## 5 Application in a Case Study: Use of MPPs as a Ceiling Acoustic Treatment
Table 6 also provides the T\({}_{30}\) mean values derived from the corresponding GA simulations to keep the comparison with the standard simulation procedure. Certainly in the GA model all of the temporal delays and acoustic behaviors due to the peculiarity of MPPs are approximated by energy parameters (absorption coefficients). With regard to the results in terms of reverberation time, the main contribution of the acoustic treatment is relevant at 500-1000 Hz, as expected, and still significant at 250 and 2000 Hz, due to the broadband performance obtained with DL configurations. The effects of treating a lecture hall with such a material instead of a common porous material are positive not only because they cover the frequency range occupied by the human voice, but also because they compensate for the typical excessive reverberation that undermines the verbal communication in historical rooms. Figure 17: Real (**left**) and imaginary (**right**) parts of complex specific admittances from data for MPP material (solid line) along with fitted filter model (dashed line) used in the FDTD simulation. Dotted lines show underlying resonances in the filter approximation. Concerning the room criteria related to intelligibility, it is quite intuitive that in an optimal condition for the verbal communication their values should be as uniform as possible throughout the space. Therefore, the spatial behavior of sound clarity (C\({}_{50}\)) [50] is provided versus the distance between the source and the receiver (see Figure 18). The same five conditions already seen in Table 6 are reported in the graph: the measured values, the calibration outcomes and the effects of the acoustic treatment according to FDTD and GA results. On average, it is possible to quantify an increase of C\({}_{50}\) values at mid frequencies higher than 4 dB, corresponding to four times the JND of the sound clarity. The acoustic treatment allowed to significantly increase the early-to-late ratio (C\({}_{50,\text{M}}\approx 2\) dB) from a poor acoustic condition (C\({}_{50,\text{M}}\approx-2.5\) dB). Figure 18 also shows a good match among all the slopes of the linear regressions involved. This is probably due to the fact that the calibration was achieved by using an energy parameter, the reverberation time, that is derived from the energy decay curve as the sound clarity. Finally, an overview is provided about other recent works that handle the micro-perforated panels with wave-base simulation methods. Table 7 helps to place the present work in a framework of similar research studies to highlight similarities and methodological differences. Concerning the boundary conditions, it should be highlighted that only in reference [65] an extended reaction model for SL-MPP absorbers was used, including the \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline \(\mathbf{T_{30}}\) (\(s\)) & \(\mathbf{125\,Hz}\) & \(\mathbf{250\,Hz}\) & \(\mathbf{500\,Hz}\) & \(\mathbf{1000\,Hz}\) & \(\mathbf{2000\,Hz}\) & \(\mathbf{4000\,Hz}\) \\ \hline Without MPP (Meas.) & 1.48 & 1.36 & 1.60 & 1.84 & 1.83 & 1.51 \\ Without MPP (FDTD) & 1.44 & 1.25 & 1.54 & 1.73 & 1.70 & 1.45 \\ Without MPP (GA) & 1.43 & 1.25 & 1.62 & 1.79 & 1.78 & 1.52 \\ \hline With MPP (FDTD) & 1.32 & 0.92 & 0.88 & 1.22 & 1.52 & 1.44 \\ With MPP (GA) & 1.33 & 1.06 & 0.87 & 1.00 & 1.34 & 1.50 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Trend in frequency of T\({}_{30}\) mean values corresponding to the results of the measurements (“Meas”), the equivalent values derived from the calibrated FDTD model (“without MPP”) and the variations due to the introduction of MPPs (“with MPP”). The T\({}_{30}\) mean values derived from the corresponding GA simulations are provided as a term of comparison. Figure 18: Trend of sound clarity (C\({}_{50}\)) as a function of the sound source-receiver distance (values and linear regressions). “M” indicates that the values have been averaged over 500 and 1000 Hz.