Wind wave

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A man standing next to large ocean waves at Porto Covo, Portugal Porto Covo Outubro 2014-3.jpg
A man standing next to large ocean waves at Porto Covo, Portugal
Video of large waves from Hurricane Marie along the coast of Newport Beach, California

In fluid dynamics, a wind wave, water wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result from the wind blowing over the water surface. The contact distance in the direction of the wind is known as the fetch . Waves in the oceans can travel thousands of kilometres before reaching land. Wind waves on Earth range in size from small ripples, to waves over 30 m (100 ft) high, being limited by wind speed, duration, fetch, and water depth. [1]

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When directly generated and affected by local wind, a wind wave system is called a wind sea. Wind waves will travel in a great circle route after being generated – curving slightly left in the southern hemisphere and slightly right in the northern hemisphere. After moving out of the area of fetch, wind waves are called swells and can travel thousands of kilometers. A noteworthy example of this is waves generated south of Tasmania during heavy winds that will travel across the Pacific to southern California, producing desirable surfing conditions. Swell consists of wind-generated waves that are not significantly affected by the local wind at that time. They have been generated elsewhere and some time previously. [2] Wind waves in the ocean are also called ocean surface waves, and are mainly gravity waves , where gravity is the main equilibrium force.

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] [4] [5]

Formation

Aspects of a water wave Water wave diagram.jpg
Aspects of a water wave
Wave formation Sjyang waveGeneration.png
Wave formation
Water particle motion of a deep water wave Deep water wave.gif
Water particle motion of a deep water wave
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 wavefront. 2. Falling wave. 3. Trough, where the water masses of the surface layer are moving horizontally in the opposite direction of the wavefront direction. 4. Rising wave. Ocean wave phases numbered.png
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 wavefront. 2. Falling wave. 3. Trough, where the water masses of the surface layer are moving horizontally in the opposite direction of the wavefront direction. 4. Rising wave.
NOAA ship Delaware II in bad weather on Georges Bank Wea00810.jpg
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: [6]

  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 the area affected by fetch (at a 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 wave propagation are:

A fully developed sea has the maximum wave size theoretically possible for a wind of 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. [7]

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: [8]

  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 that is initiated by turbulent wind shear flows based on the inviscid Orr-Sommerfeld equation in 1957. He found the energy transfer from the wind to the 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:

  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 [9]

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

For example, [10] 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 centimeters are generated by the pressure fluctuations. (The Phillips mechanism [8] )
  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 [8] )
  3. The interactions between the waves on the surface generate longer waves [11] 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 crosswind speed (Pierson & Moskowitz [12] ).
Conditions necessary for a fully developed sea at given wind speeds, and the parameters of the resulting waves
Wind conditionsWave size
Wind speed in one directionFetchWind durationAverage heightAverage wavelengthAverage period and speed
19 km/h (12 mph)19 km (12 mi)2 hr0.27 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 hr1.5 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 hr4.1 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 hr8.5 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 hr14.8 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.

Types

Surf on a rocky irregular bottom. Porto Covo, west coast of Portugal Porto Covo pano April 2009-4.jpg
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. The Pacific Ocean is 19,800km from Indonesia to the coast of Colombia and, based on an average wavelength of 76.5m, would have ~258,824 swells over that width.

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.

The largest ever recorded wind waves are not rogue waves, but standard waves in extreme sea states. For example, 29.1 m (95 ft) high waves were recorded on the RRS Discovery in a sea with 18.5 m (61 ft) significant wave height, so the highest wave was only 1.6 times the significant wave height. [13] The biggest recorded by a buoy (as of 2011) was 32.3 m (106 ft) high during the 2007 typhoon Krosa near Taiwan. [14]

Spectrum

Classification of the spectrum of ocean waves according to wave period Munk ICCE 1950 Fig1.svg
Classification of the spectrum of ocean waves according to wave period

Ocean waves can be classified based on: the disturbing force that creates them; the extent to which the disturbing force continues to influence them after formation; the extent to which the restoring force weakens or flattens them; and their wavelength or period. Seismic sea waves have a period of about 20 minutes, and speeds of 760 km/h (470 mph). Wind waves (deep-water waves) have a period of about 20 seconds.

[16]
Wave typeTypical wavelengthDisturbing forceRestoring force
Capillary wave< 2 cmWindSurface tension
Wind wave60–150 m (200–490 ft)Wind over oceanGravity
Seiche Large, variable; a function of basin sizeChange in atmospheric pressure, storm surgeGravity
Seismic sea wave (tsunami)200 km (120 mi)Faulting of sea floor, volcanic eruption, landslideGravity
TideHalf the circumference of EarthGravitational attraction, rotation of EarthGravity

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 the 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.

