Interpretation of the Acoustic Black Hole effect based on the concept of critical coupling

Abstract : An Acoustic Black Hole (ABH) in a one-dimensional (1D) beam is a passive vibration damping device based on a local reduction of the beam thickness attached to a thin layer of attenuating material. This work aims at revisiting the ABH effect by analysing the ABH trapped modes in the complex frequency plane. This analysis relies on an analytical model based on the one-dimensional thin beam theory and the transfer matrix method which assumes that the ABH termination is discretised by constant thickness piecewise elements. The model is validated with numerical simulations by the Finite Element Method. The reflection coefficients of several ABH terminations are studied. The results show that an ABH presents an infinite number of modes associated to an infinite number of poles and zeros of the reflection coefficient, the density and quality factor of which depend on the order of the ABH profile. By considering the ABH termination as an open lossy resonator, its damping efficiency results therefore from a balance between the energy leakage of each mode and its inherent losses, known as the critical coupling condition. In particular, the broadband absorption of the vibration energy is achieved for frequencies higher than that of the mode that is critically coupled. This type of analysis is used to interpret the ABH effect. It provides the losses needed to obtain the critical coupling condition, and is suitable for the optimisation of one-dimensional ABH terminations.