Influid dynamics, atrochoidal wave orGerstner wave is an exact solution of theEuler equations forperiodicsurface gravity waves. It describes aprogressive wave of permanent form on the surface of anincompressible fluid of infinite depth. The free surface of this wave solution is an inverted (upside-down)trochoid – with sharpercrests and flat troughs. This wave solution was discovered byGerstner in 1802, and rediscovered independently byRankine in 1863.

The flow field associated with the trochoidal wave is notirrotational: it hasvorticity. The vorticity is of such a specific strength and vertical distribution that the trajectories of thefluid parcels are closed circles. This is in contrast with the usual experimental observation ofStokes drift associated with the wave motion. Also thephase speed is independent of the trochoidal wave'samplitude, unlike other nonlinear wave-theories (like those of theStokes wave andcnoidal 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.

Incomputer graphics, therendering of realistic-lookingocean waves can be done by use of so-calledGerstner waves. This is a multi-component and multi-directional extension of the traditional Gerstner wave, often usingfast Fourier transforms to make (real-time)animation feasible.^[1]

Using aLagrangian specification of the flow field, the motion of fluid parcels is – for aperiodic wave on the surface of a fluid layer of infinite depth:^[2]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 constantspeed${\displaystyle c\exp (kb).}$ Further$k=2\pi /\lambda$ is thewavenumber (and${\displaystyle \lambda }$ thewavelength), while${\displaystyle c}$ is the phase speed with which the wave propagates in the${\displaystyle x}$-direction. The phase speed satisfies thedispersion 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 forAiry'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 acusp-shaped crest. Note that the highest (irrotational)Stokes wave has acrest angle of 120°, instead of the 0° for the rotational trochoidal wave.^[3]

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

Thevorticity${\displaystyle \varpi }$ under the trochoidal wave is:^[2]$${\displaystyle \varpi (a,b,t)=-\frac{2kc{e}^{2kb}}{1-e^{2kb}},}$$varying with Lagrangian elevation${\displaystyle b}$ and diminishing rapidly with depth below the free surface.

A multi-component and multi-directional extension of theLagrangian description of the free-surface motion – as used in Gerstner's trochoidal wave – is used incomputer graphics for the simulation of ocean waves.^[1] For the classical Gerstner wave the fluid motion exactly satisfies thenonlinear,incompressible andinviscid 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 bypotential flow). This description of the ocean can be programmed very efficiently by use of thefast 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): sharpercrests and flattertroughs.

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${\displaystyle y}$. Themean level of the free surface is at${\displaystyle y=0}$ and the positive${\displaystyle y}$-direction is upward, opposing theEarth's gravity of strength${\displaystyle g.}$ The free surface is describedparametrically 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 thefluid parcels at the wavy surface orbit. The free surface is specified through${\displaystyle x=\xi (\alpha ,\beta ,t),}$${\displaystyle y=\zeta (\alpha ,\beta ,t)}$ and${\displaystyle z=\eta (\alpha ,\beta ,t)}$ with:where${\displaystyle \tanh }$ is thehyperbolic tangent function,${\displaystyle M}$ is the number of wave components considered,${\displaystyle a_m}$ is theamplitude of component${\displaystyle m=1\dots M}$ and${\displaystyle {\varphi }_m}$ itsphase. Further$k_m=\sqrt{k_{x,m}^2+k_{z,m}^2}$ is itswavenumber and${\displaystyle {\omega }_m}$ itsangular frequency. The latter two,${\displaystyle k_m}$ and${\displaystyle {\omega }_m,}$ can not be chosen independently but are related through thedispersion relation:$${\displaystyle {\omega }_m^2=g{k}_m\tanh \left(k_{m}h\right),}$$with${\displaystyle h}$ the mean water depth. In deep water (${\displaystyle h\to \mathrm{\infty }}$) 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 wavenumbervector${\displaystyle {\mathit{k}}_m}$ 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${\displaystyle {\varphi }_m}$ 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${\displaystyle {\varphi }_m}$ are chosen randomly in accord with thevariance-density spectrum of a certain desiredsea state. Finally, by FFT, the ocean surface can be constructed in such a way that it isperiodic both in space and time, enablingtiling – creating periodicity in time by slightly shifting the frequencies${\displaystyle {\omega }_m}$ such that${\displaystyle {\omega }_m=m\mathrm{\Delta }\omega }$ for${\displaystyle m=1,\dots ,M.}$

In rendering, also thenormal vector${\displaystyle \mathit{n}}$ to the surface is often needed. These can be computed using thecross product (${\displaystyle \times }$) as:$${\displaystyle \mathit{n}=\frac{\mathrm{\partial }\mathit{s}}{\mathrm{\partial }\alpha }\times \frac{\mathrm{\partial }\mathit{s}}{\mathrm{\partial }\beta }\text{with}\mathit{s}(\alpha ,\beta ,t)=\left(\begin{array}{c}\xi (\alpha ,\beta ,t)\\ \zeta (\alpha ,\beta ,t)\\ \eta (\alpha ,\beta ,t)\end{array}\right).}$$

Theunit normal vector then is${\displaystyle {\mathit{e}}_n=\mathit{n}/\Vert \mathit{n}\Vert ,}$ with${\displaystyle \Vert \mathit{n}\Vert }$ thenorm of${\displaystyle \mathit{n}.}$

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