Effect of Normal-Fluid Motion on Third Sound in Liquid-Helium Films

Some theoretical consequences are derived of a proposed viscous force on the normal-fluid component of the He II film. In contrast with earlier calculations, in which the normal fluid was assumed to be immobile, this new assumption allows large energy attenuations, such as have recently been observed in third sound in He II films and in a wave mode of He II partially clamped in narrow channels. Atkins's three equations describing third sound are modified to take account of energy and entropy transfer associated with normalfluid motion, and wave modes are obtained which simultaneously satisfy these equations and the normalfluid equation of motion with a viscous force $-R{v}_n$. The four wave variables are: $\zeta$ and $T^{\prime }$, respectively, the local fluctuations in film thickness and temperature, and $v_s$ and $v_n$, the superfluid and normal-fluid velocities. Attenuation and velocity of the wave mode corresponding to third sound are calculated at temperatures from 1.2°K through $T_{\lambda }$ for all values of the dimensionless viscosity coefficient ($\frac{R}{\omega \rho }$). The maximum calculated attenuation varies from 0.47 ${\mathrm{cm}}^{-1\mathrm{}}$ at 1.2°K to 17.3 ${\mathrm{cm}}^{-1\mathrm{}}$ at 2.1°K; at the lower temperature the observed attenuation is about 2 ${\mathrm{cm}}^{-1\mathrm{}}$. However, the anomalously rapid decrease in velocity of third sound observed by Everitt et al. could not be explained in this manner. Evidence for normal-fluid motion in the film is presented, and some feasible experiments for detecting it are described.