Spectral models

Sea state can be described by the sea wave spectrum or just wave spectrum. It is composed of a wave height spectrum (WHS) and a wave direction spectrum (WDS) . Many interesting properties about the sea state can be found from the wave spectra.

WHS describes the spectral density of wave height variance ("power") versus wave frequency, with dimension . The relationship between the spectrum and the wave amplitude for a wave component is:

[ citation needed ]

Some WHS models are listed below.

where
(The latter model has since its creation improved based on the work of Phillips and Kitaigorodskii to better model the wave height spectrum for high wavenumbers. [20] )

As for WDS, an example model of might be:

Thus the sea state is fully determined and can be recreated by the following function where is the wave elevation, is uniformly distributed between 0 and , and is randomly drawn from the directional distribution function [21]

Shoaling and refraction

Waves create ripple marks in beaches. Ripples Hierkonpolku Reila.jpg
Waves create ripple marks in beaches.

As waves travel from deep to shallow water, their shape changes (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 to slow the velocity of propagation as a function of wavelength and period. As the waves slow down in shoaling water, the crests tend to realign at a decreasing 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 while the depth decreases, and reverses if it increases again, but the wave leaving the shoal area may have changed direction considerably. 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 wavelength decreases, similar to the shoaling when the water depth decreases. [22]

Breaking

Large wave breaking Big wave breaking in Santa Cruz.jpg
Large wave breaking
Giant ocean wave Giant ocean wave.jpg
Giant ocean wave

Some waves undergo a phenomenon called "breaking". [23] 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. [24] 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 seafloor, 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 localized instability when peaks cross, and these peaks may break due to instability. (see also clapotic waves)

Physics of waves

Stokes drift in shallow water waves (Animation) Shallow water wave.png
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. [25] [26]

Stokes drift in a deeper water wave (Animation) Deep water wave.png
Stokes drift in a deeper water wave (Animation)
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 l = 6.42 ft (1.96 m), period T = 1.12 s. Orbital wave motion-Wiegel Johnson ICCE 1950 Fig 6.png
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.

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 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 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. [28] [29]

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. [30] [31]

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

where

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

where

S = surface tension of the air-water interface;
= density of the water. [32]

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. [33] 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.

Models

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. Global Wave Height Speed.jpg
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 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. [34]

Seismic signals

Ocean water waves generate land seismic waves that propagate hundreds of kilometers into the land. [35] These seismic signals usually have a 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. [36] [37]

See also

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Equatorial waves are oceanic and atmospheric waves trapped close to the equator, meaning that they decay rapidly away from the equator, but can propagate in the longitudinal and vertical directions. Wave trapping is the result of the Earth's rotation and its spherical shape which combine to cause the magnitude of the Coriolis force to increase rapidly away from the equator. Equatorial waves are present in both the tropical atmosphere and ocean and play an important role in the evolution of many climate phenomena such as El Niño. Many physical processes may excite equatorial waves including, in the case of the atmosphere, diabatic heat release associated with cloud formation, and in the case of the ocean, anomalous changes in the strength or direction of the trade winds.

<span class="mw-page-title-main">Cnoidal wave</span> Nonlinear and exact periodic wave solution of the Korteweg–de Vries equation

In fluid dynamics, a cnoidal wave is a nonlinear and exact periodic wave solution of the Korteweg–de Vries equation. These solutions are in terms of the Jacobi elliptic function cn, which is why they are coined cnoidal waves. They are used to describe surface gravity waves of fairly long wavelength, as compared to the water depth.

A Sverdrup wave is a wave in the ocean, or large lakes, which is affected by gravity and Earth's rotation.

<span class="mw-page-title-main">Trochoidal wave</span> Exact solution of the Euler equations for periodic surface gravity waves

In fluid dynamics, a trochoidal wave or Gerstner wave is an exact solution of the Euler equations for periodic surface gravity waves. It describes a progressive wave of permanent form on the surface of an incompressible fluid of infinite depth. The free surface of this wave solution is an inverted (upside-down) trochoid – with sharper crests and flat troughs. This wave solution was discovered by Gerstner in 1802, and rediscovered independently by Rankine in 1863.

In physical oceanography and fluid mechanics, the Miles-Phillips mechanism describes the generation of wind waves from a flat sea surface by two distinct mechanisms. Wind blowing over the surface generates tiny wavelets. These wavelets develop over time and become ocean surface waves by absorbing the energy transferred from the wind. The Miles-Phillips mechanism is a physical interpretation of these wind-generated surface waves.
Both mechanisms are applied to gravity-capillary waves and have in common that waves are generated by a resonance phenomenon. The Miles mechanism is based on the hypothesis that waves arise as an instability of the sea-atmosphere system. The Phillips mechanism assumes that turbulent eddies in the atmospheric boundary layer induce pressure fluctuations at the sea surface. The Phillips mechanism is generally assumed to be important in the first stages of wave growth, whereas the Miles mechanism is important in later stages where the wave growth becomes exponential in time.

