COLLECTIVE MODES IN SUPERSOLID HE-4

We derive the hydrodynamic equations of motion of solid and supersolid He-4, that describe the collective modes of these phases. In particul ar, the usual hydrodynamics is modified in such a way that it leads to the presence of a propagating instead of a diffusive defect mode. The former is appropriate for a quantum crystal and observed in recent ex periments. Furthermore, we find that in supersolid helium there are tw o additional modes associated with the superfluid degrees of freedom. The observation of these additional modes is a clear experimental sign ature of the supersolid phase.