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.
Highlights
The propagation of acoustic waves along ducts or pipes has long been of interest to scientists and engineers
This is partly because ducting systems, such as those used for heating, ventilation and air conditioning (HVAC), provide ideal channels for the transmission noise through structures such as buildings and aircraft
Acoustic scattering is a feature of ducting systems that becomes relevant whenever there is an abrupt change in geometry or material property
Summary
The propagation of acoustic waves along ducts or pipes has long been of interest to scientists and engineers. The Wiener–Hopf technique has long been a convenient tool to tackle a wide range of two-part problems, whereas analytic mode-matching has only recently been established as a viable tool for two-dimensional structures in which the boundary conditions contain higher-order derivatives. The reason for this is that the eigenfunctions for such systems do not satisfy standard orthogonality conditions, such as that for the set of functions cos(nπ y/a), n = 0, 1, 2,.
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