Quasi-periodic traveling gravity-capillary waves

Abstract

We present a numerical study of spatially quasi-periodic traveling waves on the surface of an ideal fluid of infinite depth. This is a generalization of the classic Wilton ripple problem to the case when the ratio of wave numbers satisfying the dispersion relation is irrational. We propose a conformal mapping formulation of the water wave equations that employs a quasi-periodic variant of the Hilbert transform to compute the normal velocity of the fluid from its velocity potential on the free surface. We develop a Fourier pseudo-spectral discretization of the traveling water wave equations in which one-dimensional quasi-periodic functions are represented by two-dimensional periodic functions on the torus. This leads to an overdetermined nonlinear least squares problem that we solve using a variant of the Levenberg-Marquardt method. We investigate various properties of quasi-periodic traveling waves, including Fourier resonances, time evolution in conformal space on the torus, asymmetric wave crests, capillary wave patterns that change from one gravity wave trough to the next without repeating, and the dependence of wave speed and surface tension on the amplitude parameters that describe a two-parameter family of waves.

arXiv comment35 pages, 9 figures; added a section on larger-amplitude waves, expanded introduction and added references

References in corpus (12)

• The Fermi-Pasta-Ulam problem: 50 years of progress
• Time quasi-periodic gravity water waves in finite depth
• Interacting nonlinear wave envelopes and rogue wave formation in deep water
• Traveling quasi-periodic water waves with constant vorticity
• Branch cuts of Stokes wave on deep water. Part I: Numerical solution and Padé approximation
• Quasiperiodic extended Bloch states in a surface wave experiment
• The instability of Wilton ripples
• Spatially quasi-periodic water waves of infinite depth
• Overdetermined Shooting Methods for Computing Standing Water Waves with Spectral Accuracy
• Non-Canonical Hamiltonian Structure and Poisson Bracket for 2D Hydrodynamics with Free Surface
• On the dynamics of a free surface of an ideal fluid in a bounded domain in the presence of surface tension
• Quasi-periodic traveling waves on an infinitely deep fluid under gravity

Cited by in corpus (6)

• Traveling quasi-periodic water waves with constant vorticity
• Spatially quasi-periodic water waves of infinite depth
• Spatially quasi-periodic bifurcations from periodic traveling water waves and a method for detecting bifurcations using signed singular values
• Exponential asymptotics for steady parasitic capillary ripples on steep gravity waves
• Spatially quasi-periodic water waves of finite depth
• Three-dimensional gravity-capillary standing waves: computation, resonance and instability