Power-law density metafluids to achieve the acoustic black hole effect in duct terminations

The acoustic black hole (ABH) effect could be achieved in a duct filled with a metafluid such that its density grows according to a power-law, when approaching the tube termination. We derive the equations for that phenomenon and show how the original acoustic ABH for a duct termination adjusts to that framework. That ABH consists of a retarding structure made of rigid rings, separated a distance apart, and whose inner radius decays to zero when getting close to the end of the duct. The metafluid framework allows one to establish the connection between the analytical expressions of theoretical ABHs and the corresponding approximations made with the transfer matrix method (TMM). The latter can be used to consider factors that in practice deteriorate the performance of the ABHs, like the finite number of rings and cavities, their width, or the distance between subsequent perforated plates. Developments are presented for a linear ABH for which we prove convergence of the TMM solution to the analytical one.