Phonon spectra of bicrystals with relaxed grain-boundary structure display a variety of localized modes including long-wavelength acoustic modes. 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 that depend only on bulk elastic properties do not. This demonstrates that the distinct atomic structure of the interface is a controlling factor, and it is shown how local, atomic-level properties can be incorporated into continuum analyses of interfacial phenomena.