## 5 Parametric Study and Panel Optimization  
Given the complex relationship between the characteristics of each individual layer, air gaps and the overall absorption, and while considering the good agreement observed with analytical relations, a parametric study was carried out to better understand which combinations of parameters may maximize sound absorption. The parameter chosen as the target value of the optimization was the mean of the sound absorption coefficients that were calculated over the one-third octave bands from 400 Hz to 2.5 kHz (\(\alpha_{\text{avg}}\)). No conventional single number descriptor was used to underline the complete customization of the process. Among the possible optimized solutions, the one that ensured  
Figure 7: Comparison of the predicted and measured sound absorption coefficients for (**a**) double-membrane structure: micro-perforated membrane M4 followed by a 25 mm cavity combined with M5 followed by a 25 mm cavity. (**b**) triple-membrane structure: panel D, 10 mm cavity, micro-perforated membrane M4 followed by a 15 mm cavity, combined with M5 followed by a 30 mm cavity. an average \(\alpha\) not differing by more than 2% from the maximum and having the smallest thickness was selected. The overall thickness of the multi-layer combination was arbitrarily limited to 70 mm, because beyond that threshold there are several conventional (and likely cheaper) solutions that could be used. The thickness of the perorated layers was kept constant at 5 mm in order to ensure good mechanical resistance for the external finishing, while the thickness of the microperforated layers was kept at 0.09 mm to avoid changing the available materials. First, the case of a double leaf perforated panel was considered, obtaining an \(\alpha_{\text{avg}}\) of 0.45 that resulted from a first panel with 1 mm holes with a 8 mm spacing, over a 15 mm air gap, and a second panel with 1 mm holes with a 4 mm spacing, over a second air gap 25 mm thick. The analysis of the best and worst combinations showed that the essential condition that was needed to obtain high absorption was to keep hole diameter to 1 mm for both layers, with a 4 mm spacing for the outermost layer (resulting in a 4.9% perforation rate). Conversely, the lowest average absorption (below 0.05) was obtained when the spacing between the holes of the outermost layer exceeded 14 mm, while the hole diameter was 1 mm. It can be observed (Figure 8a) that the distance between the two peaks increases by increasing the hole spacing of the innermost layer, mostly because the lowest frequency peak moves towards lower frequencies at a faster pace than the high frequency peak. An increase in the spacing of the outermost layer (Figure 8b) moves the low frequency peak towards even lower frequencies, and at the same time significantly reduces the high frequency peak, thus resulting in the observed poor performance. Similarly, by increasing the thickness of the first air gap (or by decreasing that of the second air gap) the two peaks move away and out of the desired frequency range. A second optimization exercise was carried out with reference to the combination of one perforated and one micro-perforated layer, while assuming the first to be the outermost layer. In this case the best performance yielded an \(\alpha_{\text{avg}}\) of 0.63 when the first air gap (closest to wall) was 35 mm, the distance between holes in the MPM was 5 mm, the second air gap was 25 mm, and the perforated panel had 1 mm holes with a 2 mm spacing. The analysis of the best performing configurations showed that a spacing of 5 or 6 mm in the MPM was an essential condition, while the hole diameter had to be 1 or 2 mm in the perforated layer, while their spacing had to be 2 or 6 mm, respectively. As in the previous case, the worst performance was attributed to combinations where the perforated layer had a hole diameter of 1 mm, and a spacing exceeding 14 mm. Parametric analysis showed that increasing the distance between holes in the MPM  
lower frequencies.