1.
Hermosa Beach, California
–
Hermosa Beach is a beachfront city in Los Angeles County, California, United States. Its population was 19,506 at the 2010 census, the city is located in the South Bay region of the greater Los Angeles area and is one of the three Beach Cities. Hermosa Beach is bordered by the two, Manhattan Beach to the north and Redondo Beach to the south and east. The citys beach is popular for sunbathing, beach volleyball, surfing, paddleboarding, the city itself extends only about 15 blocks from east to west and 40 blocks from north to south, with the Pacific Coast Highway running down the middle. Situated on the Pacific Ocean, Hermosas average temperature is 70°F in the summer, westerly sea breezes lessen what can be high summertime temperatures in Los Angeles and elsewhere in the county and help keep the smog away 360 days of the year. Hermosa Beach was originally part of the 1784 Rancho San Pedro Spanish land grant that became the ten-mile Ocean frontage of Rancho Sausal Redondo. In 1900 a tract of 1,500 acres was purchased for $35 per acre from A. E. Pomroy, the Spanish words Rancho Sausal Redondo mean a large circular ranch of pasture of grazing land, with a grove of willow on it. The first official survey was made in the year 1901 for the walk on the Strand, Hermosa Avenue and Santa Fe Avenue. In 1904 the first pier was built and it was constructed entirely of wood even to the pilings and it extended five hundred feet out into the ocean. The pier was constructed by the Hermosa Beach Land and Water Company, in 1913 this old pier was partly washed away and later torn down and a new one built to replace it. This pier was built of concrete 1,000 feet long, small tiled pavilions were erected at intervals along the sides to afford shade for fishermen and picnic parties. A bait stand was eventually out on the end. The Los Angeles Pacific Railway, a system, was the first railway in Hermosa Beach. A few years later it was merged with most all other companies in the region to form the new Pacific Electric Railway Company. The Santa Fe Railway was next through Hermosa Beach and it was seven blocks from the beach. The street that led to the tracks was called Santa Fe Avenue, in 1926, the Santa Fe Company built a modern stucco depot and installed Western Union telegraph service in it. The first city election for city officers was held December 24,1906, on January 14,1907, Hermosa Beach became the nineteenth incorporated city of Los Angeles County. Hermosa is a Spanish word meaning beautiful, Hermosa Beach is located at 33°51′59″N 118°23′59″W
2.
California
–
California is the most populous state in the United States and the third most extensive by area. Located on the western coast of the U. S, California is bordered by the other U. S. states of Oregon, Nevada, and Arizona and shares an international border with the Mexican state of Baja California. Los Angeles is Californias most populous city, and the second largest after New York City. The Los Angeles Area and the San Francisco Bay Area are the nations second- and fifth-most populous urban regions, California also has the nations most populous county, Los Angeles County, and its largest county by area, San Bernardino County. The Central Valley, an agricultural area, dominates the states center. What is now California was first settled by various Native American tribes before being explored by a number of European expeditions during the 16th and 17th centuries, the Spanish Empire then claimed it as part of Alta California in their New Spain colony. The area became a part of Mexico in 1821 following its war for independence. The western portion of Alta California then was organized as the State of California, the California Gold Rush starting in 1848 led to dramatic social and demographic changes, with large-scale emigration from the east and abroad with an accompanying economic boom. If it were a country, California would be the 6th largest economy in the world, fifty-eight percent of the states economy is centered on finance, government, real estate services, technology, and professional, scientific and technical business services. Although it accounts for only 1.5 percent of the states economy, the story of Calafia is recorded in a 1510 work The Adventures of Esplandián, written as a sequel to Amadis de Gaula by Spanish adventure writer Garci Rodríguez de Montalvo. The kingdom of Queen Calafia, according to Montalvo, was said to be a land inhabited by griffins and other strange beasts. This conventional wisdom that California was an island, with maps drawn to reflect this belief, shortened forms of the states name include CA, Cal. Calif. and US-CA. Settled by successive waves of arrivals during the last 10,000 years, various estimates of the native population range from 100,000 to 300,000. The Indigenous peoples of California included more than 70 distinct groups of Native Americans, ranging from large, settled populations living on the coast to groups in the interior. California groups also were diverse in their organization with bands, tribes, villages. Trade, intermarriage and military alliances fostered many social and economic relationships among the diverse groups, the first European effort to explore the coast as far north as the Russian River was a Spanish sailing expedition, led by Portuguese captain Juan Rodríguez Cabrillo, in 1542. Some 37 years later English explorer Francis Drake also explored and claimed a portion of the California coast in 1579. Spanish traders made unintended visits with the Manila galleons on their trips from the Philippines beginning in 1565
3.
Lyttelton Harbour
–
Lyttelton Harbour / Whakaraupō is one of two major inlets in Banks Peninsula, on the coast of Canterbury, New Zealand, the other is Akaroa Harbour. Diamond Harbour lies to the south and the Māori village of Rāpaki to the west, at the head of the harbour is the settlement of Governors Bay. The reserve of Quail Island is near the head and Ripapa Island is just off its south shore at the entrance to Purau Bay. The harbour provides access to a commercial port at Lyttelton which today includes a petroleum storage facility. Hectors dolphins, an endemic to New Zealand, and New Zealand fur seals live in the harbour. Māori referred to the harbour as Te Whakaraupō, with this translating as harbour of the raupō reed, the common original European name for the harbour was Port Cooper, after Daniel Cooper). The surveyors under Joseph Thomas who surveyed Canterbury in the late 1840s named the harbour Port Victoria after the monarch of the United Kingdom, the name was officially changed to Lyttelton Harbour in 1858, in honour of George William Lyttelton, who was the chairman of the Canterbury Association. The official name of the harbour was amended again to become a name by the Ngāi Tahu Claims Settlement Act 1998. Early European settlers used Quail Island as a colony in 1918–25. It is now a nature reserve, fort Jervois was built on the island of Rīpapa in 1885–95. Rīpapa was used in World War I to intern German nationals as enemy aliens, from 1907 it was worked with turbine steamships and from 1933 it was named the Steamer Express. However, in 1962 New Zealand Railways started the Interislander ferry service on the 55-nautical-mile route between Picton and Wellington and this competing service not only offered a shorter crossing but also used diesel ships that had lower running costs than the Union Companys turbine steamers. The wreck of the Steamer Express TEV Wahine in 1968 was a setback for the Lyttelton service but the Union Company introduced a new ship, TEV Rangatira, in 1972. She lost money, survived on a Ministry of Transport subsidy from 1974 and was withdrawn in 1976, media related to Lyttelton Harbour at Wikimedia Commons
4.
New Zealand
–
New Zealand /njuːˈziːlənd/ is an island nation in the southwestern Pacific Ocean. The country geographically comprises two main landmasses—the North Island, or Te Ika-a-Māui, and the South Island, or Te Waipounamu—and around 600 smaller islands. New Zealand is situated some 1,500 kilometres east of Australia across the Tasman Sea and roughly 1,000 kilometres south of the Pacific island areas of New Caledonia, Fiji, because of its remoteness, it was one of the last lands to be settled by humans. During its long period of isolation, New Zealand developed a distinct biodiversity of animal, fungal, the countrys varied topography and its sharp mountain peaks, such as the Southern Alps, owe much to the tectonic uplift of land and volcanic eruptions. New Zealands capital city is Wellington, while its most populous city is Auckland, sometime between 1250 and 1300 CE, Polynesians settled in the islands that later were named New Zealand and developed a distinctive Māori culture. In 1642, Dutch explorer Abel Tasman became the first European to sight New Zealand, in 1840, representatives of Britain and Māori chiefs signed the Treaty of Waitangi, which declared British sovereignty over the islands. In 1841, New Zealand became a colony within the British Empire, today, the majority of New Zealands population of 4.7 million is of European descent, the indigenous Māori are the largest minority, followed by Asians and Pacific Islanders. Reflecting this, New Zealands culture is derived from Māori and early British settlers. The official languages are English, Māori and New Zealand Sign Language, New Zealand is a developed country and ranks highly in international comparisons of national performance, such as health, education, economic freedom and quality of life. Since the 1980s, New Zealand has transformed from an agrarian, Queen Elizabeth II is the countrys head of state and is represented by a governor-general. In addition, New Zealand is organised into 11 regional councils and 67 territorial authorities for local government purposes, the Realm of New Zealand also includes Tokelau, the Cook Islands and Niue, and the Ross Dependency, which is New Zealands territorial claim in Antarctica. New Zealand is a member of the United Nations, Commonwealth of Nations, ANZUS, Organisation for Economic Co-operation and Development, Pacific Islands Forum, and Asia-Pacific Economic Cooperation. Dutch explorer Abel Tasman sighted New Zealand in 1642 and called it Staten Landt, in 1645, Dutch cartographers renamed the land Nova Zeelandia after the Dutch province of Zeeland. British explorer James Cook subsequently anglicised the name to New Zealand, Aotearoa is the current Māori name for New Zealand. It is unknown whether Māori had a name for the country before the arrival of Europeans. Māori had several names for the two main islands, including Te Ika-a-Māui for the North Island and Te Waipounamu or Te Waka o Aoraki for the South Island. Early European maps labelled the islands North, Middle and South, in 1830, maps began to use North and South to distinguish the two largest islands and by 1907, this was the accepted norm. The New Zealand Geographic Board discovered in 2009 that the names of the North Island and South Island had never been formalised and this set the names as North Island or Te Ika-a-Māui, and South Island or Te Waipounamu
5.
