## 2 Modeling the Sound Absorption of Micro-Perforated Panels (MPP)  
A micro-perforated panel consists of a thin panel with a specific perforation ratio made by a distribution of sub-millimeter holes backed by an air cavity and a rigid wall, as shown in Figure 1.  
The acoustic complex impedance of an MPP (Z) is the result of different contributions: the real part of the impedance that needs to be matched to the air impedance (\(Z_{0}\)) and the imaginary part of the impedance provided by the air cavity and the perforations. The values of the perforation diameter \(d\), the thickness of the panel \(t\), the porosity \(\phi\) and the air cavity thickness \(D\) are the four constitutive parameters that influence the range of frequencies absorbed and the bandwidth as well. Taking into account [PERSON]'s definition [8] and according to [PERSON]'s notation [13], it is possible to define the input complex impedance of a single-layer (SL) MPP \(Z_{1}\) as follows:  
\[Z_{1,SL}=Z_{holes}+Z_{edge}+Z_{c}. \tag{1}\]  
The impedance \(Z_{holes}\) defines the viscous dissipation within the holes, \(Z_{edge}\) the distortion of the flow in the perforation edges and \(Z_{c}\) is the resonance in the air cavity:  
\[Z_{holes}=\frac{\Delta p}{u}=i\frac{\omega\rho_{0}}{\phi}\left[1-\frac{2}{s \sqrt{-i}}\frac{J_{1}(s\sqrt{-i})}{J_{0}(s\sqrt{-1})}\right]^{-1} \tag{2}\]  
Figure 1: Schematic representation of a single-layer micro-perforated panel (MPP) and its dimensional parameters. \[Z_{edge}=R_{s}+iX_{m}=2\sqrt{2\eta\omega\rho_{0}}+i\frac{\omega\rho_{0}0.85d}{F( \epsilon)} \tag{3}\]  
where \(\Delta p\) is the pressure difference at both sides of the tubes, \(u\) is the particle velocity in the tube, \(\rho_{0}\) is the air density, \(\eta\) is the air dynamic viscosity, \(\phi\) is the porosity, \(s=d\sqrt{\frac{\omega\rho_{0}}{4\eta}}\) is the perforation constant (\(d\) being the diameter of the holes), \(J_{1}\) and \(J_{0}\) the Bessel functions of first-class and order 1 and 0, respectively, and \(F(\epsilon)\) is the Fok function, a correction factor of the mass reactance [13] and \(\epsilon=\sqrt{\phi}\). Considering all the parameters introduced so far, it is possible to study the equivalent electro-acoustic system for the MPP, as shown in Figure 2.  
In practice, multiple-layer MPPs are usually preferred because of their extended absorption band. A double-layer MPP (DL-MPP) consists of two MPPs with impedances \(Z_{MPP,1}\) and \(Z_{MPP,2}\) and two air cavities with impedances \(Z_{c1}\) and \(Z_{c2}\) (see Figure 3). Considering the sound waves passing through the DL-MPP system from left to right at normal incidence, the input impedance to the DL-MPP system is:  
\[Z_{1,DL}=Z_{holes,1}+Z_{edge,1}+Z_{0}\frac{Z_{2,DL}cos(kD_{2})+iZ_{0}sin(kD_{1 })}{Z_{0}cos(kD_{1})+iZ_{2,DL}sin(kD_{2})} \tag{4}\]  
with  
\[Z_{2,DL}=Z_{edge,2}+Z_{holes,2}+Z_{c2}. \tag{5}\]  
Therefore, the absorption of a DL-MPP globally depends on eight constitutive and geometrical parameters, four for the first layer and four for the second layer: the diameter of the holes (\(d_{1}\), \(d_{2}\)), the thickness (\(t_{1}\), \(t_{2}\)), the distance (\(D_{1}\), \(D_{2}\)), and the porosity (\(\phi_{1}\), \(\phi_{2}\)). For this reason, it is difficult to predict the acoustic performance of a DL-MPP and to find, a priori, a combination of these parameters providing the maximum absorption in a specific frequency range. Figure 3: Double-layer (DL)-MPP schematic representation (**left**) and the corresponding equivalent electrical system (**right**). Figure 2: Single-layer (SL)-MPP schematic representation (**left**) and the corresponding equivalent electrical system (**right**). [PERSON] and [PERSON] [12] introduced the so-called equivalent fluid (EF) model following the Johnson-Champoux-Allard approach with an equivalent tortuosity [5]: they assumed an MPP coupled at both sides to a semi-infinite fluid.