Scattering of sound by an infinite membrane fixed on two circular regions

A compressible fluid with wave speed c lies on both sides of an infinite plane membrane whose equilibrium position is z = 0 in a Cartesian coordinate system. The membrane is free to vibrate in response to the fluid pressure, except for two disc regions S0 and S1, each of radius a, with respective centres at (x, y, z) = (0, 0, 0) and (d, 0, 0). The membrane displacement ε(x, y) is constrained to be zero on each of the discs S0 and S1, leading to a mixed boundary-value problem with different types of conditions according as x, y ∈ S0 ∪ S1 or x, y ∉ S0 ∪ S1. The system is activated by an obliquely incident plane wave of radian frequency ω and acoustic wavenumber k = ω/c. Asymptotic results are sought in the limits of large kd and small values of the fluid loading parameters. This is achieved by reducing the problem to that of a combination of single disc problems, and asymptotic results are known for this simpler class of mixed boundary value problems.