Theoretical analysis of reactive silencers with two propagating modes

A general modeling of sound ducts with reactive (or dissipative) treatments considers one main guide and a secondary one (the treatment), with another propagation constant, including discontinuities, coupled to the main one by perforations, i.e., a shunt impedance and a series impedance [see Kergomard et al., Acta Acustica 2, 1–16 (1994)]. For a periodical system the calculation can be made analytically for a succession of N cells by diagonalization: the impedance matrix of the portion of the duct with treatment can be deduced. At low frequencies only one mode propagates in each duct and the matrices are of fourth order. From the boundary conditions for the treatment the insertion loss (IL) for the main duct is deduced. Examples are an array of coupled resonators, a lining with perforated facing, and a perforated tube muffler with partitions. A conclusion is the proof of the interest of the interferences between two propagating modes of the periodical system: resonances occur and IL is proportional to log...