Damping of nearly inviscid water waves

The complete spectrum of decaying small-amplitude water waves is described. Viscosity is shown to be responsible both for the decay of gravity-capillary waves and for the appearance of a class of viscous modes that are omitted in the standard potential formulation. For sufficiently small viscosity [as measured by the parameter C≡ν(gh³)^-1/21, where ν is the kinematic viscosity, g the acceleration due to gravity, and h the undisturbed depth of the fluid] the viscous modes decay more slowly than the gravity-capillary ones and must be included in weakly nonlinear theories. The analysis indicates that for realistic values of C second-order corrections to the decay rate of gravity-capillary waves are important and suggests a straightforward resolution of existing discrepancies between experimentally measured and theoretically calculated damping rates.