page_title: Cnoidal wave
section_title: Direct derivation from the full inviscid-flow equations
hierarchical_section_title: Cnoidal wave / Direct derivation from the full inviscid-flow equations
caption_attribution_description: English: Description at USGS website:"Undular bore and whelps near the mouth of Araguari River, Brazil. View is oblique toward mouth from airplane at approximately 100 ft. altitude. Portion published as Figure 5 in U. S. Geological Survey, Circular 1022, 1988."Note: This undular bore (locally known as pororoca) is on the Araguari River in the Amapá state in north-eastern Brazil, see w:Tidal bore#South America. The bore propagates from left to right, against the stream of the river current.
context_page_description: In fluid dynamics, a cnoidal wave is a nonlinear and exact periodic wave solution of the Korteweg–de Vries equation. These solutions are in terms of the Jacobi elliptic function cn, which is why they are coined cnoidal waves. They are used to describe surface gravity waves of fairly long wavelength, as compared to the water depth.
  The cnoidal wave solutions were derived by Korteweg and de Vries, in their 1895 paper in which they also propose their dispersive long-wave equation, now known as the Korteweg–de Vries equation. In the limit of infinite wavelength, the cnoidal wave becomes a solitary wave.
  The Benjamin–Bona–Mahony equation has improved short-wavelength behaviour, as compared to the Korteweg–de Vries equation, and is another uni-directional wave equation with cnoidal wave solutions. Further, since the Korteweg–de Vries equation is an approximation to the Boussinesq equations for the case of one-way wave propagation, cnoidal waves are approximate solutions to the Boussinesq equations.
  Cnoidal wave solutions can appear in other applications than surface gravity waves as well, for instance to describe ion acoustic waves in plasma physics.
context_section_description: Cnoidal waves can be derived directly from the inviscid, irrotational and incompressible flow equations, and expressed in terms of three invariants of the flow, as shown by Benjamin & Lighthill (1954) in their research on undular bores. In a frame of reference moving with the phase speed, in which reference frame the flow becomes a steady flow, the cnoidal wave solutions can directly be related to the mass flux, momentum flux and energy head of the flow. Following Benjamin & Lighthill (1954)—using a stream function description of this incompressible flow—the horizontal and vertical components of the flow velocity are the spatial derivatives of the stream function Ψ(ξ,z): +∂zΨ and −∂ξΨ, in the ξ and z direction respectively (ξ = x−ct). The vertical coordinate z is positive in the upward direction, opposite to the direction of the gravitational acceleration, and the zero level of z is at the impermeable lower boundary of the fluid domain. While the free surface is at z = ζ(ξ); note that ζ is the local water depth, related to the surface elevation η(ξ) as ζ = h + η with h the mean water depth.
  In this steady flow, the discharge Q through each vertical cross section is a constant independent of ξ, and because of the horizontal bed also the horizontal momentum flux S, divided by the density ρ, through each vertical cross section is conserved. Further, for this inviscid and irrotational flow, Bernoulli's principle can be applied and has the same Bernoulli constant R everywhere in the flow domain. They are defined as:
  For fairly long waves, assuming the water depth ζ is small compared to the wavelength λ, the following relation is obtained between the water depth ζ(ξ) and the three invariants Q, R and S:
  This nonlinear and first-order ordinary differential equation has cnoidal wave solutions. 
  For very long waves of infinitesimal amplitude on a fluid of depth h and with a uniform flow velocity v, the flow constants are according to the shallow water equations:
        and   
  Equation (E) can be brought into non-dimensional form by use of the discharge Q and gravitational acceleration g, and defining the critical depth hc:
  related to the critical flow demarcation between subcritical flow and supercritical flow (see also Froude number). Consequently, the non-dimensional form of the equation is
  with
           and  
caption: Undular bore and whelps near the mouth of Araguari River in north-eastern Brazil. View is oblique toward mouth from airplane at approximately 100 ft (30 m) altitude.[33]
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