Multimodal analysis of acoustic propagation in three-dimensional bends

Abstract

An exact multimodal formalism is proposed for acoustic propagation in three-dimensional rigid bends of circular cross-section. Two infinite systems of first-order differential equations are constructed for the components of the pressure and axial velocity in the bend, projected on the local transverse modes. These equations are numerically unstable, due to the presence of evanescent modes, and cannot be integrated directly. An impedance matrix is defined, which obeys a Riccati equation, numerically workable. With this nonlinear first-order differential equation, the impedance can be calculated everywhere in the bend, allowing a direct characterization of its acoustical properties or allowing the acoustic field to be integrated. An exact algebraic formulation of the reflection and transmission matrices is carried out to allow bends and more complex duct systems to be characterized. This result is applied to calculate the reflection and transmission of a typical bend, and also to obtain the resonance frequencies of closed tube systems.