  Gravito-capillary waves at free-surfaces are ubiquitous in several natural
and industrial processes involving quiescent liquid pools bounded by
cylindrical walls. These waves emanate from the relaxation of initial interface
distortions, which often take the form of a cavity (depression) centred on the
symmetry axis of the container. These surface waves reflect from the container
walls leading to a radially inward propagating wave-train converging
(focussing) onto the symmetry axis. Under the inviscid approximation and for
sufficiently shallow cavities, the relaxation is well-described by the
linearised potential-flow equations. Naturally, adding viscosity to such a
system introduces viscous dissipation that enervates energy and dampens the
oscillations at the symmetry axis.
  However, for viscous liquids and deeper cavities, these equations are
qualitatively inaccurate. In this study, we elucidate a modal approach to study
the initial-value problem for concentric gravito-capillary waves generated on a
free-surface for inviscid as well as viscous liquids. For a sufficiently deep
cavity, the inward focusing of waves results in large interfacial oscillations
at the axis, necessitating a second-order nonlinear theory. We demonstrate that
this theory effectively models the interfacial behavior and highlights the
crucial role of nonlinearity near the symmetry axis. Contrary to expectations,
the addition of slight viscosity further intensifies the oscillations at the
symmetry axis. This finding underscores the limitations of the potential flow
model and suggests avenues for more accurate modelling of such complex
free-surface flows.
