The Whitham Equation as a model for surface water waves

Daulet Moldabayev, Henrik Kalisch, Denys Dutykh

Research output: Contribution to journalArticlepeer-review

60 Scopus citations

Abstract

The Whitham equation was proposed as an alternate model equation for the simplified description of uni-directional wave motion at the surface of an inviscid fluid. As the Whitham equation incorporates the full linear dispersion relation of the water wave problem, it is thought to provide a more faithful description of shorter waves of small amplitude than traditional long wave models such as the KdV equation. In this work, we identify a scaling regime in which the Whitham equation can be derived from the Hamiltonian theory of surface water waves. A Hamiltonian system of Whitham type allowing for two-way wave propagation is also derived. The Whitham equation is integrated numerically, and it is shown that the equation gives a close approximation of inviscid free surface dynamics as described by the Euler equations. The performance of the Whitham equation as a model for free surface dynamics is also compared to different free surface models: the KdV equation, the BBM equation, and the Padé (2,2) model. It is found that in a wide parameter range of amplitudes and wavelengths, the Whitham equation performs on par with or better than the three considered models.

Original languageBritish English
Pages (from-to)99-107
Number of pages9
JournalPhysica D: Nonlinear Phenomena
Volume309
DOIs
StatePublished - 19 Aug 2015

Keywords

  • Hamiltonian models
  • Nonlinear dispersive equations
  • Nonlocal equations
  • Solitary waves
  • Surface waves

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