## 5 Parametric Study and Panel Optimization
Conversely, using the 2 mm hole with 6 mm spacing resulted in a more evenly distributed sound absorption. The third scenario that was investigated, following the previous discussion, was that both the layers could be micro-perforated. In this case, only air gap thickness and hole spacings were changed given the constraints applied to layer thickness and hole dimensions. The best result, according to previously stated criteria, was given by a first micro-perforated layer with a 5 mm spacing and a 30 mm air gap, followed by the second micro-perforated layer with a 3 mm spacing and a 25 mm air gap. The resulting \(\alpha_{\mathrm{avg}}\) was about 0.72. The analysis of the best performing combinations showed that all of them shared a 3 mm spacing for the outermost layer, while, for the other one, the spacing varied between 4 and 6 mm. The parameters that minimized average \(\alpha\) were, as for the previous cases, a large spacing (exceeding 14 mm) for the holes in the outermost layer. Parametric analysis showed that, by increasing the spacing between holes in the inner layer (Figure 10a), the two resonant peaks in the absorption curve moved towards the lower frequencies, obtaining the maximum values around 5 mm spacing. Similarly, the increase in the hole spacing for the external layer (Figure 10b) caused a more evident shift towards lower frequencies, together with a significant drop in the sound absorption coefficient pertaining to the second peak. It is interesting to point out that, starting from the optimized configuration, a change in the thickness of the air gaps caused a shift of the peaks with no significant reduction of the peaks of absorption (Figure 10 c,d). Thus, the effect on the average absorption was generally negligible, provided that the peaks fell within the desired frequency range. The final optimization was carried out by adopting a three layer scheme, with two inner micro-perforated layers and the perforated layer being used as the exterior finishing surface, to offer more protection to the MPM. The boundary conditions for the optimization were the same used in the previous cases, with the only difference being the presence of the extra layer. The best possible result that could be achieved in this case was an \(\alpha_{\mathrm{avg}}\) of about 0.81, but the overall thickness was approximately 10 cm, which was well beyond the 7 cm limit. However, accepting a 2% reduction in the mean \(\alpha\) it was nonetheless possible to obtain a suitable panel configuration with an overall thickness of 70 mm and a sound absorption coefficient never falling below 0.7 in the preferred frequency interval. The panel arrangement included a perforated layer with 4 mm holes that were spaced at 10 mm, forming a 20 mm air gap with the subsequent layer being made of a micro-perforated membrane with 4 mm spacing over a 20 mm air gap, and, finally, a second micro-perforated layer with 6 mm spacing with a 25 mm air gap. Figure 9: Sound absorption coefficient of a double layer PP-MPM panel for different parameter modifications: (**a**) distance between holes of the innermost layer (microperforated); and, (**b**) hole diameter and distance between holes of the outermost layer. The analysis of the best performing parameter combinations (Figure 11) showed that, once again, the spacing in the micro-perforated layers played a major role, as in the first one it varied between 5 and 6 mm, while in the second one it was mostly equal to 3 or 4 mm. With reference to the PP, the optimal hole diameter varied between 3 and 6 mm provided that the spacing varied accordingly between 6 and 12 mm (so that the ratio remained optimally around 2.3 in most of the cases and the perforation rate was about 13%). The air gaps had to vary between 20 and 30 mm, with the one closest to wall, which could span over a much larger interval. Conversely, the worst combinations resulted from a very small perforation rate (below 1%) for the outermost layer, independent of the other parameters.