The propagation and attenuation of sound in lined ducts containing uniform or “plug” flow

This paper is concerned with a theoretical analysis of the inviscid, perturbed or acoustic field, at a particular frequency, in an infinite, two-dimensional duct of constant cross-section in which the fluid properties, other than the mean axial velocity, are constant; one duct wall has a uniform, locally reacting, frequency dependent wall impedance, the other wall is rigid. The perturbed duct field due to an infinite, uniform line source, or the two-dimensional Green's function, is formally derived for uniform or “plug” flow in the duct, and is expressed as an infinite sum of non-orthogonal modes. The optimization of modal, axial attenuation rates is examined in some detail. Under certain conditions it is found that not necessarily all the Green's function modes in “plug” flow decay away from the source: in a particular example it is shown that one mode is spatially amplified in the downstream direction and is a modified form of the well-known temporal instability of an incompressible vortex sheet adjacent to a single flexible wall.

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