# Trochoidal wave

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

The flow field associated with the trochoidal wave is not irrotational: it has vorticity. The vorticity is of such a specific strength and vertical distribution that the trajectories of the fluid parcels are closed circles. This is in contrast with the usual experimental observation of Stokes drift associated with the wave motion. Also the phase speed is independent of the trochoidal wave's amplitude, unlike other nonlinear wave-theories (like those of the Stokes wave and cnoidal wave) and observations. For these reasons – as well as for the fact that solutions for finite fluid depth are lacking – trochoidal waves are of limited use for engineering applications.

In computer graphics, the rendering of realistic-looking ocean waves can be done by use of so-called Gerstner waves. This is a multi-component and multi-directional extension of the traditional Gerstner wave, often using fast Fourier transforms to make (real-time) animation feasible.[1]

Surface elevation of a trochoidal wave (deep blue) propagating to the right. The trajectories of free surface particles are close circles (in cyan), and the flow velocity is shown in red, for the black particles. The wave height – difference between the crest and trough elevation – is denoted as ${\displaystyle H}$, the wavelength as ${\displaystyle \lambda }$ and the phase speed as ${\displaystyle c.}$

## Description of classical trochoidal wave

Using a Lagrangian specification of the flow field, the motion of fluid parcels is – for a periodic wave on the surface of a fluid layer of infinite depth:[2]

{\displaystyle {\begin{aligned}X(a,b,t)&=a+{\frac {\mathrm {e} ^{kb}}{k}}\sin \left(k(a+ct)\right),\\Y(a,b,t)&=b-{\frac {\mathrm {e} ^{kb}}{k}}\cos \left(k(a+ct)\right),\end{aligned}}}

where ${\displaystyle x=X(a,b,t)}$ and ${\displaystyle y=Y(a,b,t)}$ are the positions of the fluid parcels in the ${\displaystyle (x,y)}$ plane at time ${\displaystyle t}$, with ${\displaystyle x}$ the horizontal coordinate and ${\displaystyle y}$ the vertical coordinate (positive upward, in the direction opposing gravity). The Lagrangian coordinates ${\displaystyle (a,b)}$ label the fluid parcels, with ${\displaystyle (x,y)=(a,b)}$ the centres of the circular orbits – around which the corresponding fluid parcel moves with constant speed ${\displaystyle c\,\exp(kb).}$ Further ${\displaystyle k=2\pi /\lambda }$ is the wavenumber (and ${\displaystyle \lambda }$ the wavelength), while ${\displaystyle c}$ is the phase speed with which the wave propagates in the ${\displaystyle x}$-direction. The phase speed satisfies the dispersion relation:

${\displaystyle c^{2}={\frac {g}{k}},}$

which is independent of the wave nonlinearity (i.e. does not depend on the wave height ${\displaystyle H}$), and this phase speed ${\displaystyle c}$ the same as for Airy's linear waves in deep water.

The free surface is a line of constant pressure, and is found to correspond with a line ${\displaystyle b=b_{s}}$, where ${\displaystyle b_{s}}$ is a (nonpositive) constant. For ${\displaystyle b_{s}=0}$ the highest waves occur, with a cusp-shaped crest. Note that the highest (irrotational) Stokes wave has a crest angle of 120°, instead of the 0° for the rotational trochoidal wave.[3]

The wave height of the trochoidal wave is ${\displaystyle H=(2/k)\exp(kb_{s}).}$ The wave is periodic in the ${\displaystyle x}$-direction, with wavelength ${\displaystyle \lambda ;}$ and also periodic in time with period ${\displaystyle T=\lambda /c={\sqrt {2\pi \lambda /g}}.}$

The vorticity ${\displaystyle \varpi }$ under the trochoidal wave is:[2]

${\displaystyle \varpi (a,b,t)=-{\frac {2\,k\,c\,\mathrm {e} ^{2kb}}{1-\mathrm {e} ^{2kb}}},}$

varying with Lagrangian elevation ${\displaystyle b}$ and diminishing rapidly with depth below the free surface.

