# Sonic Crystal Noise Barrier with Resonant Cavities for Train Brake Noise Mitigation
## 2 Materials and Methods
### Numerical 2D Model  
Finite-element method (FEM) numerical simulations with the commercial software COMSOL Multiphysics 6.1 are performed to evaluate the transmission of two-dimensional SCNBs and compare its performance with a conventional barrier (CB). The CB is a flat wall with the same dimensions as the SC ones and the SCNB consists of a set of locally resonant scatterers distributed in a square-periodic lattice with two different orientations of the resonant cavity with respect to the noise source (0\({}^{\circ}\) and 90\({}^{\circ}\)) as shown in Figure 1. By evaluating the insulation provided by the rigid flat barrier, a comparison is possible to the maximum possible level considering the border diffraction according to the SCNB dimensions. Figure 1 illustrates the 2D SCNB model under investigation, which is defined by a square-periodic structure of 4 rows and 12 columns. The dimensions of the three barriers are the same: 11.5 cm in width (X-direction) per 0.36 m in length (Y-direction). In Figure 2b, the dimensions are shown. Figure 2: 3D-printed 1:2 scaled SCNB PLA prototypes: (**a**) Front view of the scaled SCNBs with the local resonator at 0\({}^{\circ}\); (**b**) Top view of the scaled SCNBs at the 0\({}^{\circ}\) and 90\({}^{\circ}\) configurations. The noise source is located at 30 cm from the barrier at the left part of the domain and in the centre of the barrier. It is radiating 1 Pa of acoustic pressure. It is placed in a rectangular domain of 50 cm in width (X-direction) per 80 cm in length (Y-direction). In this domain, the boundaries are perfectly matched layers (PML) [44], usually applied to absorb waves propagating outward from the domain due to their non-reflective properties. In this 2D model, 3-node triangular elements were utilized to construct the mesh, ensuring convergence by using mesh elements with a maximum size able to evaluate 8 points per wavelength even at the maximum frequency of the study (4.5 kHz). The final geometry has a mesh of 257,168 domain elements and 128,366 boundary elements with 2,660,126 degrees of freedom. The local resonance of the scatterers depends on the length of the neck (L), the width of the mouth (w) and the inner surface area (S = \(\pi\)\(\pi\)\({}^{2}\)), with \(r\) being the inner radius of the surface of the locally resonant scatterer. The scatterers are assumed to be rigid, neglecting the acoustic-structure coupling. In this work, the numerical study and prototyping are carried out on a 1:2 scale model, including viscothermal losses. It is important to note that in full-scale applications, these viscothermal losses are slightly different as they are not directly scalable. Nonetheless, the research proceeds with the understanding that the variations in viscothermal losses do not critically impact the fundamental acoustic performance of the SCNB for reducing TBN. These viscothermal losses in the medium are implemented in the local resonant slit of the scatterer, according to the following equations of the complex density and the complex impedance [45]:  
\[\rho_{c}=\frac{\rho_{0}}{\Psi_{v}}\left[\frac{\mathrm{kg}}{\mathrm{m}^{3}} \right], \tag{1}\]  
\[Z_{c}{}^{2}=\frac{\left(\rho_{0}c\right)^{2}}{\Psi_{v}(\gamma-(\gamma-1)\Psi_ {t})}\left[\mathrm{Pa}\cdot\frac{\mathrm{s}}{\mathrm{m}}\right], \tag{2}\]  
where \(\gamma\) is the specific heat ratio in the air and \(\Psi_{t}\), \(\Psi_{v}\) are the analytic expressions of the mean values of the viscous and thermal fields (\(\varphi\) is either \(v\) or \(t\)) in the local resonator neck:  
\[\Psi_{\varphi}=1-\frac{\tan(k_{\varphi}\frac{\varphi}{2})}{k_{\varphi}\frac{ \varphi}{2}}, \tag{3}\]  
with \(w\) being the width of the mouth of the local resonator. For further details on the implementation of viscothermal losses in the FEM model, see Appendix A in [39].