  The physical origin of frequency dependence in elastic constants, which are
often found in an ultrasound propagation in filled skutterudites and clathrate
compounds, is investigated theoretically. This dependence arises from a
coupling between the acoustic phonon and some optical phonons, which strongly
interact with electrons. Using a self-consistent ladder approximation together
with a pseudofermion mapping of the phonon to the single site Holstein Anderson
model, a soft mode of the optical phonon at zero frequency is shown to emerge.
The temperature dependence of the spectral weight of this soft mode shows an
activation-type behavior, which is characterized by the optical phonon
frequency. These features can generate the frequency dependence and the
shoulder in the elastic constants observed in some filled skutterudites and
clathrate compounds.
