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IMA Journal of Applied Mathematics Advance Access published online on February 29, 2008
IMA Journal of Applied Mathematics, doi:10.1093/imamat/hxn006
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© The Author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

An improved multimodal approach for non-uniform acoustic waveguides

Christophe Hazard{dagger} and Eric Lunéville{ddagger}

Laboratoire de Propagation d'Ondes—Etude Mathématique et Simulation, Ecole Nationale Supérieure de Techniques Avancées/Unité de Mathématiques Appliquées/POEMS, 32 Boulevard Victor, 75015 Paris, France

{dagger} Email: christophe.hazard{at}ensta.fr

>{ddagger} Email: eric.luneville{at}ensta.fr

Received on June 26, 2007; Accepted on January 7, 2008

This paper explores from the point of view of numerical analysis a method for solving the acoustic time-harmonic wave equation in a locally non-uniform 2D waveguide. The multimodal method is based on a spectral description of the acoustic field in each transverse section of the guide, using Fourier-like series. In the case of sound-hard boundaries, the weak point of the method lies in the poor convergence of such series due to a poor approximation of the field near the boundaries. A remedy was proposed by Athanassoulis & Belibassakis (1999, J. Fluid Mech., 389, 275–301), where the convergence is improved thanks to the use of enhanced series. The present paper proposes a theoretical analysis of their idea and studies further improvements. For a general model which includes different kinds of non-uniformity (varying cross section, bend and inhomogeneous medium), a hybrid spectral/variational formulation is introduced. Error estimates are provided for a semi-discretized problem which concerns the effect of spectral truncation. These estimates are confirmed by numerical results.

Keywords: multimodal decomposition; cross-section method; Fourier series; waveguide.


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