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fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) a ...
, a wind wave, water wave, or wind-generated water wave, is a
surface wave In physics, a surface wave is a mechanical wave that propagates along the interface between differing media. A common example is gravity waves along the surface of liquids, such as ocean waves. Gravity waves can also occur within liquids, at ...
that occurs on the
free surface In physics, a free surface is the surface of a fluid that is subject to zero parallel shear stress, such as the interface between two homogeneous fluids. An example of two such homogeneous fluids would be a body of water (liquid) and the air in ...
of
bodies of water A body of water or waterbody (often spelled water body) is any significant accumulation of water on the surface of Earth or another planet. The term most often refers to oceans, seas, and lakes, but it includes smaller pools of water such as p ...
as a result from the
wind Wind is the natural movement of air or other gases relative to a planet's surface. Winds occur on a range of scales, from thunderstorm flows lasting tens of minutes, to local breezes generated by heating of land surfaces and lasting a few ...
blowing over the water surface. The contact distance in the direction of the wind is known as the ''
fetch Fetch may refer to: Books * ''Fetch'', a 2012 book by Alan MacDonald and David Roberts * ''The Fetch'', a 2006 book by Chris Humphreys * ''The Fetch'', a 2009 book by Laura Whitcomb * ''The Fetch'', a 1991 book by Robert Holdstock * ''Fazbear ...
''. Waves in the oceans can travel thousands of kilometers before reaching land. Wind waves on Earth range in size from small
ripples Ripple may refer to: Science and technology * Capillary wave, commonly known as ripple, a wave traveling along the phase boundary of a fluid ** Ripple, more generally a disturbance, for example of spacetime in gravitational waves * Ripple (electri ...
, to waves over high, being limited by wind speed, duration, fetch, and water depth. 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 Great-circle navigation or orthodromic navigation (related to orthodromic course; from the Greek ''ορθóς'', right angle, and ''δρóμος'', path) is the practice of navigating a vessel (a ship or aircraft) along a great circle. Such ro ...
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. Wind waves in the ocean are also called ocean surface waves, and are mainly ''
gravity waves In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium. An example of such an interface is that between the atmosphere ...
'', where
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
is the main equilibrium force. Wind waves have a certain amount of
randomness In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual rand ...
: subsequent waves differ in height, duration, and shape with limited predictability. They can be described as a
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that ap ...
, 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 Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
. The key
statistic A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes include estimating a population parameter, describing a sample, or evaluating a hypo ...
s of wind waves (both seas and swells) in evolving
sea state In oceanography, sea state is the general condition of the free surface on a large body of water—with respect to wind waves and Swell (ocean), swell—at a certain location and moment. A sea state is characterized by statistics, including the ...
s can be predicted with wind wave models. Although waves are usually considered in the water seas of Earth, the hydrocarbon seas of
Titan Titan most often refers to: * Titan (moon), the largest moon of Saturn * Titans, a race of deities in Greek mythology Titan or Titans may also refer to: Arts and entertainment Fictional entities Fictional locations * Titan in fiction, fictiona ...
may also have wind-driven waves.


