# Acoustic Pressure Amplification through In-Duct Sonic Black Holes
## 3 Simulation Analysis
### Slow Sound, Dissipation, and Impedance Matching Performance  
Slow sound effects have been simulated using the analytical model described in Section 2.1 and are illustrated in Figure 3 below. It can be seen that the acoustic wave travelling within the C-SBH slows down to zero and is fully trapped at \(x=L\) whereas it is only slowed down to \(c_{0}/2\) at the outlet of the O-SBH. Intermediate slow sound effects are observed at the outlet of the MPP-SBH with a phase speed ranging from \(c_{0}/2.7\) down to near-zero values when increasing the MPP hole diameter from 0.5 mm up to 3 mm. Slow sound and compressional wave effects for the O-SBH result from a progressive increase of the stiffness driven wall admittance, \(y_{\mathrm{O-SBH}}(x)\) given by Equation (3b), whereas they are caused by axially varying compressional and inertial effects at the mouths of the C-SBH and MPP-SBH cavities. Figure 4a,b compare the dissipation and impedance matching properties of the C-SBH, O-SBH, and MPP-SBH, assuming a 1.5 mm hole diameter for the MPP. It can be seen from Figure 4a that the O-SBH achieves near-unit broadband dissipation from 1400 Hz up to the duct cut-on frequency due to the merging of the individual cavity resonances with a loss-to-leakage ratio close to unity [24]. All the incident energy is therefore confined and dissipated within the cavities activated at their resonance frequencies, resulting in minute reflected and transmitted powers. It also produces an efficient coupling with the incident wave due to impedance matching, e.g., near-unit resistance and near-zero reactance within 5% from 1400 Hz, as shown in Figure 4b. Figure 3: Axial variation of the effective phase speed across inner (black) and outer (red) SBHs; also shown are the effects of MPP coating the outer SBH by a micro-perforated panel with hole diameters \(d_{h}=0.5\) mm (dotted blue), \(d_{h}=1\) mm (plain blue), and \(d_{h}=3\) mm (dashed blue). Shielding the O-SBH cavity mouths with a MPP downshifts the efficiency range of the SBH with a dissipated power that stays above 0.8 between 500 Hz and 1 kHz. This is due to the inertial effects induced by the MPP on the individual cavity resonances. Moreover, the resistivity of the MPP holes adds up to the visco-thermal dissipation inside the cavities, thereby increasing the loss-to-leakage ratio's above unity. The maximum dissipation value then decays while occurring at lower frequencies. Impedance matching is still achieved within 4% above 500 Hz. Figure 4a shows that the dissipation performance of the C-SBH builds up on distinct resonance peaks with near-unit values above 800 Hz separated by dips above 0.8 as from 1 kHz. The low frequency peaks are induced by multiple quarter-wavelength resonances along the C-SBH length, scaling on the mean effective phase speed \(\overline{c}_{x}<c_{0}\) induced by slow sound effects, such that the first resonance of the C-SBH occurs at \(\overline{c}_{x}/(4L)=83\) Hz. Above 500 Hz, these axial resonances progressively couple with the quarter-wavelength lateral cavity resonances, which essentially dominate the spectrum above 1400 Hz. Figure 4b shows that, unlike the O-SBH and MPP-SBH, efficient coupling with the incident wave is only achieved at the peak C-SBH resonance frequencies, with a noticeable impedance mismatch between the peaks below 1 kHz. Impedance matching is achieved by a suitable, gradual increase in the cavity depths. It is monitored by their axial rate of increase \(m\). In the limiting case, when \(m\) is very large, the O-SBH tends towards a locally reacting expansion chamber with narrowband dissipation and strong impedance mismatch due to high reflections at the inlet. On the other hand, if \(m\) tends to zero, the dissipation vanishes due to no reflection and full transmission, while complete impedance matching is achieved up to the duct cut-on frequency due to the absence of lateral cavities.