Phonon spectra of bicrystals with relaxed grain-boundary structure dis
play a variety of localized modes including long-wavelength acoustic m
odes. Continuum solutions for localized waves that incorporate atomic-
level elastic properties of the interface via discontinuity relations
agree well with the latter modes. In contrast, classical solutions tha
t depend only on bulk elastic properties do not. This demonstrates tha
t the distinct atomic structure of the interface is a controlling fact
or, and it is shown how local, atomic-level properties can be incorpor
ated into continuum analyses of interfacial phenomena.