Formation

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: #
Wind speed In meteorology, wind speed, or wind flow speed, is a fundamental atmospheric quantity caused by air moving from high to low pressure, usually due to changes in temperature. Wind speed is now commonly measured with an anemometer. 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 Fetch may refer to: Books * ''Fetch'', a 2012 book by Alan MacDonald and David Roberts * ''The Fetch'', a 2006 book by Chris Humphreys * ''The Fetch'', a 2009 book by Laura Whitcomb * ''The Fetch'', a 1991 book by Robert Holdstock * ''Fazbear ...
'') # Width of the area affected by fetch (at a 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
wave propagation Wave propagation is any of the ways in which waves travel. Single wave propagation can be calculated by 2nd order wave equation ( standing wavefield) or 1st order one-way wave equation. With respect to the direction of the oscillation relative ...
are: *
Wave height In fluid dynamics, the wave height of a surface wave is the difference between the elevations of a crest and a neighboring trough. ''Wave height'' is a term used by mariners, as well as in coastal, ocean and naval engineering. At sea, the te ...
(vertical distance from trough to crest) * Wave length (distance from crest to crest in the direction of propagation) *
Wave period Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is e ...
(time interval between arrival of consecutive crests at a stationary point) * Wave direction or
azimuth An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north. Mathematical ...
(predominantly driven by wind direction) 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 In physical oceanography, the significant wave height (SWH, HTSGW or ''H''s) is defined traditionally as the mean ''wave height'' ( trough to crest) of the highest third of the waves (''H''1/3). Nowadays it is usually defined as four times the ...
''. This figure represents an
average In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7 ...
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. 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: # 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 John Wilder Miles (December 1, 1920 – October 20, 2008) was a research professor emeritus of applied mechanics and geophysics at Scripps Institution of Oceanography, University of California, San Diego. He was well regarded for his pioneering w ...
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: # two-dimensional parallel shear flow # incompressible, inviscid water and wind # irrotational water # slope of the displacement of the water surface is small Generally, these wave formation mechanisms occur together on the water surface and eventually produce fully developed waves. For example, 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 centimeters are generated by the pressure fluctuations. (The Phillips mechanism) # 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) # The interactions between the waves on the surface generate longer waves 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).


Types

Three different types of wind waves develop over time: *
Capillary wave A capillary wave is a wave traveling along the phase boundary of a fluid, whose dynamics and phase velocity are dominated by the effects of surface tension. Capillary waves are common in nature, and are often referred to as ripples. The wav ...
s, or ripples, dominated by surface tension effects. *
Gravity waves In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium. An example of such an interface is that between the atmosphere ...
, dominated by gravitational and inertial forces. ** Seas, raised locally by the wind. * Swells, which have traveled away from where they were raised by the wind, and have to a greater or lesser extent dispersed. 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 Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects (e.g. water striders) t ...
. 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 The Pacific Ocean is the largest and deepest of Earth's five oceanic divisions. It extends from the Arctic Ocean in the north to the Southern Ocean (or, depending on definition, to Antarctica) in the south, and is bounded by the conti ...
is 19,800km from
Indonesia Indonesia, officially the Republic of Indonesia, is a country in Southeast Asia and Oceania between the Indian and Pacific oceans. It consists of over 17,000 islands, including Sumatra, Java, Sulawesi, and parts of Borneo and New Gui ...
to the coast of
Colombia Colombia (, ; ), officially the Republic of Colombia, is a country in South America with insular regions in North America—near Nicaragua's Caribbean coast—as well as in the Pacific Ocean. The Colombian mainland is bordered by the ...
and, based on an average wavelength of 76.5m, would have ~258,824 swells over that width. Individual "
rogue wave Rogue waves (also known as freak waves, monster waves, episodic waves, killer waves, extreme waves, and abnormal waves) are unusually large, unpredictable, and suddenly appearing surface waves that can be extremely dangerous to ships, even to lar ...
s" (also called "freak waves", "monster waves", "killer waves", and "king waves") much higher than the other waves in the
sea state In oceanography, sea state is the general condition of the free surface on a large body of water—with respect to wind waves and Swell (ocean), swell—at a certain location and moment. A sea state is characterized by statistics, including the ...
can occur. In the case of the
Draupner wave Rogue waves (also known as freak waves, monster waves, episodic waves, killer waves, extreme waves, and abnormal waves) are unusually large, unpredictable, and suddenly appearing surface waves that can be extremely dangerous to ships, even to lar ...
, its height was 2.2 times the
significant wave height In physical oceanography, the significant wave height (SWH, HTSGW or ''H''s) is defined traditionally as the mean ''wave height'' ( trough to crest) of the highest third of the waves (''H''1/3). Nowadays it is usually defined as four times the ...
. Such waves are distinct from
tide Tides are the rise and fall of sea levels caused by the combined effects of the gravitational forces exerted by the Moon (and to a much lesser extent, the Sun) and are also caused by the Earth and Moon orbiting one another. Tide tables ...
s, caused by the
Moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
and
Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
's gravitational pull,
tsunami A tsunami ( ; from ja, 津波, lit=harbour wave, ) is a series of waves in a water body caused by the displacement of a large volume of water, generally in an ocean or a large lake. Earthquakes, volcanic eruptions and other underwater exp ...
s that are caused by underwater
earthquake An earthquake (also known as a quake, tremor or temblor) is the shaking of the surface of the Earth resulting from a sudden release of energy in the Earth's lithosphere that creates seismic waves. Earthquakes can range in intensity, fr ...
s or
landslide Landslides, also known as landslips, are several forms of mass wasting that may include a wide range of ground movements, such as rockfalls, deep-seated slope failures, mudflows, and debris flows. Landslides occur in a variety of environmen ...
s, and waves generated by
underwater explosion An underwater explosion (also known as an UNDEX) is a chemical or nuclear explosion that occurs under the surface of a body of water. While useful in anti-ship and submarine warfare, underwater bombs are not as effective against coastal facilities. ...
s or the fall of
meteorite A meteorite is a solid piece of debris from an object, such as a comet, asteroid, or meteoroid, that originates in outer space and survives its passage through the atmosphere to reach the surface of a planet or moon. When the original object ...
s—all having far longer
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
s than wind waves. The largest ever recorded wind waves are not rogue waves, but standard waves in extreme sea states. For example, high waves were recorded on the RRS Discovery in a sea with significant wave height, so the highest wave was only 1.6 times the significant wave height. The biggest recorded by a buoy (as of 2011) was high during the 2007 typhoon Krosa near Taiwan.