Cross sea
–
In surface navigation, a cross sea is a sea state with two wave systems traveling at oblique angles. This may occur when waves from one weather system continue despite a shift in wind. Waves generated by the new wind run at an angle to the old, creating a shifting, two weather systems that are far from each other may create a cross sea when the waves from the systems meet, usually at a place far from either weather system. Until the older waves have dissipated, they create a sea hazard among the most perilous and this sea state is fairly common and a larger percentage of ship accidents were found to have occurred in this state. A cross swell is generated when the systems are longer period swell
6.
Waves and shallow water
–
When waves travel into areas of shallow water, they begin to be affected by the ocean bottom. The free orbital motion of the water is disrupted, and water particles in orbital motion no longer return to their original position, as the water becomes shallower, the swell becomes higher and steeper, ultimately assuming the familiar sharp-crested wave shape. After the wave breaks, it becomes a wave of translation and erosion of the ocean bottom intensifies, boussinesq equations Mild-slope equation Shallow water equations Stokes drift Wave shoaling Ursell number Ballantine Scale Exploring the World Ocean The Oceans lecture slides taught at UCSC
7.
Mechanical waves
–
A mechanical wave is a wave that is an oscillation of matter, and therefore transfers energy through a medium. While waves can move long distances, the movement of the medium of transmission—the material—is limited. Therefore, oscillating material does not move far from its equilibrium position. This energy propagates in the direction as the wave. Any kind of wave has a certain energy, mechanical waves can be produced only in media which possess elasticity and inertia. A mechanical wave requires an energy input. Once this initial energy is added, the travels through the medium until all its energy is transferred. In contrast, electromagnetic waves require no medium, but can travel through one. One important property of waves is that their amplitudes are measured in an unusual way. When this gets comparable to unity, significant nonlinear effects such as harmonic generation may occur, for example, waves on the surface of a body of water break when this dimensionless amplitude exceeds 1, resulting in a foam on the surface and turbulent mixing. Some of the most common examples of mechanical waves are waves, sound waves. There are three types of waves, transverse waves, longitudinal waves, and surface waves. Transverse waves cause the medium to vibrate at an angle to the direction of the wave or energy being carried by the medium. Transverse waves have two parts—the crest and the trough, the crest is the highest point of the wave and the trough is the lowest. The distance between a crest and a trough is half of wavelength, the wavelength is the distance from crest to crest or from trough to trough. To see an example, move an end of a Slinky to the left-and-right of the Slinky, light also has properties of a transverse wave, although it is an electromagnetic wave. Longitudinal waves cause the medium to vibrate parallel to the direction of the wave and it consists of multiple compressions and rarefactions. The rarefaction is the farthest distance apart in the longitudinal wave, the speed of the longitudinal wave is increased in higher index of refraction, due to the closer proximity of the atoms in the medium that is being compressed
8.
Gravity wave
–
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 and the ocean, which rise to wind waves. A gravity wave results when fluid is displaced from a position of equilibrium, the restoration of the fluid to equilibrium will produce a movement of the fluid back and forth, called a wave orbit. Gravity waves on an interface of the ocean are called surface gravity waves or surface waves. Wind-generated waves on the surface are examples of gravity waves, as are tsunamis. Wind-generated gravity waves on the surface of the Earths ponds, lakes, seas. Shorter waves are affected by surface tension and are called gravity–capillary waves. Alternatively, so-called infragravity waves, which are due to nonlinear wave interaction with the wind waves, have periods longer than the accompanying wind-generated waves. In the Earths atmosphere, gravity waves are a mechanism that produce the transfer of momentum from the troposphere to the stratosphere and mesosphere, Gravity waves are generated in the troposphere by frontal systems or by airflow over mountains. At first, waves propagate through the atmosphere without appreciable change in mean velocity, but as the waves reach more rarefied air at higher altitudes, their amplitude increases, and nonlinear effects cause the waves to break, transferring their momentum to the mean flow. This transfer of momentum is responsible for the forcing of the many large-scale dynamical features of the atmosphere, thus, this process plays a key role in the dynamics of the middle atmosphere. The effect of gravity waves in clouds can look like altostratus undulatus clouds, and are confused with them. The phase velocity c of a gravity wave with wavenumber k is given by the formula c = g k. When surface tension is important, this is modified to c = g k + σ k ρ, where σ is the surface tension coefficient and ρ is the density. Since c = ω / k is the speed in terms of the angular frequency ω and the wavenumber. The group velocity of a wave is given by c g = d ω d k, the group velocity is one half the phase velocity. A wave in which the group and phase velocities differ is called dispersive, Gravity waves traveling in shallow water, are nondispersive, the phase and group velocities are identical and independent of wavelength and frequency. When the water depth is h, c p = c g = g h, wind waves, as their name suggests, are generated by wind transferring energy from the atmosphere to the oceans surface, and capillary-gravity waves play an essential role in this effect
9.
Wind wave
–
In fluid dynamics, wind waves, or wind-generated waves, are surface waves that occur on the free surface of bodies of water. 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 high, when directly generated and affected by local winds, 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, wind waves in the ocean are called ocean surface waves. Wind waves have an amount of randomness, subsequent waves differ in height, duration. The key statistics of wind waves 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. The great majority of large breakers seen at a result from distant winds. Water depth All of these work together to determine the size of wind waves. Further exposure to that wind could only cause a dissipation of energy due to the breaking of wave tops. Waves in an area typically have a range of heights. For weather reporting and for analysis of wind wave statistics. This figure represents an average height of the highest one-third of the waves in a time period. The significant wave height is also the value a trained observer 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 a 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 and it is assumed that, The water is originally at rest. There is a 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
10.
Fetch (geography)
–
The fetch, also called the fetch length, is the length of water over which a given wind has blown. It also plays a part in longshore drift as well. Fetch length, along with the speed, determines the size of waves produced. The wind direction is considered constant, the longer the fetch and the faster the wind speed, the more wind energy is imparted to the water surface and the larger the resulting sea state will be. Sea state Ocean surface wave Storm surge
11.
Wavelength
–
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the waves shape repeats, and thus the inverse of the spatial frequency. Wavelength is commonly designated by the Greek letter lambda, the concept can also be applied to periodic waves of non-sinusoidal shape. The term wavelength is also applied to modulated waves. Wavelength depends on the medium that a wave travels through, examples of wave-like phenomena are sound waves, light, water waves and periodic electrical signals in a conductor. A sound wave is a variation in air pressure, while in light and other electromagnetic radiation the strength of the electric, water waves are variations in the height of a body of water. In a crystal lattice vibration, atomic positions vary, wavelength is a measure of the distance between repetitions of a shape feature such as peaks, valleys, or zero-crossings, not a measure of how far any given particle moves. For example, in waves over deep water a particle near the waters surface moves in a circle of the same diameter as the wave height. The range of wavelengths or frequencies for wave phenomena is called a spectrum, the name originated with the visible light spectrum but now can be applied to the entire electromagnetic spectrum as well as to a sound spectrum or vibration spectrum. In linear media, any pattern can be described in terms of the independent propagation of sinusoidal components. The wavelength λ of a sinusoidal waveform traveling at constant speed v is given by λ = v f, in a dispersive medium, the phase speed itself depends upon the frequency of the wave, making the relationship between wavelength and frequency nonlinear. In the case of electromagnetic radiation—such as light—in free space, the speed is the speed of light. Thus the wavelength of a 100 MHz electromagnetic wave is about, the wavelength of visible light ranges from deep red, roughly 700 nm, to violet, roughly 400 nm. For sound waves in air, the speed of sound is 343 m/s, the wavelengths of sound frequencies audible to the human ear are thus between approximately 17 m and 17 mm, respectively. Note that the wavelengths in audible sound are much longer than those in visible light, a standing wave is an undulatory motion that stays in one place. A sinusoidal standing wave includes stationary points of no motion, called nodes, the upper figure shows three standing waves in a box. The walls of the box are considered to require the wave to have nodes at the walls of the box determining which wavelengths are allowed, the stationary wave can be viewed as the sum of two traveling sinusoidal waves of oppositely directed velocities. Consequently, wavelength, period, and wave velocity are related just as for a traveling wave, for example, the speed of light can be determined from observation of standing waves in a metal box containing an ideal vacuum. In that case, the k, the magnitude of k, is still in the same relationship with wavelength as shown above
12.
