We present the first full band-structure calculations for periodic, el
astic composites. For transverse polarization of the vibrations we obt
ain a ''phononic'' band gap which extends throughout the Brillouin zon
e. A complete acoustic gap or a low density of states should have impo
rtant consequences for the suppression of zero-point motion and for th
e localization of phonons, and may lead to improvements in transducers
and in the creation of a vibrationless environment.