A numerical study of the Whitham equation as a model for steady surface water waves

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.