Spatial transient behavior in waveguides with lossy impedance boundary conditions

Abstract

Attenuation in acoustic waveguides with lossy impedance boundary conditions are associated with non-Hermitian and non-normal operators. This subject has been extensively studied in fundamental and engineering research, and it has been traditionally assumed that the attenuation behavior of total sound power can be totally captured by considering the decay of each transverse mode individually. One of the classical tools in this context is the Cremer optimum concept that aims to maximize the attenuation of the least attenuated mode. However, a typical sound field may be a superposition of a large number of transverse modes which are nonorthogonal, and the individual mode attenuation may have little to do with the total sound power attenuation. By using singular value decomposition, we link the least attenuated total sound power to the maximum singular value of the non-normal propagator. The behaviors of the least attenuated total sound power depend only on the lossy boundary conditions and frequency, but are independent of sources. The sound may be almost non-decaying along the waveguide transition region for any lossy impedance boundary conditions although all modes attenuate exponentially. This spatial transient appears particularly strongly if the impedance is close to an exceptional point of the propagator, at which a pair of adjacent modes achieve maximum attenuation predicted by Cremer optimum concept. These results are confirmed using non-modal numerical calculations and a two-by-two toy model.

Mode coalescence and the Green’s function in a two-dimensional waveguide with arbitrary admittance boundary conditions

“…Most of these research works deal with circular and annular ducts with flow and have potential application to noise suppression within ducts in aero-engines, gas turbines, blowers and various mufflers [5,6,7,8,9,10]. Additionally, [12,14] consider mode-matching for finite lined region, [13] examines the relationship between the the nature of the source and the transmitted power whilst [11] discusses the form of the additional wavefunctions required at an exceptional point. More recently, the present authors have proposed a numerical algorithm which enables them to explore the trajectories of the eigenvalues in the vicinity of an exceptional point in a systematic way [15].…”