## 2 Model
### 1D Array of Helmholtz Resonators  
We show here the modeling of the wave propagation along a 1D array built by assembling \(N\) building blocks, each one made of a Helmholtz resonator and a cavity, as shown in Figure 2. The propagation of waves in this system is modeled using the Transfer Matrix Method (TMM). We consider the reciprocal system \(n\) in which only plane waves propagate. By assembling the matrices representing each building block of the array in a total transfer matrix \(\mathbf{T}\), the pressure, \(p\), and flux velocity, \(v\), at \(x=0\) can be expressed in terms of the pressure and velocity at \(x=L\) as  
\[\left[ \begin{array}{c}p\\ v\end{array} \right]_{x=0}=\mathbf{M}\left[ \begin{array}{c}p\\ v\end{array} \right]_{x=L}=\left[ \begin{array}{cc}M_{11}&M_{12}\\ M_{21}&M_{22}\end{array} \right]\left[ \begin{array}{c}p\\ v\end{array} \right]_{x=L}. \tag{6}\]  
As illustrated in Figure 2, the complete system can be represented by \(N\) individual building blocks made of a Helmholtz resonator loaded in a slit of length \(L_{s}^{(n)}\) and a width \(w_{s}^{(n)}\), a cavity with a length \(L_{w}^{(n)}\), everything embedded in a main tube of width \(w_{t}\). In this paper we use upper-case letters for the geometric parameters outside the resonators and lower-case for those of the Helmholtz resonators. Sub-indexes represent slit (s), neck (n) cavity (c) or waveguide (w). The total length of the system can be obtained as follows:  
\[L=\sum_{n=1}^{N}L^{(n)}=\sum_{n=1}^{N}(L_{s}^{(n)}+L_{w}^{(n)}) \tag{7}\]  
Figure 2: Schematic diagram of a system of \(N\) building blocks made of a Helmholtz resonator with a cavity ending on a rigid backing. Description of the geometrical parameters of the system used in Section 2.2. The geometrical parameters corresponding to the Helmholtz resonator are indicated in red, those corresponding to the slit are indicated in blue, and those corresponding to the waveguide are indicated in green. The transfer matrix of the \(n\)-th building block, \(\textbf{M}_{b}^{(n)}\) may be written as  
\[\textbf{M}_{b}^{(n)}=M_{\Delta l_{slit}}^{(n)}M_{s}^{(n)}M_{HR}^{(n)}M_{s}^{(n)}M _{\Delta l_{slit}}^{(n)}M_{w}^{(n)}, \tag{8}\]  
where, \(M_{\Delta l_{slit}}^{(n)}\) represents the transfer matrix corresponding to the correction due to the change of section between the main tube and the Helmholtz resonator (if there is not change of section, this matrix is the identity matrix), \(M_{s}^{(n)}\) is the transfer matrix representing the propagation of waves through the slit along a distance \(\frac{L^{(n)}}{2}\), \(M_{HR}^{(n)}\) represents the transfer matrix of the Helmholtz resonators, and \(M_{w}^{(n)}\) is the transfer matrix of the propagation along the cavity. These matrices are defined as follows:  
* The transfer matrix for each step in the slit of the \(n\)-th building block, \(M_{s}^{(n)}\), reads as \[M_{s}^{(n)}=\left[ \begin{array}{cc}\cos\left(k_{s}^{(n)}\frac{L_{s}^{(n)}}{ 2}\right)&iZ_{s}\sin\left(k_{s}^{(n)}\frac{L_{s}^{(n)}}{2}\right)\\ \frac{L_{s}^{\prime}}{Z_{s}}\sin\left(k_{s}^{(n)}\frac{L_{s}^{(n)}}{2}\right) &\cos\left(k_{s}^{(n)}\frac{L_{s}^{(n)}}{2}\right)\end{array} \right],\] (9) where \(k_{s}\), \(Z_{s}\) are the wavenumber and the impedance in the slit. As it was previously mentioned, the term Ms is applied along a distance \(\frac{L^{(n)}}{2}\). Thus, this term must be considered twice in the transfer matrix of the building block (before and after the one corresponding to the Helmholtz resonator). * The transfer matrix for each resonator, \(M_{HR}^{(n)}\), is defined as \[M_{HR}^{(n)}=\left[ \begin{array}{cc}1&0\\ \frac{1}{Z_{HR}^{(n)}}&1\end{array} \right],\] (10) being \(Z_{HR}\) the acoustic impedance of the Helmholtz resonator. Notice that this definition is valid when \(\lambda>>w_{n}\), where the Helmholtz resonators can be considered as point resonators. * The transfer matrix for the radiation correction of the slit to the free space, \(M_{\Delta l_{slit}}^{(n)}\), can be defined as \[M_{\Delta l_{slit}}^{(n)}=\left[