Spectrum

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 . Wind waves (deep-water waves) have a period of about 20 seconds. 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: ::: C = / 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 L(T) of the wavelength on the period (the
dispersion relation In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given t ...
). The speed of a deep-water wave may also be approximated by: ::: C = \sqrt where g is the acceleration due to gravity, per second squared. Because g and π (3.14) are constants, the equation can be reduced to: ::: C = 1.251\sqrt 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: ::: C = \sqrt = 3.1\sqrt 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 In oceanography, sea state is the general condition of the free surface on a large body of water—with respect to wind waves and Swell (ocean), swell—at a certain location and moment. A sea state is characterized by statistics, including the ...
can be described by the sea wave spectrum or just wave spectrum S(\omega, \Theta). It is composed of a wave height spectrum (WHS) S(\omega) and a wave direction spectrum (WDS) f(\Theta). Many interesting properties about the sea state can be found from the wave spectra. WHS describes the
spectral density The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies ...
of
wave height In fluid dynamics, the wave height of a surface wave is the difference between the elevations of a crest and a neighboring trough. ''Wave height'' is a term used by mariners, as well as in coastal, ocean and naval engineering. At sea, the te ...
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
("power") versus wave frequency, with
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
\ = \. The relationship between the spectrum S(\omega_j) and the wave amplitude A_j for a wave component j is: : \frac A_j^2 = S(\omega_j)\, \Delta \omega Some WHS models are listed below. * The International Towing Tank Conference (ITTC) recommended spectrum model for fully developed sea (ISSC spectrum/modified Pierson-Moskowitz spectrum): :: \frac = \frac \left(\frac\right)^ \mathrm \left 0.44 \left(\frac\right)^ \right * ITTC recommended spectrum model for limited
fetch Fetch may refer to: Books * ''Fetch'', a 2012 book by Alan MacDonald and David Roberts * ''The Fetch'', a 2006 book by Chris Humphreys * ''The Fetch'', a 2009 book by Laura Whitcomb * ''The Fetch'', a 1991 book by Robert Holdstock * ''Fazbear ...
( JONSWAP spectrum) :: S(\omega) = 155 \frac \mathrm \left(\frac\right)(3.3)^Y, :where :: Y = \exp \left \left(\frac\right)^2\right/math> :: \sigma = \begin 0.07 & \text\omega \le 5.24 / T_1, \\ 0.09 & \text\omega > 5.24 / T_1. \end :(The latter model has since its creation improved based on the work of Phillips and Kitaigorodskii to better model the wave height spectrum for high
wavenumber In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to te ...
s.) As for WDS, an example model of f(\Theta) might be: : f(\Theta) = \frac\cos^2\Theta, \qquad -\pi/2 \le \Theta \le \pi/2 Thus the sea state is fully determined and can be recreated by the following function where \zeta is the wave elevation, \epsilon_ is uniformly distributed between 0 and 2\pi, and \Theta_j is randomly drawn from the directional distribution function \sqrt: : \zeta = \sum_^N \sqrt\; \sin(\omega_j t - k_j x \cos \Theta_j - k_j y \sin \Theta_j + \epsilon_).


