2023/08/12 by John D. Carter, Carter, John D.
Earth and Planetary Sciences · Mathematics · #Amplitude #Classical mechanics #Euler equations #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Hamiltonian (control theory) #Instability #Mathematical Physics (math-ph) #Mathematical analysis #Mathematical optimization #Mathematics #Mechanics #Ocean Waves and Remote Sensing #Oceanographic and Atmospheric Processes #Optics #Physics #Subharmonic function #Traveling wave #Tropical and Extratropical Cyclones Research #Wavelength
paper · pdf · doi:10.48550/arxiv.2308.06583
openalex publication_date 2023/08/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Whitham equation is a model for the evolution of small-amplitude, unidirectional waves of all wavelengths on shallow water. It has been shown to accurately model the evolution of waves in laboratory experiments. We compute 2π-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 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.