Figure 1: Band alignment of the Bragg and non-Bragg gaps with different topological characteristics.

(a–d) Characteristic dispersion curves (top) and simulated sound pressures at 2349 Hz (bottom). The blue and red solid lines denote the fundamental and first modes, respectively, whereas the blue bold dots depict the dispersion curves of four different waveguides. (a) NB with non-Bragg gap at approximately 2500 Hz, shown by the yellow shadow. (b–d) Bragg waveguides with Bragg gap at approximately 2500 Hz, shown by the grey shadows. The Bragg waveguides with slight difference can be identified by the cutoffs of the first mode (red lines). (e) Transmission coefficients of the non-Bragg and three Bragg waveguides, denoted by the red dashed, blue dash-dot, black dotted, and green solid lines, respectively. It is clear that the Bragg gaps are much wider than the non-Bragg one but less efficient in sound attenuation. In any case, a common gap exists for the four waveguides in the frequency range of 2200 to 2600 Hz, denoted by the light blue shadow. (f) Band gaps between 2000 and 3000 Hz vs. different geometrical parameters (the mean radius and period of the waveguides). The magenta and cyan volumes denote the different topological characteristics of the non-Bragg and Bragg gaps, respectively. The bold dots present the geometric parameters of the waveguides investigated, whereas the bold lines denote the related band widths.