Shoaling and refraction

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 In biology, any group of fish that stay together for social reasons are shoaling, and if the group is swimming in the same direction in a coordinated manner, they are schooling. In common usage, the terms are sometimes used rather loosely. Ab ...
. Wave
refraction In physics, refraction is the redirection of a wave as it passes from one medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commonly observed phenomen ...
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 speed The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for exampl ...
s, 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 Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ...
is contained—converge on local shallows and shoals. Therefore, the
wave energy Wave power is the capture of energy of wind waves to do useful work – for example, electricity generation, water desalination, or pumping water. A machine that exploits wave power is a wave energy converter (WEC). Waves are generated by win ...
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 Doppler effect or Doppler shift (or simply Doppler, when in context) is the change in frequency of a wave in relation to an observer who is moving relative to the wave source. It is named after the Austrian physicist Christian Doppler, who ...
– 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.


Breaking

Some waves undergo a
phenomenon A phenomenon ( : phenomena) is an observable event. The term came into its modern philosophical usage through Immanuel Kant, who contrasted it with the noumenon, which ''cannot'' be directly observed. Kant was heavily influenced by Gottfrie ...
called "breaking". A
breaking wave In fluid dynamics, a breaking wave or breaker is a wave whose amplitude reaches a critical level at which large amounts of wave energy transform into turbulent kinetic energy. At this point, simple physical models that describe wave dynam ...
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 In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
of the
wave height In fluid dynamics, the wave height of a surface wave is the difference between the elevations of a crest and a neighboring trough. ''Wave height'' is a term used by mariners, as well as in coastal, ocean and naval engineering. At sea, the te ...
''H'' to the
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
''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''. 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. # 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 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. # 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

