  Classical continuum elastic calculations show that the acoustic vibrational
modes of an embedded nanoparticle can be lightly damped even when the
longitudinal plane wave acoustic impedances $Z_o=\rho v_L$ of the nanoparticle
and the matrix are the same. It is not necessary for the matrix to be less
dense or softer than the nanoparticle in order to have long lived vibrational
modes. Continuum boundary conditions do not always accurately reflect the
microscropic nature of the interface between nanoparticle and matrix, and a
multi-layer model of the interface reveals the possibility of additional
reduction of mode damping.
