On acoustic propagation in three-dimensional rectangular ducts with flexible walls and porous linings

Abstract

The focus of this paper is towards the development of hybrid analytic-numerical modematching methods for model problems involving three-dimensional ducts of rectangular cross-section and with flexible walls. Such methods require firstly closed form analytic expressions for the natural fluid-structure coupled waveforms that propagate in each duct section and secondly the corresponding orthogonality relations. It is demonstrated how recent theory [Lawrie, Proc. R. Soc. A. 465, 2347-2367] may be extended to a wide class of three-dimensional ducts, for example, those with flexible walls and a porous lining (modelled as an equivalent fluid) or those with a flexible internal structure such as a membrane (the "drum-like" silencer). Two equivalent expressions for the eigenmodes of a given duct can be formulated. For the ducts considered herein, the first ansatz is dependent on the eigenvalues/eigenfunctions appropriate for wave propagation in the corresponding two-dimensional duct with flexible walls, whilst the second takes the form of a Fourier series. The latter has several advantages: in particular, no "rootfinding" is involved and the method is appropriate for ducts in which the flexible wall is orthotropic. The first ansatz, however, provides important information about the orthogonality properties of the eigenmodes.

Analytic mode-matching for acoustic scattering in three dimensional waveguides with flexible walls: Application to a triangular duct

“…Recent advances in this respect have been forged by Lawrie [18] who established much of the mathematical theory underlying acoustic propagation in a 3D rectangular duct with one flexible wall. A second article extends the theory to ducts with porous linings, internal structures or orthotropic boundaries, [19]. For each of the ducts considered in [18]- [19], the "corner conditions" applied along the length of the duct where the flexible wall meets the adjacent rigid wall dictate that the eigenmodes are non-separable in form.…”

“…A second article extends the theory to ducts with porous linings, internal structures or orthotropic boundaries, [19]. For each of the ducts considered in [18]- [19], the "corner conditions" applied along the length of the duct where the flexible wall meets the adjacent rigid wall dictate that the eigenmodes are non-separable in form. As a result only a "partial" orthogonality relation can be constructed and this, of course, complicates the mode-matching procedure.…”

“…For the sake of brevity, only fully symmetric modes will be discussed and the reader is referred to [19] where anti-symmetric modes for a duct of this class are discussed.…”

“…In section 2, the theory established in [18]- [19] is extended to a rectangular duct in which all four walls are flexible.…”

“…Following [18] and [19], an appropriate ansatz for Ψ(s, y, z), can be formulated using the eigenfunctions Y m (y), m = 0, 1, 2, . .…”