Abstract
The object of this article is the comparison of numerical solutions of the so-called Whitham equation to numerical approximations of solutions of the full Euler free-surface water-wave problem. The Whitham equation ηt+32c0h0ηηx+Kh0∗ηx=0 was proposed by Whitham (1967) as an alternative to the KdV equation for the description of wave motion at the surface of a perfect fluid by simplified evolution equations, but the accuracy of this equation as a water wave model has not been investigated to date.In order to understand whether the Whitham equation is a viable water wave model, numerical approximations of periodic solutions of the KdV and Whitham equation are compared to numerical solutions of the surface water wave problem given by the full Euler equations with free surface boundary conditions, computed by a novel Spectral Element Method technique. The bifurcation curves for these three models are compared in the phase velocity–waveheight parameter space, and wave profiles are compared for different wavelengths and waveheights. It is found that for small wavelengths, the steady Whitham waves compare more favorably to the Euler waves than the KdV waves. For larger wavelengths, the KdV waves appear to be a better approximation of the Euler waves.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.