The superharmonic instability and wave breaking in Whitham equations - CNRS - Centre national de la recherche scientifique Accéder directement au contenu
Article Dans Une Revue Physics of Fluids Année : 2023

The superharmonic instability and wave breaking in Whitham equations

John Carter
Marc Francius
Henrik Kalisch

Résumé

The Whitham equation is a model for the evolution of surface waves on shallow water that combines the unidirectional linear dispersion relation of the Euler equations with a weakly nonlinear approximation based on the Korteweg–De Vries equation. We show that large-amplitude, periodic, traveling-wave solutions to the Whitham equation and its higher-order generalization, the cubic Whitham equation, are unstable with respect to the superharmonic instability (i.e., a perturbation with the same period as the solution). The threshold between superharmonic stability and instability occurs at the maxima of the Hamiltonian and L2-norm. We examine the onset of wave breaking in traveling-wave solutions subject to the modulational and superharmonic instabilities. We present new instability results for the Euler equations in finite depth and compare them with the Whitham results. We show that the Whitham equation more accurately approximates the wave steepness threshold for the superharmonic instability of the Euler equations than does the cubic Whitham equation. However, the cubic Whitham equation more accurately approximates the wave steepness threshold for the modulational instability of the Euler equations than does the Whitham equation.
Fichier principal
Vignette du fichier
2306.11889.pdf (568.6 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-04535282 , version 1 (06-04-2024)

Identifiants

Citer

John Carter, Marc Francius, Christian Kharif, Henrik Kalisch, Malek Abid. The superharmonic instability and wave breaking in Whitham equations. Physics of Fluids, 2023, 35 (10), pp.103609. ⟨10.1063/5.0164084⟩. ⟨hal-04535282⟩
0 Consultations
0 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More