## 2 Phenomenological and Numerical Models' Descriptions
### Phenomenological Model  
In phenomenological models, the material is considered as an equivalent dissipative compressible fluid and no longer as a two-phase element consisting of a structure with air-filled pores inside which dissipation and heat exchange mechanisms occur. Unlike the empirical approach, the phenomenological model does not derive from a simple optimization of the experimental data but from a theoretical treatment that allows describing the phenomena of viscous friction and heat exchange. In many cases, the phenomenological approach appears to be a balanced compromise between the number of parameters necessary for modeling, modeling complexity, and forecasting ability [9]. [PERSON] et al. [10] proposed an extended phenomenological model based on viscosity and heat dissipation. This model requires three parameters to characterize the material: resistivity, porosity, and structure factor. The [PERSON] model is particularly suitable for materials with moderate porosity. According to this theory, the acoustic behavior of a rigid frame porous material is completely characterized by the specification of the characteristic impedance \(Z_{m}\) and of the complex wave number \(k_{m}\), as indicated in the following equations:  
\[Z_{m}=\frac{\rho_{0}c}{Y}*\sqrt{(s*\gamma)}*\sqrt{1-j\,\frac{f_{\mu}}{f}*}\sqrt {1-\frac{1-\frac{1}{\gamma}}{1-j\,\frac{f_{0}}{f}}} \tag{1}\]  
\[k_{m}=\frac{2\pi f}{c}*\sqrt{(s*\gamma)}*\sqrt{1-j\,\frac{f_{\mu}}{f}*}\sqrt{ 1-\frac{1-\frac{1}{\gamma}}{1-j\,\frac{f_{0}}{f}}} \tag{2}\]  
Here,  
* \(\rho_{0}\) is the density of air (kg/m\({}^{3}\))
* \(c\) is the sound speed (m/s)
* \(\gamma\) is the porosity
* \(s\) is the structure factor
* \(\gamma\) = cp/cv is the ratio of the specific heat at a constant pressure and volume of the air
* \(f\) is the frequency (Hz)In the previous formulas there is the structure factor \(s\); for its calculation it is necessary to adopt an indirect procedure that uses the first value of the maximum of the measured absorption coefficient. The use of this procedure is necessary because of the difficulties in directly measuring this parameter [11]. Once the trend of the absorption coefficient measured as a function of the frequency is known, we can obtain the structure factor \(s\) as follows:  
\[s=\left[\frac{c\ (2n+1)}{4\ d\ f_{2n+1}}\right]^{2} \tag{3}\]  
Here,  
* \(c\) is the sound speed (m/s)
* \(n\) is an integer, \(n=0\) corresponds to the first value of the maximum
* \(d\) is the thickness of the sample (m)
* \(f_{2n+1}\) is the corresponding frequency of the first value of the maximum of the measured absorption coefficient (Hz)  
In the previous formulas, \(f_{\mu}\) and \(f_{\theta}\) are defined as follows:  
\[f_{\mu}=\frac{R_{1}Y}{2\pi\rho_{0}s} \tag{4}\]  
\[f_{\theta}=\frac{R_{1}}{2\pi\rho_{0}N_{pr}} \tag{5}\]  
Here,  
* \(R_{1}\) is the resistivity (Ns/m\({}^{4}\))
* \(Y\) is the porosity
* \(\rho_{0}\) is the density of air (kg/m\({}^{3}\))
* \(s\) is the structure factor
* \(N_{pr}\) is the Prandtl number  
Once the characteristic impedance \(Z_{m}\) and the complex wave number \(k_{m}\) have been calculated, we can define the acoustic impedance \(Z\) for a porous material, with a thickness of d, as follows:  
\[Z=-jZ_{m}\cot(k_{m}d) \tag{6}\]  
Now we can define the acoustic impedance for unit area as follows:  
\[\zeta=\frac{Z}{\rho_{0}c} \tag{7}\]  
Finally, we can define the value of the absorption coefficient for a normal incidence as follows:  
\[\alpha=\frac{4\ Re(\zeta)}{\left|\zeta\right|^{2}+2\ Re(\zeta)+1} \tag{8}\]  
As we have said, this procedure requires prior knowledge of three parameters: resistivity, porosity, and structure factor. It will therefore be necessary to experimentally measure the values of these three parameters to estimate the absorption coefficient of the material.