Acoustic propagation in non-uniform waveguides: Revisiting Webster equation using evanescent boundary modes

The scattering of an acoustic wave propagating in a non-uniform waveguide is inspected by revisiting improved multimodal methods in which the introduction of additional modes, so-called boundary modes, allows to better satisfy the Neumann boundary conditions at the varying walls. In this paper, we show that the additional modes can be identified as evanescent modes. Although non-physical, these modes are able to tackle the evanescent part of the field omitted by the truncation and are able to restore the right boundary condition at the walls. In the low-frequency regime, the system can be solved analytically, and the solution for an incident plane wave including one or two boundary modes is shown to be an improvement of the usual Webster equation.