## In computer graphics

Animation (5 MB) of swell waves using multi-directional and multi-component Gerstner waves for the simulation of the ocean surface and POV-Ray for the 3D rendering. (The animation is periodic in time; it can be set to loop after right-clicking on it while it is playing).

A multi-component and multi-directional extension of the Lagrangian description of the free-surface motion – as used in Gerstner's trochoidal wave – is used in computer graphics for the simulation of ocean waves.[1] For the classical Gerstner wave the fluid motion exactly satisfies the nonlinear, incompressible and inviscid flow equations below the free surface. However, the extended Gerstner waves do in general not satisfy these flow equations exactly (although they satisfy them approximately, i.e. for the linearised Lagrangian description by potential flow). This description of the ocean can be programmed very efficiently by use of the fast Fourier transform (FFT). Moreover, the resulting ocean waves from this process look realistic, as a result of the nonlinear deformation of the free surface (due to the Lagrangian specification of the motion): sharper crests and flatter troughs.

The mathematical description of the free-surface in these Gerstner waves can be as follows:[1] the horizontal coordinates are denoted as ${\displaystyle x}$ and ${\displaystyle z}$, and the vertical coordinate is ${\displaystyle y}$. The mean level of the free surface is at ${\displaystyle y=0}$ and the positive ${\displaystyle y}$-direction is upward, opposing the Earth's gravity of strength ${\displaystyle g.}$ The free surface is described parametrically as a function of the parameters ${\displaystyle \alpha }$ and ${\displaystyle \beta ,}$ as well as of time ${\displaystyle t.}$ The parameters are connected to the mean-surface points ${\displaystyle (x,y,z)=(\alpha ,0,\beta )}$ around which the fluid parcels at the wavy surface orbit. The free surface is specified through ${\displaystyle x=\xi (\alpha ,\beta ,t),}$ ${\displaystyle y=\zeta (\alpha ,\beta ,t)}$ and ${\displaystyle z=\eta (\alpha ,\beta ,t)}$ with:

{\displaystyle {\begin{aligned}\xi &=\alpha -\sum _{m=1}^{M}{\frac {k_{x,m}}{k_{m}}}\,{\frac {a_{m}}{\tanh \left(k_{m}\,h\right)}}\,\sin \left(\theta _{m}\right),\\\eta &=\beta -\sum _{m=1}^{M}{\frac {k_{z,m}}{k_{m}}}\,{\frac {a_{m}}{\tanh \left(k_{m}\,h\right)}}\,\sin \left(\theta _{m}\right),\\\zeta &=\sum _{m=1}^{M}a_{m}\,\cos \left(\theta _{m}\right)\quad {\text{and}}\\\theta _{m}&=k_{x,m}\,\alpha +k_{z,m}\,\beta -\omega _{m}\,t-\phi _{m},\end{aligned}}}

where ${\displaystyle \tanh }$ is the hyperbolic tangent function, ${\displaystyle M}$ is the number of wave components considered, ${\displaystyle a_{m}}$ is the amplitude of component ${\displaystyle {m=1\dots M}}$ and ${\displaystyle \phi _{m}}$ its phase. Further ${\displaystyle {k_{m}=\scriptstyle {\sqrt {(k_{x,m}^{2}+k_{z,m}^{2})}}}}$ is its wavenumber and ${\displaystyle \omega _{m}}$ its angular frequency. The latter two, ${\displaystyle k_{m}}$ and ${\displaystyle \omega _{m},}$ can not be chosen independently but are related through the dispersion relation:

${\displaystyle \omega _{m}^{2}=g\,k_{m}\,\tanh \left(k_{m}\,h\right),}$

with ${\displaystyle h}$ the mean water depth. In deep water (${\displaystyle h\to \infty }$) the hyperbolic tangent goes to one: ${\displaystyle {\tanh(k_{m}\,h)\to 1.}}$ The components ${\displaystyle k_{x,m}}$ and ${\displaystyle k_{z,m}}$ of the horizontal wavenumber vector ${\displaystyle {\boldsymbol {k}}_{m}}$ determine the wave propagation direction of component ${\displaystyle m.}$

