# Increasing the Insertion Loss of Sonic Crystal Noise Barriers with Helmholtz Resonators
## 2 Materials and Methods
### Numerical Model  
We consider a two-dimensional SC, which consists of a set of HRs distributed in an infinite square periodic lattice. Figure 1a shows the unit cell of the two-dimensional periodic system under investigation, which is defined by a scatterer with the outer radius R and a HR inside. Periodic boundary conditions are set at the X and Y boundaries so that the infinitely extended SC is defined by a periodic structure with a square lattice and a lattice constant (a) defining the unit cell. The irreducible Brillouin zone is also shown and represents the central cell of the reciprocal lattice. In a two-dimensional system, the filling fraction is the ratio between the area occupied by the scatterer and the area of the entire unit cell. The resonator is characterized by the length of the neck (\(L\)), the width of the mouth (\(w\)), and the inner surface area (S = \(\pi r^{2}\)), where \(r\) is the inner radius of the surface of the HR. The first resonance frequency of the resonator is [24]:  
\[f_{HR}=\frac{c}{2\pi}\sqrt{\frac{w}{\left(L+\Delta\frac{w}{2}\right)S}}\, \tag{1}\]  
where \(\Delta\) is the correction factor for neck length added account for the effects of the neck-cavity junction caused by a sudden change in the circular cross-sections (usually between 1.6 and 1.8) [45]. For simplicity, the geometry of the resonator for all the simulations in this work was chosen to neglect the viscothermal losses in the medium (see Appendix A). The band structure describes the dispersion relation of waves in the SC. FDTD simulations were used to calculate the band structure. In the band structure, propagation bands and BGs can be identified. In this work, we are interested in two propagation directions: \(0^{\circ}\) and \(90^{\circ}\). The band structure is calculated in the two corresponding directions in reciprocal space or k-space: \(0^{\circ}\) (\(\Gamma\)X) and \(90^{\circ}\) (\(\Gamma\)X), where G (k\({}_{\rm x}\) = 0, k\({}_{\rm y}\) = 0), X (k\({}_{\rm x}\) = \(\pi/\)a, k\({}_{\rm y}\) = 0) and X\({}^{\prime}\) (k\({}_{\rm x}\) = 0, k\({}_{\rm y}\) = \(\pi/\)a) denote the boundaries of the first Brillouin zone. Numerical simulations using the FDTD method are performed to evaluate the transmission of two SCNBs with a design based on the unit cell defined in Figure 1a. Periodic Boundary Conditions (PBC) are considered to reproduce an infinite SC in the Y-direction but truncated to three layers of scatterers in the X-direction (see Figure 1b,c). Since our SC has square symmetry, the periodicity is the same in the \(x\)- and \(y\)-axis, and in the two transmission models, both the distance between the scatterers and the transverse dimension (\(y\)-axis) of the models are a = 17 cm, with a filling fraction ff = \(\pi\) (r/a)2 = 0.4. This means that the external radius (R) is 6.86 cm and the distance between the scatterers is a/2-R = 1.64 cm. Thus, the geometric model of the SCNBs and the unit cell have the same topology, namely a two-dimensional square-periodic structure. An incident plane wave with frequency f and acoustic pressure P\({}_{\rm i}\) propagates along the \(x\)-axis from left to right in such a way that it is aligned with the mouth of the HR in Figure 1b and perpendicular to it in Figure 1c (hereafter referred to as the \(0^{\circ}\) and \(90^{\circ}\) alignments). Perfectly Matched Layers (PMLs) are applied to both \(x\)-axis boundaries in the models to absorb waves propagating outward from the domain. The transmission of the system is analyzed in the evaluation points of a line at a distance of 1 m from the SC by evaluating the transmission coefficient (\(\tau\)) defined by:  
\[\tau=\frac{<P_{t}^{2}>}{<P_{i}^{2}>}, \tag{2}\]  
where \(<>\) stands for the averaging of both, the squared incident acoustic pressure and the squared transmitted acoustic pressure (\(P_{t}\)) at the evaluation points.