  We present a general theory for calculating the damping rate of elementary
density wave excitations in a Bose-Einstein condensate strongly coupled to a
single radiation field mode of an optical cavity. Thereby we give a detailed
derivation of the huge resonant enhancement in the Beliaev damping of a density
wave mode, predicted recently by K\'onya et al., Phys.~Rev.~A 89, 051601(R)
(2014). The given density-wave mode constitutes the polariton-like soft mode of
the self-organization phase transition. The resonant enhancement takes place,
both in the normal and ordered phases, outside the critical region. We show
that the large damping rate is accompanied by a significant frequency shift of
this polariton mode. Going beyond the Born-Markov approximation and determining
the poles of the retarded Green's function of the polariton, we reveal a strong
coupling between the polariton and a collective mode in the phonon bath formed
by the other density wave modes.
