  Acoustic phonons have recently been posited as playing an integral role in
explaining the half-quantized thermal Hall effect in $\alpha$-RuCl$_3$.
Therefore, we present much needed inelastic x-ray scattering measurements of
its acoustic phonon dispersion, along with calculations using the frozen-phonon
method. We also discuss a temperature study which conclusively shows a
first-order structural transition to a non-$C2/m$ space group at low
temperature. Together these results are an important backbone for future
theoretical and experimental studies of $\alpha$-RuCl$_3$.
