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Nonlinear sound propagation in two-dimensional curved ducts: a multimodal approach

Published online by Cambridge University Press:  19 July 2019

James P. McTavish  and
Edward J. Brambley Open the ORCID record for Edward J. Brambley [Opens in a new window]
James P. McTavish
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
Edward J. Brambley*
Affiliation:
Mathematics Institute and WMG, University of Warwick, Coventry CV4 7AL, UK
*
†Email address for correspondence: E.J.Brambley@warwick.ac.uk
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Abstract

A method for studying weakly nonlinear acoustic propagation in two-dimensional ducts of general shape – including curvature and variable width – is presented. The method is based on a local modal decomposition of the pressure and velocity in the duct. A pair of nonlinear ordinary differential equations for the modal amplitudes of the pressure and velocity modes is derived. To overcome the inherent instability of these equations, a nonlinear admittance relation between the pressure and velocity modes is presented, introducing a novel ‘nonlinear admittance’ term. Appropriate equations for the admittance are derived which are to be solved through the duct, with a radiation condition applied at the duct exit. The pressure and velocity are subsequently found by integrating an equation involving the admittance through the duct. The method is compared, both analytically and numerically, against published results and the importance of nonlinearity is demonstrated in ducts of complex geometry. Comparisons between ducts of differing geometry are also performed to illustrate the effect of geometry on nonlinear sound propagation. A new ‘nonlinear reflectance’ term is introduced, providing a more complete description of acoustic reflection that also takes into account the amplitude of the incident wave.

JFM classification

Acoustics: Acoustics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 875 , 25 September 2019 , pp. 411 - 447
Copyright
© 2019 Cambridge University Press 

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