## 2 Methods
### Analytical and Numerical Design of a Novel Noise Barrier  
After comparing different types of noise barriers, one made of sonic crystal was selected [27; 28]. Sonic crystals are structures designed to control, direct, and manipulate sound waves; their effectiveness depends not on constituent materials but on the geometric characteristics of their conformation [4]. The use of sonic crystals to make an acoustic barrier--from now on called sonic crystal acoustic barrier (SCAB)--provides an innovative solution that is still uncommon in commercial applications but effective in preventing noise traffic while allowing a breathable surface that can control solar radiation, and is achievable  
Figure 4: Measured noise levels on a representative full day. They are distinct for daytime (06:00–22:00) and nighttime (22:00–06:00) and for the four measurement points (1 to 4). The red dashed lines represent the noise limits imposed by the Italian Regulation. with materials having a low environmental impact, i.e., recycled and recyclable [3,29]. The main factors influencing the properties of sonic crystals are the geometry of the unit cell, the filling fraction, and the size of the scatterers. They control the formation of the targeted frequency bands and their width, and thus the sound-insulating capacity of the sonic crystal [3,30,31]. In the preliminary design stage, the sound absorption and sound insulation curves were taken from the literature [3,29], while the barrier height, given the target insertion loss (IL), was pre-dimensioned through [PERSON]'s method [32,33]. An acoustic dimensioning of a barrier consists of determining its position concerning the source and the receiver(s), its height, \(H\), and its length, \(L\), to achieve the target insertion loss, \(IL_{Target}\), to lower the noise level at the receiver(s) below the regulatory noise limits [34]. Thanks to meticulous experimental work with impulsive signals, [PERSON] has built a noteworthy chart that shows the absolute values of the attenuation, \(\Delta L\) in dB, produced by the edge of a thin noise barrier as a function of the Fresnel number \(N=2\delta/\lambda\), where \(\delta\) is the geometric difference between the direct propagation path in the absence of the barrier and the minimum path of the diffracted waves above the upper edge of the barrier, and \(\lambda\) is the wavelength [33]. \(\delta\) can be calculated from the distance between the barrier and the source, \(A\), the distance between the barrier and the receiver, \(B\), the height of the source, \(h_{s}\), the height of the receiver, \(h_{r}\), and the assumed height of the noise barrier, \(H\) (see Figure 5). For a point source and a long noise barrier, [PERSON] and [PERSON] [35] derived an analytical formula for the attenuation that can conveniently substitute [PERSON]'s chart:  
\[\Delta L=5+10\cdot log\left(\frac{\sqrt{2\pi N}}{\tanh(2\pi N)}\right)\,[\text {dB}] \tag{1}\]  
For a linear sound source, like a road, the angular dependence of \(\delta\) must be considered, subdividing the line source into elemental sections [35]. It is worth noting that the Fresnel number and, thus, the attenuation are a function of the wavelength, i.e., of the frequency; therefore, the calculation must be repeated for each frequency band of interest. Next, the height of the barrier, \(H\), can be determined as the sum of the minimum height, \(h_{min}\), and the effective height, \(h_{eff}\). The minimum height represents the barrier's height to obstruct the direct view between the source and the receiver (see Figure 5). The effective height is the added height to achieve the desired insertion loss, \(IL_{Design}\).