Transmission line coefficients for viscothermal acoustics in conical tubes

Abstract

Viscothermal acoustic propagation in gases contained in rigid straight or conical tubes is considered. Under the assumption that the wavelength is much larger than both the boundary layer thickness and the tube radius, pressure and flow are shown to be solutions of a pair of coupled 1D differential equations, formulated as transmission line equations involving complex loss coefficients. The derivation of these loss coefficients, which is usually accomplished in cylinders, is generalized here to conical geometries. In the well-known case of circular cylinders, the Zwikker–Kosten (ZK) theory is recovered. For circular cones, the expression of the loss coefficients is derived. It involves complex-order spherical harmonics, instead of Bessel functions for circular cylinders, and makes the hydraulic radius appear as a natural relevant geometrical parameter. We show that replacing the classical radius by the hydraulic radius in the ZK theory provides an affordable and accurate approximation of the analytic model derived for cones. The proposed formulas are used to compute the input impedance of a cone, and compared with a 3D reference. In an ideal setting, using the spherical harmonics or the hydraulic radius in the 1D method accurately approximates the full 3D method, and allows to increase accuracy by approximately two orders of magnitude compared to the ZK theory.