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This paper describes an investigation of the acoustic properties of a rigidplate with a periodic pattern of holes, in a compressible, ideal, inviscidfluid in the absence of mean flow. Leppington and Levine (J. Fluid Mech., 1973)obtained an approximation of the reflection and transmission coefficients of aplane wave incident on an infinitely thin plate with a rectangular array ofperforations, assuming that a characteristic size of the perforations isnegligible relative to that of the unit cell of the grating, itself assumed tobe negligible relative to the wavelength. One part of the present study is ofmethodological interest. It establishes that it is possible to extend theirapproach to thick plates with a skew grating of perforations, thus confirmingrecent results in Bendali et al. (SIAM J. Appl. Math., 2013), but in a muchsimpler way without using complex matched asymptotic expansions of the fullwave or to a grating of multipoles. As is well-known, effective compliances forthe plate can then be derived from the corresponding approximations of thereflection and transmission coefficients. These compliances are expressed interms of the Rayleigh conductivity of an isolated perforation. Consequently, inone other part of the present study, the methodology recently introduced inLaurens et al. (ESAIM:M2AN, 2013) to obtain sharp bounds for the Rayleighconductivity has been extended to include the case for which the openings ofthe perforations on the upper and lower sides of the plate are elliptical inshape. This not only enables the determination of these bounds and of theassociated reflection and transmission coefficients for actual plates withtilted perforations but also yields single expressions covering all usual casesof perforations: straight or tilted with a circular or an ellipticalcross-section.

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