Sound propagation in lined ducts with slowly varying boundaries

  • Thomas White

Student thesis: Phd


In this thesis we consider the propagation of acoustic waves through slowly varying lined axisymmetric ducts in the presence of background flow via a Wentzel-Kramers-Brillouin (WKB) approach. Lined ducts are often of interest in industrial problems, such as jet engines that are lined with honeycomb arrays to absorb unwanted noise. Therefore, the accurate modelling of such systems is imperative in designing the optimal liner for a given duct. Here we shall describe the system by considering the inviscid, linear acoustic model. If a duct has slowly varying boundaries, i.e. an axisymmetric duct with boundaries whose radial position vary slowly along the duct axis, then we can apply a WKB approximation. The approximation is based on an ansatz that assumes that there exists slowly varying acoustic modes that are analogous to their straight duct counterparts, with parameters varying slowly along the duct. The leading order behaviour of these slowly varying modes has been studied extensively both with (Rienstra, 1999) and without (Nayfeh and Telionis, 1973) background flow. We extend the results, by increasing the order of accuracy of the approximation, defining the solution to first order in the small parameter, epsilon, and validate our theory against full finite-element simulations. When there are multiple possible propagating modes inside a duct, we observe, via full finite element simulations, that exciting the duct with a single mode can lead to multiple modes propagating along the duct. This coupling phenomenon cannot be described by the WKB approach discussed above. We address this by posing a solution that accounts for the coupling of slowly varying, cut-on, acoustic modes, bridging the gap between the physically motivated WKB ansatz and the more numerical, modal projection methods in the literature. The theory works in the cases of lined ducts without flow, and rigid-walled ducts with background flow, however, we have not yet found a way to accommodate lined ducts with background flow.
Date of Award1 Aug 2024
Original languageEnglish
Awarding Institution
  • The University of Manchester
SupervisorWilliam Parnell (Supervisor) & Raphael Assier (Supervisor)


  • modal coupling
  • slowly-varying geometry
  • duct acoustics
  • WKB approximation

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