# A Metawindow with Optimised Acoustic and
## 2 Materials and Methods
### Numerical Analysis Setup  
For the first stage, namely the numerical simulation, the FEM model is used. The 3D domain is filled with air, where air density (\(\rho=1.215\) kg/m\({}^{3}\)) and sound speed (\(c=343\) m/s) at room temperature (\(T=20\)\({}^{\circ}\)C) are used. The outdoor boundary is characterised by a plane wave radiating towards the receiving environment with a pressure amplitude of 1 Pa and an airspeed of the sound of 343 m/s, as shown in Figure 2a. The simulation study is performed using frequency domain analysis covering 300 to 5000 Hz with a step size of 100 Hz. The lower bound of the frequency range is limited by the anechoic chamber's cut-off frequency of 300 Hz used in the experimental analysis. As it can be seen from the  
Figure 1: (**a**) Schematic representation of the AMW unit structure (**left**) with an exploded view (**right**) where \(W\) is the acoustic metawindow (AMW) unit width (\(W=0.13\) m). (**b**) Details of the AMW unit and a unit cell, including their geometries and dimensions. Here, \(L\) (\(L=0.4\) m) and \(W\) are the AMW unit constants, length and width, respectively. The characterising measurements of the ventilation cavities are respectively \(B=0.095\) m and \(C=0.067\) m. Other geometrical parameters are the number of space cavities (\(N\)), length of each cavity (\(l\)), rotation angle (\(\alpha\)), and channel air gap (\(g\)) (i.e., distance between one flank and the other), and their dimensions are \(N=4\); \(l=0.066\), \(0.05\), \(0.033\), and \(0.016\) m; \(\alpha=55\)°; and \(g=0.024\) m. This geometrical configuration of the AMW unit and the related measures specified here are used for the numerical and experimental analyses.  
geometrical characterisation and measures in Figure 1, the 3D model is complex, and the required mesh size to reach 5000 Hz is 343/6/5000 = 0.011 m. Such fine mesh results in very computationally intensive models. For this reason, a slightly coarse minimum mesh size of 0.023 m is used, which has no problem in meeting the convergence criterion. The simulation accuracy is later validated experimentally.  
The acoustic performance is evaluated by measuring the insertion loss (_IL_) at three receiving points (A, B, and C), respectively, where A and B are in front of the AMW unit surface (0.2 and 0.5 m away) and C is at the side of it (0.2 m away) (Figure 2b). The choice of these three specific points instead of a wider spatially averaged pressure level on the radiating side is mainly due to the lack of sound power measurement equipment.  
Figure 2: (**a**) Schematic of boundary conditions used in numerical acoustics simulations; (**b**) 2D representation of experimental settings with \(m\) = 2 m being the length of the anechoic chamber’s inner area; (**c**) photograph of the experimental setup in the anechoic chamber.  
The _IL_ is calculated by the difference between the sound pressure level (_SPL_) at each of these points with and without the AMW unit applied. Therefore, the _IL_ of the AMW unit is calculated as  
\[IL_{AMW}=SPL_{w\alpha AMW}-SPL_{w\alpha AMW}\ (\mathrm{dB}) \tag{1}\]  
where:  
* \(SPL_{w\alpha AMW}\) = A-weighted _SPL_ of experimental configuration without the AMW unit (dB);
* \(SPL_{w\alpha MW}\) = A-weighted _SPL_ of experimental configuration with the AMW unit (dB).