Transfer matrices to characterize linear and quadratic acoustic black holes in duct terminations

The acoustic black hole (ABH) effect for sound reduction in duct terminations can be accomplished by means of retarding structures. The latter act as waveguides and their performance relies on two factors. First, a power-law decay of the duct radius and, second, an appropriate dependence of the wall admittance with the duct radius. In theory, the ABH can be achieved placing a set of rigid rings inside the duct, with inner radii and inter-spacing decreasing to zero as the tube end section is approached. In this work we focus on the linear and quadratic inner radius decay cases, referred to as the linear and quadratic ABHs. To begin with, analytical expressions are derived for the quadratic ABH and compared to those of the linear one. In both cases the solutions become singular at the final section of the duct. The wall admittance manifests the same behavior. Therefore, one has to deal with imperfect ABHs ending before the singularity, even in the best case scenario. Yet in practice, one may encounter further factors that deteriorate the ABH behavior. The number of rings and cavities between them is finite as it is the ring thicknesses. Damping also plays an important role. It is herein proposed to analyze the influence of all these factors on the reflection coefficient of the ABHs by means of the transfer matrix method (TMM). Transfer matrices are presented which allow one to relate the acoustic pressure and acoustic particle velocity between different sections of the retarding structure. They constitute a quick and valuable tool for an initial design of ABHs.