# Sonic Crystal Noise Barrier with Resonant Cavities for Train Brake Noise Mitigation
## 3 Results
### Numerical Results  
Since the main purpose of SCNBs is to achieve a similar insulation level that a CB could provide, the comparison is necessary to obtain a more global picture of the case under study. Figure 4 shows the numerical results of the FEM simulations obtained from the 2D model explained in Section 2.1. With the same dimensions as the proposed SCNB, a CB can achieve 20 to 25 dB of IL in the studied TBN frequency range (2 kHz-4.5 kHz). The numerical IL is calculated at the evaluation points from the acoustic pressure with and without the barrier for each case, following from (4). An open barrier such as the SCNB has lower levels but is competitive at some frequencies. One of the most important points to consider is the orientation of the mouth of the resonant cavity according to the noise source. With this rotation, a high reduction is observed when the mouth is set to 90\({}^{\circ}\). In a large part of the target frequency range (2.85  
Figure 4: Insertion loss of numerical FEM models with 0\({}^{\circ}\) SCNB (blue), 90\({}^{\circ}\) SCNB (red) and CB (black). Figure 3: Experimental set up measurement system: (**a**) SCNB measurement; (**b**) CB measurement. to 4.2 kHz, with colour gradient) of the TBN (green colour), differences of up to 27 dB in the IL are observed when the scatterers are rotated by 90\({}^{\circ}\) compared to the 0\({}^{\circ}\) case. This reduction results in a smaller Bragg-BG and the destructive interaction of the LR-BG with the Bragg-BG. This illustrates how important it is to correctly localize the aperture of local resonators with respect to the incident noise. In addition, from 2.85 kHz to 3.7 kHz, the 0\({}^{\circ}\) SCNB produces similar levels of insulation compared with the CB with two peaks of maximums in the IL at approximately 3.5 kHz. To gain a deeper insight into the noise control mechanisms that influence both 0\({}^{\circ}\) SCNB and 90\({}^{\circ}\) SCNB, the band structure is calculated with FEM simulation. With a unit-cell model, we compute the dispersion curves diagram and BG structure of two configurations of scatterers. We utilized the conventional representation using wave-vectors, where the directions \(\Gamma\)X correspond to specific pairs of (k\({}_{\rm x}\), k\({}_{\rm y}\)) values, namely (0, 0), (\(\pi\)/a, 0) and (\(\pi\)/a, \(\pi\)/a), where k\({}_{\rm x}\) and k\({}_{\rm y}\) represent the wavenumbers in the two orthogonal directions of an infinite periodic structure with a lattice constant (\(a\)). In Figure 5, the 0\({}^{\circ}\) SCNB and 90\({}^{\circ}\) SCNB configurations have two complete band gaps (grey colour). The first one is related to the Bragg-BG centred at 3.1 kHz, and the second one is related to the LR-BG, located at 4.37 kHz. Examining the BG structures, computed for both the 0\({}^{\circ}\) SCNB and 90\({}^{\circ}\) SCNB, significant differences in the BG structure are observed. Specifically, the analysis of the 0\({}^{\circ}\) SCNB reveals the increase in the pseudo-gap around 3.1 kHz in \(\Gamma\)X (blue colour). For the propagation directions of interest in our experimental measurements and simulations, the relevant part of this band gap structure is between the directions \(\Gamma\)X. In the case of the 0\({}^{\circ}\) SCNB, a pseudo-band gap is observed between 1.54 kHz and 4.28 kHz, corresponding to the target frequency range of TBN, while for 90\({}^{\circ}\) SCNB, it occurs between 1.73 kHz and 3.47 kHz, not covering the full range of interest. In this section, the results of the experimental measurements are analysed and the two different configurations of SCNBs at 0\({}^{\circ}\) and 90\({}^{\circ}\) are compared. Figure 6 shows a significant difference between the configurations in the lower frequency range (1 to 1.8 kHz), while the CB reaches levels of IL around 15 dB, and in the 0\({}^{\circ}\) and 90\({}^{\circ}\) cases, peaks about 6 and 9 dB are obtained, respectively.