Optimal attenuation caused by boundary impedance in acoustic waveguides

Abstract

Non-Hermitian systems can exhibit “exceptional points” (EPs) at which modes coalesce. The connection between EPs and acoustic damping goes back to the observation of Cremer (1953) that optimal attenuation in a duct occurs when the two lowest modes have equal complex-valued eigenvalues, although the physical basis for this effect remains unclear. In an attempt to understand Cremer’s observation, we consider the model case of a two-dimensional waveguide with different impedance conditions on the two boundaries. This allows us to determine the complete set of all possible pairs of passive impedance conditions that give rise to EPs, and from these to select impedances appropriate to a particular frequency bands. The non-separable, and generally non-symmetric, mode shapes are described. The theoretical findings are linked to realistic passive impedance values based on various models for boundary impedance, such as perforated and porous panels. These comparisons are discussed to illustrate the feasibility of optimized wall impedances in absorbing sound passing through ducts. [Work supported by NSF.]