## 5 Discussion  
The absorption performances of the different materials in backed or unbacked configurations are summarized in Table 3 with \(\overline{\alpha}_{\text{unb}},\overline{\alpha}_{55}\) and \(\overline{\alpha}_{20}\), the total absorptions calculated between 80 Hz and \(f_{\text{cut-on}}\) for the unbacked case and for cases in which the materials are backed by a rigid cavity with depths of 55 mm and 20 mm, respectively. In each case,  
Figure 11: Simulated (dotted) and measured (plain) absorption properties of MCP1 (red), MCP2 (magenta), UMPP (black) and MPP (blue) backed by an open cavity with a depth of 152 mm, assuming normal incidence. corresponds to the bandwidth-to-thickness ratio of the absorber, where \(\overline{D}\) is the overall thickness of the absorber (\(\overline{D}=t\) for the unbacked case and \(\overline{D}=t+D\approx D\) for the backed case). \(\Delta\lambda\) is defined as  
\[\Delta\lambda=\lambda_{\text{max}}-\lambda_{\text{min}}=c_{0}\bigg{(}\frac{1}{ f_{\text{min}}}-\frac{1}{f_{\text{max}}}\bigg{)} \tag{11}\]  
The frequency bounds \(f_{\text{min}}\) and \(f_{\text{max}}\) exist such that the absorption exceeds 0.8 between these bounds. If the absorption stays greater than 0.8 above the first duct cut-on frequency \(f_{\text{cut-on}}\), then \(f_{\text{max}}\) is equal to \(f_{\text{cut-on}}\) because the total absorption is evaluated up to \(f_{\text{cut-on}}\). If the absorption stays greater than 0.8 at very low frequencies, then \(f_{\text{min}}\) is equal to \(f_{\text{low}}\) (80 Hz), the lowest frequency limit for the Kundt tube measurements. If the absorption stays below 0.8 up to \(f_{\text{cut-on}}\), then \(\Delta\lambda\) cannot be calculated. A possible indicator of the high broadband acoustic efficiency of the absorbers is given by large values of the index \(\overline{\alpha}\)\(\Delta\lambda/\overline{D}\). This corresponds to the bold numbers in Table 3.  
Regarding rigidly backed micro-perforated absorbers, the causality principle imposes an upper bound on their bandwidth-to-thickness ratio [20], given by  
\[\frac{\Delta\lambda}{D}\leq\frac{4\pi^{2}}{\left|\log(1-\alpha_{0})\right|}, \tag{12}\]  
stating that a large absorption \(\alpha_{0}\) can only be achieved over a limited bandwidth \(\Delta\lambda\) or that it requires a deep cavity. By choosing \(\alpha_{0}=0.8\), Equation (12) leads to the ultimate bandwidth-to-thickness ratio, \(\Delta\lambda/D\leq 24.5\). This is verified in Table 3 through the values of \(\Delta\lambda/\overline{D}\big{|}_{\text{55}}\) or \(\Delta\lambda/\overline{D}\big{|}_{\text{20}}\) not exceeding 24.5. Regarding unbacked absorbers, to the authors' knowledge, no similar criterion has yet been derived, and the ratio \(\Delta\lambda/\overline{D}\) can reach quite high values, as seen by \(\Delta\lambda/\overline{D}\big{|}_{\text{mb.}}\). In Table 3, the highest acoustic efficiency for unbacked absorbers is reached by UMPP, being slightly above that of MCP1, due to the ability of the UMPP to reach similar absorption values compared to MCP1, albeit over a reduced bandwidth but with a thickness that is half less than that of MCP1. However, given the minute thicknesses involved, more weight should be given to \(\overline{\alpha}\) and \(\Delta\lambda\) in the unbacked case, with respect to \(\overline{D}\). MCP1 then outperforms the UMPP with absorption values exceeding 0.8 over 5.5 octaves, leading to a total absorption of 0.84. This is due to a near-optimal channel radius, 14 \(\upmu\)m, for MCP1. This leads to a specific transfer resistance, equal to 1.9 (Equation (5) and Figure 3a) over an ultra-broad frequency range. This value is nearly equal to the theoretical optimal resistance value for an unbacked MCP, equal to 2 [8]. Such resistance is achieved over a maximum bandwidth controlled by the Shear number, Sh, which should stay below 1 as a necessary condition for the VBL to occupy the full channel radius and efficiently dissipate the acoustic energy.