# Acoustic Pressure Amplification through In-Duct Sonic Black Holes
## 5 Discussion  
The PG dynamic ranges of the three types of dissipative SBHs described in Section 3 are summarized in Table 1 together with their efficiency ranges when used as sensors, but also as silencers, e.g., when more than 80% of the incident energy is confined within the metamaterial. The greatest sensor performance is achieved by the C-SBH which produces a PG as high as 64 (SNR = 18 dB) with a wide ER covering 1630 Hz. Ultimate slow sound effects occur within the C-SBH with the axial phase speed decaying to zero at the sensor termination leading to wave trapping, compression, and amplification at frequency-dependent axial positions. The C-SBH exhibits the highest PG peak values above 1 kHz at specific frequencies where the sensor efficiently couples with the incident wave (full impedance matching) and when the cavity resonances are critically coupled, e.g., when the losses equal the leakages. However, the inner C-SBH sensor cannot accommodate ducted flow conditions. Figure 13: Axial spectrograms of the pressure gains measured at the back cavities of a 3D-printed outer S-MPP-SBH sensor (**a**) in the no-flow case and (**b**) under a low-speed flow with bulk velocity 30 ms\({}^{-1}\); the incident wave is a white noise signal up to 2 kHz; the red dots spot the pressure gain peak values at the axial locations of the cavities and at their resonance frequencies. The PG results obtained with the C-SBH can be compared against one-port air-metal anisotropic sensors studied in [14; 15], with graded cavity depths opened at their mouths and set in a free-field environment. FEM simulations in [14] (Figure 3) showed, in accordance with Figures 6b and 11a, a gradual increase of the PG and bandwidth from the deepest to the shallowest cavities, e.g., from the lowest to the highest activated frequencies, for a range of cavity width \(w\) between 0.25 mm and 2 mm. An optimal width \(w\) was found that maximized the PG for each activated cavity. Experiments performed in [14] (Figure 4) with cavity widths \(w=1.4\) mm separated by 2 mm ribs showed a wave compression effect and pressure enhancement of the incident pulse components detected between 7 kHz and 10 kHz with a PG between 12 and 20. These values are consistent with those given in Table 1 for the C-SBH, albeit lower due to narrower cavities. An extended design was proposed in [15] with coiled cavities able to enhance the PG up to 85 given a coiling width of 3 mm. A maximum PG of 64 was reached by the C-SBH with straight cavities of width 3.5 mm, as indicated in Table 1.  
The outer O-SBH sensor is a flow-compliant alternative, albeit less efficient than the C-SBH as it only produces a maximum PG of 14 (SNR = 11.5 dB) over a narrower bandwidth of 526 Hz above 1500 Hz. Also, it yields a mildly slow sound effect with a halved axial phase speed at the outlet but achieves excellent impedance matching and provides a whole set of critically coupled resonances merged over its ER. Increasing the cavity width is the most influential factor in enhancing the PG of O-SBH sensors. Indeed, at the largest cavity width (upper bounded by \(L/N\)), the maximum PG can reach half of the C-SBH highest value (here PG = 32). In order to enhance sensing of the incident pulse low-frequency components while lowering the flow drag induced by opened cavity mouths as well as the risk of aero-acoustic instabilities, an approach is to cover the cavity mouths with a (micro-)perforated layer that brings effective inertia to the SBH cavity impedances while reducing the static pressure drop and therefore the flow friction factor. It can be seen from Table 1 that a MPP-SBH sensor can produce a PG of the same order, and even higher by a factor of 1.3, than the O-SBH sensor, when increasing the hole diameter or the perforation ratio.