The physical origin of frequency dependence in elastic constants, which areoften found in an ultrasound propagation in filled skutterudites and clathratecompounds, is investigated theoretically. This dependence arises from acoupling between the acoustic phonon and some optical phonons, which stronglyinteract with electrons. Using a self-consistent ladder approximation togetherwith a pseudofermion mapping of the phonon to the single site Holstein Andersonmodel, 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 anactivation-type behavior, which is characterized by the optical phononfrequency. These features can generate the frequency dependence and theshoulder in the elastic constants observed in some filled skutterudites andclathrate compounds.
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