On the bandgap mechanism of periodic acoustic black holes

Abstract

Acoustic black holes (ABHs) are primarily intended to eliminate the propagation of bending waves in structures although the same physical principles have been used to reduce the propagation of acoustic waves in ducts. The latter are usually referred to as sonic black holes (SBHs). ABHs (and also SBHs) only work well above a certain cut-on frequency which depends on their size compared to the incident wavelength. This limits their effectiveness to the high frequency range. To overcome this problem, several works have proposed the design of periodic ABHs to generate stopbands for low frequencies and improve their operating range. The purpose of this communication is to shed some light on the nature of such bandgaps. The k(ω) method is used to calculate the complex dispersion curves of various periodic ABH configurations and it is shown that, contrary to what is suggested in many works, the generation of bandgaps is essentially due to Bragg scattering and not to local resonances for embedded ABHs, although the latter play a role in some cases. The opposite occurs for additive ABHs. It is also observed that the complex dispersion curves of periodic ABHs present an intriguing behaviour compared to those usually found in metamaterials, since they tend to disappear at high frequencies, where the local wave intensity is stronger. An explanation of all these facts is given in the case of a periodic SBH, single-leaf and double-leaf embedded ABHs, and pillar and single-leaf additive ABHs.