Influence of viscosity on the reflection and transmission of an acoustic wave by a periodic array of screens: The general 3-D problem

An analysis is presented of the diffraction of a pressure wave by a periodic grating including the influence of the air viscosity. The direction of the incoming pressure wave is arbitrary. As opposed to the classical nonviscous case, the problem cannot be reduced to a plane problem having a definite 3-D character. The system of partial differential equations used for solving the problem consists of the compressible Navier-Stokes equations associated with no-slip boundary conditions on solid surfaces. The problem is reduced to a system of two hypersingular integral equations for determining the velocity components in the slits' plane and a hypersingular integral equation for the normal component of velocity. These equations are solved by using Galerkin's method with some special trial functions. The results can be applied in designing protective screens for miniature microphones realized in MEMS technology. In this case, the physical dimensions of the device are on the order of the viscous boundary layer so that the viscosity cannot be neglected. The analysis indicates that the openings in the screen should be on the order of 10 μm in order to avoid excessive attenuation of the signal. This paper also provides the variation of the transmission coefficient with frequency in the acoustical domain.