## 4 Acoustic Properties of MCPs and MPPs Resonant Absorbers  
Measurements were carried out for the absorption performance of the four types of materials listed in Table 1 when backed by a rigid cavity with a variable depth, as illustrated in Figure 4 (right). Table 2 summarizes the absorption performance of the individual resonators with \(\overline{\alpha}\), the total absorption evaluated between 80 Hz and \(f_{\text{cut-on}}\); \(\alpha_{\text{max. }}\), the value of the first absorption peak; and \(f_{1},\ f_{2}\), the lower and upper frequencies delimiting the bandwidth at 50% of the maximum of the first absorption peak. Figure 8 presents the measured and simulated absorption results, assuming a cavity depth of 55 mm. Both the simulation and measurements show the effects of the two first acoustic resonances on the absorption performance of the rigidly backed materials up to 6500 Hz. The \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material—Cavity Depth (mm)—Backing Type** & \(\left|f_{1}-f_{2}\right|\) (Hz) & \(\mathbf{\pi_{\text{max.}}}\) & \(\overline{\mathbf{\pi}}\) \\ \hline MCP1—55—Closed & 300–3000 & 0.92 & 0.71 \\ MCP1—35—Closed & 435–4800 & 0.91 & 0.70 \\ MCP1—25—Closed & **480–6500** & 0.86 & **0.76** \\ MCP1—20—Closed & 512–6500 & 0.87 & 0.75 \\ MCP1—10—Closed & 700–6500 & 0.82 & 0.69 \\ MCP1—152—Open & 85–530 & 0.83 & 0.73 \\ MCP2—55—Closed & 540–2760 & 0.48 & 0.45 \\ MCP2—152—Open & 80–520 & 0.43 & 0.73 \\ UMPP—55—Closed & 330–2160 & **0.99** & 0.51 \\ UMPP—20—Closed & 800–3000 & **0.99** & 0.54 \\ UMPP—152—Open & 80–526 & 0.97 & 0.55 \\ MPP—55—Closed & 730–2300 & 0.40 & 0.42 \\ MPP\(-\)152—Open & 80–520 & 0.22 & 0.58 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Measured absorption performance of individual resonators. Bold values represent the best performance for each indicator. Figure 8: Simulated (dotted) and measured (plain) absorption properties of MCP1 (red), MCP2 (magenta), UMPP (black) and MPP (blue) backed by a rigid cavity with a depth of 55 mm, assuming normal incidence. highly dissipative MCP1 provides the greatest performance for a 55 mm cavity depth with a total frequency-averaged absorption of 0.71 up to 6500 Hz, with maximum absorption values of up to 0.92 and with the broadest bandwidth that extends to 50% of the maximum over a range of 2700 Hz around each resonance (see Table 2). The bandwidth of MCP1 resonances spans three octaves. The performance of MCP2, which is less dissipative, is more moderate than that of MCP1 with a total absorption of 0.45, a maximum absorption of 0.48 and a bandwidth that spans over one octave around the resonances. It should be noted that the correlation between the measured and predicted absorption values of MCP2 degrade at and around the resonances, whereas an acceptable correlation is observed for the other materials. Indeed, Figure 2b shows that the channels of MCP2 are not exactly circular but rather hexagonal. Moreover, on a larger scale (not shown), it can be observed that the channels are grouped into hexagonal patches whose boundaries present some micro-porosity at the junctions, which is not accounted for in the model. These defects contribute to the discrepancies observed in Figure 8 for MCP2. The absorption performance of MCP2 is similar to that of the MPP resonant absorber, which produces a total absorption of 0.42 and a maximum absorption of 0.6 at the second resonance, but a narrower efficiency bandwidth (less than one octave). Although the UMPP absorber achieves a mere total absorption of 0.51, it reaches 800 Hz, a near-unit absorption value with a resonance bandwidth of 1830 Hz, thereby achieving a so-called critically coupled condition [19] at a frequency for which all of the incident energy is dissipated by the UMPP. A second resonance can also be observed at 3650 Hz, reaching a high absorption value of 0.95 but with a narrow bandwidth of 450 Hz. Regarding the unbacked case, Table 2 shows that MCP1 and UMPPs still appear to be the most performant materials when backed by a rigid cavity.