The sound field due to a periodic array of forced wave-bearing strips
D. F. Popple, F. G. Leppington · IMA Journal of Applied Mathematics · 1995
A compressible fluid in a two-dimensional half-space (y > 0) is bounded by a plane surface (y = 0) which is acoustically hard except for a set of periodically arranged strips Sn given by nd − a 0, with ∂φ/∂y = 0 on the plane y = 0, x ∉ Sn. The boundary condition on the pistons Sn is taken to have the form where the prescribed forcing function V(x) is the same on each strip, so that V(x + nd) = V(x), and the operators L and M are polynomial functions of the second derivative ∂2/∂x2. This boundary condition includes the possibilities of an elastic plate, a membrane, or an impedance surface for Sn. When the separation distance d is much greater than the strip width 2a and wavelength 2π/k, the problem is reduced to that of finding the potential φp due to a single piston So set in a rigid baffle, together with a potential φc subject to a similar condition with forcing functions exp (±ikx) in place of V(x). The problem is generalized to allow for the possibility of a phased forcing function V(x), such that V(x + nd) = exp (iβnd)V(x), where β is a given constant.