  We consider a simple class of holographic massive gravity models for which
the dual field theories break translational invariance spontaneously. We study,
in detail, the longitudinal sector of the quasi-normal modes at zero charge
density. We identify three hydrodynamic modes in this sector: a pair of sound
modes and one diffusion mode. We numerically compute the dispersion relations
of the hydrodynamic modes. The obtained speed and the attenuation of the sound
modes are in agreement with the hydrodynamic predictions. On the contrary, we
surprisingly find disagreement in the case of the diffusive mode; its diffusion
constant extracted from the quasi-normal mode data does not agree with the
expectations from hydrodynamics. We confirm our numerical results using
analytic tools in the decoupling limit and we comment on some possible reasons
behind the disagreement. Finally, we extend the analysis of the collective
longitudinal modes beyond the hydrodynamic limit by displaying the dynamics of
the higher quasi-normal modes at large frequencies and momenta.
