  Arbitrary waves incident on a solid embedded nanoparticle are studied. The
acoustic vibrational frequencies are shown to correspond to the poles of the
scattering cross section in the complex frequency plane. The location of the
poles is unchanged even if the incident wave is nonplanar. A second approach
approximating the infinite matrix as a very large shell surrounding the
nanoparticle provides an alternate way of predicting the mode frequencies. The
wave function of the vibration is also provided.
