The muffler elements that use perforated elements for acoustic attenuation are common in practice. In typical commercial mufflers perforated elements are used involving two, three, four or more interacting ducts. Analysis of such configurations involves writing down the basic governing equations of mass continuity, momentum balance, etc., and then elimination of velocity variables to obtain the coupled ordinary differential equations in terms of acoustic pressure variables. Mathematical modelling and the consequent analytical derivation of the transmission loss for these multi-duct acoustical elements become increasingly tedious, as just not the number of ducts, but also their relative arrangement along with the boundary conditions dictate the analysis considerably. In the present paper, authors have proposed a generalization and thus an algebraic algorithm to directly produce the system matrix, eliminating the tedium of writing the basic governing equations and elimination of velocity variables. Also, a convenient approach for applying the boundary conditions is outlined here.
The concept of perforate impedance and its exploitation for acoustic attenuation through several elements including concentric tube resonators, is common in practice. Variable cross-sectional area ducts are often used for better performance at lower frequencies (like in horns), whereas concentric tube resonators are often used to provide attenuation at relatively higher frequencies. A combination of the two leads to conical concentric-tube resonators. Using a one-dimensional control volume approach, a mathematical model is presented that accounts for waves in an incompressible mean flow in the center tube, wave propagation in the cavity, and an acoustic coupling between the two due to the impedance of the perforate. The matrizant model results have been validated for self consistency. In the sections dealing with discussion and parametric study, the effect of the moving medium has been neglected so as to bring out clearly the physical effect of the variable area ducts. A few notable effects have been found that include an effective length shorter than the geometric length for an inhomogeneous duct. Some useful features like the absence of pass bands are noticed in the transmission loss spectrum. Finally, results of a parametric study are presented for use by the noise control engineers.
Wave coupling exists in the wave propagation in multiple interacting ducts within a waveguide. One may use the segmentation approach, decoupling approach, eigenvalue approach, or the matrizant approach to derive the overall transfer matrix for the muffler section with interacting ducts, and then apply the terminal boundary conditions to obtain a two-by-two transfer matrix. In such instances, a boundary condition applied to a vector is given as a linear combination of its components. Spatial dimensions along with parameters like impedance of the perforated interface may yield numerical instability during computation leading to inaccurate prediction of the acoustic performance of mufflers. Here, an inherently stable boundary-condition-transfer approach is discussed to analyze the plane wave propagation in suchlike mufflers and applied to waveguides of variable cross-sectional area. The concept of pseudo boundary conditions applied to the state vector at an intermediate point is outlined. The method is checked for self-consistency and shown to be stable even for extreme geometries.
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