# Increasing the Insertion Loss of Sonic Crystal Noise Barriers with Helmholtz Resonators
## 4 SCNB Scatterer with Two HRs  
As a result of the interaction between the Bragg resonance and the local resonances, the increase in insulation of the SCNB depends strongly on the alignment of the mouth of the HR. The relative value of the local resonance (f\({}_{\rm HR}\)) with respect to the central frequency of the Bragg-BG (f\({}_{\rm c}\)) is an important factor as it determines the interaction between the two phenomena. The use of resonators in SCNBs does not necessarily increase their sound insulation. Therefore, careful attention should be paid to the alignment of the mouths and the resonant frequencies of the HRs in the scatterers of SCNBs. Figure 5 shows a scatterer for a SCNB consisting of two HR aligned at 0\({}^{\circ}\) and 90\({}^{\circ}\), with variable and coupled surfaces. The inner moving wall can be rotated by an angle \(\theta\), and simultaneously changes the resonances of the two HRs. Both HR have identical mouths w = 3 cm and L = 1.5 cm. As in Section 3, IL is computed numerically in a SCNB consisting of scatterers arranged in a two-dimensional square lattice with a filling fraction ff = 0.4 and the central frequency of the Bragg-BG is f\({}_{\rm c}\) = 1000 Hz. Figure 4: Insertion loss as a function of frequency of SCNBs with f\({}_{\rm HR}\) < f\({}_{\rm c}\) (red) and f\({}_{\rm HR}\) > f\({}_{\rm c}\) (blue) with (**left**) 0\({}^{\circ}\) alignment and (**right**) 90\({}^{\circ}\) alignment. The colored area graphically represents the variance of the IL of the SCNB with HRs with respect to the SCNB-CC (dashed black). Figure 5: Cylindrical scatterer consisting of two HRs with identical mouth and variable coupled surfaces through a rotating inner wall. Figure 6 shows the behavior of a SCNB with cylinders with two HRs and the influence on the IL due to the movement of the inner moving wall. Figure 6 left shows the IL in color scale for the SCNB with scatterers with coupled HRs as a function of the rotation angle \(\theta\) of the wall analyzed with the same numerical model as in Figure 1b,c. The change in SCNB insulation behavior results from \(\theta\)-driven surfaces changes in the HRs. Thus, if the HR with \(0^{\circ}\) alignment has a smaller surface (\(\theta=0^{\circ}\)), the HR with \(90^{\circ}\) alignment has the largest surface and the IL in the BG-Bragg is high. However, the surfaces ratio of the HRs is changed, IL varies. When the surfaces of the HRs are inverted (\(\theta=180^{\circ}\)), the IL in the BG-Bragg decreases drastically and only the resonances of the respective HRs are observed. HRs do not behave the same with identical surfaces due to the uneven alignment of their mouths in relation to the incident wave. This asymmetry favors the increase of the insulation of the structure when frequencies of the HRs are properly tuned for each alignment. Figure 6 right shows the IL as a function of frequency for the cases \(\theta=0^{\circ}\), \(\theta=180^{\circ}\) and the SCNB-CC. As seen in Section 3, the IL of SCNB at the Bragg-BD increases in \(0^{\circ}\) alignment for \(f_{\rm HR}<f_{\rm c}\) and decreases in \(90^{\circ}\) alignment for \(f_{\rm HR}>f_{\rm c}\). This condition is fulfilled for both coupled-surface HRs when \(\theta=0^{\circ}\). In this case, the surface of the HR with \(0^{\circ}\) alignment is actually minimum and the surface of the \(90^{\circ}\) alignment is maximum. It can be seen that the frequency range corresponding to the Bragg-BG (between 500 Hz and 1300 Hz), the IL increases by about 20 dB compared to the SCNB-CC due to the interaction between Bragg and local resonances. The central frequencies of HR-BG of both HRs appear as local maxima in the IL at f = 610 Hz and f = 1165 Hz for the HR aligned at \(90^{\circ}\). In this configuration, the resonators have resonance frequencies at the edges of the Bragg-BG. The interaction between the local resonances and the Bragg resonance leads to a reduction in the transmission of the SCNB or, in other words, an increase in the insertion loss.