A predicted standing wave pressure and phase angle profile for a hard wall rectangular duct with a region of converging-diverging area variation is compared to published experimental measurements in a study of sound propagation without flow. The factor of 1/2 area variation used is sufficient magnitude to produce large reflections. The prediction is based on a transmission matrix approach developed for the analysis of sound propagation in a variable area duct with and without flow. The agreement between the measured and predicted results is shown to be excellent

Miles, J. H.

English

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NASA Technical Memorandum 82741
a^
r
J. H. Miles
Lewis Research Center
Cleveland, Ohio
r ^
4AS4
•	 Verification of an Acoustic Transmission Matrix
Analysis of Sound Propagation in a Variable
Area Duct Without Flow
(i:ASA-TM-82741)	 VERIEICATICN Of AN ACOUSTIC
	 N82-12891
TRANSMISSION MAiuIY ANALYSIS CF SCUND
PROPAGATION IN A VARIALLE AnEA LUCT ImITHOUT
FLUW (N4SA)
	
15 F HC AU:/MF A01	 CSCL 20A	 Unclas
63/71 08411
Prepared for the ^`.A^-j
One-hundred-second Meeting of the Acoustical Society of America
Miami Beach, Florida, December 1-4, 1981
T
I
Verification of an acoustic transmission matrix dnalysis of
f	 sound propdgdtton ► n a variable area duct without slow
by J. 11. hliles
National Aeronautics dnd space Administration
Lewis Research Lenter
Llevelana, Uhio 4413b
Abi I RAL T
A predicted standiny wave press-ire and phase any le protIle for a hdrd
wail rectanyuldr duct with d region of converging-aiveryiny area variation
is compared to published experimental measurements in a study of sound
propagation without flow.	 The factor of 1/,^ area variation used is of
suttiCient md Lill ► tuLie to proliuce ldrge retIections.	 the prediction is based
un d transm ► ssrun matrix approach cleveloped for the analysis of sound propa-
ydtion in a variable area uuct with and without flow. the agreement between
thi s
 measured anu preuiited resull, ► s shown to be excellent.
NUML NLLAI URt
matrix
ared,	 n
co ► sent-opic	 speed of	 sounu,	 m/s
r t requenl:y,	 ht
t. duct height	 equat ► .m parameter,	 ui
I unit matrix
i (-1)1j2
V. wave	 number,	 W/c	 111-1
0 Lis
L duct	 length,	 m
L' length of	 drtd	 vdr ► dtiOn,	 m
III d	 ln(N)/dx,	 m
P transformation matrix
p pressure,	 Pa
k pressure reflection facLor
i transmission matrix
t t ime,
	 s
u velocity,	 m/s
x Cartesian coordinate, 	 m
Y acoustic state vector
y Cuct height, m
L acoustic	 impedance, mks kayles
IN- x region	 length,	 m
X eigenvalue
p o gas aensity,	 kg/m
W angular frequency,	 radians/s
Superscripts ar.a	 Subscripts:
(	 )' scaled quantity
(	 ) 0 reference state quantity
(	 ) 1 perturbed quantity
INIKUOUCIIUN
An acoustic transmission riatrix analysis o
variable area uuct with and witkout How is ais
once 1 is concernea only with Lheoretical solut
equations. This paper concentrates on the expe
analysis aiscussed in ket. I.
2
The analysis is compared with an experimental standing wave pressure
and phase angle profile obtaineo in a study of sound propagation without
flow in a long haro wall rectangular duct containing a region of converging-
diverging area variation (Ref. [).
This experimental data provides a good test of the theory since the
gradients of the area variation in the region of the converging-diverging
area variation are large enough to produce a reflection much greater than
the reflection produced at the duct exit.
In the present paper, first a brief review of the theory is presented.
Next, the experimental data is discussed. Then the analytical results are
compared to the measurements and conclusions are drawn.
ThtUkY
The tolluw1ny assuPuptions are made.	 The duct has an area profile
A(x). The acoustic equations are described by first order linearized, per-
turbed equations of continuity momentum, energy and state for isentropic
sound propagation.
State variable equation
The perturben state variables are the pressure and velocity. Ihest
quantities are assumed to vary with time as exp (-iwt). From ket. 1, the
Fourier transtorm of the resulting equations is
	
dp l /dx	 0	 POc0 ik o 	pl
dY 
_	 = b^f	 (1)
ox	 tko	 _ d In (A)
	
