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

Abstract

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 microns 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.

Boundary-Layer Effects on Acoustic Transmission Through Narrow Slit Cavities

“…He states that Kirchhoff's tube equations are still valid for slits, with a substitution of the slit width w for the tube radius r. Stinson [20] simplified Kirchhoff's solutions when applied to tube cross sections of arbitrary shape, including an infinitely long slit cavity, finding agreement with Lord Rayleigh. More recently, Homentcovschi and Miles studied a nonresonant system of periodic, very thin screens (L ≪ λ 0 ) with slit widths of the order of the boundary-layer thickness, investigating acoustic attenuation due to viscosity and its effect on diffraction [21,22]. However, none of this research explored the effect of the boundary-layer perturbation on a resonant slit system, which is the focus of the current study.…”