•The transfer matrix method is used to characterize built-up acoustic black hole duct terminations.•A differential equation describes wave propagation inside ideal acoustic black holes.•Acoustic waves in a metafluid with a power-law increasing density obey the differential acoustic black hole equation.•An acoustic black hole made of rings and cavities corresponds to a realization of the metafluid.•The transfer matrix solution in the metafluid formally tends to the solution of the differential acoustic black hole equation. The transfer matrix method has been proposed to analyze the acoustic black hole effect in duct terminations. The latter is achieved by placing a retarding waveguide structure inside the duct, which consists in a set of rings whose inner radii decrease to zero following a power law. The rings are separated by thin air cavities. If the number of ring-cavity ensembles is large enough, wave propagation inside the waveguide can be treated as a continuous problem. A governing differential equation can be derived for the acoustic black hole which intrinsically relies on assumptions from transfer matrix theory. However, no formal demonstration exists showing that the transfer matrix solution is consistent and formally tends to the solution of the continuous problem. Proving such consistency is the main goal of the paper and an original option has been adopted to this purpose. First, we prove the differential equation for the acoustic black hole is identical to the wave equation for a metafluid with a power-law varying density. Transfer matrices are then applied to solve wave propagation in a discretization of the metafluid into layers of constant density. It is shown that when the number of layers tends to infinity and their thicknesses to zero, the transfer matrix solution satisfies the metafluid equation and therefore the acoustic black hole one. The transfer matrices for the metafluid discrete layers take a particularly simple form, which is a great advantage. This work allows one to interpret the retarding waveguide structure as a particular realization of the metafluid.