Frequency
–
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as frequency, which emphasizes the contrast to spatial frequency. The period is the duration of time of one cycle in a repeating event, for example, if a newborn babys heart beats at a frequency of 120 times a minute, its period—the time interval between beats—is half a second. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as vibrations, audio signals, radio waves. For cyclical processes, such as rotation, oscillations, or waves, in physics and engineering disciplines, such as optics, acoustics, and radio, frequency is usually denoted by a Latin letter f or by the Greek letter ν or ν. For a simple motion, the relation between the frequency and the period T is given by f =1 T. The SI unit of frequency is the hertz, named after the German physicist Heinrich Hertz, a previous name for this unit was cycles per second. The SI unit for period is the second, a traditional unit of measure used with rotating mechanical devices is revolutions per minute, abbreviated r/min or rpm. As a matter of convenience, longer and slower waves, such as ocean surface waves, short and fast waves, like audio and radio, are usually described by their frequency instead of period. Spatial frequency is analogous to temporal frequency, but the axis is replaced by one or more spatial displacement axes. Y = sin = sin d θ d x = k Wavenumber, in the case of more than one spatial dimension, wavenumber is a vector quantity. For periodic waves in nondispersive media, frequency has a relationship to the wavelength. Even in dispersive media, the frequency f of a wave is equal to the phase velocity v of the wave divided by the wavelength λ of the wave. In the special case of electromagnetic waves moving through a vacuum, then v = c, where c is the speed of light in a vacuum, and this expression becomes, f = c λ. When waves from a monochrome source travel from one medium to another, their remains the same—only their wavelength. For example, if 71 events occur within 15 seconds the frequency is, the latter method introduces a random error into the count of between zero and one count, so on average half a count. This is called gating error and causes an error in the calculated frequency of Δf = 1/, or a fractional error of Δf / f = 1/ where Tm is the timing interval. This error decreases with frequency, so it is a problem at low frequencies where the number of counts N is small, an older method of measuring the frequency of rotating or vibrating objects is to use a stroboscope
13.
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. The process of wind-wave dissipation can be explained by applying energy spectrum theory in a manner as for the formation of wind-waves. By past and present observations and derived theories, the physics of the ocean-wave dissipation can be categorized by its passing regions along to water depth. In deep water, wave dissipation occurs by the actions of friction or drag forces such as opposite-directed winds or viscous forces generated by turbulent flows—usually nonlinear forces, in shallow water, the behaviors of wave dissipations are mostly types of shore wave breaking. Some of simple descriptions of wind-wave dissipation were proposed when we consider only ocean surface waves such as wind waves. By means of the simple, the interactions of waves with the structure of the upper layers of the ocean are ignored for simplified theory in many proposed mechanisms. 1) dissipation by wave breaking Wind-wave breaking at coastal area is a source of the wind-wave dissipation. The wind waves lose their energy to the shore or sometimes back to the ocean when those break at the shore, 2) dissipation by wave–turbulence interaction The turbulent wind flows and viscous eddies inside waves can both affect wave dissipation. In the very early understandings, the viscosity can barely affect on the waves so that the dissipation of the swells by viscosity also barely considered. However, recent weather forecasting models begin considering “wave-turbulence interaction” for the wave modeling, 3) dissipation by wave-wave modulation Wave–wave interactions can affect to the wave dissipation. In the early eras, the ideas that a wave breaking can take energy from the long waves through the modulation were proposed by Phillips. Most cases of the swell dissipations are due to this dissipation type, when wind waves approach to coast area from deep water, the waves change their heights and lengths. The wave height becomes higher and the wavelength becomes shorter as the velocity is slowed when ocean waves approach to the shore. If the water depth is shallow, the wave crest become steeper and the trough gets broader and shallower, finally. The motions of wave breaking are different with along to the steepness of shores and waves, • Spilling breaker With lower shore slope, the waves lose energy slowly as approaching to the shore. The waves spill sea water down the front of the waves when those are breaking, • Plunging breaker With moderately steep shore slope, the wave loses energy quickly. If the shore slope is steep enough, the crest of wave moves faster than the trough, the crest curls over front of the wave, and after the crest plunges sea water to the trough. • Surging breaker With highly steep shore slope, if the shore steepness is very high, the waves climb along through the shore slope, and release energy to the backward from the shore
14.
Wind speed
–
Wind speed, or wind flow velocity, is a fundamental atmospheric quantity. Wind speed is caused by air moving from high pressure to low pressure, Wind speed affects weather forecasting, aircraft and maritime operations, construction projects, growth and metabolism rate of many plant species, and countless other implications. Wind speed is now commonly measured with an anemometer but can also be classified using the older Beaufort scale which is based on observation of specifically defined wind effects. Wind speed is affected by a number of factors and situations and these include the pressure gradient, Rossby waves and jet streams, and local weather conditions. There are also links to be found between wind speed and wind direction, notably with the gradient and terrain conditions. Pressure gradient is a term to describe the difference in air pressure between two points in the atmosphere or on the surface of the Earth and it is vital to wind speed, because the greater the difference in pressure, the faster the wind flows to balance out the variation. The pressure gradient, when combined with the Coriolis effect and friction, Rossby waves are strong winds in the upper troposphere. These operate on a scale and move from West to East. The Rossby waves are themselves a different wind speed from what we experience in the lower troposphere, the wind gust was evaluated by the WMO Evaluation Panel who found that the anemometer was mechanically sound and the gust was within statistical probability and ratified the measurement in 2010. The anemometer was mounted 10 m above ground level, the pattern and scales of the gusts suggest that a mesovortex was embedded in the already strong eyewall of the cyclone. The now second highest surface wind speed ever recorded is 372 km/h at the Mount Washington Observatory,6,288 ft above sea level in the US on 12 April 1934. The anemometer, specifically designed for use on Mount Washington was later tested by the US National Weather Bureau, Wind speeds within certain atmospheric phenomena may greatly exceed these values but have never been accurately measured. Directly measuring these tornadic winds is rarely done as the violent wind would destroy the instruments, another method of estimating is to use Doppler on Wheels to sense the wind speeds remotely. The figure of 486 km/h during the 1999 Bridge Creek–Moore tornado in Oklahoma on 3 May 1999 is often quoted as the surface wind speed. However, another figure of 512 kilometres per hour has also quoted for the same tornado. Yet another number used by the Center for Severe Weather Research for that measurement is 486 ±32 km/h, however, speeds measured by Doppler radar are not considered official records. Wind speed is a factor in the design of structures. It is often the governing factor in the required strength of a structures design
15.
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 used by mariners, as well as in coastal, ocean. At sea, the 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 corresponds to what a mariner observes when estimating visually the average wave height. For a sine wave, the wave height H is twice the amplitude, for a periodic wave it is simply the difference between the maximum and minimum of the surface elevation z = η, H = max − min, with cp the phase speed of the wave. The sine wave is a case of a periodic wave. For this to be possible, it is necessary to first split the time series of the surface elevation into individual waves. Commonly, a wave is denoted as the time interval between two successive downward-crossings through the average surface elevation. Then the individual wave height of each wave is again the difference between maximum and minimum elevation in the interval of the wave under consideration. Only the highest one-third is used, since this corresponds best with visual observations of experienced mariners, eyes, Significant wave height Hm0, defined in the frequency domain, is used both for measured and forecasted wave variance spectra. In case of a measurement, the standard deviation ση is the easiest and most accurate statistic to be used, sea state Significant wave height Wind wave
16.
Crest (physics)
–
A crest is the point on a wave with the maximum value or upward displacement within a cycle. A crest is a point on the wave where the displacement of the medium is at a maximum, a trough is the opposite of a crest, so the minimum or lowest point in a cycle. When in antiphase – 180° out of phase – the result is destructive interference, superposition principle Wave Kinsman, Blair, Wind Waves, Their Generation and Propagation on the Ocean Surface, Dover Publications, ISBN 0-486-49511-6,704 pages
17.
