Abstract

The interaction of a plane wave with a periodic array of slits is an important problem in fluid dynamics, electromagnetism and solid mechanics. In particular, such an arrangement is commonly used as a model for turbomachinery noise. Previous work has been restricted to the case where the slits possess a Neumann (no-flux) boundary condition. Consequently, in this work we consider “complex” boundary conditions including Robin (e.g. compliance), oblique derivatives (porosity) and generalised Cauchy conditions (impedance). We employ generalised derivatives and Fourier transforms to recast the Helmholtz equation as an integral equation amenable to the Wiener—Hopf method. Although the slits are of finite length, we are able to avoid a true matrix Wiener—Hopf problem by assuming the structure of the scattered field. Since the Wiener—Hopf kernel is meromorphic, the Fourier transform may be inverted analytically to obtain the scattered field. The Wiener—Hopf analysis shows that an effect of modifying the boundary conditions is to perturb the zeros of the kernel function, which correspond to the “duct modes” in the near field. In aeroacoustic applications, this result shows that blade porosity can dramatically reduce the unsteady lift, which has implications for turbomachinery design.