  We calculate the phonon density of states (DOS) for strongly amorphous
materials with a short-ranged interatomic potential. Exponentially decaying and
abruptly truncated interatomic potentials are examined. Thermally excited mean
square deviations from equilibrium are calculated with rapid increases noted as
the average number of neighbors is reduced. The Inverse Participation Ratio
(IPR) is used to characterize the phonon states and identify localized phonon
modes as the bonding range (and hence the average number of neighbors per atom)
is diminished. For the truncated potential, the characteristics of the IPR
histogram change qualitatively below $n_{\mathrm{neigh}}$ with the appearance
of localized phonon modes below $n_{\mathrm{neigh}} = 6.0$.
