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 pl ane 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 Sg and S-1, each of radius a, with respective c entres at (x, y, z) = (0, 0, 0) and (d, 0, 0). The membrane displacement et a(x, y) is constrained to be zero on each of the discs S-0 and S-1, leading to a mixed boundary-value problem with different types of conditions accor ding as x, y is an element of S-0 boolean OR S-1 or x, y is not an element of S-0 boolean OR S-1. The system is activated by an obliquely incident pla ne wave of radian frequency omega and acoustic wavenumber k = omega/c. Asym ptotic results are sought in the limits of large kd and small values of the fluid loading parameter. 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.