## 4 Discussion  
Sonic black holes may represent an intriguing and unconventional means for achieving the anechoic termination of waveguides. The primary motivation for their use lies in the avoidance of fibrous materials, which may present challenges in certain situations. It has been demonstrated that if an SBH possesses a coarse internal structure--characterised by a rib density of approximately \(n\sim 1\,\mathrm{cm}^{-1}\) and rib thickness \(h_{\mathrm{r}}\sim 1\,\mathrm{mm}\), parameters that are compatible with conventional manufacturing techniques--its absorbing performance is insufficient for most practical applications. To enhance the absorbing performance of SBHs, we introduced the mean value of the absorption coefficient as a performance metric for optimisation purposes. It should be noted, however, that in some applications, alternative approaches may be more appropriate; for instance, the absorption coefficient spectrum could be frequency-weighted to improve low-frequency absorption. Nevertheless, the approaches adopted in this work can be readily adapted to suit any chosen performance metric. One of the parameters that strongly influences SBH performance is its profile. In this work, the profile was defined using a dimensionless shape function constructed with three control points interconnected by cubic splines. This method allows for the construction of a wide variety of smooth profiles using a five-dimensional parameter vector. We conducted an extensive parametric study, the results of which indicate the following: the optimal SBH shape features an elongated narrow section adjacent to the apex of the structure, as well as a cross-sectional jump at the SBH opening. This shape differs significantly from those previously studied. The SBH acts as an efficient absorber, particularly when its internal structure is fine, with a rib density of \(n\gtrsim 20\,\mathrm{cm}^{-1}\). A high number of ribs forming the SBH structure provides sufficient losses in the acoustic boundary layer adjacent to the solid-air interface in the slits. The requirement for a high rib density can be partially alleviated by minimising the maximum contraction at the apex of the SBH and, especially, by using very narrow ribs. It has been identified that the acoustic black hole effect plays a significant role in low-frequency absorption, such that increasing the SBH length results in an increase in the mean absorption coefficient. Figure 14: Absorption coefficient spectrum of the optimized MPP-SBH from Figure 10 (\(A_{\mathrm{min}}=0.25\,\mathrm{mm}\), \(n\,=\,1\,\mathrm{cm}^{-1}\), \(h_{\mathrm{r}}\,=\,1\,\mathrm{mm}\)) and the same MPP-SBH with the internal ribs removed. Figure 15: Panel (**a**): shape functions of MPP-SBHs optimized for different lengths \(L\); in all the cases, \(A_{\mathrm{min}}\,=\,0.25\,\mathrm{mm}\), \(n\,=\,1\,\mathrm{cm}^{-1}\), and \(h_{\mathrm{r}}=1\,\mathrm{mm}\). Panel (**b**): the absorption coefficient spectra of the individual MPP-SBHs. This parametric study revealed that, although SBHs can serve as effective acoustic absorbers, their fine internal structure--resembling an anisotropic porous material--presvents manufacturing challenges, even with contemporary 3D printing techniques. This issue can be circumvented by introducing an alternative loss mechanism into the SBH structure. For this reason, micro-perforated plates attached to the ends of the ribs were utilised. Although this idea has been proposed in previous works, it has not been applied to a rectangular SBH, where the profile can be easily shaped even with the attached MPP. Our extensive parametric study revealed that MPP-SBHs can function as effective acoustic absorbers even with a low rib density of \(n\sim 1\,\mathrm{cm}^{-1}\), making these structures easy to manufacture. The optimal shapes of MPP-SBHs are similar to those of SBHs. This study also indicated that the mean absorption coefficient of MPP-SBHs is not very sensitive to the value of the maximum contraction, in contrast to the low-frequency spectrum of the absorption coefficient.