Wiegel, R. L.; Johnson, J. W. (1950). "Proceedings 1st International Conference on Coastal Engineering". Long Beach, California: ASCE: 5–21.
  21. ^ For the particle trajectories within the framework of linear wave theory, see for instance:
    Phillips (1977), page 44.
    Lamb, H. (1994). Hydrodynamics (6th ed.). Cambridge University Press. ISBN 978-0-521-45868-9. Originally published in 1879, the 6th extended edition appeared first in 1932. See §229, page 367.
    L. D. Landau and E. M. Lifshitz (1986). Fluid mechanics. Course of Theoretical Physics. 6 (Second revised ed.). Pergamon Press. ISBN 0-08-033932-8. See page 33.
  22. ^ A good illustration of the wave motion according to linear theory is given by Prof. Robert Dalrymple's Java applet.
  23. ^ For nonlinear waves, the particle paths are not closed, as found by George Gabriel Stokes in 1847, see the original paper by Stokes. Or in Phillips (1977), page 44: "To this order, it is evident that the particle paths are not exactly closed ... pointed out by Stokes (1847) in his classical investigation".
  24. ^ Solutions of the particle trajectories in fully nonlinear periodic waves and the Lagrangian wave period they experience can for instance be found in:
    J. M. Williams (1981). "Limiting gravity waves in water of finite depth". Philosophical Transactions of the Royal Society A. 302 (1466): 139–188. Bibcode:1981RSPTA.302..139W. doi:10.1098/rsta.1981.0159.
    J. M. Williams (1985). Tables of progressive gravity waves. Pitman. ISBN 978-0-273-08733-5.
  25. ^ Carl Nordling, Jonny Östermalm (2006). Physics Handbook for Science and Engineering (Eight ed.). Studentliteratur. p. 263. ISBN 978-91-44-04453-8.
  26. ^ In deep water, the group velocity is half the phase velocity, as is shown here. Another reference is [1].
  27. ^ Wood, AMM & Fleming, CA 1981, Coastal hydraulics, John Wiley & Sons, New York
  28. ^ Peter Bormann. Seismic Signals and Noise
  29. ^ Bernard, P. (1941). "Sur certaines proprietes de la boule etudiees a l'aide des enregistrements seismographiques". Bulletin de l'Institut océanographique de Monaco. 800: 1–19.
  30. ^ Longuet-Higgins, M. S. (1950). "A theory of the origin of microseisms". Philosophical Transactions of the Royal Society A. 243 (857): 1–35. Bibcode:1950RSPTA.243....1L. doi:10.1098/rsta.1950.0012.


  • G. G. Stokes (1880). Mathematical and Physical Papers, Volume I. Cambridge University Press. pp. 197–229.
  • Phillips, O. M. (1977). The dynamics of the upper ocean (2nd ed.). Cambridge University Press. ISBN 0-521-29801-6.
  • Holthuijsen, Leo H. (2007). Waves in oceanic and coastal waters. Cambridge University Press. ISBN 0-521-86028-8.
  • Janssen, Peter (2004). The interaction of ocean waves and wind. Cambridge University Press. ISBN 978-0-521-46540-3.


  • Rousmaniere, John (1989). The Annapolis Book of Seamanship (2nd revised ed.). Simon & Schuster. ISBN 0-671-67447-1.
  • Carr, Michael (October 1998). "Understanding Waves". Sail. pp. 38–45.

External links


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Hawaiian scale

Hawaiian scale is an expression of the height of a wind wave affecting water. It is the expression conventionally used by surfers in Hawaii and is also used in Australia and parts of South Africa.

The expression, always given in feet, is a scaled figure corresponding to roughly half the actual measured or estimated height of a wave's face. Thus, a "3-foot" wave is roughly six feet high, i.e., head-high to a 6-foot person; a "2-foot" wave is roughly four feet high, i.e., chest-high to such a person; and a "6- to 8-foot" wave would be 2 to approaching 3 times head-high to such a person. As wave height increases, however, so does the difficulty of judging that height, and as wave height approaches 20 feet, the range of absolute wave heights corresponding to a given scaled expression tends to widen.

