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Effective conditions for the reflection of an acoustic wave by low-porosity perforated plates

Published online by Cambridge University Press:  05 March 2014

S. Laurens ,
E. Piot ,
A. Bendali ,
M’B. Fares  and
S. Tordeux
S. Laurens*
Affiliation:
IMT, University of Toulouse, INSA, 135 avenue de Rangueil, F-31077, Toulouse, France CERFACS, 42 avenue Gaspard Coriolis, F-31100, Toulouse, France
E. Piot
Affiliation:
ONERA – The French Aerospace Lab, F-31055, Toulouse, France
A. Bendali
Affiliation:
IMT, University of Toulouse, INSA, 135 avenue de Rangueil, F-31077, Toulouse, France CERFACS, 42 avenue Gaspard Coriolis, F-31100, Toulouse, France
M’B. Fares
Affiliation:
CERFACS, 42 avenue Gaspard Coriolis, F-31100, Toulouse, France
S. Tordeux
Affiliation:
INRIA and University of Pau, LMA, avenue de l’Université, F-64000, Pau, France
*
†Email address for correspondence: sophie.laurens@cerfacs.fr
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Abstract

This paper describes an investigation of the acoustic properties of a rigid plate with a periodic pattern of holes, in a compressible, ideal, inviscid fluid in the absence of mean flow. Leppington & Levine (J. Fluid Mech., vol. 61, 1973, pp. 109–127) obtained an approximation of the reflection and transmission coefficients of a plane wave incident on an infinitely thin plate with a rectangular array of perforations, assuming that a characteristic size of the perforations is negligible relative to that of the unit cell of the grating, itself assumed to be negligible relative to the wavelength. One part of the present study is of methodological interest. It establishes that it is possible to extend their approach to thick plates with a skew grating of perforations, thus confirming recent results of Bendali et al. (SIAM J. Appl. Math., vol. 73 (1), 2013, pp. 438–459), but in a much simpler way without using complex matched asymptotic expansions of the full wave or to a grating of multipoles. As is well-known, effective compliances for the plate can then be derived from the corresponding approximations of the reflection and transmission coefficients. These compliances are expressed in terms of the Rayleigh conductivity of an isolated perforation. Consequently, in one other part of the present study, the methodology recently introduced by Laurens et al. (ESAIM, Math. Model. Numer. Anal., vol. 47 (6), 2013, pp. 1691–1712) to obtain sharp bounds for the Rayleigh conductivity has been extended to include the case for which the openings of the perforations on the upper and lower sides of the plate are elliptical in shape. This not only enables the determination of these bounds and of the associated reflection and transmission coefficients for actual plates with tilted perforations but also yields single expressions covering all usual cases of perforations: straight or tilted with a circular or an elliptical cross-section.

JFM classification

Acoustics: Acoustics Mathematical Foundations: Mathematical Foundations Waves/Free-surface Flows: Wave scattering
Type
Papers
Information
Journal of Fluid Mechanics , Volume 743 , 25 March 2014 , pp. 448 - 480
Copyright
© 2014 Cambridge University Press 

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