Analytical modelling of sound transmission in a lined duct

Abstract

The focus of this thesis is on the prediction of sound attenuation through a lined duct, based on a mathematical model. Ducts with a single section as well as multi-segmented sections are discussed. The duct of interest has a rectangular cross-section as normally used for ventilation purposes. The mean flow in a ventilation duct is very low and can be neglected. In this thesis, two-dimensional analytical models are developed for sound transmission in a series of different duct configurations. Two models of the lining behaviour are considered, either locally-reacting
or bulk-reacting.

The models are used first to obtain the transverse and axial wavenumbers of various modes of the duct. The required finite numbers of wavenumbers are tracked using Müller’s method. The wavenumbers are traced from a very low frequency to high frequency using small frequency steps. It is found that, for a duct with a bulk-reacting lining, the number of modes with a transverse wavenumber below a particular value may exceed the corresponding number of modes in a duct with a locally-reacting lining. These additional modes are termed lining modes. The number of lining modes depends on the lining thickness. Dispersion curves are presented for both types of lining. The transmission of sound through the duct is then calculated using the mode-matching technique. The mode-matching model allows analysis of multi-modal wave propagation in the duct. The model is first developed for an infinitely long rigid duct with a finite length of lined insert. The estimation from the locally reacting model, that is widely available in the literature, is compared with the estimation from the newly developed bulk-reacting model. Although the locally reacting model often overestimates the performance of a bulk-reacting lining it is found that this is not always the case, especially for a small lining thickness and at lower frequencies where the locally reacting model may under-estimate the performance.

The analytical model is then extended to a multi-segmented lining where the lined section is uniformly segmented with rigid walled sections in a periodic manner. For a bulk-reacting lining, the segmented arrangement renders the lining more similar to the behaviour of a duct with a locally-reacting lining and improves the peak attenuation. Little improvement is found in the case of a multi-segmented locally-reacting lining. The effect of duct height, lining thickness and lining flow resistivity on sound attenuation is studied using the analytical model. Experiments are presented which validate the analytical model.

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