Propagation of second sound near T_λ

Superfluid hydrodynamics for second sound, expanded to first order in ∇ρ_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 λ 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 T_λ is included, a significant improvement can be obtained. The critical exponent ζ, describing the divergence of ρ_s, was found to be ζ=0.6708±0.0004, in good agreement with the renormalization-group prediction 0.672±0.002. The range for the reduced temperature parameter was extended to ɛ=2×10^-7, substantially closer to the transition than in the previous analysis of this data. The shift in T_λ can be considered acceptable if the data very near T_λ are reinterpreted. The effect of the ∇ρ_s term is shown to be important for ɛ<10^-6.