  We use the amplitude expansion in the phase field crystal framework to
formulate an approach where the fields describing the microscopic structure of
the material are coupled to a hydrodynamic velocity field. The model is shown
to reduce to the well known macroscopic theories in appropriate limits,
including compressible Navier-Stokes and wave equations. Moreover, we show that
the dynamics proposed allows for long wavelength phonon modes and demonstrate
the theory numerically showing that the elastic excitations in the system are
relaxed through phonon emission.
