Abstract
Abstract
Highlights
Travelling water waves have long played a central role in the field of fluid mechanics
To date, spatially quasi-periodic water waves have only been investigated through weakly nonlinear models (Zakharov 1968; Bridges & Dias 1996; Janssen 2003; Ablowitz & Horikis 2015) or through a Fourier–Bloch stability analysis in which the eigenfunctions of the linearization about a Stokes wave have a different period than the Stokes wave (Longuet-Higgins 1978; Deconinck & Oliveras 2011; Trichtchenko, Deconinck & Wilkening 2016)
Generalizing Wilton’s work to the case in which the linear dispersion relation supports two irrationally related wavenumbers that travel at the same speed, Bridges & Dias (1996) used a spatial Hamiltonian structure to construct weakly nonlinear approximations of spatially quasi-periodic travelling gravity–capillary waves for two special cases: deep water and shallow water
Summary
Travelling water waves have long played a central role in the field of fluid mechanics. No methods currently exist to study the long-time evolution of unstable subharmonic perturbations under the full water wave equations nor to compute quasi-periodic travelling waves beyond the weakly nonlinear regime. Generalizing Wilton’s work to the case in which the linear dispersion relation supports two irrationally related wavenumbers that travel at the same speed, Bridges & Dias (1996) used a spatial Hamiltonian structure to construct weakly nonlinear approximations of spatially quasi-periodic travelling gravity–capillary waves for two special cases: deep water and shallow water The existence of such waves in the fully nonlinear setting is still an open problem. In Appendix A, we study the dynamics of quasi-periodic travelling waves and show that the waves maintain a permanent form but generally travel at a non-uniform speed in conformal space in order to travel at constant speed in physical space
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
