Dispersion relation for phonon second-sound waves in superfluid helium

The velocity and attenuation of phonon second-sound waves in superfluid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ at saturated vapor pressure have been calculated, as a function of sound-wave frequency $\omega$, over the entire frequency range at a single temperature 0.25°K. Second sound is obtained as a wavelike normal mode of a model phonon Boltzmann equation containing, in addition to the lifetime ${\tau }_{\parallel }$ of a single thermal phonon due to small-angle scattering, a sequence of longer lifetimes characterizing wide-angle scattering of phonons with anomalous dispersion. The calculated second-sound phase velocity shows a dispersion spread out over four orders of magnitude in frequency in the range $\omega {\tau }_{\parallel }\lesssim 1$. Moreover, there is a wide frequency range satisfying $\omega {\tau }_{\parallel }\gg 1$ in which a second-sound collective mode still propagates, with the same velocity as a thermal phonon but with an attenuation length much longer than the thermal-phonon mean free path. The existence of a collective mode in the regime $\omega {\tau }_{\parallel }\gg 1$, due to small-angle scattering, supports Maris's proposed explanation of resonancelike dispersion in the first-sound velocity, and also implies that the transition from collective to ballistic propagation in heat-pulse experiments is more complicated than previously supposed.