Homogenization of thin-structured surfaces for acoustics in the presence of a two-dimensional low Mach potential flow

A surface homogenization method for acoustic waves over thin microstructured surfaces in the presence of a fluid in a potential flow is presented. Sound hard surfaces are considered, the flow is considered two-dimensional and slow and a low Mach approximation is introduced. We consider acoustic waves with a typical wavelength 1 / k much larger than the array spacing h and thickness e . Owing to the small parameter ε = k h , with e / h = O ( 1 ) , a matched asymptotic expansion technique is applied to the low Mach potential wave equation in the frequency domain. A boundary condition is obtained on an equivalent flat wall, which links the acoustic velocity to its normal and tangential derivatives (of the Myers type). The accuracy of the effective model is tested numerically for various periodic shapes and the accuracy of the model in O ( ε 2 ) is validated.