  An explanation for the glass-like anomaly observed in the low-temperature
specific heat of incommensurate phases is proposed. The key point of this
explanation is the proper account for the phason damping when computing the
thermodynamic magnitudes. The low-temperature specific heat of the
incommensurate phases is discussed within three possible scenarios for the
phason dynamics: no phason gap, static phason gap and a phason gap of dynamical
origin. Existing NMR and inelastic scattering data indicate that these
scenarios are possible in biphenyl, blue bronze $\rm K_{0.30}MoO_3$ and BCPS
respectively. Estimates of the corresponding low-temperature specific heat are
in reasonable agreement with the experiments.
