Abstract: Using the plane-waves expansion method, we study the propagation of elastic waves through the 2-D periodic composite, constructed from two different materials. We reveal the existence of several very large complete band gaps in the band structure of a material such as Pb cylinders reinforced epoxy composite. Within these gaps the propagation of elastic wave is forbidden. The influence of the geometry of the cylinders and the effect of the composition of the composite material on the band structure are studied. The complete band gaps are observed in the cases of Pb cylinders in epoxy or C cylinders in epoxy, but not in the opposite situation.