Stokes drift
–
For a pure wave motion in fluid dynamics, the Stokes drift velocity is the average velocity when following a specific fluid parcel as it travels with the fluid flow. For instance, a particle floating at the surface of water waves, experiences a net Stokes drift velocity in the direction of wave propagation. More generally, the Stokes drift velocity is the difference between the average Lagrangian flow velocity of a parcel, and the average Eulerian flow velocity of the fluid at a fixed position. This nonlinear phenomenon is named after George Gabriel Stokes, who derived expressions for this drift in his 1847 study of water waves. The Stokes drift is the difference in end positions, after an amount of time. The end position in the Lagrangian description is obtained by following a specific fluid parcel during the time interval, the Stokes drift velocity equals the Stokes drift divided by the considered time interval. Often, the Stokes drift velocity is referred to as Stokes drift. Stokes drift may occur in all instances of oscillatory flow which are inhomogeneous in space, for instance in water waves, tides and atmospheric waves. In the Lagrangian description, fluid parcels may drift far from their initial positions, as a result, the unambiguous definition of an average Lagrangian velocity and Stokes drift velocity, which can be attributed to a certain fixed position, is by no means a trivial task. However, such a description is provided by the Generalized Lagrangian Mean theory of Andrews. The Stokes drift is important for the transfer of all kind of materials. Further the Stokes drift is important for the generation of Langmuir circulations, for nonlinear and periodic water waves, accurate results on the Stokes drift have been computed and tabulated. Often, the Lagrangian coordinates α are chosen to coincide with the Eulerian coordinates x at the time t = t0. But also other ways of labeling the fluid parcels are possible, different definitions of the average may be used, depending on the subject of study, see ergodic theory, time average, space average, ensemble average and phase average. Now, the Stokes drift velocity ūS equals u ¯ S = u ¯ L − u ¯ E, in many situations, the mapping of average quantities from some Eulerian position x to a corresponding Lagrangian position α forms a problem. Since a fluid parcel with label α traverses along a path of many different Eulerian positions x, a mathematically sound basis for an unambiguous mapping between average Lagrangian and Eulerian quantities is provided by the theory of the Generalized Lagrangian Mean by Andrews and McIntyre. Here the last term describes the Stokes drift 12 k u ^2 / ω, the Stokes drift was formulated for water waves by George Gabriel Stokes in 1847. Further, the Stokes drift velocity decays exponentially with depth, at a depth of a wavelength, z = -¼ λ, it is about 4% of its value at the mean free surface
18.
Significant wave height
–
In physical oceanography, the significant wave height is defined traditionally as the mean wave height of the highest third of the waves. Nowadays it is defined as four times the standard deviation of the surface elevation – or equivalently as four times the square root of the zeroth-order moment of the wave spectrum. The symbol Hm0 is usually used for that latter definition, the significant wave height may thus refer to Hm0 or H1/3, the difference in magnitude between the two definitions is only a few percent. The original definition resulted from work by the oceanographer Walter Munk during World War II, the significant wave height was intended to mathematically express the height estimated by a trained observer. It is commonly used as a measure of the height of ocean waves, significant wave height, scientifically represented as Hs or Hsig, is an important parameter for the statistical distribution of ocean waves. The most common waves are less in height than Hs and this implies that encountering the significant wave is not too frequent. However, statistically, it is possible to encounter a wave that is higher than the significant wave. Generally, the distribution of the individual wave heights is well approximated by a Rayleigh distribution. However, in changing conditions, the disparity between the significant wave height and the largest individual waves might be even larger. Other statistical measures of the wave height are also widely used, the RMS wave height, which is defined as square root of the average of the squares of all wave heights, is approximately equal to Hs divided by 1.4. For example, according to the Irish Marine Institute, … at midnight on 9/12/2007 a record significant wave height was recorded of 17. 2m at with a period of 14 seconds. The maximum ever measured wave height from a satellite is 20. 1m during a North Atlantic storm in 2011, the World Meteorological Organization stipulates that certain countries are responsible for providing weather forecasts for the worlds oceans. These respective countries meteorological offices are called Regional Specialized Meteorological Centers, in their weather products, they give ocean wave height forecasts in significant wave height. In the United States, NOAAs National Weather Service is the RSMC for a portion of the North Atlantic, the Ocean Prediction Center and the Tropical Prediction Centers Tropical Analysis and Forecast Branch issue these forecasts. RSMCs use wind-wave models as tools to predict the sea conditions. In the U. S. NOAAs WAVEWATCH III model is used heavily, a significant wave height is also defined similarly, from the wave spectrum, for the different systems that make up the sea. We then have a significant wave height for the wind-sea or for a particular swell
19.
Average
–
In colloquial language, an average is the sum of a list of numbers divided by the number of numbers in the list. In mathematics and statistics, this would be called the arithmetic mean, in statistics, mean, median, and mode are all known as measures of central tendency. The most common type of average is the arithmetic mean, one may find that A = /2 =5. Switching the order of 2 and 8 to read 8 and 2 does not change the value obtained for A. The mean 5 is not less than the minimum 2 nor greater than the maximum 8. If we increase the number of terms in the list to 2,8, and 11, one finds that A = /3 =7. Along with the arithmetic mean above, the mean and the harmonic mean are known collectively as the Pythagorean means. The geometric mean of n numbers is obtained by multiplying them all together. See Inequality of arithmetic and geometric means, thus for the above harmonic mean example, AM =50, GM ≈49, and HM =48 km/h. The mode, the median, and the mid-range are often used in addition to the mean as estimates of central tendency in descriptive statistics, the most frequently occurring number in a list is called the mode. For example, the mode of the list is 3 and it may happen that there are two or more numbers which occur equally often and more often than any other number. In this case there is no agreed definition of mode, some authors say they are all modes and some say there is no mode. The median is the number of the group when they are ranked in order. Thus to find the median, order the list according to its elements magnitude, if exactly one value is left, it is the median, if two values, the median is the arithmetic mean of these two. This method takes the list 1,7,3,13, then the 1 and 13 are removed to obtain the list 3,7. Since there are two elements in this remaining list, the median is their arithmetic mean, /2 =5, the table of mathematical symbols explains the symbols used below. Other more sophisticated averages are, trimean, trimedian, and normalized mean, one can create ones own average metric using the generalized f-mean, y = f −1 where f is any invertible function. The harmonic mean is an example of this using f = 1/x, however, this method for generating means is not general enough to capture all averages
20.
Nautical mile
–
A nautical mile is a unit of measurement defined as exactly 1852 meters. Historically, it was defined as one minute of latitude, which is equivalent to one sixtieth of a degree of latitude. Today it is an SI derived unit, being rounded to a number of meters. The derived unit of speed is the knot, defined as one mile per hour. The geographical mile is the length of one minute of longitude along the Equator, there is no internationally agreed symbol. M is used as the abbreviation for the mile by the International Hydrographic Organization and by the International Bureau of Weights. NM is used by the International Civil Aviation Organization, nm is used by the U. S. National Oceanic and Atmospheric Administration. Nmi is used by the Institute of Electrical and Electronics Engineers, the word mile is from the Latin word for a thousand paces, mīlia. In 1617 the Dutch scientist Snell assessed the circumference of the Earth at 24,630 Roman miles, around that time British mathematician Edmund Gunter improved navigational tools including a new quadrant to determine latitude at sea. He reasoned that the lines of latitude could be used as the basis for a unit of measurement for distance, as one degree is 1/360 of a circle, one minute of arc is 1/21600 of a circle. These sexagesimal units originated in Babylonian astronomy, Gunter used Snells circumference to define a nautical mile as 6,080 feet, the length of one minute of arc at 48 degrees latitude.3 metres. Other countries measure the minute of arc at 45 degrees latitude, in 1929, the international nautical mile was defined by the First International Extraordinary Hydrographic Conference in Monaco as 1,852 meters. Imperial units and United States customary units used a definition of the nautical mile based on the Clarke Spheroid, the United States nautical mile was defined as 6,080.20 feet based in the Mendenhall Order foot of 1893. It was abandoned in favour of the nautical mile in 1954.181 meters. It was abandoned in 1970 and, legally, references to the unit are now converted to 1,853 meters. Conversion of units Orders of magnitude
21.
Spectral density
–
The power spectrum S x x of a time series x 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, the statistical average of a certain signal or sort of signal as analyzed in terms of its frequency content, is called its spectrum. When the energy of the signal is concentrated around a time interval, especially if its total energy is finite. More commonly used is the spectral density, which applies to signals existing over all time. The power spectral density then refers to the energy distribution that would be found per unit time. Summation or integration of the spectral components yields the total power or variance, identical to what would be obtained by integrating x 2 over the time domain, the spectrum of a physical process x often contains essential information about the nature of x. For instance, the pitch and timbre of an instrument are immediately determined from a spectral analysis. The color of a source is determined by the spectrum of the electromagnetic waves electric field E as it fluctuates at an extremely high frequency. Obtaining a spectrum from time series such as these involves the Fourier transform, however this article concentrates on situations in which the time series is known or directly measured. The power spectrum is important in signal processing and in the statistical study of stochastic processes, as well as in many other branches of physics. Typically the process is a function of time but one can similarly discuss data in the domain being decomposed in terms of spatial frequency. Any signal that can be represented as an amplitude that varies in time has a frequency spectrum. This includes familiar entities such as light, musical notes, radio/TV. When these signals are viewed in the form of a frequency spectrum, in some cases the frequency spectrum may include a distinct peak corresponding to a sine wave component. And additionally there may be corresponding to harmonics of a fundamental peak. In physics, the signal might be a wave, such as an electromagnetic wave, the power spectral density of the signal describes the power present in the signal as a function of frequency, per unit frequency. Power spectral density is expressed in watts per hertz. When a signal is defined in terms only of a voltage, for instance, in this case power is simply reckoned in terms of the square of the signal, as this would always be proportional to the actual power delivered by that signal into a given impedance
22.
