In
fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) a ...
, a trochoidal wave or Gerstner wave is an exact solution of the
Euler equations for
periodic surface 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 ...
s. It describes a
progressive wave
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
of permanent form on the surface of an
incompressible fluid
In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An eq ...
of infinite depth. The free surface of this wave solution is an inverted (upside-down)
trochoid
In geometry, a trochoid () is a roulette curve formed by a circle rolling along a line. It is the curve traced out by a point fixed to a circle (where the point may be on, inside, or outside the circle) as it rolls along a straight line. If the ...
– with sharper
crests and flat troughs. This wave solution was discovered by
Gerstner in 1802, and rediscovered independently by
Rankine Rankine is a surname. Notable people with the surname include:
* William Rankine (1820–1872), Scottish engineer and physicist
** Rankine body an elliptical shape of significance in fluid dynamics, named for Rankine
** Rankine scale, an absolute-t ...
in 1863.
The flow field associated with the trochoidal wave is not
irrotational: it has
vorticity
In continuum mechanics, vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along wi ...
. The vorticity is of such a specific strength and vertical distribution that the trajectories of the
fluid parcel In fluid dynamics, within the framework of continuum mechanics, a fluid parcel is a very small amount of fluid, identifiable throughout its dynamic history while moving with the fluid flow. As it moves, the mass of a fluid parcel remains constant, ...
s are closed circles. This is in contrast with the usual experimental observation of
Stokes drift associated with the wave motion. Also the
phase speed
The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for exampl ...
is independent of the trochoidal wave's
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
, unlike other nonlinear wave-theories (like those of the
Stokes wave
In fluid dynamics, a Stokes wave is a nonlinear and periodic surface wave on an inviscid fluid layer of constant mean depth.
This type of modelling has its origins in the mid 19th century when Sir George Stokes – using a perturbation series ...
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
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
, the
rendering of realistic-looking
ocean wave
In fluid dynamics, a wind wave, water wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result from the wind blowing over the water surface. The contact distance in the direction of t ...
s 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 transform
A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in ...
s to make (real-time)
animation
Animation is a method by which still figures are manipulated to appear as moving images. In traditional animation, images are drawn or painted by hand on transparent celluloid sheets to be photographed and exhibited on film. Today, most ani ...
feasible.
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:
where
and
are the positions of the fluid parcels in the
plane at time
, with
the horizontal coordinate and
the vertical coordinate (positive upward, in the direction opposing gravity). The Lagrangian coordinates
label the fluid parcels, with
the centres of the circular orbits – around which the corresponding fluid parcel moves with constant
speed
In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a scalar quant ...
Further
is the
wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to te ...
(and
the
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
), while
is the phase speed with which the wave propagates in the
-direction. The phase speed satisfies the
dispersion
Dispersion may refer to:
Economics and finance
*Dispersion (finance), a measure for the statistical distribution of portfolio returns
*Price dispersion, a variation in prices across sellers of the same item
*Wage dispersion, the amount of variatio ...
relation:
which is independent of the wave nonlinearity (i.e. does not depend on the wave height
), and this phase speed
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
, where
is a (nonpositive) constant. For
the highest waves occur, with a
cusp
A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth.
Cusp or CUSP may also refer to:
Mathematics
* Cusp (singularity), a singular point of a curve
* Cusp catastrophe, a branch of bifurc ...
-shaped crest. Note that the highest (irrotational)
Stokes wave
In fluid dynamics, a Stokes wave is a nonlinear and periodic surface wave on an inviscid fluid layer of constant mean depth.
This type of modelling has its origins in the mid 19th century when Sir George Stokes – using a perturbation series ...
has a
crest angle of 120°, instead of the 0° for the rotational trochoidal wave.
The
wave height
In fluid dynamics, the wave height of a surface wave is the difference between the elevations of a crest and a neighboring trough. ''Wave height'' is a term used by mariners, as well as in coastal, ocean and naval engineering.
At sea, the te ...
of the trochoidal wave is
The wave is periodic in the
-direction, with wavelength
and also periodic in time with
period
The
vorticity
In continuum mechanics, vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along wi ...
under the trochoidal wave is:
[
varying with Lagrangian elevation and diminishing rapidly with depth below the free surface.
]
In computer graphics
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
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
for the simulation of ocean waves.[ For the classical Gerstner wave the fluid motion exactly satisfies the ]nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many oth ...
, incompressible
In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An eq ...
and inviscid
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the in ...
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
In fluid dynamics, potential flow (or ideal flow) describes the velocity field as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field, which is a valid app ...
). This description of the ocean can be programmed very efficiently by use of the fast Fourier transform
A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in ...
(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:[ the horizontal coordinates are denoted as and , and the vertical coordinate is . The ]mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set.
For a data set, the '' ar ...
level of the free surface is at and the positive -direction is upward, opposing the Earth's gravity
The gravity of Earth, denoted by , is the net acceleration that is imparted to objects due to the combined effect of gravitation (from mass distribution within Earth) and the centrifugal force (from the Earth's rotation).
It is a vector qua ...
of strength The free surface is described parametrically as a function of the parameters and as well as of time The parameters are connected to the mean-surface points around which the fluid parcel In fluid dynamics, within the framework of continuum mechanics, a fluid parcel is a very small amount of fluid, identifiable throughout its dynamic history while moving with the fluid flow. As it moves, the mass of a fluid parcel remains constant, ...
s at the wavy surface orbit. The free surface is specified through and with:
where is the hyperbolic tangent
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the ...
function, is the number of wave components considered, is the amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
of component and its phase. Further is its wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to te ...
and its angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...
. The latter two, and can not be chosen independently but are related through the dispersion relation
In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given t ...
:
with the mean water depth. In deep water () the hyperbolic tangent goes to one: The components and of the horizontal wavenumber vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
determine the wave propagation direction of component
The choice of the various parameters and for and a certain mean depth 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. for a description how to do this. Most often, the wavenumbers are chosen on a regular grid in -space. Thereafter, the amplitudes and phases are chosen randomly in accord with the variance-density spectrum of a certain desired sea state
In oceanography, sea state is the general condition of the free surface on a large body of water—with respect to wind waves and Swell (ocean), swell—at a certain location and moment. A sea state is characterized by statistics, including the ...
. Finally, by FFT, the ocean surface can be constructed in such a way that it is periodic both in space and time, enabling tiling
Tiling may refer to:
*The physical act of laying tiles
* Tessellations
Computing
*The compiler optimization of loop tiling
*Tiled rendering, the process of subdividing an image by regular grid
*Tiling window manager
People
*Heinrich Sylvester T ...
– creating periodicity in time by slightly shifting the frequencies such that for
In rendering, also the normal vector
In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve ...
to the surface is often needed. These can be computed using the cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
() as:
The unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (a ...
normal vector then is with the norm of
Notes
References
*. Reprinted in: ''Annalen der Physik'' 32(8), pp. 412–445, 1809.
*
* Originally published in 1879, the 6th extended edition appeared first in 1932.
*
*
{{DEFAULTSORT:Trochoidal wave
Water waves
Wave mechanics
Physical oceanography
3D computer graphics
Articles containing video clips
Oceanographical terminology