Sound propagation in slowly varying duct

This paper analyses the propagation of acoustic waves through slowly varying lined ducts via a Wentzel–Kramers–Brillouin (WKB) approach. It is assumed that the curvature of the duct boundary can be related to a small parameter, ϵ, and slowly varying modes can be defined as solutions to the Helmholtz equation. The duct lining is approximated by a surface impedance boundary condition applied at the fluid-duct interface. It is assumed that slowly varying modes exist in the duct and do not reflect or change eigenstate as they propagate. Posing a WKB ansatz, the known leading order asymptotic solution is recovered and corrected to first order, more accurately representing the normal boundary condition, and addressing errors in the pressure field. It is found that, when compared with numerical simulations performed using COMSOL, this solution is unsatisfactory when there are multiple highly cut-on modes for a given frequency. This is due to modal coupling, which is especially prevalent at low frequencies. It is possible to account for the coupling whilst retaining a WKB approach, by defining a coupled mode solution, where the acoustic field is projected over a set of propagating modes which satisfy the true boundary condition. This approach is implemented to O(ϵ) and validated against full numerical simulations performed in COMSOL.