Instability of near-extreme solutions to the Whitham equation

The Whitham equation is a model for the evolution of small-amplitude, unidirectional waves of all wavelengths in shallow water. It has been shown to accurately model the evolution of waves in laboratory experiments. We compute 2 pi-periodic traveling wave solutions of the Whitham equation and numerically study their stability with a focus on solutions with large steepness.We show that the Hamiltonian oscillates at least twice as a function of wave steepness when the solutions are sufficiently steep. We show that a superharmonic instability is created at each extremum of the Hamiltonian and that between each extremum the stability spectra undergo similar bifurcations. Finally, we compare these results with those from the Euler equations.

Automated summary

This study investigates the traveling wave solutions and stability of the Whitham equation, revealing that the Hamiltonian oscillates at least twice when the solutions are sufficiently steep and a superharmonic instability is created at each extremum of the Hamiltonian. The stability spectra also undergo similar bifurcations between each extremum. Furthermore, a comparison with the results from the Euler equations is presented.