Gaussian series solution for sonic black holes in duct terminations

Abstract

Acoustic black holes (ABHs) at the end of duct terminations slow down impinging waves by means of a set of rings separated by cavities, whose inner radii diminish following a power law profile. Energy tends to concentrate at the end of the waveguide and is dissipated by visco-thermal losses, resulting in a very low reflection coefficient. Such anechoic behavior is governed by a modified Webster equation that takes into account the wave propagation inside the duct of variable cross section and wall admittance. To date, only analytical solutions have been found for the linear and quadratic profiles. In this paper, we show that the generalized Webster equation can be transformed into a Helmholtz-type equation with non-constant wavenumber that can be solved analytically by the Bremmer series. Therefore, a general analytical solution is obtained which is valid for any ABH waveguide profile.