  We study sound in Galilean invariant systems of one-dimensional fermions. At
low temperatures, we find a broad range of frequencies in which in addition to
the waves of density there is a second sound corresponding to ballistic
propagation of heat in the system. The damping of the second sound mode is
weak, provided the frequency is large compared to a relaxation rate that is
exponentially small at low temperatures. At lower frequencies the second sound
mode is damped, and the propagation of heat is diffusive.
