Modal solutions in circular and annular ducts with locally or bulk reacting lining

摘要

In circular and annular lined ducts there exist modes of the sound field in both azimutal and radial directions. There are two distinct tasks to be solved for technical applications. In the first task, one is interested in the attenuation of the least attenuated mode because this mode determines the sound reduction under general conditions of excitation. The second task aims at the computation of a set of modes. The solutions of this task are needed in a modal analysis of sound fields in structured round silencers, the modes being orthogonal to each other. It is sufficient in the first task to look for the least attenuated mode among the two first radial modes. A set of modes for a modal analysis must be unique in the sense that no mode solution appears twice in the set of solutions and that no mode is missing in a set extending up to a given order. Otherwise the systems of equations for the amplitudes of the modes in the further analysis would become singular. Therefore, the problem of the limits between the modes must be solved and/or safe algorithms for modal solutions should be available. Both problems will be tackled in the present paper. The numerical difficulties increase at the transitions from locally reacting linings to bulk reacting linings and from circular to annular ducts. The present analysis assumes no mean flow in the duct. The solutions under this condition serve as start solutions in numerical algorithms for the modal solutions with flow. [References: 33]