rspa.royalsocietypublishing.org Research Cite this article: Mercier J-F, Maurel A. 2013 Acoustic propagation in non-uniform waveguides: revisiting Webster equation using evanescent boundary modes. Proc R Soc A 469: 20130186. http://dx.doi.org/10.1098/rspa.2013.0186 Received: 9 April 2013 Accepted: 22 May 2013 Subject Areas: acoustics, mathematical physics Keywords: non-uniform waveguide, scalar waves, modal method, boundary modes, Webster equation Author for correspondence: Jean-François Mercier e-mail: jean-francois.mercier@ensta.fr Acoustic propagation in non-uniform waveguides: revisiting Webster equation using evanescent boundary modes Jean-François Mercier 1 and Agnès Maurel 2 1 Laboratoire POEMS, ENSTA ParisTech, UMR CNRS 7231, 828 Boulevard des Maréchaux, 91762 Palaiseau Cedex, France 2 LOA/Institut Langevin, ESPCI, UMR CNRS, rue Jussieu, 75005 Paris, France 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. 1. Introduction Since the pioneering work of Stevenson [1,2], the multimodal propagation method has been widely used to describe propagation in non-uniform waveguides in acoustics [3–7] and in elasticity [8,9]. In the two- dimensional acoustic case, the pressure p satisfies the Helmholtz equation: ( + k 2 )p(x, y) = 0, (1.1) in a waveguide of height h(x) (located in y ∈ [0, h(x)]) with boundary conditions ∂ y p(x, 0) = 0 and ∂ y p(x, h) = h ′ (x)∂ x p(x, h). (1.2) 2013 The Author(s) Published by the Royal Society. All rights reserved.