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.
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 formula_2 and formula_3 are the positions of the fluid parcels in the formula_4 plane at time formula_5, with formula_6 the horizontal coordinate and formula_7 the vertical coordinate (positive upward, in the direction opposing gravity). The Lagrangian coordinates formula_8 label the fluid parcels, with formula_9 the centres of the circular orbits – around which the corresponding fluid parcel moves with constant speed formula_10 Further formula_11 is the wavenumber (and formula_12 the wavelength), while formula_13 is the phase speed with which the wave propagates in the formula_6-direction. The phase speed satisfies the dispersion relation:
which is independent of the wave nonlinearity (i.e. does not depend on the wave height formula_16), and this phase speed formula_13 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 formula_18, where formula_19 is a (nonpositive) constant. For formula_20 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.
The wave height of the trochoidal wave is formula_21 The wave is periodic in the formula_6-direction, with wavelength formula_23 and also periodic in time with period formula_24
The vorticity formula_25 under the trochoidal wave is:
varying with Lagrangian elevation formula_27 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 for the simulation of ocean waves. 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: the horizontal coordinates are denoted as formula_6 and formula_29, and the vertical coordinate is formula_7. The mean level of the free surface is at formula_31 and the positive formula_7-direction is upward, opposing the Earth's gravity of strength formula_33 The free surface is described parametrically as a function of the parameters formula_34 and formula_35 as well as of time formula_36 The parameters are connected to the mean-surface points formula_37 around which the fluid parcels at the wavy surface orbit. The free surface is specified through formula_38 formula_39 and formula_40 with:
where formula_42 is the hyperbolic tangent function, formula_43 is the number of wave components considered, formula_44 is the amplitude of component formula_45 and formula_46 its phase. Further formula_47 is its wavenumber and formula_48 its angular frequency. The latter two, formula_49 and formula_50 can not be chosen independently but are related through the dispersion relation:
with formula_52 the mean water depth. In deep water (formula_53) the hyperbolic tangent goes to one: formula_54 The components formula_55 and formula_56 of the horizontal wavenumber vector formula_57 determine the wave propagation direction of component formula_58
The choice of the various parameters formula_59 and formula_46 for formula_61 and a certain mean depth formula_52 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 formula_63-space. Thereafter, the amplitudes formula_44 and phases formula_46 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 formula_48 such that formula_67 for formula_68
In rendering, also the normal vector formula_69 to the surface is often needed. These can be computed using the cross product (formula_70) as:
The unit normal vector then is formula_72 with formula_73 the norm of formula_74