## 2 Materials, Modeling and Characterization
### Modeling
Solving Equations (1) and (3) for \(v\), dividing by the perforation ratio and assuming a narrow channel behavior, \(\text{Sh}<1\), provide the following expression for the effective transfer impedance of an MCP with a micrometric channel diameter [8]:  
\[Z_{\text{MCP}}=\frac{8t\mu}{\sigma a^{2}(1-4B_{v})}+\text{j}\omega\frac{4\rho _{0}\,t}{3\sigma}\approx\frac{8t\mu}{\sigma a^{2}(1-4B_{v})}. \tag{5}\]  
It reduces to a pure acoustic resistance, with a constant value over a broad frequency range, and minute reactance due to a high perforation ratio, as illustrated in Figure 3a,b, respectively. Unlike MPPs, the outer correction terms of MCPs can be neglected due to their micrometric channel diameters and large perforation ratios. Equation (5) shows that the constant resistance of an MCP is controlled by the channel radius following an inverse-power law dependence with an order of 4. Conversely, Figure 3 shows that MPPs exhibit a much lower resistance than MCP1 as well as a greater reactance, which both increase with the frequency. Note that Equations (4) and (5) describing the effective transfer impedance of MCPs and MPPs assume a sufficient number of holes/channels per acoustic wavelength \(\lambda\), typically \(\Lambda<\lambda/4\), which is clearly verified for MCPs. Different load conditions can be applied behind the porous materials. In unbacked configurations, a simple model assumes a plane wave anechoic load, such that the overall input impedance over the material surface is given by \(Z_{\text{in}}=Z_{\text{MPP,MCP}}+Z_{0}\) with \(Z_{\text{MPP,MCP}}\) given by Equations (4) or (5) and \(Z_{0}=\rho_{0}c_{0}\), the plane wave impedance. From this model, Figure 3c shows that MCP1 is more dissipative than an MPP, which is essentially reactive over the whole frequency range. In practice, porous samples may be inserted at the open end of a Kundt tube to achieve an anechoic termination, as described in Section 2.3. Therefore, a cost-efficient realistic model of the radiation impedance of a flanged termination is required. The model established by [PERSON] et al. (2015) is used in the unbacked configuration. It has been verified using measurements behind micro-porous materials (see Section 3), assumed to be circular with radius \(R\). The radiation impedance is then modeled as \(Z_{\text{RAD}}=(1+R_{\text{RAD}})(1-R_{\text{RAD}})^{-1}\) with  
\[R_{\text{RAD}}=-\left(1+\text{j}\frac{k_{0}R}{c_{1}}\right)^{-(c_{2}+1)}, \tag{6}\]  
\(k_{0}=\omega/c_{0}\), \(c_{1}=0.8216\) and \(c_{2}=0.35\). The input impedance of the unbacked materials is then calculated as \(Z_{\text{in}}=Z_{\text{MPP,MCP}}+Z_{\text{RAD}}\). Backing the porous samples with a rigidly closed cavity with depth \(D\) leads to resonant absorbers whose input impedance is given by  
\[Z_{\text{in}}=Z_{\text{MPP,MCP}}+\text{j}Z_{0}\cot\left(\widetilde{k}_{0,R}D \right), \tag{7}\]  
with \(\widetilde{k}_{0,R}\), a complex wavenumber that describes viscothermal attenuation effects over the cavity walls ([PERSON], 2013). The imaginary part of \(Z_{\text{in}}\) vanishes at the resonance frequencies of the absorber. In the case of a rigidly backed MPP, the first zero is associated with a Helmholtz-type hole-cavity resonance that occurs at  
\[f_{\text{HC,MPP}}=\frac{1}{2\pi}\sqrt{\frac{\rho_{0}c_{0}^{2}}{mD}}, \tag{8}\]  
Figure 3: (**a**) Simulated specific transfer resistance, \(\text{Real}(Z_{\text{MPP,MCP}}/Z_{0})\); (**b**) Simulated specific transfer reactance, \(\text{Imag}(Z_{\text{MPP,MCP}}/Z_{0})\); (**c**) Simulated dissipation of MCP1 (red) and MPP (blue) assuming normal incidence and plane wave anechoic load; \(Z_{\text{MPP}}\) and \(Z_{\text{MCP}}\) are given by Equations (4) and (5), respectively. with \(m\approx\rho_{0}t/\sigma\), the MPP effective mass. It is followed by high-order quarter-wavelength resonances downshifted by a large effective mass due to a low perforation ratio. In the case of a rigidly backed MCP, all the resonances are of quarter-wavelength types that occur at  
\[v_{\Sigma}=0, \tag{2}\]0  
with \(n\), an integer.