  The object of this study is to investigate the effect of viscosity on
propagation of free-surface waves in an incompressible viscous fluid layer of
arbitrary depth. While we provide a more detailed study of properties of linear
surface waves, the description of fully nonlinear waves in terms of KdV-like
equations is discussed. In the linear case, we find that in shallow enough
fluids, no surface waves can propagate. Even in any thicker fluid layers,
propagation of very short and very long waves is forbidden. When wave
propagation is possible, only a single propagating mode exists for any given
horizontal wave number. The numerical results show that there can be two types
of non-propagating modes. One type is always present, and there exist always
infinitely many of such modes at the same parameters. In contrast, there can be
zero, one or two modes belonging to the other type. Another significant feature
is that KdV-like equations describing propagating nonlinear viscous surface
waves may not exist since viscosity gives rise to a new wave number that cannot
be small at the same time as the original one. Nonetheless, we propose a
reasonable nonlinear description in terms of 1+1 variate functions that makes
possible successive approximations.

Fluid Dynamics