An exact solution for one-dimensional acoustic fields in ducts with an axial temperature gradient

Abstract

In this paper an approach is presented for obtaining exact analytical solutions for sound propagation in ducts with an axial mean temperature gradient. The one-dimensional wave equation for ducts with an axial mean temperature gradient is derived. The analysis neglects the effects of mean flow, and therefore the solutions obtained are valid only for mean Mach numbers that are less than 0.1. The derived wave equation is then transformed to the mean temperature space. It is shown that by use of suitable transformations, the derived wave equation can be reduced to analytically solvable equations (e.g., Bessel’s differential equation). The applicability of the developed technique is demonstrated by obtaining a solution for ducts with a linear temperature profile. This solution is then applied to investigate the dependence of sound propagation in a quarter wave tube upon the mean temperature profile. Furthermore, the developed analytical solution is used to extend the classical impedance tube technique to the determination of admittances of high temperature systems (e.g., flames). The results obtained using the developed analytical solution are in excellent agreement with experimental as well as numerical results. Analytical solutions were also obtained for a duct with an exponential temperature profile and also for a temperature profile that corresponds to a constant convective heat transfer coefficient at the wall.