Numerical Solution of the Transport Equation for Sound Propagation in He II

Assuming an anomalous phonon dispersion, the velocity shift and the attenuation of ultrasound in liquid ${\mathrm{He}}^4$ below 0.5 K are calculated by a direct numerical solution of the Boltzmann equation for the phonon distribution function. In order to account for the hydrodynamic regime the collision invariants are treated separately. The results are in qualitative agreement with the experiments and show a nonmonotonic increase of the velocity shift as a function of the applied frequency, but they are different from an approximate solution using relaxation times. The method developed to solve the Boltzmann equation can also be applied directly to calculate transport properties of phonons in solids.