Periodic elastic composites can give rise to complete band gaps in whi
ch vibrations, sound, and phonons are forbidden. This is of interest f
or applications such as creation of vibrationless environment and impr
ovements in transducers, as well as for pure physics related to locali
zed vibrations and phonons. We present rudimentary ideas of the ''band
-gap engineering'' necessary for designing suitable frequency gaps. Th
is is illustrated by means of acoustic band structures, computed for p
eriodic, binary composites of long, elastic cylinders that form a hexa
gonal lattice.