Mode damping of layered He-3-He-4 films over a solid substrate

The dispersion relation for coupled oscillations of liquid interfaces in th e layered He-3-He-4 films on solid substrate is calculated by taking into a ccount viscous dissipation and effects of the van der Waals interaction bet ween the liquid phases and the solid substrate and among themselves. We pro vide a detailed analysis of the layer thickness dependence of the mode spec trum within the hydrodynamical approach by solving the linearized Navier-St okes equation. For actual helium viscosities, we find that a mode with weak dissipation, corresponding to flexural displacements of the upper phase, d oes exist at large wave numbers. In the approximation of ideal fluids, we r eproduce previous results obtained from different methods. We find a wide i nterval of wave numbers for weakly damped modes when the layer thickness of a given phase is significantly smaller than the other one. We interpret ou r results for weak and strong dissipation by comparing the viscous penetrat ion depth of the modes with characteristic lengths of the system. We discus s the implications of our results for experiments to measure mode frequency and transport properties of surface electrons over the layered film.