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:
X ( a , b , t ) = a + e k b k sin ( k ( a + c t ) ) , Y ( a , b , t ) = b − e k b k cos ( k ( a + c t ) ) , {\displaystyle {\begin{aligned}X(a,b,t)&=a+{\frac {e^{kb}}{k}}\sin \left(k(a+ct)\right),\\Y(a,b,t)&=b-{\frac {e^{kb}}{k}}\cos \left(k(a+ct)\right),\end{aligned}}}
where x = X ( a , b , t ) {\displaystyle x=X(a,b,t)} and y = Y ( a , b , t ) {\displaystyle y=Y(a,b,t)} are the positions of the fluid parcels in the ( x , y ) {\displaystyle (x,y)} plane at time t {\displaystyle t} , with x {\displaystyle x} the horizontal coordinate and y {\displaystyle y} the vertical coordinate (positive upward, in the direction opposing gravity). The Lagrangian coordinates ( a , b ) {\displaystyle (a,b)} label the fluid parcels, with ( x , y ) = ( a , b ) {\displaystyle (x,y)=(a,b)} the centres of the circular orbits – around which the corresponding fluid parcel moves with constant speed c exp ( k b ) . {\displaystyle c\,\exp(kb).} Further k = 2 π / λ {\textstyle k=2\pi /\lambda } is the wavenumber (and λ {\displaystyle \lambda } the wavelength), while c {\displaystyle c} is the phase speed with which the wave propagates in the x {\displaystyle x} -direction. The phase speed satisfies the dispersion relation:
c 2 = g k , {\displaystyle c^{2}={\frac {g}{k}},}
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