The choice of the various parameters ${\displaystyle a_{m},k_{x,m},k_{z,m}}$ and ${\displaystyle \phi _{m}}$ for ${\displaystyle {m=1,\dots ,{M},}}$ and a certain mean depth ${\displaystyle h}$ determines the form of the ocean surface. A clever choice is needed in order to exploit the possibility of fast computation by means of the FFT. See e.g. Tessendorf (2001) for a description how to do this. Most often, the wavenumbers are chosen on a regular grid in ${\displaystyle (k_{x},k_{z})}$-space. Thereafter, the amplitudes ${\displaystyle a_{m}}$ and phases ${\displaystyle \phi _{m}}$ are chosen randomly in accord with the variance-density spectrum of a certain desired sea state. Finally, by FFT, the ocean surface can be constructed in such a way that it is periodic both in space and time, enabling tiling – creating periodicity in time by slightly shifting the frequencies ${\displaystyle \omega _{m}}$ such that ${\displaystyle \omega _{m}=m\,\Delta \omega }$ for ${\displaystyle {m=1,\dots ,{M}.}}$

In rendering, also the normal vector ${\displaystyle {\boldsymbol {n}}}$ to the surface is often needed. These can be computed using the cross product (${\displaystyle \times }$) as:

${\displaystyle {\boldsymbol {n}}={\frac {\partial {\boldsymbol {s}}}{\partial \alpha }}\times {\frac {\partial {\boldsymbol {s}}}{\partial \beta }}\quad {\text{with}}\quad {\boldsymbol {s}}(\alpha ,\beta ,t)={\begin{pmatrix}\xi (\alpha ,\beta ,t)\\\zeta (\alpha ,\beta ,t)\\\eta (\alpha ,\beta ,t)\end{pmatrix}}.}$

The unit normal vector then is ${\displaystyle {\boldsymbol {e}}_{n}={\boldsymbol {n}}/\|{\boldsymbol {n}}\|,}$ with ${\displaystyle \|{\boldsymbol {n}}\|}$ the norm of ${\displaystyle {\boldsymbol {n}}.}$

## Notes

1. ^ a b c Tessendorf (2001)
2. ^ a b Lamb (1994, §251)
3. ^ Stokes, G.G. (1880), "Supplement to a paper on the theory of oscillatory waves", Mathematical and Physical Papers, Volume I, Cambridge University Press, pp. 314–326, OCLC 314316422

## References

Airy wave theory

In fluid dynamics, Airy wave theory (often referred to as linear 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 fluid layer has a uniform mean depth, and that the fluid flow is inviscid, incompressible and irrotational. This theory was first published, in correct form, by George Biddell Airy in the 19th century.Airy wave theory is often applied in ocean engineering and coastal engineering for the modelling of random sea states – giving a description of the wave kinematics and dynamics of high-enough accuracy for many purposes. Further, several second-order nonlinear properties of surface gravity waves, and their propagation, can be estimated from its results. Airy wave theory is also a good approximation for tsunami waves in the ocean, before they steepen near the coast.

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 wave height to water depth (for waves in shallow water), and wave height to wavelength (for waves in deep water).

Bahama Banks

The Bahama Banks are the submerged carbonate platforms that make up much of the Bahama Archipelago. The term is usually applied in referring to either the Great Bahama Bank around Andros Island, or the Little Bahama Bank of Grand Bahama Island and Great Abaco, which are the largest of the platforms, and the Cay Sal Bank north of Cuba. The islands of these banks are politically part of the Bahamas. Other banks are the three banks of the Turks and Caicos Islands, namely the Caicos Bank of the Caicos Islands, the bank of the Turks Islands, and wholly submerged Mouchoir Bank. Further southeast are the equally wholly submerged Silver Bank and Navidad Bank north of the Dominican Republic.

