Scattering of sound by an infinite membrane fixed on two circular regions
AbstractA 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 S 0 and S 1 , each of radius a , with respective centres at ( x, y, z ) = (0, 0, 0) and ( d, 0, 0). The membrane displacement ॉ( 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 according as x, y ∈ S 0 ∪ S 1 or x, y ∉ S 0 ∪ S 1 . The system is activated by an obliquely incident plane wave of radian frequency ३ and acoustic wavenumber k = ३/ 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.