  We re-examine the effect of long-range Coulomb interactions on the collective
amplitude and phase modes in the incommensurate charge-density-wave ground
state of quasi-one-dimensional conductors. Using an effective action approach
we show that the longitudinal acoustic phonon protects the gapless linear
dispersion of the lowest phase mode in the presence of long-range Coulomb
interactions. Moreover, in Gaussian approximation, amplitude fluctuations are
not affected by long-range Coulomb interactions. We also calculate the
collective mode dispersions at finite temperatures and compare our results with
the measured energies of amplitude and phase modes in K$_{0.3}$MoO$_3$. With
the exception of the lowest phase mode, the temperature dependence of the
measured mode energies can be quantitatively described within a multi-phonon
Fr\"{o}hlich model for generic electron-phonon interactions neglecting
long-range Coulomb interactions.
