The Whitham Equation with Surface Tension
Abstract
The viability of the Whitham equation as a nonlocal model for capillarygravity waves at the surface of an inviscid incompressible fluid is under study. A nonlocal Hamiltonian system of model equations is derived using the Hamiltonian structure of the free surface water wave problem and the DirichletNeumann operator. The system features gravitational and capillary effects, and when restricted to oneway propagation, the system reduces to the capillary Whitham equation. It is shown numerically that in various scaling regimes the Whitham equation gives a more accurate approximation of the freesurface problem for the Euler system than other models like the KdV, and Kawahara equation. In the case of relatively strong capillarity considered here, the KdV and Kawahara equations outperform the Whitham equation with surface tension only for very long waves with negative polarity.
 Publication:

arXiv eprints
 Pub Date:
 February 2020
 arXiv:
 arXiv:2002.09946
 Bibcode:
 2020arXiv200209946D
 Keywords:

 Physics  Fluid Dynamics;
 Mathematics  Analysis of PDEs;
 Mathematics  Numerical Analysis;
 Physics  Atmospheric and Oceanic Physics;
 Physics  Computational Physics
 EPrint:
 19 pages, 5 figures, 1 table, 36 references. Other author's papers can be downloaded at http://www.denysdutykh.com/. arXiv admin note: text overlap with arXiv:1410.8299