Orthogonality relations for fluid-structural waves in a three-dimensional, rectangular duct with flexible walls

Abstract

An exact expression for the fluid-coupled structural waves that propagate in a three-dimensional, rectangular waveguide with elastic walls is presented in terms of the non-separable eigenfunctions ψ n ( y , z ). It is proved that these eigenfunctions are linearly dependent and that an eigenfunction expansion representation of a suitably smooth function f ( y , z ) converges point-wise to that function. Orthogonality results for the derivatives ψ n y ( a , z ) are derived which, together with a partial orthogonality relation for ψ n ( y , z ), enable the solution of a wide range of acoustic scattering problems. Two prototype problems, of the type typically encountered in two-part scattering problems, are solved, and numerical results showing the displacement of the elastic walls are presented.