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Viscosity effect on capillary waves at liquid interfaces

We discuss the viscosity damping effect on capillary waves in a binary-liquid system, using the linearized Navier-Stokes equation. The damping correction for the dispersion relation depends on the wave vector k as well as the interfacial tension. The calculated k-dependence of damping is characterized by a critical capillary-wave-number value , which separates the regions of weak and strong damping. The surface and interfacial roughnesses of a binary liquid system with large liquid depths are calculated and compared to experiments. Although the analysis has been restricted to the classical, macroscopic level, we obtain a noticeable modification from earlier hydrodynamic results for capillary wave damping and liquid-liquid interfacial roughness.