  Motivated by recent studies of three-state Potts nematic states in
magic-angle twisted bilayer graphene and doped-Bi$_2$Se$_3$, we analyze the
impact of critical nematic fluctuations on the low energy properties of
phonons. In this study we propose how to identify the three-state Potts nematic
fluctuations by ultrasound attenuation. The Gaussian fluctuation analysis shows
that the Landau damping term becomes isotropic due to fluctuations of the
$C_{3}$-breaking bond-order, and the nemato-elastic coupling is also shown to
be isotropic. These two features lead to an isotropic divergence of the
transverse sound attenuation coefficient and an isotropic lattice softening, in
contrast to the case of the $C_4$-breaking bond-order which shows the strong
anisotropy. Moreover, we use a mean-field approximation and discuss the
impurity effects. The transition temperature takes its maximum near the filling
of the van-Hove singularity, and the large density of states favors the nematic
phase transition. It turns out that the phase transition is of weak first-order
in the wide range of filling and, with increasing the impurity scattering, the
first order transition line at low temperatures gradually shifts towards the
second-order line, rendering the transition a weak first-order in a wider range
of parameters. Furthermore, it is confirmed that the enhancement of the
ultrasound attenuation coefficient will be clearly observed in experiments in
the case of a weak first-order phase transition.
