Propagation of Waves in Cylindrical Hard-Walled Ducts with Generally Weak Undulations
The method of multiple scales is utilized to analyze the wave propagation in cylindrical hard-walled ducts having weak undulations which need not be periodic. Results are presented for two and three interacting modes. In the case of modes traveling in the same direction in a uniform duct, two interacting, spinning or nonspinning modes propagate unattenuated in an undulated duct. Moreover, neither of them can exist without strongly exciting the other. On the other hand, in the case of modes propagating in opposite directions, they may be cut off as a result of the interaction. OR two-dimension al ducts, straightforward expansions of the form (f>0 + e 7 were obtained by Isakovitch ] for the case of a waveguide with only one sinusoidally undulating wall, by Samuels 2 for the case of a waveguide with inphase wall undulations, and by Salant 3 for the general problem. Nayfeh4 showed that the above expansions are not uniform near the resonant frequencies because the correction term e 7 dominates the first term 0. He determined a uniform expansion for waves propagating in a two-dimensional hardwalled duct with sinusoidally perturbed walls by using the method multiple scales.5 In this paper, we extend the latter analysis to the case of linear waves propagating in a cylinderical hard-walled duct whose wall has weak undulations which need not be periodic. The gas is assumed to be inviscid, irrotational, and nonheat conducting. Dimensionless quantities are introduced by using the mean radius of the duct r0, the undisturbed speed of sound c, and the time r0/c as reference quantities. The dimensionless radius of the duct is assumed to have the form