## 3 Methods
### Analytical Prediction Formulation  
In order to calculate the overall acoustic impedance, and hence the corresponding sound absorption coefficient, the transfer matrix method was used [10], according to which, if the bottom of layer \(i\) has an impedance of \(z_{\text{si}}\), and the layer \(i\) has a characteristic acoustic impedance \(z_{\text{i}}\), and then the impedance at the bottom of layer \(i+1\) is:  
\[z_{\text{si+1}}=\frac{-jz_{\text{si}}z_{\text{i}}\cot(k_{\text{xi}}d_{\text{i} })+z_{\text{i}}^{2}}{z_{\text{si}}-jz_{\text{i}}\cot(k_{\text{xi}}d_{\text{i} })}, \tag{1}\]  
where \(j\) is the imaginary unit, \(k_{\text{xi}}\) is the wavenumber for layer \(i\), and \(d_{\text{i}}\) the corresponding thickness. The above formula can be recursively applied to calculate the surface impedance of a multi-layered absorbent, as many researchers have demonstrated [10; 25]. In the subsequent sections, its application to the specific case of a series of perforated and micro-perforated panels will be further validated. Obviously, as the final value of the impedance results from the recursive application of Equation (1), it is not possible to obtain a simple and straightforward expression to relate sound absorption and the different input parameters. The impedance for the different layers can be calculated according to theoretical expressions briefly summarized below. For an air layer (with thickness \(d\)), in front of a rigid surface (for which \(z_{\text{si}}\) is infinite), Equation (1) yields:  
\[z_{\text{si+1}}=-j\;pc\;\cot(kd), \tag{2}\]  
where \(\rho\) is the air density (assumed equal to 1.21 kg/m\({}^{3}\)) and \(c\) is the speed of sound in air (assumed equal to 340 m/s). The presence of a perforated panel behaving as a Helmholtz resonator in the front of the air cavity alters its impedance by the addition of mass (\(j\omega m\)) and resistance (\(r_{\text{m}}\)) terms, so that:  
\[z_{\text{p}}=r_{\text{m}}+j[\omega m-\rho c\cot(kd)], \tag{3}\]  
\(m\) being the mass of the vibrating plug of air within the perforations and \(\omega\) the angular frequency. The mass of the vibrating plug must take the effect of the radiation impedance and of the mutual interactions between neighboring elements into account, resulting in the addition of an \"end correction\" to the thickness of the panel. A typical expression for the mass is [10]:  
\[m=\rho/\varepsilon\;[t+2\delta a+(8v/\omega\;(1+t/2a))^{0.5}], \tag{4}\]  
where \(\varepsilon\) is the perforation rate, \(t\) is the panel thickness, \(a\) is the hole radius, \(v\) is the kinematic viscosity of air (assumed equal to 1.5\(\cdot\)10\({}^{-7}\) m\({}^{2}\)/s), and \(\delta\) is the end correction factor. The latter can be calculated by means of different formulas, among which one that is suitable for more open structures [10] assumes that \(\delta=0.8(1-1.47\varepsilon^{1/2}+0.47\varepsilon^{3/2})\). The resistance term \(r_{\text{m}}\) is responsible for the losses within the device and, consequently, of the absorption as a function of frequency. For normal holes (not sub-millimeter in size), and if no additional porous layers are used, the term can be expressed as:  
\[r_{\text{m}}=\rho/\varepsilon\;(8v\omega)^{0.5}\;(1+t/2a), \tag{5}\]  
Finally, for microperforated panels the losses occur because of viscous boundary layer effects in the perforations, provided that the diameter is sub-millimeter and comparable to the layer thickness.