Group velocity
–
The group velocity of a wave is the velocity with which the overall shape of the waves amplitudes—known as the modulation or envelope of the wave—propagates through space. For example, if a stone is thrown into the middle of a very still pond, the expanding ring of waves is the wave group, within which one can discern individual wavelets of differing wavelengths traveling at different speeds. The longer waves travel faster than the group as a whole, the shorter waves travel more slowly, and their amplitudes diminish as they emerge from the trailing boundary of the group. The group velocity vg is defined by the equation, v g ≡ ∂ ω ∂ k where ω is the angular frequency. The phase velocity is, vp = ω / k, the function ω, which gives ω as a function of k, is known as the dispersion relation. If ω is directly proportional to k, then the velocity is exactly equal to the phase velocity. A wave of any shape will travel undistorted at this velocity, if ω is a linear function of k, but not directly proportional, then the group velocity and phase velocity are different. The envelope of a packet will travel at the group velocity, while the individual peaks. If ω is not a function of k, the envelope of a wave packet will become distorted as it travels. Since a wave packet contains a range of different frequencies, the group velocity ∂ω/∂k will be different for different values of k, therefore, the envelope does not move at a single velocity, but its wavenumber components move at different velocities, distorting the envelope. If the wavepacket has a range of frequencies, and ω is approximately linear over that narrow range. For example, for deep water gravity waves, ω=√gk, and this underlies the Kelvin wake pattern for the bow wave of all ships and swimming objects. Regardless of how fast they are moving, as long as their velocity is constant, one derivation of the formula for group velocity is as follows. Consider a wave packet as a function of x and time t, α. Let A be its Fourier transform at time t=0, α = ∫ − ∞ ∞ d k A e i k x. By the superposition principle, the wavepacket at any time t is α = ∫ − ∞ ∞ d k A e i, assume that the wave packet α is almost monochromatic, so that A is sharply peaked around a central wavenumber k0. Then, linearization gives ω ≈ ω0 + ω0 ′ where ω0 = ω and ω0 ′ = ∂ ω ∂ k | k = k 0, then, after some algebra, α = e i ∫ − ∞ ∞ d k A e i. There are two factors in this expression, the first factor, e i, describes a perfect monochromatic wave with wavevector k0, with peaks and troughs moving at the phase velocity ω0 / k 0 within the envelope of the wavepacket
23.
Infragravity wave
–
Infragravity waves are ocean surface gravity waves generated by ocean waves of shorter periods. The amplitude of infragravity waves is most relevant in shallow water, in particular along coastlines hit by high amplitude and long period wind waves, wind waves and ocean swells are shorter, with typical dominant periods of 1 to 25 s. This distinguishes infragravity waves from normal oceanic gravity waves, which are created by wind acting on the surface of the sea, whatever the details of their generation mechanism, discussed below, infragravity waves are these subharmonics of the impinging gravity waves. Technically infragravity waves are simply a subcategory of gravity waves and refer to all gravity waves with greater than 30 s. This could include such as tides and oceanic Rossby waves. The term infragravity wave appears to have coined by Walter Munk in 1950. Two main processes can explain the transfer of energy from the wind waves to the long infragravity waves. The most common process is the interaction of trains of wind waves which was first observed by Munk and Tucker and explained by Longuet-Higgins. Because wind waves are not monochromatic they form groups, the Stokes drift induced by these groupy waves transports more water where the waves are highest. The waves also push the water around in a way that can be interpreted as a force, combining mass and momentum conservation, Longuet-Higgins and Stewart give, with three different methods, the now well-known result. Namely, the sea level oscillates with a wavelength that is equal to the length of the group, with a low level where the wind waves are highest. This oscillation of the sea surface is proportional to the square of the wave amplitude. Another process was proposed later by Graham Symonds and his collaborators and it appears that this is probably a good explanation for infragravity wave generation on a reef. In the case of coral reefs, the infragravity periods are established by resonances with the reef itself, infragravity waves generated along the Pacific coast of North America have been observed to propagate transoceanically to Antarctica and there to impinge on the Ross Ice Shelf. Their frequencies more closely couple with the ice shelf natural frequencies, further, they are not damped by sea ice as normal ocean swell is. As a result, they flex floating ice shelves such as the Ross Ice Shelf, media related to Gravity waves at Wikimedia Commons
24.
Signal processing
–
According to Alan V. Oppenheim and Ronald W. Schafer, the principles of signal processing can be found in the classical numerical analysis techniques of the 17th century. Oppenheim and Schafer further state that the digitalization or digital refinement of techniques can be found in the digital control systems of the 1940s and 1950s. Feature extraction, such as understanding and speech recognition. Quality improvement, such as reduction, image enhancement. Including audio compression, image compression, and video compression and this involves linear electronic circuits as well as non-linear ones. The former are, for instance, passive filters, active filters, additive mixers, integrators, non-linear circuits include compandors, multiplicators, voltage-controlled filters, voltage-controlled oscillators and phase-locked loops. Continuous-time signal processing is for signals that vary with the change of continuous domain, the methods of signal processing include time domain, frequency domain, and complex frequency domain. This technology was a predecessor of digital processing, and is still used in advanced processing of gigahertz signals. Digital signal processing is the processing of digitized discrete-time sampled signals, processing is done by general-purpose computers or by digital circuits such as ASICs, field-programmable gate arrays or specialized digital signal processors. Typical arithmetical operations include fixed-point and floating-point, real-valued and complex-valued, other typical operations supported by the hardware are circular buffers and look-up tables. Examples of algorithms are the Fast Fourier transform, finite impulse response filter, Infinite impulse response filter, nonlinear signal processing involves the analysis and processing of signals produced from nonlinear systems and can be in the time, frequency, or spatio-temporal domains. Nonlinear systems can produce complex behaviors including bifurcations, chaos, harmonics
25.
Polynesia
–
Polynesia is a subregion of Oceania, made up of over 1,000 islands scattered over the central and southern Pacific Ocean. The indigenous people who inhabit the islands of Polynesia are termed Polynesians and they share many similar traits including the language family, culture, and beliefs. Historically, they were experienced sailors who used stars to navigate at night, the term Polynesia was first used in 1756 by French writer Charles de Brosses, and originally applied to all the islands of the Pacific. In 1831, Jules Dumont dUrville proposed a restriction on its use during a lecture to the Geographical Society of Paris, historically, these islands have also been referred to as the South Sea Islands. Polynesia is characterized by an amount of land spread over a very large portion of the mid. Most Polynesian islands and archipelagos, including the Hawaiian Islands and Samoa, are composed of volcanic islands built by hotspots, New Zealand, Norfolk Island, and Ouvéa, the Polynesian outlier near New Caledonia, are the unsubmerged portions of the largely sunken continent of Zealandia. At first, the Pacific plate was subducted under the Australian plate, the Alpine Fault that traverses the South Island is currently a transform fault while the convergent plate boundary from the North Island northwards is called the Kermadec-Tonga Subduction Zone. The volcanism associated with subduction zone is the origin of the Kermadec. Out of approximately 300,000 or 310,000 square kilometres of land, over 270,000 km2 are within New Zealand, the Zealandia continent has approximately 3,600,000 km2 of continental shelf. The oldest rocks in the region are found in New Zealand and are believed to be about 510 million years old, the oldest Polynesian rocks outside of Zealandia are to be found in the Hawaiian Emperor Seamount Chain, and are 80 million years old. Polynesia is generally defined as the islands within the Polynesian Triangle, geographically, the Polynesian Triangle is drawn by connecting the points of Hawaii, New Zealand and Easter Island. The other main island groups located within the Polynesian Triangle are Samoa, Tonga, there are also small Polynesian settlements in Papua New Guinea, the Solomon Islands, the Caroline Islands, and in Vanuatu. An island group with strong Polynesian cultural traits outside of this triangle is Rotuma. The people of Rotuma have many common Polynesian traits but speak a non-Polynesian language, some of the Lau Islands to the southeast of Fiji have strong historic and cultural links with Tonga. However, in essence, Polynesia is a term referring to one of the three parts of Oceania. Some islands of Polynesian origin are outside the triangle that geographically defines the region. The Phoenix Islands and Line Islands, most of which are part of Kiribati, are geographically Polynesian islands, tracing Polynesian languages places their prehistoric origins in the Malay Archipelago, and ultimately, in Taiwan. Between about 3000 and 1000 BC speakers of Austronesian languages began spreading from Taiwan into Island Southeast Asia, there are three theories regarding the spread of humans across the Pacific to Polynesia
26.
International Standard Book Number
–
The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
27.
