downloading Waves Complete 14 (17.07.23)

The light-gray shading gives the range extension by numerical approximations using fifth-order stream-function theory, for high waves ( H > ¼ H breaking). 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. Background Korteweg–de Vries, and Benjamin–Bona–Mahony equations Validity of several theories for periodic water waves, according to Le Méhauté (1976). Photo taken from Phares des Baleines (Whale Lighthouse) at the western point of Île de Ré (Isle of Rhé), France, in the Atlantic Ocean. Crossing swells, consisting of near-cnoidal wave trains. For the shown case, the elliptic parameter is m = 0.9. A cnoidal wave, characterised by sharper crests and flatter troughs than in a sine wave. 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.Ĭnoidal wave solutions can appear in other applications than surface gravity waves as well, for instance to describe ion acoustic waves in plasma physics. 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. In the limit of infinite wavelength, the cnoidal wave becomes a solitary wave. 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. They are used to describe surface gravity waves of fairly long wavelength, as compared to the water depth. These solutions are in terms of the Jacobi elliptic function cn, which is why they are coined cnoidal waves. In fluid dynamics, a cnoidal wave is a nonlinear and exact periodic wave solution of the Korteweg–de Vries equation. The sharp crests and very flat troughs are characteristic for cnoidal waves. Nonlinear and exact periodic wave solution of the Korteweg–de Vries equation US Army bombers flying over near-periodic swell in shallow water, close to the Panama coast (1933).

Downloading Waves Complete 14 (17.07.23)