  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.