Physical oceanography
–
Physical oceanography is the study of physical conditions and physical processes within the ocean, especially the motions and physical properties of ocean waters. Physical oceanography is one of several sub-domains into which oceanography is divided, others include biological, chemical and geological oceanography. Physical oceanography may be subdivided into descriptive and dynamical physical oceanography, descriptive physical oceanography seeks to research the ocean through observations and complex numerical models, which describe the fluid motions as precise as possible. Dynamical physical oceanography focuses primarily upon the processes that govern the motion of fluids with emphasis upon theoretical research and these are part of the large field of Geophysical Fluid Dynamics that is shared together with meteorology. The fundamental role of the oceans in shaping Earth is acknowledged by ecologists, geologists, meteorologists, climatologists, an Earth without oceans would truly be unrecognizable. Roughly 97% of the water is in its oceans. The tremendous heat capacity of the oceans moderates the planets climate, the oceans influence extends even to the composition of volcanic rocks through seafloor metamorphism, as well as to that of volcanic gases and magmas created at subduction zones. Though this apparent discrepancy is great, for land and sea, the respective extremes such as mountains and trenches are rare. Because the vast majority of the oceans volume is deep water. The same percentage falls in a salinity range between 34–35 ppt, there is still quite a bit of variation, however. Surface temperatures can range from below freezing near the poles to 35 °C in restricted tropical seas, in terms of temperature, the oceans layers are highly latitude-dependent, the thermocline is pronounced in the tropics, but nonexistent in polar waters. The halocline usually lies near the surface, where evaporation raises salinity in the tropics and these variations of salinity and temperature with depth change the density of the seawater, creating the pycnocline. Energy for the ocean circulation comes from solar radiation and gravitational energy from the sun, perhaps three quarters of this heat is carried in the atmosphere, the rest is carried in the ocean. The atmosphere is heated from below, which leads to convection, by contrast the ocean is heated from above, which tends to suppress convection. Instead ocean deep water is formed in regions where cold salty waters sink in fairly restricted areas. This is the beginning of the thermohaline circulation, oceanic currents are largely driven by the surface wind stress, hence the large-scale atmospheric circulation is important to understanding the ocean circulation. The Hadley circulation leads to Easterly winds in the tropics and Westerlies in mid-latitudes and this leads to slow equatorward flow throughout most of a subtropical ocean basin. The return flow occurs in an intense, narrow, poleward western boundary current, like the atmosphere, the ocean is far wider than it is deep, and hence horizontal motion is in general much faster than vertical motion
28.
Airy wave theory
–
In fluid dynamics, Airy wave theory gives a linearised description of the propagation of gravity waves on the surface of a homogeneous fluid layer. The theory assumes that the layer has a uniform mean depth. This theory was first published, in form, by George Biddell Airy in the 19th century. Further, several second-order nonlinear properties of gravity waves, and their propagation. Airy wave theory is also a good approximation for tsunami waves in the ocean and this linear theory is often used to get a quick and rough estimate of wave characteristics and their effects. This approximation is accurate for small ratios of the height to water depth. Airy wave theory uses a potential approach to describe the motion of gravity waves on a fluid surface. This is due to the fact that for the part of the fluid motion. Airy wave theory is used in ocean engineering and coastal engineering. Diffraction is one of the effects which can be described with Airy wave theory. Further, by using the WKBJ approximation, wave shoaling and refraction can be predicted, earlier attempts to describe surface gravity waves using potential flow were made by, among others, Laplace, Poisson, Cauchy and Kelland. But Airy was the first to publish the correct derivation and formulation in 1841, soon after, in 1847, the linear theory of Airy was extended by Stokes for non-linear wave motion – known as Stokes wave theory – correct up to third order in the wave steepness. Even before Airys linear theory, Gerstner derived a nonlinear wave theory in 1802. Airy wave theory is a theory for the propagation of waves on the surface of a potential flow. The waves propagate along the surface with the phase speed cp. The angular wavenumber k and frequency ω are not independent parameters, surface gravity waves on a fluid are dispersive waves – exhibiting frequency dispersion – meaning that each wavenumber has its own frequency and phase speed. Note that in engineering the wave height H – the difference in elevation between crest and trough – is often used, H =2 a and a =12 H, underneath the surface, there is a fluid motion associated with the free surface motion. While the surface shows a propagating wave, the fluid particles are in an orbital motion
29.
Ballantine scale
–
The Ballantine scale is a biologically defined scale for measuring the degree of exposure level of wave action on a rocky shore. The species present in the littoral zone therefore indicate the degree of the shores exposure, an abbreviated summary of the scale is given below. The scale runs from 1) an extremely exposed shore, to 8) an extremely sheltered shore, the littoral zone generally is the zone between low and high tides. The supra-littoral is above the barnacle line, the eulittoral zone is dominated by barnacles and limpets with a coralline belt in the very low littoral along with other Rhodophyta and Alaria in the upper sublittoral. Exposed shores show a Verrucaria belt mainly above the tide, with Porphyra. The mid shore is dominated by barnacles, limpets and some Fucus, Himanthalia and some Rhodophyta such as Mastocarpus and Corallina are found in the low littorral with Himanthalia, Alaria and Laminaria digitata dominant in the upper sublittoral. Semi-exposed shores show a Verrucaria belt just above high tide with clear Pelvetia in the upper-littoral, limpets, barnacles and short Fucus vesiculosus midshore. Laminaria and Saccorhiza polyschides and small algae common in the sublittoral, sheltered shores show a narrow Verrucaria zone at high water and a full sequence of fucoids, Pelvetia, Fucus spiralis, Fucus vesiculosus, Fucus serratus, Ascophyllum nodosum. Laminaria digitata is dominant the upper sublittoral, very sheltered shores show a very narrow zone of Verrucaria, the dominance of the littoral by a full sequence of the fucoids and Ascophyllum covering the rocks. Laminaria saccharina, Halidrys, Chondrus and or Furcellaria, a Biologically-defined Exposure Scale for the Comparative Description of Rocky Shores
30.
Modulational instability
–
The phenomenon was first discovered − and modelled − for periodic surface gravity waves on deep water by T. Brooke Benjamin and Jim E. Feir, in 1967. Therefore, it is known as the Benjamin−Feir instability. It is a mechanism for the generation of rogue waves. Modulation instability only happens under certain circumstances, the most important condition is anomalous group velocity dispersion, whereby pulses with shorter wavelengths travel with higher group velocity than pulses with longer wavelength. The instability is strongly dependent on the frequency of the perturbation, at certain frequencies, a perturbation will have little effect, whilst at other frequencies, a perturbation will grow exponentially. The overall gain spectrum can be derived analytically, as is shown below, random perturbations will generally contain a broad range of frequency components, and so will cause the generation of spectral sidebands which reflect the underlying gain spectrum. The tendency of a signal to grow makes modulation instability a form of amplification. By tuning an input signal to a peak of the gain spectrum, the imaginary unit i satisfies i 2 = −1. The model includes group velocity dispersion described by the parameter β2, a periodic waveform of constant power P is assumed. The beginning of instability can be investigated by perturbing this solution as A = e i γ P z, the complex conjugate of ε is denoted as ε ∗. Instability can now be discovered by searching for solutions of the equation which grow exponentially. The nonlinear Schrödinger equation is constructed by removing the carrier wave of the light being modelled, therefore, ω m and k m dont represent absolute frequencies and wavenumbers, but the difference between these and those of the initial beam of light. It can be shown that the function is valid, provided c 2 = c 1 ∗. Therefore, instability will occur when β22 ω m 2 +2 γ P β2 <0 and this condition describes the requirement for anomalous dispersion. The gain spectrum can be described by defining a gain parameter as g ≡2 | ℑ |, the growth rate is maximum for ω2 = − γ P / β2. Modulation instability of optical fields has been observed in systems, namely. Modulation instability occurs owing to inherent optical nonlinearity of the due to photoreaction-induced changes in the refractive index. Ostrovsky, L. A. Modulation instability, The beginning
31.
Boussinesq approximation (water waves)
–
In fluid dynamics, the Boussinesq approximation for water waves is an approximation valid for weakly non-linear and fairly long waves. The approximation is named after Joseph Boussinesq, who first derived them in response to the observation by John Scott Russell of the wave of translation, the 1872 paper of Boussinesq introduces the equations now known as the Boussinesq equations. The Boussinesq approximation for water waves takes into account the structure of the horizontal and vertical flow velocity. This results in partial differential equations, called Boussinesq-type equations. In coastal engineering, Boussinesq-type equations are used in computer models for the simulation of water waves in shallow seas. While the Boussinesq approximation is applicable to fairly long waves – that is and this is useful because the waves propagate in the horizontal plane and have a different behaviour in the vertical direction. Often, as in Boussinesqs case, the interest is primarily in the wave propagation and this elimination of the vertical coordinate was first done by Joseph Boussinesq in 1871, to construct an approximate solution for the solitary wave. Subsequently, in 1872, Boussinesq derived the equations known nowadays as the Boussinesq equations, thereafter, the Boussinesq approximation is applied to the remaining flow equations, in order to eliminate the dependence on the vertical coordinate. As a result, the partial differential equations are in terms of functions of the horizontal coordinates. As an example, consider potential flow over a bed in the plane, with x the horizontal. The bed is located at z = −h, where h is the water depth. Invoking Laplaces equation for φ, as valid for incompressible flow, gives, φ = + = and this series may subsequently be truncated to a finite number of terms. Now the Boussinesq approximation for the velocity potential φ, as given above, is applied in these boundary conditions, further, in the resulting equations only the linear and quadratic terms with respect to η and ub are retained. The cubic and higher terms are assumed to be negligible. This set of equations has been derived for a horizontal bed. When the right-hand sides of the equations are set to zero. From the terms between brackets, the importance of nonlinearity of the equation can be expressed in terms of the Ursell number, for the case of infinitesimal wave amplitude, the terminology is linear frequency dispersion. The frequency dispersion characteristics of a Boussinesq-type of equation can be used to determine the range of wave lengths, for which it is a valid approximation
32.
