page_title: Trochoidal wave
hierarchical_section_title: Trochoidal wave
context_page_description: 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 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 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.
context_section_description: 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.
caption: 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 
    
      
        
          H
        
      
      {\displaystyle H}
    
  , the wavelength as 
    
      
        
          λ
        
      
      {\displaystyle \lambda }
    
   and the phase speed as 
    
      
        
          c
          .
        
      
      {\displaystyle c.}