The origin of the scale is obscure. Commentator Neal Miyake has proposed the following candidates:

Hawaiian lifeguards' announcement of smaller wave sizes in an effort to minimize casual tourists' interest in a surf break (albeit at the risk of attracting novice surfers aware of their own limitations who might be deterred by an announcement of a larger wave size)

A possible local practice of taking measurements "from the back" of the wave, i.e., from mean sea level (in technical terms, of measuring not the "peak-to-peak amplitude" of the wave or the height of the wave's face, the latter of which is increased by the wave's drawing water from in front of it as it breaks, but rather the wave's "semi-amplitude" or "peak amplitude"), or from wave buoy readings

Modesty or false modesty in early surfers' reports of their own accomplishmentsIn Australia, which otherwise uses the metric system, surfers and surfer-oriented media such as Australia's Surfing Life and Tracks magazines still measure and describe waves in terms of feet. Some journalists and media outlets that provide information to surfers but are not staffed by insiders to the sport express wave size in metric units using direct conversion from a literal interpretation of the scale's output, e.g., labeling as "1-metre" a wave that insiders would describe as "3-foot" or slightly larger. Such an attempt, however, is unsatisfactory both to surfers who do not use the converted units and to non-surfers and novices who do not realize that the trough-to-crest wave height is twice the figure quoted.

Hendrik Tolman

Hendrik Lieuwe Tolman (born 16 January 1961 in Leeuwarden) is the original developer of the WAVEWATCH III (tm) wind wave model.He has been the branch chief for the Marine Modeling and Analysis Branch in the US National Weather Service of NOAA since 2007, having joined the branch in 1992.

ISLES project

The ISLES (Irish–Scottish Links on Energy Study) was a project to facilitate the development of offshore renewable energy sources, such as wind, wave and tidal energy, and renewable energy trade between Scotland, Republic of Ireland and Northern Ireland. It assessed the feasibility and developed a conception of creating an integrated offshore transmission network connecting the renewable energy projects' sites located off the west coast of Scotland, north and east coasts of Northern Ireland, west coast of the Republic of Ireland and in the Irish Sea with onshore grids. It was a joint project between the governments of Scotland, Ireland, and Northern Ireland, which is partly funded by the European Union's INTERREG IVA Programme. The funding from INTEREG was about €2 million.The ISLES project was announced at the meeting of Minister for Enterprise, Energy and Tourism of Scotland Jim Mather and Minister for Communications, Energy and Natural Resources of Ireland Eamon Ryan in Glasgow on 7 June 2008. The project was managed under the Special EU Programmes Body. The contract to undertake the feasibility study was awarded to a consortium led by RPS Group and its final reports were delivered to the inter-governmental steering group in late-2011. The findings were disseminated at the conference in Glasgow on 23 November 2011. According to these findings, there are no technological barriers to ISLES and the project is feasible, although landfall points throughout the three jurisdictions have significant constraints due to environmental issues. It would cost about £1 million per each MW of installed capacity.

Kamen Rider Girls

Kamen Rider Girls (仮面ライダーGIRLS, Kamen Raidā Gāruzu) are an idol group developed by Avex Trax and Ishimori Productions to commemorate the Kamen Rider Series' 40th anniversary in 2011. Each of the group's current members represent one of the protagonists of the Kamen Rider Series. The group made their premiere at an event featuring members of Columbia Music Entertainment's Project.R group, the musical collaboration who provides music for the Super Sentai series. The group's debut songs were "Koi no Rider Kick" (恋のライダーキック, Koi no Raidā Kikku, "Love's Rider Kick") and "Heart no Henshin Belt" (♡(ハート)の変身ベルト, Hāto no Henshin Beruto, "The Heart's Transformation Belt"). The group's debut songs were played on the DJ HURRY KENN Ride the Groove Internet radio program, the successor to the Wind Wave radio programs from the Kamen Rider W series. The group's debut single is "Let's Go RiderKick 2011", serving as the theme song for OOO, Den-O, All Riders: Let's Go Kamen Riders with "Koi no Rider Kick" and "Heart no Henshin Belt" as the single's B-sides. A second single titled "KAMEN RIDER V3" was released on August 3, 2011 with the PV having a cameo by Hiroshi Tanahashi.

Kamen Rider W

Kamen Rider W (仮面ライダーW (ダブル), Kamen Raidā Daburu, also called Kamen Rider Double), is a 2009-2010 Japanese tokusatsu drama, the eleventh series in the Heisei period run of the Kamen Rider Series. It premiered following the finale of Kamen Rider Decade on September 6, 2009, and aired alongside Samurai Sentai Shinkenger in TV Asahi's Super Hero Time programming block. Following Shinkenger's finale, it aired alongside Tensou Sentai Goseiger, until W concluded on August 29, 2010. The series is described as the "Heisei Kamen Rider 10th Anniversary Project: Fall Campaign" (平成仮面ライダー10周年プロジェクト 秋の陣, Heisei Kamen Raidā Jusshūnen Purojekuto: Aki no Jin). In the first episode of Kamen Rider Fourze, W is revealed to be in the same continuity as the original Showa timeline, making it the first series to do so since Kamen Rider Agito.

