  In this paper we study a finite-depth layer of viscous incompressible fluid
in dimension $n \ge 2$, modeled by the Navier-Stokes equations. The fluid is
assumed to be bounded below by a flat rigid surface and above by a free, moving
interface. A uniform gravitational field acts perpendicularly to the flat
surface, and we consider the cases with and without surface tension acting on
the free interface. In addition to these gravity-capillary effects, we allow
for a second force field in the bulk and an external stress tensor on the free
interface, both of which are posited to be in traveling wave form, i.e.
time-independent when viewed in a coordinate system moving at a constant
velocity parallel to the rigid lower boundary. We prove that, with surface
tension in dimension $n \ge 2$ and without surface tension in dimension $n=2$,
for every nontrivial traveling velocity there exists a nonempty open set of
force and stress data that give rise to traveling wave solutions. While the
existence of inviscid traveling waves is well known, to the best of our
knowledge this is the first construction of viscous traveling wave solutions.
  Our proof involves a number of novel analytic ingredients, including: the
study of an over-determined Stokes problem and its under-determined adjoint, a
delicate asymptotic development of the symbol for a normal-stress to
normal-Dirichlet map defined via the Stokes operator, a new scale of
specialized anisotropic Sobolev spaces, and the study of a pseudodifferential
operator that synthesizes the various operators acting on the free surface
functions.
