Transfer matrices to analyze the acoustic black hole effect in duct terminations

The acoustic black hole (ABH) effect for sound absorption in duct terminations can be achieved through a combination of two features. First, a power-law decay of the duct radius when approaching the tube end, and second, an appropriate decay of the wall admittance with the duct radius. The latter can be achieved by inserting a set of rigid rings with inner area decreasing to zero at the end of the tube, which constitute the ABH. The corresponding wall admittance can be modeled by means of a lumped compliance per unit surface at the continuum level. Yet in practice, one will have to deal with an imperfect ABH consisting of a finite amount of cavities separated by rings of finite thickness. The performance of such a retarding structure can be analyzed by means of the transfer matrix method for acoustic filters. In this work we do so for an ABH-like tube termination with linear decaying internal radius. The effects of varying the number and distribution of cavities, as well as the thickness of the separating rings are investigated.