The three-dimensional acoustic field in cylindrical and annular ducts with an axially varying mean temperature

Abstract

This paper presents analytical solutions for the three dimensional acoustic field in cylindrical and annular ducts with dependence of mean temperature on axial position. A wave equation for the pressure perturbation is constructed in cylindrical coordinates, applying a zero mean flow condition. Separation of variables is used to express the pressure perturbation as a product of functions which vary only axially, radially and circumferentially. The axial dependence of the mean temperature means that a general analytical solution for the axial second order ordinary differential equation (ODE) cannot be obtained. Variable transformation is applied, yielding a standard second order ODE with known solutions for linear and quadratic axial mean temperature dependence. The acoustic field and resonant frequencies for an annular duct with linear/quadratic axial mean temperature variation predicted using these solutions match perfectly with those calculated using the linearised Euler equations. The analytical solution for the linear mean temperature profile is applied to more complicated profiles in a piecewise linear manner, axially segmenting the temperature profile into regions that can be approximated as linear. The acoustic field and resonant frequency are predicted very accurately even when very few axial segments are used.