# Acoustic Pressure Amplification through In-Duct Sonic Black Holes
## 5 Discussion
This is linked to the enhanced slow sound effect, as shown in Figure 3. Meanwhile, Table 1 shows that a 2% perforation ratio enables to downshift by 1 kHz the minimum frequency that can be enhanced by the MPP-SBH sensor, which also corresponds to the frequency above which near-unit impedance matching is achieved, as seen in Figure 4b. A S-MPP-SBH with a higher 2.8% perforation ratio leads to greater maximum PG, as confirmed by the measurements, but the effective inertia brought by the larger perforations is reduced, thereby increasing the minimum frequency components that can be amplified in the cavities (here from 500 Hz to 800 Hz). Thus, it appears that a balance has to be sought on the MPP parameters of MPP-SBH sensors to achieve both a sufficient PG and to be efficient at low frequencies. Measurements at low Mach number 0.09 showed that the pressure gain performance of the S-MPP-SBH sensor is resilient to low-speed flow conditions, but further increases in the flow convection velocity or variations in the MPP parameters will modify the slow sound properties and the amount of flow resistance and reactance brought by the flow-holes interaction. Indeed, downstream (resp. upstream) propagation conditions will decrease \begin{table}
\begin{tabular}{c c c c} \hline
**SBH Type** & **Minimum PG–Maximum PG** & **Sensor ER** & **Silencer ER** \\ \hline C-SBH & 14–64 & 330–1960 & 1000–2000 \\ \hline O-SBH & 3.5–14 & 1537–2063 & 1500–2142 \\ \hline MPP-SBH & 1.2–12 & 500–1655 & 500–1020 \\ \hline S-MPP-SBH (no flow, exp.) & 5.5–15 & 800–1084 & 630–1050 \\ \hline S-MPP-SBH (flow, exp.) & 6.8–13.5 & 712–1103 & 650–1070 \\ \hline \end{tabular}
\end{table} 
Table 1: PG dynamic range and efficiency ranges (ER in Hz) of the graded dissipative SBHs. (resp. increase) the axial wavelength, and so the axial phase speed within the MPP-SBH sensor. Hence, downstream convection conditions, as is the case in our experiment, tend to impede the occurrence of slow sound, rather favored by upstream propagation conditions. Another effect is the increase in perforated wall resistance when the Mach number increases, whatever the flow direction. An overly large resistance involves a decrease in the PG due to the high dissipation of the incident wave through the shear layer at the inlet of the holes. In order to maintain a minimum PG, it should be balanced by a decrease in the cavity or MPP inner resistance, which can be achieved by increasing the cavity width or the perforation ratio. A further risk is the occurrence of flow-induced tone over the perforated facing. It may occur at a frequency of the order \(U/\Lambda\) with \(\Lambda\) the holes pitch, namely 4000 Hz in the experiment, which is well beyond the plane wave regime. This risk can be tamed by lowering the quality factor of the MPP-cavity resonators. Comparing the efficiency ranges of the graded metamaterials in Table 1, they typically show a narrower ER when used as sensors rather than silencers, except for the C-SBH which already provides a PG of 14 at 330 Hz. Slow sound and impedance matching conditions are prerequisites to achieving both high sensing and attenuation performance. Critical coupling conditions that ensure a balance between losses and leakages also favor sensing performance as well as near-unit dissipation. This last condition can, however, be relaxed for pressure amplification, as observed for MPP-SBH sensors still able to achieve low-frequency PG performance despite over-resistive conditions. Finally, this study shows the ability of MPP-SBH silencers, initially designed for noise control in duct systems, to amplify a number \(N\) of the incident spectral components that activate the SBH cavities, and which could be readily measured by an array of small-sized, low consumption and resilient MEMS (micro-electro mechanical system) microphones [30] located at the back of the cavities. The spectrum of such signals could be used to monitor fan defaults in HVAC circuits.