Viscous Effects on Small-Amplitude Surface Waves
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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.