Carbonate platform

A carbonate platform is a sedimentary body which possesses topographic relief, and is composed of autochthonic calcareous deposits. Platform growth is mediated by sessile organisms whose skeletons build up the reef or by organisms (usually microbes) which induce carbonate precipitation through their metabolism. Therefore, carbonate platforms can not grow up everywhere: they are not present in places where limiting factors to the life of reef-building organisms exist. Such limiting factors are, among others: light, water temperature, transparency and pH-Value. For example, carbonate sedimentation along the Atlantic South American coasts takes place everywhere but at the mouth of the Amazon River, because of the intense turbidity of the water there. Spectacular examples of present-day carbonate platforms are the Bahama Banks under which the platform is roughly 8 km thick, the Yucatan Peninsula which is up to 2 km thick, the Florida platform, the platform on which the Great Barrier Reef is growing, and the Maldive atolls. All these carbonate platforms and their associated reefs are confined to tropical latitudes. Today's reefs are built mainly by scleractinian corals, but in the distant past other organisms, like archaeocyatha (during the Cambrian) or extinct cnidaria (tabulata and rugosa) were important reef builders.

Clapotis

In hydrodynamics, a clapotis (from French for "lapping of water") is a non-breaking standing wave pattern, caused for example, by the reflection of a traveling surface wave train from a near vertical shoreline like a breakwater, seawall or steep cliff.

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, and can cause severe damage to shore structures. The term was coined in 1877 by French mathematician and physicist Joseph Valentin Boussinesq who called these waves ‘le clapotis’ meaning ‘’the lapping".In the idealized case of "full clapotis" where a purely monotonic incoming wave is completely reflected normal to a solid vertical wall,

the standing wave height is twice the height of the incoming waves at a distance of one half wavelength from the wall.

In this case, the circular orbits of the water particles in the deep-water wave are converted to purely linear motion, with vertical velocities at the antinodes, and horizontal velocities at the nodes.

The standing waves alternately rise and fall in a mirror image pattern, as kinetic energy is converted to potential energy, and vice versa.

In his 1907 text, Naval Architecture, Cecil Peabody described this phenomenon:

At any instant the profile of the water surface is like that of a trochoidal wave, but the profile instead of appearing to run to the right or left, will grow from a horizontal surface, attain a maximum development, and then flatten out till the surface is again horizontal; immediately another wave profile will form with its crests where the hollows formerly were, will grow and flatten out, etc. If attention is concentrated on a certain crest, it will be seen to grow to its greatest height, die away, and be succeeded in the same place by a hollow, and the interval of time between the successive formations of crests at a given place will be the same as the time of one of the component waves.

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 this context, are waves propagating on the water surface, with gravity and surface tension as the restoring forces. As a result, water with a free surface is generally considered to be a dispersive medium.

For a certain water depth, surface gravity waves – i.e. waves occurring at the air–water interface and gravity as the only force restoring it to flatness – propagate faster with increasing wavelength. On the other hand, for a given (fixed) 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 (i.e. only forced by surface tension) propagate faster for shorter wavelengths.

Besides frequency dispersion, water waves also exhibit amplitude dispersion. This is a nonlinear effect, by which waves of larger amplitude have a different phase speed from small-amplitude waves.

František Josef Gerstner

Franz Josef von Gerstner (German: Franz Josef von Gerstner, Czech: František Josef Gerstner; 23 February 1756 – 25 July 1832) was a German-Bohemian physicist and engineer.

List of submarine volcanoes

A list of active and extinct submarine volcanoes and seamounts located under the world's oceans. There are estimated to be 40,000 to 55,000 seamounts in the global oceans. Almost all are not well-mapped and many may not have been identified at all. Most are unnamed and unexplored. This list is therefore confined to seamounts that are notable enough to have been named and/or explored.

Oceanic plateau

An oceanic or submarine plateau is a large, relatively flat elevation that is higher than the surrounding relief with one or more relatively steep sides.There are 184 oceanic plateaus covering an area of 18,486,600 km2 (7,137,700 sq mi), or about 5.11% of the oceans. The South Pacific region around Australia and New Zealand contains the greatest number of oceanic plateaus (see map).

