# Acoustic Pressure Amplification through In-Duct Sonic Black Holes
## 3 Simulation Analysis
### Optimization Study  
Particle swarm optimization [26] of the O-SBH parameters \(w_{p}\) and \(m\) has been performed to maximize the pressure gain peak values, evaluated from the TMM and averaged over all the cavity positions, for a given spectral resolution fixed by the axial density of  
Figure 10: Variations of the PG peak values simulated by TMM for a fully opened SBH (\(L=0.15\) m) without coating (\(m=2\), \(N=20\), plain red with bullets; \(m=5\), \(N=20\), dotted red; \(m=2\), \(N=30\), dashed red) as a function of the cavity width \(w\) and when coated by a perforated panel (\(\sigma_{h}=2\%\), \(t_{h}=1\) mm, plain blue with bullets) as a function of the panel hole diameter \(d_{h}\) up to \(d_{h,\text{max}}=w=3.6\) mm, the uncoated SBH cavity width.  
resonators \(L/N\), related to length \(L=0.15\) m and \(N=20\) cavities. The optimal sensing parameters are given by \(w_{p}=0.5\) mm and \(m=3.1\), compared to \(w_{p}=4\) mm and \(m=2\) for the optimal silencer. The optimal ring thickness obtained when maximizing the sensing performance corresponds to the lower bound imposed by manufacturing constraints and set for the \(w_{p}\) variable in the optimization process. It coincides with the one resulting from the parametric study in Section 3.4, corresponding to a cavity width \(w=7\) mm, which is twice as large as that associated with the optimized O-SBH silencer. In accordance with Figure 10, a higher axial rate of increase of the cavity depths tends to enhance the PG, but this is a second-order effect with respect to the influence of the cavity width.  
Figure 11a,b clearly show enhanced PG peak values for the optimized O-SBH sensor (\(\text{PG}_{\text{max}}=38\)) with respect to the optimized O-SBH silencer (\(\text{PG}_{\text{max}}=14\)) by a factor of 2.7. Such enhancement is mostly prominent above 1600 Hz. It corresponds to the frequency range over which under-damped cavities are resonant, as it can be seen from the individual dissipation peaks activated above 1600 Hz in Figure 11c,d for the optimized O-SBH sensor.  
Figure 11: Axial spectrograms of the pressure gains at the cavity backings of O-SBHs whose mean-squared pressures [(**a**), circles] and total dissipation [(**b**), squares] have been optimized; dissipation [(**c**), red], transmission loss [(**c**), black], specific input resistance [(**d**), red] and reactance [(**d**), black] spectra associated to O-SBHs whose mean-squared pressures [(**c**,**d**), circles] and total dissipation [(**c**,**d**), squares] have been optimized; all simulations achieved with TMM.  
Due to the broader width of the sensor rings, the pressure inside the lateral cavities is less attenuated by visco-thermal effects, thereby contributing to a larger PG. This is achieved at the expense of lower dissipation performance, which essentially translates into larger back-reflections above 1600 Hz and a significantly higher transmission loss [Figure 11c]. Hence, the optimized O-SBH sensor is also an efficient attenuator, but it does not lead to minute reflections, unlike the optimized O-SBH silencer, whose critically coupled cavity resonances were able to achieve full wave trapping. Figure 11d also shows that the optimized O-SBH silencer couples more efficiently with the incident wave and over a broader bandwidth than the optimized O-SBH sensor, which provides a larger impedance mismatch increasing with frequency above 1600 Hz. It shows that the impedance matching constraint is stronger to achieve axial noise reduction by the O-SBH (\(m=2\)) than lateral pressure amplification (\(m=3.1\)).  
Optimization of the MPP-SBH sensing performance led to the same optimal cavity width despite lower PG peak values below 10 due to the added damping brought by the MPP. A higher increase rate of the cavity depths is obtained (\(m=4.2\)) due to the large weight given by the MPP added reactance to the amplification of the low-frequency components in the deepest cavities.