Power station

A power station, also referred to as a power plant or powerhouse and sometimes generating station or generating plant, is an industrial facility for the generation of electric power. Most power stations contain one or more generators, a rotating machine that converts mechanical power into electrical power. The relative motion between a magnetic field and a conductor creates an electrical current. The energy source harnessed to turn the generator varies widely. Most power stations in the world burn fossil fuels such as coal, oil, and natural gas to generate electricity. Others use nuclear power, but there is an increasing use of cleaner renewable sources such as solar, wind, wave and hydroelectric.


RenewableUK, formerly known as the 'British Wind Energy Association' (BWEA), is the trade association for wind power, wave power and tidal power industries in the United Kingdom. RenewableUK has over 660 corporate members, from wind, wave and tidal stream power generation and associated industries.

The association carries out research, and co-ordinates statistics and intelligence on marine and wind power in the UK and its waters. It also represents its members internationally, and to Government, regional bodies and local authorities in the UK.

Spar (platform)

A spar is a type of floating oil platform typically used in very deep waters, and is named for logs used as buoys in shipping that are moored in place vertically. Spar production platforms have been developed as an alternative to conventional platforms. The deep draft design of spars makes them less affected by wind, wave and currents and allows for both dry tree and subsea production. Spars are most prevalent in the US Gulf of Mexico; however, there are also spars located offshore Malaysia and Norway.A spar platform consists of a large-diameter, single vertical cylinder supporting a deck. The cylinder is weighted at the bottom by a chamber filled with a material that is denser than water (to lower the center of gravity of the platform and provide stability). Additionally, the spar hull is encircled by helical strakes to mitigate the effects of vortex-induced motion. Spars are permanently anchored to the seabed by way of a spread mooring system composed of either a chain-wire-chain or chain-polyester-chain configuration.There are three primary types of spars; the classic spar, truss spar, and cell spar. The classic spar consists of the cylindrical hull noted above, with heavy ballast tanks located at the bottom of the cylinder.

A truss spar has a shorter cylindrical "hard tank" than a classic spar and has a truss structure connected to the bottom of the hard tank. This truss structure consists of four large orthogonal "leg" members with X-braces between each of the legs and heave plates at intermediate depths to provide damping. At the bottom of the truss structure, there is a relatively small keel, or soft tank, that houses the heavy ballasting material. Soft tanks are typically rectangular in shape but have also been round to accommodate specific construction concerns. The majority of spars are of this type.A third type of spar, the cell spar, has a large central cylinder surrounded by smaller cylinders of alternating lengths. At the bottom of the longer cylinders is the soft tank housing the heavy ballasting material, similar to a truss spar. The cell spar design was only ever used for one platform, the Red Hawk spar, which was decommissioned in 2014 under the Bureau of Safety and Environmental Enforcement's "Rigs-to-Reefs" program. At the time of its decommissioning it was the deepest floating platform to ever be decommissioned.The first spar was the Brent Spar, a platform designed for storage and offloading of crude oil products. It was installed in the Brent Field in June 1976. The attempted deep sea disposal of the platform in the 1990s created a huge backlash by Greenpeace. The spar was eventually dismantled and pieces were used as a foundation for a quay in Norway.The first spar designed for production was the Neptune spar, located in the Gulf of Mexico, and was installed in September 1996 by Kerr McGee (now Anadarko).The world's deepest production platform is Perdido, a truss spar in the Gulf of Mexico, with a mean water depth of 2,438 meters. It is operated by Royal Dutch Shell and was built at a cost of $3 billion.

Surf zone

As ocean surface waves come closer to shore they break, forming the foamy, bubbly surface called surf. The region of breaking waves defines the surf zone. After breaking in the surf zone, the waves (now reduced in height) continue to move in, and they run up onto the sloping front of the beach, forming an uprush of water called swash. The water then runs back again as backswash. The nearshore zone where wave water comes onto the beach is the surf zone. The water in the surf zone, or breaker zone, is shallow, usually between 5 and 10 m (16 and 33 ft) deep; this causes the waves to be unstable.

