We consider the scattering of dark matter particles from superfluid liquid He-4, which has been proposed as a target for their direct detection. Focusing on dark matter masses below similar to 1 MeV, we demonstrate from sum-rule arguments the importance of the production of single phonons with energies. omega less than or similar to 1 meV. We show further that the anomalous dispersion of phonons in liquid He-4 at low pressures [i.e., d(2)omega(q)/dq(2) > 0, where q and omega(q) are the phonon momentum and energy] has the important consequence that a single phonon will decay over a relatively short distance into a shower of lower-energy phonons centered on the direction of the original phonon. Thus, the experimental challenge in this regime is to detect a shower of low-energy phonons, not just a single phonon. Additional information from the distribution of phonons in such a shower could enhance the determination of the dark matter mass.