Oceanic plateaus produced by large igneous provinces are often associated with hotspots, mantle plumes, and volcanic islands — such as Iceland, Hawaii, Cape Verde, and Kerguelen. The three largest plateaus, the Caribbean, Ontong Java, and Mid-Pacific Mountains, are located on thermal swells. Other oceanic plateaus, however, are made of rifted continental crust, for example Falkland Plateau, Lord Howe Rise, and parts of Kerguelen, Seychelles, and Arctic ridges.

Plateaus formed by large igneous provinces were formed by the equivalent of continental flood basalts such as the Deccan Traps in India and the Snake River Plain in the United States.

In contrast to continental flood basalts, most igneous oceanic plateaus erupt through young and thin (6–7 km (3.7–4.3 mi)) mafic or ultra-mafic crust and are therefore uncontaminated by felsic crust and representative for their mantle sources.

These plateaus often rise 2–3 km (1.2–1.9 mi) above the surrounding ocean floor and are more buoyant than oceanic crust. They therefore tend to withstand subduction, more-so when thick and when reaching subduction zones shortly after their formations. As a consequence, they tend to "dock" to continental margins and be preserved as accreted terranes. Such terranes are often better preserved than the exposed parts of continental flood basalts and are therefore a better record of large-scale volcanic eruptions throughout Earth's history. This "docking" also means that oceanic plateaus are important contributors to the growth of continental crust. Their formations often had a dramatic impact on global climate, such as the most recent plateaus formed, the three, large, Cretaceous oceanic plateaus in the Pacific and Indian Ocean: Ontong Java, Kerguelen, and Caribbean.

Outline of oceanography

The following outline is provided as an overview of and introduction to Oceanography.

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 precisely as possible.

Dynamical physical oceanography focuses primarily upon the processes that govern the motion of fluids with emphasis upon theoretical research and numerical models. These are part of the large field of Geophysical Fluid Dynamics (GFD) that is shared together with meteorology. GFD is a sub field of Fluid dynamics describing flows occurring on spatial and temporal scales that are greatly influenced by the Coriolis force.

Trochoid

A trochoid (from the Greek word for wheel, "trochos") is a roulette formed by a circle rolling along a line. In other words, it is the curve traced out by a point fixed to a circle (where the point may be on, inside, or outside the circle) as it rolls along a straight line. If the point is on the circle, the trochoid is called common (also known as a cycloid); if the point is inside the circle, the trochoid is called curtate; and if the point is outside the circle, the trochoid is called prolate. The word "trochoid" was coined by Gilles de Roberval.

Undersea mountain range

Undersea mountain ranges are mountain ranges that are mostly or entirely underwater, and specifically under the surface of an ocean. If originated from current tectonic forces, they are often referred to as a mid-ocean ridge. In contrast, if formed by past above-water volcanism, they are known as a seamount chain. The largest and best known undersea mountain range is a mid-ocean ridge, the Mid-Atlantic Ridge. It has been observed that, "similar to those on land, the undersea mountain ranges are the loci of frequent volcanic and earthquake activity".

Wave base

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

Wind wave

In fluid dynamics, wind waves, or wind-generated waves, are water surface waves that occur on the free surface of the oceans and other bodies (like lakes, rivers, canals, puddles or ponds). They result from the wind blowing over an area of fluid surface. Waves in the oceans can travel thousands of miles before reaching land. Wind waves on Earth range in size from small ripples, to waves over 100 ft (30 m) high.When directly generated and affected by local waters, a wind wave system is called a wind sea. After the wind ceases to blow, wind waves are called swells. More generally, a swell consists of wind-generated waves that are not significantly affected by the local wind at that time. They have been generated elsewhere or some time ago. Wind waves in the ocean are called ocean surface waves.

Wind waves have a certain amount of randomness: subsequent waves differ in height, duration, and shape with limited predictability. They can be described as a stochastic process, in combination with the physics governing their generation, growth, propagation, and decay—as well as governing the interdependence between flow quantities such as: the water surface movements, flow velocities and water pressure. The key statistics of wind waves (both seas and swells) in evolving sea states can be predicted with wind wave models.

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

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