Swell (ocean)

A swell, in the context of an ocean, sea or lake, is a series of mechanical waves that propagate along the interface between water and air and thus are often referred to as surface gravity waves. These series of surface gravity waves are not wind waves, which are generated by the immediate local wind, but instead are generated by distant weather systems, where wind blows for a duration of time over a fetch of water. More generally, a swell consists of wind-generated waves that are not—or are hardly—affected by the local wind at that time. Swell waves often have a long wavelength, but this varies due to the size, strength and duration of the weather system responsible for the swell and the size of the water body. Swell wavelength also varies from event to event. Occasionally, swells which are longer than 700 m occur as a result of the most severe storms. Swells have a narrower range of frequencies and directions than locally generated wind waves, because swell waves have dispersed from their generation area, have dissipated and therefore lost an amount of randomness, taking on a more defined shape and direction. Swell direction is the direction from which the swell is coming. It is measured in degrees (as on a compass), and often referred to in general directions, such as a NNW or SW swell.


WAM or Wam may refer to:

Wam, Pakistan, a village of Ziarat District

WAM (Emirates news agency), a news agency in the United Arab Emirates

WAM!, a cable television channel for children

Warren Abstract Machine, an abstract machine for the execution of Prolog

Web audience measurement, a tool that measures Internet usage in India

Web access management, a form of identity management

Weekender Records or Weekender Artist Management

Weisman Art Museum, in Minneapolis

Wet and messy fetishism

Wide area multilateration, a surveillance technology for air traffic management

Wireless asset monitoring system

Worcester Art Museum, in Worcester, Massachusetts

Wolfgang Amadeus Mozart (1756–1791), musician

WAM, a kind of wind wave model

Wave action (continuum mechanics)

In continuum mechanics, wave action refers to a conservable measure of the wave part of a motion. For small-amplitude and slowly varying waves, the wave action density is:

where is the intrinsic wave energy and is the intrinsic frequency of the slowly modulated waves – intrinsic here implying: as observed in a frame of reference moving with the mean velocity of the motion.

The action of a wave was introduced by Sturrock (1962) in the study of the (pseudo) energy and momentum of waves in plasmas. Whitham (1965) derived the conservation of wave action – identified as an adiabatic invariant – from an averaged Lagrangian description of slowly varying nonlinear wave trains in inhomogeneous media:

where is the wave-action density flux and is the divergence of . The description of waves in inhomogeneous and moving media was further elaborated by Bretherton & Garrett (1968) for the case of small-amplitude waves; they also called the quantity wave action (by which name it has been referred to subsequently). For small-amplitude waves the conservation of wave action becomes:

  using     and  

where is the group velocity and the mean velocity of the inhomogeneous moving medium. While the total energy (the sum of the energies of the mean motion and of the wave motion) is conserved for a non-dissipative system, the energy of the wave motion is not conserved, since in general there can be an exchange of energy with the mean motion. However, wave action is a quantity which is conserved for the wave-part of the motion.

The equation for the conservation of wave action is for instance used extensively in wind wave models to forecast sea states as needed by mariners, the offshore industry and for coastal defense. Also in plasma physics and acoustics the concept of wave action is used.

The derivation of an exact wave-action equation for more general wave motion – not limited to slowly modulated waves, small-amplitude waves or (non-dissipative) conservative systems – was provided and analysed by Andrews & McIntyre (1978) using the framework of the generalised Lagrangian mean for the separation of wave and mean motion.

Wave base

The wave base, in physical oceanography, is the maximum depth at which a water wave's passage causes significant water motion. For water depths deeper than the wave base, bottom sediments and the seafloor are no longer stirred by the wave motion above.

Wave height

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.

Wave model (disambiguation)

Wave model is a concept of language development in historical linguistics.

A wave model is a theoretical concept comparing a phenomenon of any type to any part of a physical wave.

Wave model can refer to:

Wind wave model, a mathematical model of sea waves

Density wave model, a mathematical model of a spiral galaxy

Kinematic wave model, an explanation of traffic flow

Wind-wave dissipation

Wind-wave dissipation or "swell dissipation" is process in which a wave generated via a weather system loses its mechanical energy transferred from the atmosphere via wind. Wind waves, as their name suggests, are generated by wind transferring energy from the atmosphere to the ocean's surface, capillary gravity waves play an essential role in this effect, "wind waves" or "swell" are also known as surface gravity waves.

Wind wave model

In fluid dynamics, wind wave modeling describes the effort to depict the sea state and predict the evolution of the energy of wind waves using numerical techniques. These simulations consider atmospheric wind forcing, nonlinear wave interactions, and frictional dissipation, and they output statistics describing wave heights, periods, and propagation directions for regional seas or global oceans. Such wave hindcasts and wave forecasts are extremely important for commercial interests on the high seas. For example, the shipping industry requires guidance for operational planning and tactical seakeeping purposes.For the specific case of predicting wind wave statistics on the ocean, the term ocean surface wave model is used.

Other applications, in particular coastal engineering, have led to the developments of wind wave models specifically designed for coastal applications.

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