Acoustic Waves in a Cylindrical Duct of Finite Length

Abstract

The propagation of acoustic waves in a hard-walled circular cylindrical duct of finite length is studied theoretically. Acoustic waves are generated at one end of the duct by an imposed time-periodic excitation, and the duct is terminated at the other end by an arbitrary acoustic impedance. The steady-state solution in the interior of the duct is represented as a sum over various modes, which are either “propagating” or “nonpropagating.” The amplitude of a nonpropagating mode varies axially down the duct (amplitude of a propagating mode remains constant) and is the sum of negative and positive exponential terms in distance from the source. Both propagating and nonpropagating modes can transfer acoustic energy down and out the end of a finite-length duct (in contrast to the result for an infinite-length duct). For each mode, an expression is derived for the characteristic impedance, and an approximation is given for the terminating impedance when the duct is open-ended. The results (especially with “spinning-waves”) have possible application to the study of noise from the inlet ducts of turbo-fan engines.