Sonic black holes without rainbow trapping: Separating two distinct mechanisms

The sonic black hole (SBH) effect in duct terminations refers to the progressive slowing down of propagating waves, accompanied by a reduction in wavelength and an increase in amplitude. In practice, it is realised by inserting a series of concentric rings separated by cavities into the duct, forming an SBH waveguide. Depending on the frequency range and the SBH dimensions, local cavity resonances may be excited, leading to rainbow trapping. The objective of the present work is to distinguish and clarify the roles of the SBH and rainbow-trapping effects in SBH waveguides, which have been a source of some misunderstandings regarding their functioning. To this end, three SBH configurations with square cross-sections are considered: an ideal metafluid model, in which only the SBH effect occurs, and two discrete designs, a standard SBH and a plug-in SBH, both comprising a finite number of square rings, where both mechanisms may coexist. We analyze the admittances of the cavity openings that control wave propagation inside the SBHs by deriving semi-analytical expressions for the standard SBH and analytical ones for the plug-in SBH. These developments, together with the associated dispersion curves, allow us to identify the frequency ranges in which the SBH effect dominates and those in which rainbow trapping is triggered. Finally, we show that rainbow trapping can be suppressed by subdividing an SBH into smaller units, ensuring that wave propagation inside the waveguide is entirely governed by the SBH effect. This approach enables broadband, near-perfect sound absorption over the frequency range of interest.