The propagation of longitudinal elastic waves in quasi one-dimensional structure consisting of harmonic oscillators periodically jointed on a slender beam is studied. Sub-frequency locally resonant band gap with highly asymmetric attenuation is observed in both theoretical and experimental results, and both results match well. The stiffness and mass ratios are found analytically as two factors that influence the actual attenuation in the band gap of the locally resonant phononic crystals. The study on the weights of the two factors shows that the stiffness ratio is the key one. Thus, the reason for the mismatch between the regions of the sharp attenuation and the theoretical band gap in the locally resonant phononic crystals is discovered.
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