Propagation and localization of acoustic waves in Fibonacci phononic circuits

Abstract

A theoretical investigation is made of acoustic wave propagation in one-dimensionalphononic bandgap structures made of slender tube loops pasted together with slender tubesof finite length according to a Fibonacci sequence. The band structure and transmissionspectrum is studied for two particular cases. (i) Symmetric loop structures, which are shownto be equivalent to diameter-modulated slender tubes. In this case, it is found that besidesthe existence of extended and forbidden modes, some narrow frequency bands appear in thetransmission spectra inside the gaps as defect modes. The spatial localization of the modeslying in the middle of the bands and at their edges is examined by means of the localdensity of states. The dependence of the bandgap structure on the slender tubediameters is presented. An analysis of the transmission phase time enables us toderive the group velocity as well as the density of states in these structures. Inparticular, the stop bands (localized modes) may give rise to unusual (strong normal)dispersion in the gaps, yielding fast (slow) group velocities above (below) the speedof sound. (ii) Asymmetric tube loop structures, where the loops play the role ofresonators that may introduce transmission zeros and hence new gaps unnoticedin the case of simple diameter-modulated slender tubes. The Fibonacci scalingproperty has been checked for both cases (i) and (ii), and it holds for a periodicity ofthree or six depending on the nature of the substrates surrounding the structure.