  We prove that traveling waves in viscous compressible liquids are a generic
phenomenon. The setting for our result is a horizontally infinite, finite depth
layer of compressible, barotropic, viscous fluid, modeled by the free boundary
compressible Navier-Stokes equations in dimension $n \ge 2$. The bottom
boundary of the fluid is flat and rigid, while the top is a moving free
boundary. A constant gravitational field acts normal to the flat bottom. We
allow external forces to act in the fluid's bulk and external stresses to act
on its free surface. These are posited to be in traveling wave form, i.e.
time-independent when viewed in a coordinate system moving at a constant,
nontrivial velocity parallel to the lower rigid boundary.
  In the absence of such external sources of stress and force, the fluid system
reverts to equilibrium, which corresponds to a flat, quiescent fluid layer with
vertically stratified density. In contrast, when such sources of stress or
force are present, the system admits traveling wave solutions. We establish a
small data well-posedness theory for this problem by proving that for every
nontrivial traveling wave speed there exists a nonempty open set of stress and
forcing data that give rise to unique traveling wave solutions, and that these
solutions depend continuously on the data and the wave speed. When $n \ge 3$ we
prove this with surface tension accounted for at the free boundary, while in
the case $n=2$ we prove this with or without surface tension. To the best of
our knowledge, this result constitutes the first general construction of
traveling wave solutions to any free boundary compressible fluid equations.
