Viscosity effect on the surface and internal waves in a two-layer fluid with a free surface

Analytical expressions for the decay rates of surface (on the free boundary) and internal (on the interface) waves in a two-layer fluid are derived in the small kinematic viscosity asymptotics. The internal wave decay rates are found to be approximately two orders smaller than those of the surface waves, which is attributable to the difference in their frequencies, all other factors being the same. The decay rates depend on the kinematic viscosities of both media but, when the viscosity of one of the media is small, they are chiefly determined by the viscosity of the other medium, which allows one to use the model of the contact between viscous and inviscid media. The frequencies of the capillary surface and internal waves in the capillary counterpart of the "dead water" effect considerably differ in value which has an effect on the large difference in the decay rates.