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.

中文翻译：

---

粘度对周期性屏幕阵列反射和传输声波的影响：一般 3-D 问题

分析了周期性光栅对压力波的衍射，包括空气粘度的影响。传入压力波的方向是任意的。与经典的非粘性情况相反，该问题不能简化为具有明确 3-D 特征的平面问题。用于求解该问题的偏微分方程组由与固体表面无滑移边界条件相关的可压缩 Navier-Stokes 方程组成。该问题被简化为一个由两个超奇异积分方程组成的系统，用于确定狭缝平面中的速度分量，以及一个用于速度法向分量的超奇异积分方程。这些方程是通过使用具有一些特殊试验函数的 Galerkin 方法求解的。研究结果可用于设计采用 MEMS 技术实现的微型麦克风的保护屏。在这种情况下，设备的物理尺寸在粘性边界层的数量级上，因此粘性不能被忽略。分析表明，屏幕上的开口应在 10 微米左右，以避免信号过度衰减。本文还提供了在声域中传输系数随频率的变化。分析表明，屏幕上的开口应在 10 微米左右，以避免信号过度衰减。本文还提供了在声域中传输系数随频率的变化。分析表明，屏幕上的开口应在 10 微米左右，以避免信号过度衰减。本文还提供了在声域中传输系数随频率的变化。