An impurity atom substituting for a host atom near the surface of a discrete semiinfinite lattice can give, rise to local oscillation. In general the conditions for local oscillation are low defect mass and strong binding of the impurity to its ligands. Results have been computed for a monatomic cubic crystal for both impurity atoms at and near the surface and are shown in Figs. 3, 4 and 5, of the text. Rotational invariance requires that force constants at the surface be different from bulk force constants. These surface forces give, rise to surface waves, even where they have been reported absent in monatomic cubic crystals with nearest-neighbor forces only. These forces also influence the dynamics of local modes of surface impurities. It is shown that surface forces can be taken into account quantitatively as a perturbation of an ideal semiinfinite crystal if the eigenfrequency of the local mode is large enough, say twice the lattice frequency. For very large frequencies and small impurity mass the limit of the Einstein model is reached. An infrared absorption experiment is proposed to detect local oscillation of surface impurities. © 1971.