# Increasing the Insertion Loss of Sonic Crystal Noise Barriers with Helmholtz Resonators
## 3 Bragg and Helmholtz Resonances Interaction
Figure 2-right shows the case where the resonance frequency of the HR has been tuned above the Bragg-BG (higher surface). The scatterers of the SCNB have a HR with a mouth w = 1 cm, a neck L = 1 cm, and r = 1.2 cm, corresponding to f\({}_{\rm HR}\) = 1400 Hz > fc. Again, the reduction of the transmission coefficient can be observed in the frequency range where the local resonance and the Bragg resonance interact. In Supplementary Materials an animation shows the interaction of Bragg diffraction and the local resonance with the f\({}_{\rm HR}\) in the range from 400 Hz to 1400 Hz. Figure 2: Transmission coefficient of the SCNB and band structure of the corresponding SC with (**left**) f\({}_{\rm HR}\) < fc and (**right**) f\({}_{\rm HR}\) > fc of the configurations with 0\({}^{\circ}\) alignment (blue) and 90\({}^{\circ}\) alignment (red). The SCNB-CC case is plotted in black for reference. Figure 3 shows the numerical results of IL for several SCNBs consisting of arrays of HRs tuned to f\({}_{\rm c}\) = 1000 Hz (a = 0.17 m) in a color scale. Each sub-figure contains the results of IL for 60 numerical simulations of arrays of HRs with different resonant frequencies. The resonant frequency of the HR is plotted on the \(y\)-axis and the frequency of the propagating waves on the \(x\)-axis. For illustration, the lower and upper limits of the Bragg-BG of the SCNB-CC are drawn with vertical white dashed lines at f = 713 Hz and f = 1092 Hz, respectively (criterion IL = 18 dB). Since the phenomenon of local resonance does not occur in an SCNB-CC, the IL is identical for all SCNBs along the entire \(y\)-axis. The white dotted line corresponds to the value of the theoretical resonance frequency of the HR, calculated from (1). Figure 3 left shows the case of the SCNB with 0\({}^{\circ}\) alignment. The IL depends strongly on the resonance frequency of the HRs. The nature of the interaction between Bragg-BG and HR-BG is determined by their relative frequency resonances. When f\({}_{\rm HR}\) is far from f\({}_{\rm c}\) (f\({}_{\rm HR}\) < 600 Hz and f\({}_{\rm HR}\) > 1200 Hz), both resonances are well separated. When the resonances are closed (600 Hz \(<\) f\({}_{\rm HR}\) < 1200 Hz), the interaction is significant and important changes in the IL are observed with respect SCNB-CC. The IL has an asymmetric shape because the transmission is much higher at frequencies f \(<\) f\({}_{\rm HR}\) than at frequencies f \(>\) f\({}_{\rm HR}\). For example, at f\({}_{\rm HR}\) = 1400 Hz, the Bragg-BG is similar to that of the SCNB-CC at the same frequency, which agrees quite well with the white vertical dashed lines. The interaction between the resonances of the IL of the SCNB with 90\({}^{\circ}\) alignment is shown in Figure 3 right. It is also asymmetric with respect to the local resonant frequency, but the insulation increase occurs at frequencies below the Bragg frequency. The increase of IL due to the interaction between the Bragg resonance and the local resonance thus strongly depends on the specific orientation of the HRs of the SCNB with respect to the incident plane wave. Figure 4 shows the IL as a function of frequency for specific cases with f\({}_{\rm HR}\) = 400 Hz (f\({}_{\rm HR}\) < f\({}_{\rm c}\)) and f\({}_{\rm HR}\) = 1400 Hz (f\({}_{\rm HR}\) > f\({}_{\rm c}\)) in red and blue respectively. Figure 4 left shows the 0\({}^{\circ}\) alignment. The SCNB-CC is shown in dashed black. The colored area visually shows the change of the IL of the SCNB with HRs with respect to the SCNB-CC. It can be observed that in the case where f\({}_{\rm HR}\) is above the Bragg-BG, the IL increases compared to the SCNB-CC. If the resonance frequency is below, the opposite is true: the insulation of the SCNB decreases. Instead, the IL decreases compared to the SCNB-CC when the f\({}_{\rm HR}\) is above the Bragg-BG, as can be seen in Figure 4 right for a 90\({}^{\circ}\) alignment. The IL of the SCNB, on the other hand, increases when the resonance is below.