## 3 Results
### Optimized SBHs  
An obvious way how to increase losses in the SBHs is increasing the rib density \(n\), which increases the area of the solid-fluid interface with the adjacent acoustic boundary layer. In the following example, \(A_{\text{min}}=0.25\,\text{mm}\), the rib density is increased to \(n\ =\ 20\,\text{cm}^{-1}\) (\(N=500\)), and \(\zeta\ =\ 0.5\,(h_{\text{r}}\ =\ h_{\text{s}}\ =\ 0.25\,\text{mm})\). Figure 5a shows the shape function of the optimized SBH, together with the linear and quadratic one for the sake of comparison. It can be observed that the optimized shape is characteristic with small values of its slopes at the extremities, and along the-most-contracted part. What is also interesting is a large cross-sectional jump at the SBH's opening, where \(G(1)\ =\ 0.601\). The numerical experiments indicate that this cross-sectional jump helps to enhance the low-frequency performance of the SBH. If the binding condition \(G(1)\ =\ 1\) is enforced during optimisation, the low-frequency absorption peak occurs at a higher frequency, and a lower value of \(\langle\mathcal{A}\rangle\) is achieved. Figure 5b shows the absorption coefficient spectra of the individual SBHs, where the solid lines show the results of the 1D Riccati model, and the dashed lines correspond to the 2D LNSE-FEM's results. It can be observed that the agreement between both the models is very good, which validates the predictions of the simplified Riccati model. It can be seen in Figure 5b that according to the mean absorption coefficient, the optimized SBH performs the best (\(\langle\mathcal{A}\rangle\ =\ 0.856\)), the quadratic SBH performs somewhat worse (\(\langle\mathcal{A}\rangle\ =\ 0.829\)), and the linear SBH performs the worst (\(\langle\mathcal{A}\rangle\ =\ 0.709\)). Nevertheless, in all the cases, the performance is much better that in the case of SBHs with a coarse internal structure, as seen in Figure 4.  
The absorption coefficient spectrum of the optimized SBH is characteristic with an absorption peak (\(\mathcal{A}\ =\ 0.739\)) at very low frequency of \(120\,\text{Hz}\). The existence of an absorption peak at this low frequency (as will be shown later) is caused by the resonance along the SBH structure (global resonance). As it has recently been shown [15; 26], the wave speed in the most contracted part of the SBH is significantly reduced, so the long contraction of the optimized SBH results in this low-frequency peak. The absorption coefficient of the optimized SBH has the value \(\mathcal{A}>0.9\) for frequencies \(f>250\,\text{Hz}\). The absorption coefficient spectrum of the quadratic SBH also possesses a resonant peak with \(\mathcal{A}\ =\ 0.998\) at the frequency of \(207\,\text{Hz}\), followed by a dip with \(\mathcal{A}=0.714\) at the frequency of \(272\,\text{Hz}\). Again, the existence of this low-frequency peak is caused by the longer contracted part of the profile, which is not as long as in the case of the optimized SBH. The absorption coefficient spectrum of the linear SBH does not display any low-frequency  
Figure 5: Panel (**a**): shape functions. Panel (**b**): absorption coefficient spectra of SBHs with linear, quadratic, and optimized profiles. Solid lines—prediction by the Riccati model. Dashed lines—results from the LNSE-FEM model. In all the cases, \(A_{\text{min}}=0.25\,\text{mm}\), \(n\ =\ 20\,\text{cm}^{-1}\), and \(\zeta\ =\ 0.5\). absorption peaks; the absorption coefficient has the value \(\mathcal{A}>0.9\) only for frequencies \(f>620\,\mathrm{Hz}\). The distribution of the acoustic pressure amplitude in the optimized SBH discussed above at four different frequencies is shown in Figure 6. In all the cases, acoustic wave with amplitude \(1\,\mathrm{Pa}\) impinges on the SBH from the right. It can be observed that at the lowest two frequencies of \(120\,\mathrm{Hz}\) and \(270\,\mathrm{Hz}\), the acoustic field is more or less independent on the \(y\) coordinate and it has the highest amplitude at \(x=0\).