  $N$-layer transition metal dichalcogenides provide a unique platform to
investigate the evolution of the physical properties between the bulk (three
dimensional) and monolayer (quasi two-dimensional) limits. Here, using
high-resolution micro-Raman spectroscopy, we report a unified experimental
description of the $\Gamma$-point optical phonons in $N$-layer $2H$-molybdenum
ditelluride (MoTe$_2$). We observe a series of $N$-dependent low-frequency
interlayer shear and breathing modes (below $40~\rm cm^{-1}$, denoted LSM and
LBM) and well-defined Davydov splittings of the mid-frequency modes (in the
range $100-200~\rm cm^{-1}$, denoted iX and oX), which solely involve
displacements of the chalcogen atoms. In contrast, the high-frequency modes (in
the range $200-300~\rm cm^{-1}$, denoted iMX and oMX), arising from
displacements of both the metal and chalcogen atoms, exhibit considerably
reduced splittings. The manifold of phonon modes associated with the in-plane
and out-of-plane displacements are quantitatively described by a force constant
model, including interactions up to the second nearest neighbor and surface
effects as fitting parameters. The splittings for the iX and oX modes observed
in $N$-layer crystals are directly correlated to the corresponding bulk Davydov
splittings between the $E_{2u}/E_{1g}$ and $B_{1u}/A_{1g}$ modes, respectively,
and provide a measurement of the frequencies of the bulk silent $E_{2u}$ and
$B_{1u}$ optical phonon modes. Our analysis could readily be generalized to
other layered crystals.
