## 3 Acoustic Properties of Unbacked MCPs and MPPs
The UMPP and MPP have perforation ratios that are considerably lower than those of MCP1 and MCP2, so they have much larger reactance values than those of MCP1 and MCP2 as the frequency increases. This is verified in Figure 3b when comparing the reactance of MCP1 to that of the MPP. Hence, the UMPP and MPP tend to be reflective at high frequencies, especially the UMPP, whose absorption curve already decays from 400 Hz down to 0.4 at 6500 Hz (Figure 5 (black curve)). Acoustic reflections on the UMPP are more pronounced than those on the MPP due to the perforation ratio of the UMPP that is about four times lower than that of the MPP, as well as due to smaller hole diameters. At low frequencies, it is of interest to observe that UMPPs achieve absorption values that are similar to those of MCP1 (around 0.9) but over a smaller bandwidth (only two octaves) that ranges between 100 Hz and 400 Hz. A larger acoustic flow distortion [18] occurs at the inlet/outlet of the holes of UMPPs with respect to the MPPs due to a hole pitch that is larger than that of MPPs. Hence, the resistance of UMPPs is 15 times greater than that of MPPs. It therefore produces a sufficient amount of dissipation and transmission to achieve high absorption values up to 400 Hz, a frequency at which the viscous boundary layer (VBL) occupies half of the radius of the holes of UMPPs. Beyond 400 Hz, the VBL thickness further decreases, and the dissipative effects tend to be overwhelmed by the reflection so that the frequency increases. On the transmitting side, it can be seen from Figure 6 that, amongst the four materials, only those with the lowest porosity, the UMPP and MPP, exhibit the largest discrepancy (up to 20%) with respect to the radiation impedance of a flanged open termination. However, these differences do not drastically impede the correlation between the simulated and measured absorptions, as can be seen in Figure 5. Assuming rigid panels, the sound field radiated by the materials is the result of the sound fields radiated either by a discrete number of apertures, e.g., by 25 holes per cm\({}^{2}\) for the UMPP and MPP, or by a continuum of apertures, e.g., by thousands of parallel micro-channels per cm\({}^{2}\) for MCP1 and MCP2. In the latter case, the high porosity and unit acoustic flow tortuosity across the channels explain that one can assimilate the radiation of micro-capillary materials to that of an open flanged termination. Figure 7 shows how the highly porous materials (MCP1 and MCP2) modify the radiation properties of the open termination (green arrows), which is highly reflective at low frequencies (Figure 7a) and highly transmissive at high frequencies (Figure 7b). At high frequencies, the intensity vectors are essentially parallel and point out toward a direction that is normal to the open termination (Figure 7b). This corresponds to plane wave propagation conditions where pressure and axial velocity are in-phase. At low frequencies, monopole-type radiation conditions induce tangential intensity components that are preponderant toward the edge of the open termination (Figure 7a). This is due to out-of-phase relationships between the pressure and the axial velocity that are associated with near-field effects. Closing the open termination for MCP1 or MCP2 has rather distinctive effects. MCP1 (red arrows) is highly dissipative over the whole frequency range. At low frequencies, it therefore blocks sound transmission (similar to the reflective open termination). At high frequencies, the dissipative effects of MCP1 also reduce sound transmission. This effect is clearly seen in Figure 7b with a lower amplitude of the intensity vectors of MCP1 with respect to those of the transmissive open termination. At low frequencies, MCP2 (magenta arrows) is more transmissive than MCP1 and the open termination. This results in larger amplitudes of the radiated intensity vectors, especially toward the grazing angles, as can be seen in Figure 7a. At high frequencies (typically above 2000 Hz), MCP2 transmits as much sound as the open termination.