## 3 Results
### Parametric Study  
Within this section, the dependence of optimized MPP-SBHs' absorbing performance on various geometrical parameters is studied.  
Figure 12 shows the absorption coefficient spectra of MPP-SBHs optimized for various fixed values of \(A_{\mathrm{min}}\); in all the cases, \(n~{}=~{}1\,\mathrm{cm}^{-1}\), and \(h_{\mathrm{r}}~{}=~{}1\,\mathrm{mm}\). It can be observed that the low-frequency absorption spectrum is very much dependent on the value of \(A_{\mathrm{min}}\), whereas the mean absorption coefficient \(\langle\mathcal{A}\rangle\) (see the figure legend) is, more or less, unaffected. This behaviour could be advantageously used for shaping the absorption spectra of MPP-SBHs.  
Figure 11: Distribution of the acoustic pressure amplitude in optimized the MPP-SBH at four particular frequencies; acoustic wave with amplitude \(1\,\mathrm{Pa}\) impinges from the right.  
Figure 13 shows the absorption coefficient spectra of MPP-SBHs optimized for various fixed values of \(h_{\rm r}\); in all the cases, \(n~{}=~{}1\,\)cm\({}^{-1}\), and \(A_{\rm min}~{}=~{}0.25\,\)mm. The thickness of the ribs influences the spectrum of the absorption coefficient only at low frequencies; there is a general trend that the smaller the value of \(h_{\rm r}\), the lower the frequency at which the absorption coefficient starts to increase from the zero value. Moreover, the smaller the value of \(h_{\rm r}\), the higher the mean absorption coefficient. Simply put, the ribs only serve to introduce a locally compliant wall with smoothly changing admittance, the most of the absorption takes place in the MPP, and the space occupied by the ribs reduces the active surface of the MPP. Similarly, and for the same reason, if the thickness of the ribs is kept constant and the density of ribs \(n\) increases, the mean absorption coefficient decreases. Of course, this does not mean that the ribs are unimportant. If the ribs are removed, the absorbing performance of the structure decreases significantly.  
Figure 14 shows the absorption coefficient spectrum of the optimized MPP-SBH presented in Figure 10 (\(A_{\rm min}=0.25\,\)mm, \(n~{}=~{}1\,\)cm\({}^{-1}\), \(h_{\rm r}~{}=~{}1\,\)mm) and the same MPP-SBH with all the internal ribs removed (only the first rib positioned at \(x=L\) is kept). It is evident that the absorbing performance of the resulting structure is really poor. Without the ribs, the MPP and the cavity behind does not provide the conditions for wideband and efficient acoustic energy absorption; moreover, longitudinal modes are excited in the cavity behind the MPP, which are otherwise suppressed by the presence of the ribs.  
As it can be expected, an important parameter which strongly influences the performance of a MPP-SBH is its length \(L\). Figure 15a shows the shape functions of MPP-SBHs optimized for various lengths in intervals from \(L~{}=~{}21\,\)cm to \(L~{}=~{}29\,\)cm; in all the cases, \(A_{\rm min}~{}=~{}0.25\,\)mm, \(n~{}=~{}1\,\)cm\({}^{-1}\), and \(h_{\rm r}=1\,\)mm. It can be observed in Panel (a) of the figure, similarly as in the case of the optimized SBH, that the individual shape functions are very similar near the structure's apex, whereas at the opening, the higher the \(L\), the higher the value of \(G(1)\). At the same time, as the low-frequency absorption peak is caused by the acoustic black hole effect, the longer the structure, the lower the frequency of this peak and the higher the mean absorption coefficient.  
Figure 12: Absorption coefficient spectra of optimized MPP-SBHs for various values of \(A_{\rm min}\); in all the cases, \(n~{}=~{}1\,\)cm\({}^{-1}\), and \(h_{\rm r}~{}=~{}1\,\)mm.  
Figure 13: Absorption coefficient spectra of optimized MPP-SBHs for various values of \(h_{\rm r}\); in all the cases, \(n~{}=~{}1\,\)cm\({}^{-1}\), and \(A_{\rm min}~{}=~{}0.25\,\)mm.