Unidimensional models of acoustic propagation in axisymmetric waveguides

This paper presents a model of the linear acoustic propagation in axisymmetric waveguides, the pressure depending on a single space variable. The approach consists of writing the wave equation and the boundary conditions for a coordinate system rectifying the isobaric map at each time. The two-dimensional dependence of the problem is thus transferred from the pressure to the coefficients of the wave equation. From this result, an exclusively geometrical necessary condition is deduced for the admissibility of isobaric maps. However, the knowledge of the waveguide geometry is not sufficient to separate the pressure and the isobaric map solutions. In order to develop a unidimensional wave equation, a geometrical hypothesis is discussed. For lossless and motionless rigid waveguides, the deduced equation leads to exact results for tubes and cones. It may be interpreted as a Webster equation for a particular coordinate system so that the particular profiles for which analytical solutions of the pressure exist are redefined. The wave equation is also established for pipes with visco-thermal losses and, more generally, for mobile walls having a small admittance. The compatibility of the geometrical hypothesis with the exact model is specified for this general case.