Trochoidal wave

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.

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.

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] 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]

the vertical coordinate (positive upward, in the direction opposing gravity).

the centres of the circular orbits – around which the corresponding fluid parcel moves with constant speed

The phase speed satisfies the dispersion relation:

the same as for Airy's linear waves in deep water.

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.

and diminishing rapidly with depth below the free surface.

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

-direction is upward, opposing the Earth's gravity of strength

The free surface is described parametrically as a function of the parameters

around which the fluid parcels at the wavy surface orbit.

determine the wave propagation direction of component

A clever choice is needed in order to exploit the possibility of fast computation by means of the FFT.

Most often, the wavenumbers are chosen on a regular grid in

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

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 , the wavelength as and the phase speed as
Vector contributions from the gravitational force (medium gray) and the gradient of the pressure (black) come together in an amazing way to produce the uniform circular motion of the fluid particles. For uniform circular motion, the net force (light gray) has constant magnitude and always points towards the center of the circle. The fluid particles are colored according to their values. Since the pressure is a function only of , the animation illustrates how the pressure gradient vectors are always perpendicular to the color bands, and their magnitudes are larger when the color bands are closer together.
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).