Breaking wave
–
At this point, simple physical models that describe wave dynamics often become invalid, particularly those that assume linear behaviour. The most generally familiar sort of breaking wave is the breaking of surface waves on a coastline. Wave breaking generally occurs where the amplitude reaches the point that the crest of the wave actually overturns—though the types of breaking water waves are discussed in more detail below. Certain other effects in fluid dynamics have also been termed breaking waves, wave breaking also occurs in plasmas, when the particle velocities exceed the waves phase speed. Breaking of water surface waves may occur anywhere that the amplitude is sufficient, however, it is particularly common on beaches because wave heights are amplified in the region of shallower water. See also waves and shallow water, there are four basic types of breaking water waves. They are spilling, plunging, collapsing, and surging, when the ocean floor has a gradual slope, the wave will steepen until the crest becomes unstable, resulting in turbulent whitewater spilling down the face of the wave. This continues as the approaches the shore, and the waves energy is slowly dissipated in the whitewater. Because of this, spilling waves break for a time than other waves. Onshore wind conditions make spillers more likely, a plunging wave occurs when the ocean floor is steep or has sudden depth changes, such as from a reef or sandbar. A plunging wave breaks with more energy than a significantly larger spilling wave, the wave can trap and compress the air under the lip, which creates the crashing sound associated with waves. With large waves, this crash can be felt by beachgoers on land, offshore wind conditions can make plungers more likely. This is the tube that is so highly sought after by surfers, the surfer tries to stay near or under the crashing lip, often trying to stay as deep in the tube as possible while still being able to shoot forward and exit the barrel before it closes. A plunging wave that is parallel to the beach can break along its length at once, rendering it unrideable. Surfers refer to waves as closed out. Collapsing waves are a cross between plunging and surging, in which the crest never fully breaks, yet the bottom face of the wave gets steeper and collapses, surging breakers originate from long period, low steepness waves and/or steep beach profiles. The outcome is the movement of the base of the wave up the swash slope. The front face and crest of the wave remain relatively smooth with little foam or bubbles, resulting in a very narrow surf zone, or no breaking waves at all
33.
Clapotis
–
The resulting clapotic wave does not travel horizontally, but has a fixed pattern of nodes and antinodes. These waves promote erosion at the toe of the wall, the term was coined in 1877 by French mathematician and physicist Joseph Valentin Boussinesq who called these waves ‘le clapotis’ meaning ‘’the lapping. The standing waves alternately rise and fall in a mirror image pattern, as energy is converted to potential energy. This may also occur at sea between two different wave trains of near equal wavelength moving in opposite directions, but with unequal amplitudes, in partial clapotis the wave envelope contains some vertical motion at the nodes. When a wave strikes a wall at an oblique angle. In this situation, the individual crests formed at the intersection of the incident and this wave motion, when combined with the resultant vortices, can erode material from the seabed and transport it along the wall, undermining the structure until it fails. Clapotic waves on the sea surface also radiate infrasonic microbaroms into the atmosphere, clapotis has been called the bane and the pleasure of Sea kayaking. Rogue wave Boussinesq, J. Théorie des ondes liquides périodiques, mémoires présentés par divers savants à lAcadémie des Sciences. Boussinesq, J. Essai sur la théorie des eaux courantes, mémoires présentés par divers savants à lAcadémie des Sciences. Clapotis and Wave Reflection, With an Application to Vertical Breakwater Design, clapotis Wave Action – via YouTube
34.
Cnoidal wave
–
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 and they are used to describe surface gravity waves of fairly long wavelength, as compared to the water depth. The cnoidal wave solutions were derived by Korteweg and de Vries, in their 1895 paper in which they also propose their dispersive long-wave equation, in the limit of infinite wavelength, the cnoidal wave becomes a solitary wave. The Benjamin–Bona–Mahony equation has improved short-wavelength behaviour, as compared to the Korteweg–de Vries equation, cnoidal wave solutions can appear in other applications than surface gravity waves as well, for instance to describe ion acoustic waves in plasma physics. The KdV equation is a wave equation, including both frequency dispersion and amplitude dispersion effects. Shallow water equations — are also nonlinear and do have amplitude dispersion, Boussinesq equations — have the same range of validity as the KdV equation, but allow for wave propagation in arbitrary directions, so not only forward-propagating waves. The drawback is that the Boussinesq equations are more difficult to solve than the KdV equation. Airy wave theory — has full frequency dispersion, so valid for arbitrary depth and wavelength, however, for long waves the Boussinesq approach—as also applied in the KdV equation—is often preferred. This is because in shallow water the Stokes perturbation series needs many terms before convergence towards the solution, due to the peaked crests, while the KdV or Boussinesq models give good approximations for these long nonlinear waves. The KdV equation can be derived from the Boussinesq equations, further improvements in short-wave performance can be obtained by starting to derive a one-way wave equation from a modern improved Boussinesq model, valid for even shorter wavelengths. The cnoidal wave solutions of the KdV equation were presented by Korteweg and de Vries in their 1895 paper, solitary wave solutions for nonlinear and dispersive long waves had been found earlier by Boussinesq in 1872, and Rayleigh in 1876. The search for these solutions was triggered by the observations of this solitary wave by Russell, cnoidal wave solutions of the KdV equation are stable with respect to small perturbations. Further cn is one of the Jacobi elliptic functions and K is the elliptic integral of the first kind. The latter, m, determines the shape of the cnoidal wave, for m equal to zero the cnoidal wave becomes a cosine function, while for values close to one the cnoidal wave gets peaked crests and flat troughs. For values of m less than 0.95, the function can be approximated with trigonometric functions. An important dimensionless parameter for nonlinear long waves is the Ursell parameter, for small values of U, say U <5, a linear theory can be used, and at higher values nonlinear theories have to be used, like cnoidal wave theory. The demarcation zone between—third or fifth order—Stokes and cnoidal wave theories is in the range 10–25 of the Ursell parameter. Based on the analysis of the nonlinear problem of surface gravity waves within potential flow theory
35.
Dispersion (water waves)
–
In fluid dynamics, dispersion of water waves generally refers to frequency dispersion, which means that waves of different wavelengths travel at different phase speeds. Water waves, in context, are waves propagating on the water surface, with gravity. As a result, water with a surface is generally considered to be a dispersive medium. For a certain depth, surface gravity waves – i. e. waves occurring at the air–water interface. On the other hand, for a wavelength, gravity waves in deeper water have a larger phase speed than in shallower water. In contrast with the behavior of gravity waves, capillary waves propagate faster for shorter wavelengths, besides frequency dispersion, water waves also exhibit amplitude dispersion. This is an effect, by which waves of larger amplitude have a different phase speed from small-amplitude waves. This section is about frequency dispersion for waves on a fluid layer forced by gravity, for surface tension effects on frequency dispersion, see surface tension effects in Airy wave theory and capillary wave. The simplest propagating wave of unchanging form is a sine wave. Characteristic phases of a wave are, the upward zero-crossing at θ =0, the wave crest at θ = ½ π, the downward zero-crossing at θ = π. A certain phase repeats itself after an integer m multiple of 2π, the dispersion relation has two solutions, ω = +Ω and ω = −Ω, corresponding to waves travelling in the positive or negative x–direction. The dispersion relation will in general depend on other parameters in addition to the wavenumber k. For gravity waves, according to theory, these are the acceleration by gravity g. The dispersion relation for these waves is, an equation with tanh denoting the hyperbolic tangent function. An initial wave phase θ = θ0 propagates as a function of space and its subsequent position is given by, x = λ T t + λ2 π θ0 = ω k t + θ0 k. This shows that the moves with the velocity, c p = λ T = ω k = Ω k. A sinusoidal wave, of small amplitude and with a constant wavelength, propagates with the phase velocity. While the phase velocity is a vector and has an associated direction, according to linear theory for waves forced by gravity, the phase speed depends on the wavelength and the water depth
36.
