# Acoustic Pressure Amplification through In-Duct Sonic Black Holes
## 3 Simulation Analysis
### Pressure Gain Performance  
Figures 5-7 compare the pressure field distribution simulated by TMM throughout the C-SBH, O-SBH, and MPP-SBH together with the pressure gain (PG) performance achieved by these silencers, when used as sensors, at each cavity position as a function of frequency. An incident Gaussian pulse with a center frequency and half-bandwidth, \(f_{c}/2=1071\) Hz, is assumed to broadly cover the frequency range where impedance matching conditions are met for the three types of sensors. Figure 4: (**a**) Dissipation spectra and (**b**) specific input impedance curves associated to inner (black) and outer (red) SBHs; effects of coating the outer SBH by a MPP are shown in blue. Figure 5a shows amplification of the in-duct pressure field by a factor up to 5 at the axial locations and at the resonance frequencies of the C-SBH cavities coupled to the main duct. It is followed by a strong attenuation of the acoustic pressure beyond these axial locations, showing confinement of the amplified pressure to frequency-dependent axial locations. It can be seen in Figure 5b that this is accompanied by a substantial amplification of the pulse frequency components at the back of the cavities, ranging from PG = 14 for the deepest cavity (\(n=N=20\)) at 330 Hz up to PG = 64 for the cavity \(n=10\) at 1960 Hz. A steep increase in the PG values is observed in Figure 5b above 1 kHz due to impedance matching and full wave trapping satisfied above this frequency [Figure 4]. Note that the in-duct pressure drops when impedance matching conditions are not fulfilled, e.g., between two successive duct-cavity resonance frequencies. Figure 5: (a) Axial spectrogram of the sound pressure level upstream and within a C-SBH sensor (the red patch shows the axial sensor location); (b) Axial spectrogram of the PG simulated with TMM at the back of the C-SBH cavities (the red dots spot the peak levels at the cavities axial locations). Figure 6: (a) Axial spectrogram of the sound pressure level upstream and within an outer O-SBH sensor (the red patch shows the axial sensor location); (b) Axial spectrogram of the PG simulated with TMM at the back of the O-SBH cavities (the red dots spot the peak levels at the cavities axial locations). Figure 6a shows near-zero transmission and full trapping of the incident pulse components falling within the O-SBH sensor efficiency range, e.g., above 1500 Hz. One observes from Figure 6b that these components are amplified by a factor of PG = 3.5 at 1540 Hz for the cavity \(n=20\) up to PG = 14 at 1945 Hz for the cavity \(n=12\). However, the most energetic pulse components comprised between 500 Hz and 1500 Hz are not amplified as they fall below the efficiency range of the O-SBH sensor, 1500 Hz\(-f_{c}\), within which impedance matching occurs and cavity resonances are activated. In order to amplify and detect these components, an approach is to downshift the efficiency range of the sensor by covering the cavity mouths of the O-SBH with a MPP skin, leading to a MPP-SBH sensor with added mass brought by the micro-perforations. Figure 7a shows that the MPP-SBH provides a substantial dissipation of the incident pressure over its efficiency range, e.g., between 500 Hz and 1 kHz, with less attenuation above 1 kHz due to a transmission coefficient reaching 0.6 at 1700 Hz. Meanwhile, the MPP-SBH produces an amplification of the incident pulse by a factor of PG = 8 for the cavity \(n=7\) at 776 Hz up to PG = 12 for the cavity \(n=5\) at 1072 Hz, as seen in Figure 7b. Below 700 Hz, the resonances of the MPP-SBH deepest cavities are in over-resistive regime due to the too large resistance brought by the MPP. The losses then exceed the leakage: they are efficient at dissipating the incident pulse through the MPP, but they induce a low PG at the back of the deepest cavities. Above 700 Hz, the losses decrease with respect to the leakage, leading to an amplification of the incident pulse by a factor of 12. Further decrease of the PG to 7 for the cavity \(n=2\) at 1664 Hz is due to overly large transmission leakages.