Viscous effects on small-amplitude surface waves

Abstract

The transient behavior of small-amplitude standing waves on the plane surface of an infinitely deep viscous liquid is investigated. An integro-differential equation of motion for the free surface is derived and solved exactly. The small-time behavior of the solution agrees with that computed in the approximation of irrotational flow, and the large-time one with the results for the discrete spectrum obtained by means of the standard normal-mode analysis. In between these two asymptotic regimes, however, the exact solution is significantly different from either approximation, except when the effect of viscosity is very small.