Equatorial wave
–
Equatorial waves are ocean 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 Earths rotation and its 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 atmosphere and ocean and play an important role in the evolution of many climate phenomena such as El Niño. Equatorial waves may be separated into a series of subclasses depending on their fundamental dynamics, at shortest periods are the equatorial gravity waves while the longest periods are associated with the equatorial Rossby waves. In addition to these two subclasses, there are two special subclasses of equatorial waves known as the mixed Rossby-gravity wave and the equatorial Kelvin wave. The latter two share the characteristics that they can have any period and also that they may carry only in an eastward direction. The remainder of this article discusses the relationship between the period of waves, their wavelength in the zonal direction and their speeds for a simplified ocean. Rossby-gravity waves, first observed in the stratosphere by M. Yanai, but, oddly, their crests and troughs may propagate westward if their periods are long enough. The eastward speed of propagation of waves can be derived for an inviscid slowly moving layer of fluid of uniform depth H. Because the Coriolis parameter vanishes at 0 degrees latitude, the “equatorial beta plane” approximation must be made. This approximation states that “f” is approximately equal to βy, where “y” is the distance from the equator and β is the variation of the coriolis parameter with latitude, ∂ f ∂ y = β. Once the frequency relation is formulated in terms of ω, the angular frequency and these three solutions correspond to the equatorial gravity waves, the equatorially trapped Rossby waves and the mixed Rossby-gravity wave. Equatorial gravity waves can be either westward- or eastward-propagating, the governing equations for these equatorial waves are similar to those presented above, except that there is no meridional velocity component. The continuity equation, ∂ ϕ ∂ t + c 2 ∂ u ∂ x =0 the u-momentum equation, ∂ u ∂ t = − ∂ ϕ ∂ x the v-momentum equation, u β y = − ∂ ϕ ∂ y. The solution to these equations yields the following phase speed, c2 = gH, also, these Kelvin waves only propagate towards the east. Kelvin waves have been connected to El Niño in recent years in terms of precursors to this atmospheric and oceanic phenomenon, the weak low pressure in the Indian Ocean typically propagates eastward into the North Pacific Ocean and can produce easterly winds. This wave can be observed at the surface by a rise in sea surface height of about 8 cm. If the Kelvin wave hits the South American coast, its water gets transferred upward
37.
Internal wave
–
Internal waves are gravity waves that oscillate within a fluid medium, rather than on its surface. To exist, the fluid must be stratified, the density must decrease continuously or discontinuously with depth/height due to changes, for example, If the density changes continuously, the waves can propagate vertically as well as horizontally through the fluid. Internal waves, also called gravity waves, go by many other names depending upon the fluid stratification, generation mechanism, amplitude. If propagating horizontally along an interface where the density decreases with height. If the interfacial waves are large amplitude they are called solitary waves or internal solitons. If moving vertically through the atmosphere where substantial changes in air density influences their dynamics, If generated by flow over topography, they are called Lee waves or mountain waves. If the mountain waves break aloft, they can result in strong winds at the ground known as Chinook winds or Foehn winds. If generated in the ocean by tidal flow over submarine ridges or the continental shelf, If they evolve slowly compared to the Earths rotational frequency so that their dynamics are influenced by the Coriolis effect, they are called inertia gravity waves or, simply, inertial waves. Internal waves are usually distinguished from Rossby waves, which are influenced by the change of Coriolis frequency with latitude. An internal wave can readily be observed in the kitchen by slowly tilting back, clouds that reveal internal waves launched by flow over hills are called lenticular clouds because of their lens-like appearance. Less dramatically, a train of waves can be visualized by rippled cloud patterns described as herringbone sky or mackerel sky. The outflow of air from a thunderstorm can launch large amplitude internal solitary waves at an atmospheric inversion. In northern Australia, these result in Morning Glory clouds, used by some daredevils to glide along like a surfer riding an ocean wave, satellites over Australia and elsewhere reveal these waves can span many hundreds of kilometers. According to Archimedes principle, the weight of an object is reduced by the weight of fluid it displaces. This holds for a parcel of density ρ surrounded by an ambient fluid of density ρ0. Its weight per volume is g, in which g is the acceleration of gravity. Dividing by a density, ρ00, gives the definition of the reduced gravity. Because water is more dense than air, the displacement of water by air from a surface gravity wave feels nearly the full force of gravity
38.
Kinematic wave
–
These waves are also applied to model the motion of highway traffic flows. In these flows, mass and momentum equations can be combined to yield a kinematic wave equation, depending on the flow configurations, the kinematic wave can be linear or non-linear, which depends on whether the wave celerity is a constant or a variable. In general, the wave can be advecting and diffusing, however, in simple situation, the kinematic wave is mainly advecting. For F = h 2 /2, this reduces to the Burgers equation
39.
Longshore drift
–
Longshore drift is a geological process that consists of the transportation of sediments along a coast parallel to the shoreline, which is dependent on oblique incoming wind direction. Oblique incoming wind squeezes water along the coast, and so generates a current which moves parallel to the coast. Longshore drift is simply the sediment moved by the longshore current and this current and sediment movement occurs within the surf zone. Beach sand is moved on such oblique wind days, due to the swash and backwash of water on the beach. Breaking surf sends water up the beach at an oblique angle, thus beach sand can move downbeach in a zig zag fashion many tens of meters per day. This process is called beach drift but some regard it as simply part of longshore drift because of the overall movement of sand parallel to the coast. Longshore drift affects numerous sediment sizes as it works in different ways depending on the sediment. Sand is largely affected by the force of breaking waves. There are numerous calculations that take into consideration the factors that produce longshore drift, some of these are, Geological changes, e. g. erosion, backshore changes and emergence of headlands. Change in hydrodynamic forces, e. g. change in wave diffraction in headland, change to hydrodynamic influences, e. g. the influence of new tidal inlets and deltas on drift. Alterations of the sediment budget, e. g. switch of shorelines from drift to swash alignment, the intervention of humans, e. g. cliff protection, groynes, detached breakwaters. The sediment budget takes into consideration sediment sources and sinks within a system, a good example of the sediment budget and longshore drift working together in the coastal system is inlet ebb-tidal shoals, which store sand that has been transported by long shore transport. As well as storing sand these systems may also transfer or by pass sand into other systems, therefore inlet ebb-tidal systems provide a good sources. Long shore occurs in a 90 to 80 degree backwash so it would be presented as an angle with the wave line. This section consists of features of long shore drift that occur on a coast where long shore drift occurs uninterrupted by man-made structures, spits are formed when longshore drift travels past a point where the dominant drift direction and shoreline do not veer in the same direction. As well as dominant drift direction, spits are affected by the strength of wave driven current, wave angle, spits are landforms that have two important features, with the first feature being the region at the up-drift end or proximal end. The proximal end is attached to land and may form a slight “barrier” between the sea and an estuary or lagoon. As an example, the New Brighton spit in Canterbury, New Zealand, was created by longshore drift of sediment from the Waimakariri River to the north and this spit system is currently in equilibrium but undergoes phases of deposition and erosion
40.
Luke's variational principle
–
In fluid dynamics, Lukes variational principle is a Lagrangian variational description of the motion of surface waves on a fluid with a free surface, under the action of gravity. This principle is named after J. C. Luke, who published it in 1967, Lukes Lagrangian formulation can also be recast into a Hamiltonian formulation in terms of the surface elevation and velocity potential at the free surface. This is often used when modelling the spectral density evolution of the free-surface in a sea state, both the Lagrangian and Hamiltonian formulations can be extended to include surface tension effects, and by using Clebsch potentials to include vorticity. Lukes Lagrangian formulation is for surface gravity waves on an—incompressible. The Lagrangian L, as given by Luke, is, L = − ∫ t 0 t 1 d t, from Bernoullis principle, this Lagrangian can be seen to be the integral of the fluid pressure over the whole time-dependent fluid domain V. This is in agreement with the principles for inviscid flow without a free surface. This may also include moving wavemaker walls and ship motion. For the case of an unbounded domain with the free fluid surface at z=η. The bed-level term proportional to h2 in the energy has been neglected, since it is a constant. Below, Lukes variational principle is used to arrive at the equations for non-linear surface gravity waves on a potential flow. The variation δ L =0 in the Lagrangian with respect to variations in the velocity potential Φ, consider a small variation δΦ in the velocity potential Φ. Then the resulting variation in the Lagrangian is, δ Φ L = L − L = − ∫ t 0 t 1 ∬ d x d t. The first integral on the right-hand side integrates out to the boundaries, in x and t, of the integration domain and is zero since the variations δΦ are taken to be zero at these boundaries. Similarly, variations δΦ only non-zero at the bottom z = -h result in the kinematic bed condition, ∇ Φ ⋅ ∇ h + ∂ Φ ∂ z =0 at z = − h. Considering the variation of the Lagrangian with respect to small changes δη gives, the Hamiltonian structure of surface gravity waves on a potential flow was discovered by Vladimir E. The Hamiltonian H is the sum of the kinetic and potential energy of the fluid and this is expressed by the Dirichlet-to-Neumann operator D, acting linearly on φ. The Hamiltonian density can also be written as, H =12 ρ φ +12 ρ g η2, as a result, the Hamiltonian is a quadratic functional of the surface potential φ. Also the potential energy part of the Hamiltonian is quadratic, the source of non-linearity in surface gravity waves is through the kinetic energy depending non-linear on the free surface shape η
