An improved multimodal approach for non-uniform acoustic waveguides

• Published in: IMA journal of applied mathematics Vol. 73; no. 4; pp. 668 - 690
• Main Authors: Hazard, Christophe, Lunéville, Eric
• Format: Journal Article
• Language: English
• Published: Oxford Oxford University Press 01-08-2008; Oxford Publishing Limited (England); Oxford University Press (OUP)
• Subjects: Acoustics; cross-section method; Engineering Sciences; Exact sciences and technology; Fourier analysis; Fourier series; Mathematical analysis; Mathematics; Mechanics; multimodal decomposition; Numerical Analysis; Partial differential equations; Physics; Sciences and techniques of general use; waveguide; Fourier series; multimodal decomposition; cross-section method; waveguide; Wave equation; Approximation; Error estimation; Inhomogeneous medium; Acoustic wave; Fourier analysis; Acoustics; Numerical method; Convergence; Truncation; Numerical analysis; Applied mathematics
• ISSN: 0272-4960; 1464-3634
• DOI: 10.1093/imamat/hxn006

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.