  We report a persistent low-energy phonon broadening around $q_{B} \sim 0.28$
r.l.u. along the Cu-O bond direction in the high-$T_c$ cuprate
Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ (Bi-2212). We show that such broadening
exists both inside and outside the conventional charge density wave (CDW)
phase, via temperature dependent measurements in both underdoped and heavily
overdoped samples. Combining inelastic hard x-ray scattering, diffuse
scattering, angle-resolved photoemission spectroscopy, and resonant soft x-ray
scattering at the Cu $L_3$-edge, we exclude the presence of a CDW in the
heavily overdoped Bi-2212 similar to that observed in the underdoped systems.
Finally, we discuss the origin of such anisotropic low-energy phonon
broadening, and its potential precursory role to the CDW phase in the
underdoped region.
