  This work presents a detailed study of the dispersion of capillary waves with
small amplitude in viscous fluids using an analytically derived solution to the
initial value problem of a small-amplitude capillary wave as well as direct
numerical simulation. A rational parametrization for the dispersion of
capillary waves in the underdamped regime is proposed, including predictions
for the wavenumber of critical damping based on a harmonic oscillator model.
The scaling resulting from this parametrization leads to a self-similar
solution of the frequency dispersion of capillary waves that covers the entire
underdamped regime, which allows an accurate evaluation of the frequency at a
given wavenumber, irrespective of the fluid properties. This similarity also
reveals characteristic features of capillary waves, for instance that critical
damping occurs when the characteristic timescales of dispersive and dissipative
mechanisms are balanced. In addition, the presented results suggest that the
widely adopted hydrodynamic theory for damped capillary waves does not
accurately predict the dispersion when viscous damping is significant and a new
definition of the damping rate, which provides consistent accuracy in the
underdamped regime, is presented.
