Propagation of second sound near T lambda.

Abstract

Superfluid hydrodynamics for second sound, expanded to first order in \ensuremath{\nabla}${\mathrm{\ensuremath{\rho}}}_{\mathit{s}}$ and including second-sound damping and finite-amplitude effects, are cast into a boundary-value-problem format, suitable for calculating the resonant frequency in a second-sound cavity operating near the \ensuremath{\lambda} point. This model is applied to the data of Marek, Lipa, and Philips, which showed deviations from a simpler model in the region close to the transition. We find that our model by itself cannot explain the deviations, but if a shift in the estimated location of ${\mathit{T}}_{\ensuremath{\lambda}}$ is included, a significant improvement can be obtained. The critical exponent \ensuremath{\zeta}, describing the divergence of ${\mathrm{\ensuremath{\rho}}}_{\mathit{s}}$, was found to be \ensuremath{\zeta}=0.6708\ifmmode\pm\else\textpm\fi{}0.0004, in good agreement with the renormalization-group prediction 0.672\ifmmode\pm\else\textpm\fi{}0.002. The range for the reduced temperature parameter was extended to \ensuremath{\varepsilon}=2\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}7}$, substantially closer to the transition than in the previous analysis of this data. The shift in ${\mathit{T}}_{\ensuremath{\lambda}}$ can be considered acceptable if the data very near ${\mathit{T}}_{\ensuremath{\lambda}}$ are reinterpreted. The effect of the \ensuremath{\nabla}${\mathrm{\ensuremath{\rho}}}_{\mathit{s}}$ term is shown to be important for \ensuremath{\varepsilon}${10}^{\mathrm{\ensuremath{-}}6}$.