The Whitham Equation as a Model for Surface Water Waves
arXiv:1410.8299 · doi:10.1016/j.physd.2015.07.010
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. 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 two standard free surface models: the KdV and the BBM equation. It is found that in a wide parameter range of amplitudes and wavelengths, the Whitham equation performs on par with or better than both the KdV and BBM equations.
14 pages, 4 figures
References in corpus (2)
Cited by in corpus (22)
- On Whitham's conjecture of a highest cusped wave for a nonlocal dispersive equation
- Symmetry and decay of traveling wave solutions to the Whitham equation
- Dispersive shallow water wave modelling. Part I: Model derivation on a globally flat space
- The Whitham Equation with Surface Tension
- A comparative study of bi-directional Whitham systems
- Expansion shock waves in regularised shallow water theory
- Numerical Bifurcation for the Capillary Whitham Equation
- Solitary wave solutions to a class of Whitham-Boussinesq systems
- Solitary wave solutions to a class of modified Green-Naghdi systems
- Numerical Study of Nonlinear Dispersive Wave Models with SpecTraVVave
- Symmetric solutions of evolutionary partial differential equations
- On the Stability of Solitary Wave Solutions for a Generalized Fractional Benjamin-Bona-Mahony Equation
- Non-uniform dependence on initial data for the Whitham equation
- Water wave problem with inclined walls
- Numerical bifurcation and stability for the capillary-gravity Whitham equation
- Linear Whitham-Boussinesq modes in channels of constant cross-section
- Whitham modulation theory for generalized Whitham equations and a general criterion for modulational instability
- Rigorous derivation of the Whitham equations from the water waves equations in the shallow water regime
- A Regularized Shallow Water System
- Stability of Periodic, Traveling-Wave Solutions to the Capillary-Whitham Equation
- Operator splitting for dispersion-generalized Benjamin-Ono equations
- Linear guided modes and Whitham-Boussinesq model for variable topogra