The Whitham Equation as a Model for Surface Water Waves

TL;DR

The paper demonstrates that the Whitham equation effectively models surface water wave dynamics, capturing short wave behavior more accurately than traditional models like KdV and BBM, through derivation, numerical integration, and comparison.

Contribution

It derives the Whitham equation from Hamiltonian theory and compares its performance to standard models, showing its superior accuracy in certain regimes.

Findings

Whitham equation closely approximates Euler water wave dynamics.

Performs better than KdV and BBM in various amplitude and wavelength ranges.

Validated through numerical simulations against inviscid free surface data.

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.