On the propagation of third sound in 4He films

Abstract

A detailed theory on the propagation of third sound along planar 4He films is presented. The treatment follows the lines set out by Bergman, later also pursued by van Beelen and Bannink, in taking the accompanying waves in vapour and substrate fully into account but omitting the non-linear regime near the onset points of superfluidity of the films. The theory results in a basic set of five linear homogeneous equations for five unknown amplitudes, three of the film variables and two of the vapour. The set is solved numerically for given frequency and substrate properties, yielding values for the speed, attenuation and amplitude ratios of the third sound waves, as functions of film thickness and temperature. The theory is applied to the parallel-plate geometry, in which case each equation can be split into two, yielding two independent sets, one for the symmetric and one for the antisymmetric mode of propagation. The role of the spacing between the plates is illustrated numerically in some examples. It appears that for small spacings the sound wave in the vapour plays an important role for the third-sound propagation, a feature that was not recognized by Bergman. It is also demonstrated that for a narrow spacing, increasing the temperature causes a cross-over from the so-called modified Putterman limit to the vapour dominated limit, which might explain the observed behaviour of the amplitude ratio of film thickness and temperature, recently reported by Laheurte et al. It is finally mentioned that the described treatment could be extended to other geometries by introducing the appropriate geometry-dependent coefficients into the basic set of linear equations.