# Increasing the Insertion Loss of Sonic Crystal Noise Barriers with Helmholtz Resonators
## 3 Bragg and Helmholtz Resonances Interaction  
The effect of multiple scattering interference significantly reduces the transmission of a SCNB in the BGs and increases its insulation. In general, this phenomenon applies not only to the Bragg-BG, but also to the HR-BG [20]. In the model proposed here, the scatterers have a local resonator. Depending on the central frequencies and the width of their BGs, both Bragg-BG and HR-BG can overlap, leading to a different type of interaction between the phenomena [33,34]. Next, the transmission of an incident plane wave from a SCNB is investigated numerically using the models shown in Figure 1b for the 0\({}^{\circ}\) alignment and Figure 1c for the 90\({}^{\circ}\) alignment. The aim is to analyze the physical interaction between the first Bragg  
Figure 1: **(a)** Unit cell of the two-dimensional scatterer with HR, which forms the periodic system defining the SC and irreducible Brillouin zone of the square lattice SC based on the Bloch–Floquet theory for the calculation of the band structure. Numerical models for the SNCBs with 3-rwo scatterers for the evaluation of the IL with (**b**) 0\({}^{\circ}\) aligned and (**c**) 90\({}^{\circ}\) aligned HRs. resonance and the first local resonance and their mutual interaction. The value of the specific central frequency of the HR-BG influences the width and the insulation of the Bragg-BG. Two SCNBs are studied with their HR-BG below and above the Bragg-BG with a fixed lattice constant (a = 0.17 m), i.e., a central Bragg frequency of fc = 1000 Hz. According to (1), the HR resonance frequency depends on the neck, the mouth, and the surface of the two-dimensional resonator. The resonant frequency of the HR could be changed by suitably varying each of these parameters according to (1). However, due to geometric constraints in the design of the resonator inside the scatterer, it is appropriate to vary the surface of the resonator to cover a wide range of HR resonant frequencies with realistic geometries. Figure 2 shows results of FDTD simulations of the transmission coefficient of a SCNB in a solid line based on the models shown in Figure 1b,c together with the band structure of the SC, with circles with 0\({}^{\circ}\) alignment in red and 90\({}^{\circ}\) alignment in blue for TX and \(\Gamma\)X', respectively. For reference, the SCNB is shown with Closed Cylinders, SCNB-CC, i.e., without local resonance, in black. The scatterers of the SCNB have a HR with a mouth w = 1 cm, a neck L = 1 cm, and an r = 4.5 cm, corresponding to a resonant frequency f\({}_{\rm HR}\) = 400 Hz < fc. The frequency range includes the first Bragg-BG with the central frequency f\({}_{\rm c}\) and the first HR-BG, with resonance frequency f\({}_{\rm HR}\). Both phenomena have higher resonances, but they are far from interacting with the first resonances and their mutual interactions are beyond the scope of this work. Although the band structure was obtained from an infinite periodic SC and the transmission loss from the SCNB with an array of three layers of resonators, the agreement between the presence of BGs and transmission minima is significant. Figure 2-left shows the case of a HR with a small surface and, thus, a resonant frequency lower than the central frequency of the Bragg-BG (f\({}_{\rm HR}\) < fc). At low frequencies, below and around the local resonance, oscillations are observed in the transmission, which are due to Fabry-Perot interference phenomena [46] due to the multiple reflections of the waves at the truncated ends of the SC. This is thus a phenomenon caused by the finite length of the crystal and cannot be explained by the band structure of the infinite SC. The interaction between the two BGs is visible in the frequency range of the HR-BG below the Bragg-BG, where the greatest difference from the case of the SCNB-CC is observed. A significant reduction in transmission is observed for both 0\({}^{\circ}\) alignment and 90\({}^{\circ}\) alignment due to the local resonance.