## 5 Parametric Study and Panel Optimization
The effect of their variations was investigated starting from the selected best performing combination of parameters. The increase in hole spacing in the first MPM (Figure 12a) yielded a general shifting of the peaks towards lower frequencies, with significant changes in their amplitude. In particular, a higher perforation rate made the high frequency peaks less effective, while decreasing the perforation rate (by increasing the distance) made the low frequency peak less effective. Similarly, increasing hole spacing in the second MPM (Figure 12b) resulted in shifting the peaks towards lower frequency (except the central peak which remained substantially stable around 1.25 kHz), with the interesting effect that, when the distance was larger than 4 mm, the third peak started merging with the second one. The variations in hole diameter and spacing for the outermost layer were analyzed as a function of hole diameter and, consequently, in terms of the perforation rate (Figure 12c), although viscous losses due to actual hole dimension may also affect the result. A low perforation rate implies a shifting of the peaks towards low frequencies and a significant depression of the second and third peaks, thus dramatically reducing the average sound absorption coefficient, as shown in Figure 3c, in agreement with the results of the optimization. Conversely, an increase in perforation rate beyond  
Figure 10: Comparison of the predicted sound absorption coefficients for double-membrane structure. (**a**) distance between holes of the innermost layer; (**b**) distance between holes of the outermost layer; (**c**) thickness of first air gap; and, (**d**) thickness of second air gap. the optimal value only affects the high frequency peak, while the absorption due to the MPM remains substantially unchanged. Finally, the effect of the air gaps was investigated (Figure 12d). Increasing the thickness of the layer closest to the rigid surface mostly shifted the first peak towards the lowest frequencies, without affecting the maximum absorption, except for a small reduction of the value in the \"valley\" between the two peaks. It is interesting to notice that the high frequency peak remained unchanged, independent of the air gap value. When the intermediate air gap was increased, it was the second peak (and to  
Figure 11: Plot of best 200 combinations of the seven input parameters returned by the optimization and yielding an average absorption coefficient (calculated over the one-third octave bands from 400 Hz to 2.5 kHz) within 2% of the maximum, equal to 0.81. Figure 12: Comparison of the predicted sound absorption coefficients for triple-membrane structure as a function of individual parameter changes: (**a**) spacing of the first MPM; (**b**) spacing of the second MPM; (**c**) perforation rate of PP; and, (**d**) air gaps thickness. Reference parameters were: 25 mm back cavity; 1 st MPP with 6 mm spacing and 0.27 mm holes; 20 mm cavity; 2 nd MPP with 4 mm spacing and 0.27 holes; 20 mm cavity; PP with 4 mm holes with 10 mm spacing. a smaller extent the third one) that showed the largest variations, moving towards low frequencies and changing amplitude. A too thin air gap caused the peaks to drop dramatically, thus making the absorber ineffective. However, a reduction in the absorption values also appeared when the air gap was too large. Finally, an increase in thickness of the air gap closer to the exterior layer had no influence at all on the first peak, while it shifted the second and particularly the third one towards lower frequencies. Again, a variation in peaks amplitude appeared, but they remained around and above 0.8, with larger variations appearing in the valleys. Once the best configuration was identified through the optimization procedure, the resulting information was used to assemble the panel (Figure 13a) and measure its sound absorption coefficient in the impedance tube according to the previously described methods. The results showed (Figure 13b) fairly good agreement between measurements and predictions.