Viscous dissipation in steep capillary-gravity waves

Some simple but exact general expressions are derived for the viscous stresses required at the surface of irrotational capillary-gravity waves of periodic or solitary type on deep water in order to maintain them in steady motion. These expressions are applied to nonlinear capillary waves, and to capillary-gravity waves of solitary type on deep water. In the case of pure capillary waves some algebraic expressions are found for the work done by the surface stresses, from which it is possible to infer the viscous rate of decay of free, nonlinear capillary waves. Similar calculations are carried out for capillary-gravity waves of solitary type on deep water. It is shown that the limiting rate of decay of a solitary wave at low amplitudes is just twice that for linear, periodic waves. This is due to the spreading out of the wave envelope at low wave steepnesses. At large wave steepnesses the dissipation increases by an order of magnitude, owing to the sharply increased curvature in the wave troughs. The calculated rates of decay are in agreement with recent observations.