  Transition metal dichalcogenides (TMDs) are an exciting family of 2D
materials; a member of this family, MoS$_2$, became the first measured
monolayer semiconductor. In this article, a generalized phenomenological
continuum model for the optical vibrations of the monolayer TMDs valid in the
long-wavelength limit is developed. Non-polar oscillations involve differential
equations for the phonon displacement vector that describe phonon dispersion up
to a quadratic approximation. On the other hand, the polar modes satisfy
coupled differential equations for the displacement vectors and the inner
electric field. The two-dimensional phonon dispersion curves for in-plane and
out-of-plane oscillations are thoroughly analyzed. This model provides an
efficient approach to obtain the phonon dispersion curves at the $\Gamma$-point
of the Brillouin zone of the whole family of TMD monolayers. The model
parameters are fitted from density functional perturbation theory calculations.
A detailed evaluation of the intravalley Pekar-Fr\"ohlich (P-F) and the
$A_1$-homopolar mode deformation potential (Dp) coupling mechanisms is
performed. The effects of metal ions and chalcogen atoms on polaron mass and
binding energy are studied, considering these two contributions, the
short-range Dp and P-F. It is argued that both mechanisms must be considered
for a correct analysis of the polaron properties.
