We study theoretically the interaction of phonons with surface vibrational modes in a finite-size superlattice with a free surface. A phonon incident normally on the superlattice from a substrate is perfectly reflected, i.e., the reflection rate is unity irrespective of frequency. However, it comes back to the substrate with a large time delay when the frequency coincides with an eigenfrequency of the surface mode. This result is attributable to the resonant interaction of incident phonons with a vibrational mode localized near the surface