## 4 Numerical and Experimental Results
### From Periodic to Fibonacci Sequence and Vice Versa  
Before giving the transmission spectra of a given FS composed of A and B blocks, it will be interesting to understand first the dispersion curves of the periodic structures made of only A blocks or only B blocks. The dispersion relation of an infinite periodic comb-like structure constituted by a segment of length \(d_{1}\) and a stub of length \(d_{m}\) (m = 2 or 3) is given by [67]  
\[\cos(k_{B}d_{1})=\cos(\omega d_{1}\sqrt{\varepsilon}/c)+0.5\cos(\omega d_{m} \sqrt{\varepsilon}/c)\sin(\omega d_{1}\sqrt{\varepsilon}/c)/\sin(\omega d_{m} \sqrt{\varepsilon}/c), \tag{14}\]  
where \(k_{B}\) is the Bloch wave-vector. In the particular case of a periodic structure composed of B blocks (i.e., \(d_{m}=d_{3}=d_{1}/2=0.5\) m), then Equation (14) becomes simply \(\cos(k_{B}d_{1})=3\cos(\omega d_{1}\sqrt{\varepsilon}/c)/2+1/2\). Therefore, the band edges are given by \(\omega d_{1}\sqrt{\varepsilon}/c=\arccos(1/3)\) at the center of the Brillouin zone (BZ) and \(\omega d_{1}\sqrt{\varepsilon}/c=\pi\) [mod. \(2\pi\)] at the limit of the BZ. This result shows that the allowed bands close at the limit of the BZ, whereas the limits of the bands at the center of the BZ are given by \(f=38\) MHz and \(f=140\) MHz. In addition, the band gaps are periodic every \(\omega d_{1}\sqrt{\varepsilon}/c=2\pi\) (i.e., \(f=196\) MHz). These results are in accordance with the dispersion curves sketched in Figure 2a inside the first BZ. The transmission spectra through a finite periodic structure composed of 13 B blocks is plotted in Figure 2b. A wide allowed band in the frequency region [38; 39; 40] MHz and two gaps at the edges of this band can clearly be observed. The solid (dashed) curves represent the theoretical results with (without) taking into account the dissipation, whereas open circles correspond to the experimental data. Despite the small number of B blocks, the allowed (forbidden) transmission spectra coincide perfectly with the bands (gaps) of the infinite system (Figure 2a). The theoretical results with dissipation (solid curves) are in good agreement with experimental results (open circles). Let us notice that the attenuation inside the cables induces a decay in the transmission amplitude, especially at high frequencies. Similarly, in the case of a periodic structure composed of A blocks (i.e., \(d_{m}=d_{2}=d_{1}=1\) m), then Equation (14) becomes simply \(\cos(k_{B}d_{1})=3\cos(\omega d_{1}\sqrt{\varepsilon}/c)/2\). The band-gap edges are given by \(\cos(k_{B}d_{1})=\pm 1\) or equivalently \(\cos(\omega d_{1}\sqrt{\varepsilon}/c)=\pm 2/3\). Therefore, the band edges are given by \(\omega d_{1}\sqrt{\varepsilon}/c=\arccos(\pm 2/3)\) [mod. \(2\pi\)]. The first four band limits fall at \(f=26\) MHz, \(f=72\) MHz, \(f=124\) MHz and \(f=170\) MHz, in accordance with the dispersion curves sketched in Figure 2i inside the first Brillouin zone. The transmission spectra through a finite periodic structure composed of 13 A blocks is plotted in Figure 2h. Here also, the allowed (forbidden) transmission spectra coincide perfectly with the bands (gaps) of the infinite system (Figure 2i) and the band gaps are periodic every \(\omega d_{1}\sqrt{\varepsilon}/c=2\pi\) (i.e., \(f=196\) MHz). Figure 2: (**a,i**) Dispersion curves of the periodic structures composed of only B or only A blocks respectively. (**b–h**): Variation of the transmittance versus the frequency for different finite structures composed of 13 blocks \(A\) and \(B\). (**e**) Transmission spectrum for generation \(S_{6}\) of the Fibonacci structure. The solid (dashed) curves represent the theoretical results with (without) dissipation, whereas open circles correspond to the experimental data. Now, we start perturbing the periodic structure composed of 13 B blocks (Figure 2b) by replacing the first \(B\) block of the system with an \(A\) block.