References

  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.{{cite book}}: CS1 maint: location (link)
  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 (1): 468–475. Bibcode:2012Icar..219..468L. doi:10.1016/j.icarus.2012.03.002.
  4. Barnes, Jason W.; Sotin, Christophe; Soderblom, Jason M.; Brown, Robert H.; Hayes, Alexander G.; Donelan, Mark; Rodriguez, Sebastien; Mouélic, Stéphane Le; Baines, Kevin H.; McCord, Thomas B. (2014-08-21). "Cassini/VIMS observes rough surfaces on Titan's Punga Mare in specular reflection". Planetary Science. 3 (1): 3. Bibcode:2014PlSci...3....3B. doi:10.1186/s13535-014-0003-4. ISSN   2191-2521. PMC   4959132 . PMID   27512619.
  5. Heslar, Michael F.; Barnes, Jason W.; Soderblom, Jason M.; Seignovert, Benoît; Dhingra, Rajani D.; Sotin, Christophe (2020-08-14). "Tidal Currents Detected in Kraken Mare Straits from Cassini VIMS Sun Glitter Observations". The Planetary Science Journal. 1 (2): 35. arXiv: 2007.00804 . Bibcode:2020PSJ.....1...35H. doi:10.3847/PSJ/aba191. ISSN   2632-3338. S2CID   220301577.
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  8. 1 2 3 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. S2CID   116675962.
  9. 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. S2CID   119795395.
  10. "Chapter 16, Ocean Waves". Archived from the original on 2016-05-11. Retrieved 2013-11-12.
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  12. 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.
  13. 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 .
  14. 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". Annales Geophysicae. 26 (6): 1327–1329. Bibcode:2008AnGeo..26.1327L. doi: 10.5194/angeo-26-1327-2008 .
  15. Munk, Walter H. (1950). "Proceedings 1st International Conference on Coastal Engineering". Long Beach, California: ASCE: 1–4.{{cite journal}}: Cite journal requires |journal= (help)
  16. Tom Garrison (2009). Oceanography: An Invitation to Marine Science (7th ed.). Yolanda Cossio. ISBN   978-0495391937.
  17. International Towing Tank Conference (ITTC) , retrieved 11 November 2010
  18. International Ship and Offshore Structures Congress
  19. Pierson, W. J.; Moscowitz, L. (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
  20. Elfouhaily, T.; Chapron, B.; Katsaros, K.; Vandemark, D. (July 15, 1997). "A unified directional spectrum for long and short wind-driven waves" (PDF). Journal of Geophysical Research . 102 (C7): 15781–15796. Bibcode:1997JGR...10215781E. doi: 10.1029/97jc00467 .
  21. Jefferys, E. R. (1987), "Directional seas should be ergodic", Applied Ocean Research, 9 (4): 186–191, doi:10.1016/0141-1187(87)90001-0
  22. 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:1964DSRA...11..529L. doi:10.1016/0011-7471(64)90001-4.
  23. Gulrez, Tauseef; Hassanien, Aboul Ella (2011-11-13). Advances in Robotics and Virtual Reality. Springer Science & Business Media. ISBN   9783642233630.
  24. R.J. Dean and R.A. Dalrymple (2002). Coastal processes with engineering applications. Cambridge University Press. ISBN   978-0-521-60275-4. p. 96–97.
  25. 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. S2CID   116675962.
  26. 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. S2CID   119795395.
  27. Figure 6 from: Wiegel, R. L.; Johnson, J. W. (1950). "Proceedings 1st International Conference on Coastal Engineering". Long Beach, California: ASCE: 5–21.{{cite journal}}: Cite journal requires |journal= (help)
  28. 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. Vol. 6 (Second revised ed.). Pergamon Press. ISBN   978-0-08-033932-0. See page 33.
  29. A good illustration of the wave motion according to linear theory is given by Prof. Robert Dalrymple's Java applet Archived 2017-11-14 at the Wayback Machine .
  30. 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".
  31. 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. S2CID   122673867.
    J. M. Williams (1985). Tables of progressive gravity waves. Pitman. ISBN   978-0-273-08733-5.
  32. Carl Nordling, Jonny Östermalm (2006). Physics Handbook for Science and Engineering (Eight ed.). Studentliteratur. p. 263. ISBN   978-91-44-04453-8.
  33. In deep water, the group velocity is half the phase velocity, as is shown here. Another reference is .
  34. Wood, AMM & Fleming, CA 1981, Coastal hydraulics, John Wiley & Sons, New York
  35. "Peter Bormann. Seismic Signals and Noise" (PDF). Archived from the original (PDF) on 2016-03-04. Retrieved 2012-03-08.
  36. 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.
  37. 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. S2CID   31828394.

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