Resonance damping of gravity–capillary waves on water covered with a visco-elastic film of finite thickness: A reappraisal

Abstract

A new approach to the problem of damping of gravity–capillary waves (GCW) on water covered with a layer of viscous liquid (a film) of finite thickness with two elastic boundaries is developed. It is shown that the rotational component of GCW can be described formally as a “forced” longitudinal or Marangoni wave (MW), and the potential component of GCW plays a role of the “external force.” The resonance-like excitation of the forced MW is demonstrated when the GCW and MW frequencies and wave numbers are approximately close to each other. For a film that is thinner than the viscous boundary layers in film, a single forced MW exists that is located within the boundary layer beneath the water surface. For a thick film, the forced MW is characterized by the existence of two spatially separated MW modes: one is localized in the boundary layer below the upper, air–film interface and another within the boundary layers in the vicinity of the water–film interface. Then, at different elasticities of the interfaces, a double peak dependence of the GCW damping coefficient on wave number can occur due to the resonance with the two forced MW modes. The dependence of the damping coefficient on film thickness is characterized by a strong maximum appearing when the film and boundary layer thickness values are comparable to each other. The developed theory is consistent with existing numerical studies and experiment.