In recent years,there is a growing interest in the study of wave propagation in periodic structures,as they exhibit excellent physical properties such as band gap[1-7] and negative refraction.[8-10] These properties make them have potential applications as acoustic or vibration devices such as acoustic filters,acoustic lenses,transducers,and vibration insulators.As is known,two primary mechanisms,i.e.,Bragg scattering and local resonance,are responsible for the forming of the band gap of phononic crystals.Bragg scattering phononic crystals can exhibit a band gap in the frequency corresponding to the wave length in the order of the structural period,i.e.,the lattice constant,and the band gap relates to the lattice symmetry.Thus the band gap at very low frequency can only be achieved by using structures with very large periodicity or materials with very slow elastic wave velocity.A phononic crystal would have to be several meters in size in order to shield environmental noise in the audible frequency range.To overcome this limitation,locally resonant phononic crystals[11,12] are proposed to exhibit a band gap in the frequency in two orders of magnitude lower than the expected one by Bragg scattering.