Wave parameters of an acoustic black hole beam from exact wave-like solutions

Abstract

Acoustic black hole (ABH) structures have garnered significant interest due to their unique wave characteristics, encompassing wave velocity reduction, wavelength compression, amplitude augmentation, and energy concentration. Existing wave parameters characterizing ABH features are derived from the geometrical acoustics theory based on local uniformity assumption. Despite their widespread use, they are not rigorously exact and their applicable range remains unknown. Leveraging a tactic variable substitution technique, this paper derives the exact solutions of Bernoulli-Euler ABH beams, cast in a wave-like form. Different from the existing exact solutions, the wave-like form of the solutions allows clear separation of different wave components, thus leading to a full set of explicit analytical expressions of ABH-specific wave parameters including phase velocity, group velocity, wavelength, energy distribution, and energy transport velocity. Valid for the entire frequency range, this new set of exact wave-like solutions allows for the re-assessment of the existing wave parameters based on uniformity assumption to determine their validation range. Meanwhile, the exact wave parameters given by this paper allow for accurate quantification of the ABH effects and inherent physical interpretation of wave motion within an ABH beam, which provides the benchmark and reference solutions for ABH-related research and applications.