  A model for ripple formation on liquid surfaces exposed to an external laser
or particle beam and a variable ground is developed. The external incident beam
is hereby mechanically coupled to the liquid surface due to surface roughness.
Starting from the Navier Stokes equation the coupled equations for the velocity
potential and the surface height are derived in a shallow-water approximation
with special attention to viscosity. The resulting equations obey conservation
laws for volume and momentum where characteristic potentials for gravitation
and surface tension are identified analogously to conservative forces. The
approximate solutions are discussed in the context of ripple formation in laser
materials processing involving melting of a surface by a laser beam. Linear
stability analysis provides the formation of a damped wave modified by an
interplay between the external beam, the viscosity, and the surface tension.
The limit of small viscosity leads to damped gravitational and the limit of
high viscosity to capillary waves. The resulting wavelengths are in the order
of the ripples occurring in laser welding experiments hinting to the
involvement of hydrodynamic processes in their origin. By discussing the
response of the system to external periodic excitations with the help of
Floquet multipliers, we show that the ripple formation could be triggered by a
a periodically modulated external beam, e.g. appropriate repetition rates of an
incident laser beam. The weak nonlinear stability analysis provides ranges
where hexagonal or stripe structures can appear. The orientation of stripe
structures and ripples are shown to be dependent on the incident angle of the
laser or particle beam where a minimal angle is reported. Numerical simulations
confirm the findings and allow to describe the influence of variable grounds.
