AN EXACT SOLUTION FOR ONE-DIMENSIONAL ACOUSTIC FIELDS IN DUCTS WITH AN AXIAL TEMPERATURE GRADIENT
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 e}ects of mean ~ow\ and therefore the solutions obtained are valid only for mean Mach
numbers that are less than 9=0[ 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 di}erential
equation#[ The applicability of the developed technique is demonstrated by obtaining a
solution for ducts with a linear temperature pro_le[ This solution is then applied to
investigate the dependence of sound propagation in a quarter wave tube upon the mean
temperature pro_le[ 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[\ ~ames#[ 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 pro_le and also for a
temperature pro_le that corresponds to a constant convective heat transfer coe.cient at
the wall[
7 0884 Academic Press Limited
INTRODUCTION
The behavior of one!dimensional acoustic _elds in ducts with a mean temperature gradient
is a problem of considerable scienti_c and practical interest[ For instance\ there is a need
to develop an understanding of the manner in which a mean axial temperature gradient
"caused\ for example\ by heat transfer to or from the walls# a}ects the propagation of
sound waves and the stability of small amplitude disturbances in a duct[ Such understand!
ing will improve existing capabilities for controlling combustion instabilities in propulsion
and power generating systems\ designing pulse combustors and automotive mu/ers\
analyzing the behavior of resonating thermal systems\ and measuring impedances of high
temperature systems "e[g[\ ~ames#[ Consequently\ there exists a need for obtaining exact
analytical solutions that describe one!dimensional wave systems in ducts with axial
temperature gradients[
A physical description of the e}ect of a mean temperature gradient upon wave
propagation in a duct may be obtained by assuming that the gas in the duct consists of
in_nitesimally thin gas layers\ each at a di}erent "constant# temperature\ that are in contact
with one another[ In this case\ propagation of sound from one layer to another is
accompanied by wave transmission and re~ection\ which modi_es the wave structure in
the duct[ To date\ considerable e}orts have been expended on the development of an
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289 R[ I[ SUJITH ET AL[
understanding of wave propagation in such ducts and improving existing capabilities for
controlling the behavior of practical systems where such phenomena occur[
The behavior of one!dimensional waves in ducts with an axial temperature gradient is
described by the solutions of a second order wave equation with variable coe.cients 0[
Cummings 0 has developed an approximate analytical solution for ducts with arbitrary
temperature gradients in the absence of mean ~ows[ Munjal and Prasad 1 and Peat 2
have developed exact solutions for ducts with small temperature gradients\ in the presence
of mean ~ows[ Also\ Kapur et al[ 3 obtained numerical solutions for sound propagation
in ducts with axial temperature gradients\ in the absence of mean ~ows\ by integrating the
wave equation using a Runge
Kutta method[ The same approach was followed by Zinn
and co!workers 4
7 who developed the impedance tube technique for high temperature
systems\ in the presence of mean ~ows[ While these numerical methods yields accurate
answers\ they often do not provide adequate insight into the physics of the problem[
Moreover\ since they cannot be implemented without the availabity of an e.cient
computer\ it is often di.cult to incorporate these solution approaches into practical design
procedures[ These di.culties would be alleviated if analytical solutions of the equations
that describe one!dimensional acoustic oscillations in ducts with a temperature gradient
were available[
Robins 8 developed an approach for obtaining analytical solutions of a related
problem that is of interest in underwater acoustics^ that is\ wave propagation in water
where the density and speed of sound vary with distance owing to changes in water
depth[ Robins used an inverse approach to obtain density pro_les that reduce the wave
equation to a standard di}erential equation whose exact solutions are known[ A
signi_cantly di}erent analytical approach for solving such problems is presented in this
paper[
A method for obtaining an exact solution that describes the behavior of one!dimensional
oscillations in a duct with an arbitrary axial temperature pro_le is outlined in this paper[
The developed analytical solution procedure consists of several steps[ First\ the one!
dimensional wave equation for a constant area duct with an arbitrary axial temperature
gradient is derived for a perfect\ inviscid and non!heat!conducting gas[ Next\ assuming
periodic solutions\ the derived wave equation is reduced to a second order ordinary
di}erential equation with variable coe.cients[ Since the equation has variable coe.cients\
exact solutions of this equation for an arbitrary temperature pro_le cannot be obtained[
To obtain an exact solution\ the derived di}erential equation is transformed from the
physical\ x\ space to the mean temperature\ T\ space[ Using appropriate transformations\
the resulting equation is then reduced to a standard di}erential equation "e[g[\ Bessel|s
di}erential equation#\ whose form depends upon the speci_c mean temperature pro_le
in the duct[ The application of the developed solution technique is demonstrated in this
paper by obtaining analytical solutions for wave propagation in ducts with linear and
exponential temperature pro_les and a temperature pro_le that corresponds to a constant
heat transfer coe.cient\ and utilizing the developed solution in high temperature admit!
tance measurements[ The analysis neglects the e}ects of mean ~ow\ which limits the
applicability of the developed solutions to mean ~ows with Mach numbers that are less
than 9=0

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