Acoustic transmission in curved ducts with varying cross-sections

Abstract

Methods for predicting the acoustic propagation properties of ducts are developed and analysed for ducts that are asymptotically straight with constant cross-section. With application to splitter silencers in mind, the first part of the analysis is devoted to two-dimensional ducts, where area changes can be large. By the so-called Building Block Method, propagation properties for complicated ducts are synthesized from corresponding results for more simple ducts. Conformal mapping techniques are then applied to transform the simple duct to a straight, constant-cross-sectional duct with variable acoustic refractive index. Stable formulations for the scattering operators in terms of ordinary differential equations are formulated by wave splitting using an invariant embedding technique. The numerical method consists of a standard Matlab ordinary differential equation solver, and it is proved that the numerical scheme converges with increasing truncation. Numerical results are presented and analysed for a smooth area contraction and an L-bend. Finally, the theory is generalized to a few classes of three-dimensional ducts.

Improved multimodal admittance method in varying cross section waveguides

“…with the boundary mode (improved method) 8) and the converged field, considered as an exact solution To analyse the convergence of the pressure field, the error |p − p ex | is split into three parts 10) and with the boundary mode 11) with N p + 1 the number of propagating modes. 1 by using the behaviour of p ex n for large n, namely p ex n ∝ 1/n 2 and p ex n − (χ , ϕ n )p −1 ∝ 1/n 4 .…”

“…As any method used to solve the coupled mode equations, the multimodal admittance method has a rate of convergence with respect to the number of local transverse modes in the series expansion of the pressure at each location on the axis of the waveguide. In the literature, several propositions have been made to improve this rate of convergence [7][8][9][10][11]. Similar to the classical attachment mode used in structural mechanics [12][13][14], all these techniques use a boundary mode which is not a local transverse mode but that encapsulates the less convergent part of the series.…”

Propagation in waveguides with varying cross section and curvature: a new light on the role of supplementary modes in multi-modal methods

“…The transformation is chosen for its flexibility, which renders the presented method accessible to a large class of waveguide geometry. In previous studies [19][20][21][22], angle-preserving transformations (as conformal mapping) were preferred, and this is discussed. Next, in §3, the multi-modal formulation is derived, equation (3.3).…”

“…Although they remain valid for many configurations, they require an iterative, numerical, process to get the local coordinate, which may be involved. Alternatively, conformal mappings have been proposed [19,20] which ensure ∂ x X = ∂ y Y and ∂ x Y = −∂ y X (the Cauchy-Riemann conditions for holomorphic transformations). Thus, H = I, which means that (i) the transformation is orthogonal and (ii) the structure of the Helmholtz equation is preserved, that is, the wavefield satisfies p + (k 2 / det J)p = 0, which describes wave propagation in a medium with variable index.…”

Scattering in Quantum Tubes

“…Nilsson [3] treats a general method for the acoustic transmission in curved ducts with varying cross-sections. Wellposedness, i.e.…”