Propagation of Acoustic Waves in Two Waveguides Coupled by Perforations. II. Analysis of Periodic Lattices of Finite Length

The paper deals with the generic problem of two waveguides coupled by perforations, which can be perforated tube mufflers without or with partitions, possibly with absorbing materials. Other examples are ducts with branched resonators of honeycomb cavities, which can be coupled or not, and splitter silencers. Assuming low frequencies, only one mode is considered in each guide. The propagation in the two waveguides can be very different, thanks e.g. to the presence of constrictions. The model is a discrete, periodic one, based upon 4thorder impedance matrices and their diagonalization. All the calculation is analytical, thanks to the partition of the matrices in 2nd-order matrices, and allows the treatment of a very wide types of problems. Several aspects are investigated: the local or non-local character of the reaction of one guide to the other; the definition of a coupling coefficient; the effect of finite size when a lattice with n cells in inserted into an infinite guide; the relationship between the transmission coefficient and the dispersion. The assumptions are as follows: linear acoustics, no mean flow, rigid wall. However the effect of the series impedance of the perforations, which is generally ignored, is taken into account, and is discussed. When there are no losses, it is shown that, for symmetry reasons, the cut-off frequencies depend on either the series impedance or the shunt admittance, and are the eigenfrequencies o f the cells of the lattice, with zero-pressure or zero-velocity at the ends of the cells.