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.
Van der Waals-Zeeman Institute, Institute of Physics, University of Amsterdam（阿姆斯特丹大学范德瓦尔斯-泽曼研究所物理学院）；Department of Applied Mathematics and Theoretical Physics, University of Cambridge（剑桥大学应用数学与理论物理系）
Perturbation of Traveling Boussinesq Solitons by Periodic Bathymetry
Embry-Riddle Aeronautical University（埃姆布里-里德尔航空大学）；Naval Research Laboratory（海军研究实验室）；National Institute of Physics and Nuclear Engineering（物理与核工程国家研究所）
