## 3 Results
### Parametric Study  
It has been demonstrated in the previous section that an increase in the rib density \(n\) can increase the SBH's absorbing performance. Within this section, the dependence of the optimized SBH's absorbing properties on other parameters is studied. Figure 7 shows the mean value of the absorption coefficient \(\langle\mathcal{A}\rangle\) of an optimized SBH as a function of the rib density \(n\) and the profile contraction at its apex \(A_{\text{min}}\), whereas the wall porosity \(\zeta=0.5\) is kept constant. It can be observed that the maximum value of \(\langle\mathcal{A}\rangle\) slightly increases with the decreasing value of \(A_{\text{min}}\), and for each value of \(A_{\text{min}}\), there is an optimal rib density \(n\). For example, for \(A_{\text{min}}\ =\ 1.00\,\mathrm{mm}\), maximum \(\langle\mathcal{A}\rangle\ =\ 0.858\) for \(n=60\,\mathrm{cm}^{-1}\), whereas for \(A_{\text{min}}\ =\ 0.25\,\mathrm{mm}\), maximum \(\langle\mathcal{A}\rangle\ =\ 0.861\) for \(n=50\,\mathrm{cm}^{-1}\). More importantly, the numerical results show that with a lower value of \(A_{\text{min}}\), the absorbing performance decreases less rapidly with the decreasing rib density \(n\). Figure 6: Distribution of the acoustic pressure amplitude in optimized SBH at four particular frequencies; acoustic wave with amplitude \(1\,\mathrm{Pa}\) impinges from right. Figure 8a shows the mean value of the absorption coefficient \(\langle\mathcal{A}\rangle\) as a function of the wall porosity \(\zeta\) and the rib density \(n\); in all the cases, \(A_{\text{min}}~{}=~{}0.25\,\text{mm}\). It can be observed that \(\langle\mathcal{A}\rangle\) strongly depends on \(\zeta\): the higher the wall porosity \(\zeta\), the higher the value of the maximum \(\langle\mathcal{A}\rangle\). It is also interesting that for the highest value of \(\zeta=0.75\), the dependence of \(\langle\mathcal{A}\rangle\) on the rib density \(n\) is only weak. Figure 8b shows the dependence of the absorption coefficient spectrum on the value of the rib density \(n\); in all the cases, \(\zeta=0.75\) and \(A_{\text{min}}~{}=~{}0.25\,\text{mm}\). It can be observed [see also the black line in Figure 8a] that, whereas the mean absorption coefficient is almost constant, the low-frequency behaviour of the SBHs is very different. For lower values of \(n\), there is a low-frequency absorption peak in the absorption coefficient spectrum, and the lower the \(n\), the lower its frequency. The presence of this absorption peak is caused by the fact that with the lower value of \(n\), the slits become wider and there is less dissipation in the structure, which makes the spectrum more oscillatory. Another parameter which strongly influences the performance of an SBH is its length \(L\). Figure 9a shows the shape functions of SBHs optimized for various lengths in intervals from \(L~{}=~{}21\,\text{cm}\) to \(L~{}=~{}29\,\text{cm}\); in all the cases, \(A_{\text{min}}~{}=~{}0.25\,\text{mm}\), \(n~{}=~{}20\,\text{cm}^{-1}\), and \(\zeta=0.5\). The optimal shape functions are very similar, almost indistinguishable at the SBH's apex up to ca. \(X<0.2L\), and at the SBH opening, the larger the SBH's length \(L\), the higher the value \(G(1)\). Figure 9b shows the absorption coefficient spectra for these optimized SBHs with different lengths. The figure legend shows that the mean absorption coefficient increases with the SBHs' length. All the absorption spectra are very similar, and there is one low-frequency absorption peak whose frequency decreases with the increasing SBHs' length \(L\). This behaviour is quite plausible. As it has been demonstrated earlier, this absorption peak is due to the acoustic black hole effect--the global resonance along the SBH structure. As the SBHs' length \(L\) increases, the length of the most contracted part of the SBHs with the slowest wave speed also increases--see Figure 9a.