Cnoidal wave
（重定向自Cnoidal）
![US Army bombers flying over near-periodic swell in shallow water, close to the Panama coast (1933). The sharp crests and very flat troughs are characteristic for cnoidal waves.](/Images/godic/202501/05/Periodic_waves_in_shallow_water0940.jpg")
![Crossing swells, consisting of near-cnoidal wave trains. Photo taken from Phares des Baleines (Whale Lighthouse) at the western point of Île de Ré (Isle of Rhé), France, in the Atlantic Ocean.](/Images/godic/202501/05/Ile_de_ré0940.JPG")
![Validity of several theories for periodic water waves, according to Le Méhauté (1976).[2] The light-blue area gives the range of validity of cnoidal wave theory; light-yellow for Airy wave theory; and the dashed blue lines demarcate between the required order in Stokes' wave theory. The light-gray shading gives the range extension by numerical approximations using fifth-order stream-function theory, for high waves (H > ¼ Hbreaking).](/Images/godic/202501/05/Water_wave_theories.svg0940.png")
![Cnoidal wave profiles for three values of the elliptic parameter m.
blue : m = 0,
red : m = 0.9 and
black : m = 0.99999.](/Images/godic/202501/05/Cnoidal_wave_profiles.svg0940.png")
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.