## 3 MPP Samples
### Models Compensation  
The experimental results for the MPP configurations showed some discrepancies between the measured data and both theoretical models decribed in Section 2. There were two important differences in common for each configuration:  
* A small frequency shift of the sound absorption peaks;
* Absorption bandwidths larger than expected.  
The first issue is connected to the samples' mounting procedure (Figure 9).  
All samples were rounded and had a diameter slightly smaller than the sample holder. This did not guarantee the samples to be firmly mounted inside the tube. In order to avoid the side gaps between the specimen and the tube and to respect the mechanical boundary conditions, small strips of adhesive tape were applied to clamp the edges as much as possible. This caused small vibrations of the samples and a part of the energy was dissipated through the adhesive strips, increasing the absorption bandwidth.  
The absorption bandwidth is due to the variability of the geometrical parameters of the samples: mainly, the irregular shapes and sizes of the perforations are the reasons why the measured data bandwidths appear to be larger than the predicted ones [33; 34]. In particular, the smaller the perforations were, the bigger were the discrepancies. The perforations' perimeter was larger than expected and, consequently, a bigger absorption bandwidth occured. Instead of acting on the sample mounting procedure, a new compensating impedance has been added to the models, taking into consideration the discrepancies.  
Figure 8: Data flow: excitation signal generation, acquisition of the impulse responses \(h_{1}(t)\) and \(h_{2}(t)\), processing, comparison with models, output of results.  
Figure 9: Details of the sample MPP02 mounted inside the impedance tube.  
Considering the electro-acoustic analogy of the single-layer MPPs, the following changes were applied:  
* \(R_{comp}\), a resistance in series with the MPP impedance, associated to the dissipative losses due to the irregular perforations;
* \(L_{comp}\), a new inductance in parallel with the MPP surface impedance, associated with the displacement of air along the boundary of the sample and the sample vibration [35].  
This is still valid for the double-layer configurations: furthermore, the mounting inaccuracy is emphasized by the doubling of the MPP layer. In a DL configuration, two different resistances in series with the MPP impedance and two inductances in parallel with the MPP impedances have been considered, one for every MPP layer. Thus, in order to consider the geometrical dimension issues of the steel samples and the main problem connected to the sample mounting procedure, some changes to the equivalent circuits were made, as shown in Figure 10.  
The explicit equation of the surface impedance in the case of a DL-MPP configuration after compensation is the following:  
\[Z_{1,EF,comp}=(Z_{MPP1,EF}L_{comp,1})/(Z_{MPP1,EF}+L_{comp,1})+R_{comp,1}+Z_{c} \tag{14}\]  
\[Z_{2,EF,comp}=(Z_{MPP2,EF}L_{comp,2})/(Z_{MPP2,EF}+L_{comp,2})+R_{comp,2}+Z_{c} \tag{15}\]  
\[Z_{EF,comp}=Z_{1,EF,comp}+Z_{0}\frac{Z_{2,EF,comp}cos(kD_{2})+iZ_{0}sin(kD_{1})} {Z_{0}cos(kD_{1})+iZ_{2,EF,comp}sin(kD_{2})} \tag{16}\]  
In the present work, the values of \(R_{comp}\) and \(L_{comp}\) for the DL-MPP combinations were estimated by trial and error, trying to match the experimental data.