Separation of Traveling and Standing Waves in a Rigid-Walled Circular Duct Containing an Intermediate Impedance Discontinuity

Abstract

In this paper, we study the phenomenon of separation of traveling and standing waves in a one-dimensional rigid-walled circular duct. The underlying mechanism for separation, mode complexity, is linear and introduced here by a damped side branch representing an impedance discontinuity. The left end of the duct is driven at a single frequency by a harmonic acoustic source, and the right end is a rigid termination. The position and impedance of the side branch are independent parameters in the analysis. Sufficient conditions for acoustic wave separation in the duct are derived analytically and employed in a three-dimensional finite element analysis to verify the theoretical result. A physical experiment, consisting of a circular duct with a damped side branch, was constructed based on analytical predictions, the physical parameters were measured or identified, and its performance was documented. These experimental parameters were employed in a second three-dimensional finite element analysis to obtain a direct comparison with experimental results. The comparison reveals the extent to which higher-order (unmodeled) effects degrade the separation phenomenon. It is demonstrated that an intermediate damped side branch used as a nonresonant device can be predictively designed to achieve nearly ideal separation of traveling and standing waves in a rigid-walled circular duct in order to direct and control acoustic energy transmission through the duct system.

Analytical Solutions for Side-branch Impedance Producing Spatial Localization of Acoustic Waves in Ducts with Varying Cross Sections

“…Recently, a nonresonant damped side branch has been used for spatial separation (localization) of acoustic traveling and standing waves in a duct with constant cross-sectional area (CSA), where the acoustic impedance of the side branch required to produce this localization is found analytically as a function of wavenumber and side-branch location. [1][2][3] In structural dynamics, spatial separation of traveling and standing waves has also been studied for a non-dispersive taut string with an attached spring-dashpot support and a vibration absorber, 4,5 a dispersive taut string on a partial viscoelastic foundation, 6 and an Euler-Bernoulli beam with one or two spring-dashpot supports. 7 The ducts, strings and beams stud-ied for localization of waves were all assumed to have constant geometries, as in the case of the duct 1,2 shown in Fig.…”