Formation of longitudinal wave band structures in one-dimensional phononic crystals

Existing research on the mechanism of frequency bands in phononic crystals were without exception concerned with the band gaps (stop-bands). This paper reports, for the first time, the formation mechanism of the band structures, including both the phase constant and the attenuation constant spectra, of longitudinal waves in one-dimensional phononic crystals based on the periodic ternary rod model. Closed-form dispersion relation is obtained by our proposed method of reverberation-ray matrix. The band structures of characteristic longitudinal wave in the infinite one-dimensional phononic crystal are found to be generated from the dispersion curves of equivalent longitudinal wave in the unit cell due to zone folding effect and wave interference phenomenon. The characteristic times of longitudinal wave traversing the individual components of the unit cell as well as traversing the unit cell itself and the contrast of characteristic impedances of components are revealed to be the essential parameters determining the band structures. The deep understanding of band-structure formation provides direct scheme for adjusting frequency bands and propagation characteristics of elastic waves in one-dimensional phononic crystals. It will facilitate the design and optimization of phononic crystals to satisfy specific requirements for wave guiding/filtering and vibration control/isolation applications.