Semi-analytical and numerical characterization of acoustic black holes in duct terminations

In recent times, the acoustic black hole (ABH) effect has been studied for noise control in ducts, while it originates for structural waves propagating in beams and plates. The comprehension of an ABH can be tackled by different methods: mathematical and analytical techniques, transfer matrix method (TMM) and finite element method (FEM) simulations, alongside with experiments. In this thesis, the three aforementioned methods are considered. A new theoretical framework, based on Gaussian discretization of the variational formulation of Helmholtz equation is proposed. The problem considered involves a rigid residual surface at the termination of the ABH and the proposed approach allows to compute the ABH modes through an eigenvalue problem. Therefore, this theoretical approach is validated against FEM results, showing a very strong agreement. Then, the transfer matrix method is introduced and applied to the ABH problem. It is shown that the TMM solution formally tends to the solution of the ABH equation in the limit case of number of rings tending to infinity. In order to do that, the concept of a metamaterial is used and the analogy between an acoustic wave propagating in an ABH and a wave propagating in a duct filled with a metafluid with particular physical properties is discussed. Finally, some preliminary FEM results are obtained and discussed. The influence of many parameters, such as number of rings and ABH order, on the ABH performance, expressed in terms of its reflection coefficient, is discussed. FEM results are the most expensive (as for computational cost) and closest to reality. In fact, FEM allows to visualize and understand the physics inside the ABH termination.