The propagation of surface waves – that is ‘third’ sound – on superfluid helium is considered. The fluid is treated as a continuum, using the two-fluid model of Landau, and incorporating the effects of healing, relaxation, thermal conductivity and Newtonian viscosity. Under the assumptions only of a small amplitude and long wavelength, a linear theory is developed which includes some discussion of the matching to the outer regions of the vapour. This results in a comprehensive propagation speed for linear waves, although a few properties of the flow are left undetermined at this order. A nonlinear theory is then outlined (without dwelling on the details) which leads to the Burgers equation in an appropriate far field, and enables the leading-order theory to be concluded.
Some numerical results, for two temperatures, are presented by first recording the Helmholtz free energy as a polynomial in densities. Owing to the inaccuracies inherent in this procedure, only the equilibrium state can be satisfactorily reproduced, but this is sufficient to predict the onset phenomenon. The propagation speed, as a function of film thickness, is roughly estimated by using the earlier results of Johnson (without healing) suitably doctored to incorporate the new computed superfluid density distributions. The looked-for reduction in the predicted speeds is evident, but the magnitude of this reduction is too large for very thin films. However, it is hoped that the analytical results presented here will prove more effective when a complete and accurate description of the Helmholtz free energy is available.