Wind waves are mechanical
wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
s that propagate along the interface between
water Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as ...
and
air The atmosphere of Earth is the layer of gases, known collectively as air, retained by Earth's gravity that surrounds the planet and forms its planetary atmosphere. The atmosphere of Earth protects life on Earth by creating pressure allowing f ...
; the restoring force is provided by gravity, and so they are often referred to as surface gravity waves. As the
wind Wind is the natural movement of air or other gases relative to a planet's surface. Winds occur on a range of scales, from thunderstorm flows lasting tens of minutes, to local breezes generated by heating of land surfaces and lasting a few ...
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 The mile, sometimes the international mile or statute mile to distinguish it from other miles, is a British imperial unit and United States customary unit of distance; both are based on the older English unit of length equal to 5,280 Engli ...
, also in 1957. 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 A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the '' sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ...
, but more a
trochoid In geometry, a trochoid () is a roulette curve formed by a circle rolling along a line. It is the curve traced out by a point fixed to a circle (where the point may be on, inside, or outside the circle) as it rolls along a straight line. If the ...
with the sharper curves upwards—as modeled in trochoidal wave theory. Wind waves are thus a combination of transversal and
longitudinal Longitudinal is a geometric term of location which may refer to: * Longitude ** Line of longitude, also called a meridian * Longitudinal engine, an internal combustion engine in which the crankshaft is oriented along the long axis of the vehicle, ...
waves. When waves propagate in shallow water, (where the depth is less than half the wavelength) the particle trajectories are compressed into
ellipse In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
s. In reality, for
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...
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. As the depth below the free surface increases, the radius of the circular motion decreases. At a depth equal to half the
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
λ, the orbital movement has decayed to less than 5% of its value at the surface. The
phase speed The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for exampl ...
(also called the celerity) of a surface gravity wave is – for pure periodic wave motion of small-
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
waves – well approximated by :c=\sqrt where :''c'' =
phase speed The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for exampl ...
; :''λ'' =
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
; :''d'' = water depth; :''g'' = acceleration due to gravity at the Earth's surface. In deep water, where d \ge \frac\lambda, so \frac \ge \pi and the hyperbolic tangent approaches 1, the speed c approximates :c_\text=\sqrt. In SI units, with c_\text in m/s, c_\text \approx 1.25\sqrt\lambda, when \lambda 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 The shallow-water equations (SWE) are a set of hyperbolic partial differential equations (or parabolic if viscous shear is considered) that describe the flow below a pressure surface in a fluid (sometimes, but not necessarily, a free surface). ...
can be used. If the wavelength is very long compared to the water depth, the phase speed (by taking the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
of when the wavelength approaches infinity) can be approximated by :c_\text = \lim_ c = \sqrt. On the other hand, for very short wavelengths,
surface tension Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects (e.g. water striders) t ...
plays an important role and the phase speed of these gravity-capillary waves can (in deep water) be approximated by :c_\text=\sqrt where :''S'' =
surface tension Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects (e.g. water striders) t ...
of the air-water interface; :\rho =
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
of the water. 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 The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope'' of the wave—propagates through space. For example, if a stone is thrown into the middl ...
which is half of the
phase speed The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for exampl ...
. 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 d decreases towards the
coast The coast, also known as the coastline or seashore, is defined as the area where land meets the ocean, or as a line that forms the boundary between the land and the coastline. The Earth has around of coastline. Coasts are important zones in n ...
, this will have an effect: wave height changes due to wave shoaling and
refraction In physics, refraction is the redirection of a wave as it passes from one medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commonly observed phenomen ...
. 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 Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
\rho, gravity acceleration g and the wave height H (which, for regular waves, is equal to twice the
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
, a): :E=\frac\rho g H^2=\frac\rho g a^2. The velocity of propagation of this energy is the
group velocity The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope'' of the wave—propagates through space. For example, if a stone is thrown into the middl ...
.


Models

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 Numerical weather prediction (NWP) uses mathematical models of the atmosphere and oceans to predict the weather based on current weather conditions. Though first attempted in the 1920s, it was not until the advent of computer simulation in t ...
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 Beach nourishment (also referred to as beach renourishment, beach replenishment, or sand replenishment) describes a process by which sediment, usually sand, lost through longshore drift or erosion is replaced from other sources. A wider beach ...
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.


Seismic signals

Ocean water waves generate land seismic waves that propagate hundreds of kilometers into the land. 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 Michael Selwyn Longuet-Higgins FRS (8 December 1925 – 26 February 2016) was a mathematician and oceanographer at the Department of Applied Mathematics and Theoretical Physics (DAMTP), Cambridge University, England and Institute for Nonlinea ...
in 1950 after in 1941 Pierre Bernard suggested this relation with standing waves on the basis of observations.


See also

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References


Scientific

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Other

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External links


Current global map of peak wave periodsCurrent global map of significant wave heights
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