Propagation and localization of acoustic waves in Fibonacci phononic circuits

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