Phonons in an inhomogeneous continuum: Vibrations of an embedded nanoparticle

The spectrum of acoustic vibrational modes of an inhomogeneous elastic continuum is analyzed with application to a spherical nanoparticle embedded in an infinite glass block. The relationship of these modes to the discrete vibrational spectrum of a free sphere is studied. The vibrational modes of a sphere with a fixed surface are relevant in some situations. Comparisons are also made to calculations of mode frequency and damping based on complex-valued frequency.