`uu 1' dx	 o0 c0	 dx	 t^l
3
Method of solution
The method of solution used in this study is presented in ket. 1. for
clarity, it is briefly discussed. 	 The tirst step in obtaining a solution to
Eq. (1) is to divide the duct into a number of regions or subsections. The
region size is selected so that the area variation in each region can be
approximated by an exponential area variation. Lonsequently, in a given
region
N = A 
Enix
0
With this assumption the value of n ln(H) /dx is a constant in the region
given by
	
m = d ln(N)/dx
	
(.$)
The b matrix in Eq. (1) is now inuepenuent of x in the region where it
is to be evaluated.	 Thus, the solution to Eq. (1) for the jth region is
f(x j + o x ) = (ebAx)l f(xj )	 l4)
where x is the length of the jth region ano exp (bax) is known as the
matrix exponential.	 The value of exp (bex) is oefineo by
e bex = 1 + (bax) + 	 — + (bAx )' + ...	 (5)
C.	 ^.
The matrix exp (bex) is the transmission matrix of the jth region (1)j,
so that
(box).	 (box)R	 111
l l )^ = e	 J -	 —a:1	 lb)
1 ^1	 12t
4-U
4
IConsequently, the transmission matrix, T, for the variable area duct is
touna from
N
Y(x} x-L _	 e(trex)d 
y ( x )I x 	= 1 Y(x) x=0
	
(1)
I	 I
j=1
n -	 where
N
L =	 (ex)^
j=1
Evaluation of matrix exponential
While the series given in Eq. (5) refines the matrix exponential, it
was not used for computer calculations since the convergence of the series
i
may be slow. The matrix exponential was evaluated from
	
e bex = C-1 N eUex p 1 C	 (g) I
where
	
U = P-1 .a/ p 	 (10)
	1 	 U
C =	 (lc}
	
U	
^ODU
'^L L	 ^ 1	 'AB C L	 a 2
(13)
	
- 'i0^21	 -'^[ 1
and the a are the eigenvalues of the ,V matrix given by the roots of
(8)
5
	a2+ ba + c=U	 (14)
where
	
b = - (.v/11 + A( 22 )	 (15)
and
C
	 ''/11 
'"(2 2 - s"/12 "fZ 1	 (16)
H more complete discussion of the evaluation of the matrix exponential can
be found in Ret. 1.
boundary condition
The standing wave profile is calculated as follows. iir,t, the pres-
sure at the duct exit (identified as location one) is assumed to be 1 Na.
H duct exit impedance is then used to calculate the velocity
u 
	 = 1/1 1
	(17)
where the duct exit impedance is related to the duct exit pressure reelec-
tion tactor by
1+k
0 oc o = T_-_T
Next, the transfer matrices, T n , between each of the measurement points is
calculated. The state vector at the last measurement point is
	
pm	 1
= Tm-lTm-2 '•• 1
3 1 2 1 1	 (ly)
	
um	 1/11
and the impedance at this point is
Zm
 = Pm/u111	 (20)
U
Note, that this is an initial value problem where the impedance and
pressure are specified at the exit which differs from the boundary value
approach used by two dimensional numerical solutions.
The pressure at the point m-1 due to a unit pressure at the point m
is obtainea from
[Ti-p 	 11	 (21)
u	 = Tnr-1 	 1/Z
m-1
	
m
This procedure is continued at each point. Finally, the pressure at the
duct exit is
pl	 1
-1	 -1	 -1	 (22)u	 = T 1 T213 
	 Trn-^ 1/Z
1	 m
The pressure magnitude o f the standing wave pressure profile is given by
	
Ipml = (prT)pm)1
/2 	 (23)
and the phase angle of the stdnding wave phase profile is given by
Pm
e = tan-1 lm (L4)
heprn
However, the pressure ma gnituae plotted is scaled with repect to the mea-
sured pressure magnitude at the duct exit p(1)
exit
(1) ,/P ( 1)
p m - pin pexi 	 (r5)
tion of the ouct t
begins propagating
The phase angle is given with respect to the angle at x = U. 	 In addition
since the experimental measurements of kef. 2 are made assuming a time fac-
tor of exp ( * iwt) while the analysis is based on using exp (-iwt), the
sign of the angle calculatea is reversed. Consequently,
em = e(U) - em	(2b)
THE EXPERIMENTAL DATA
The general acoustic duct system described in Ref. 3 was modified into
the simple no flow apparatus shown ir. Fig. 1(a) and used to measure the data
discussed in kef. 2.
In the experiment discussed in kef. z the duct temperature was 27 C so
that the speed of sound was 348 m/s. The souno traveled in a 0.1 m by
U.U381 in hara wall rectangular test section which was 1.42 m long. It was
constructed so that the 8 top plates and 8 bottom plates were detachable.
The variable area test piece was inserted in elate of the bottom piece at
location b.
The variable area test piece is shown in Fig. 1(b). 	 The surface pro-
file is prescribed by a fourth degree polynomial
Y/H = 16(x/L*) 2 - 32(x/L*) 3 * 16(x/L * ) 4	(27)
where L* is U.U76[ m anu H is 0.0191 m which is just half the auct height.
The sound source used to obtain the data discussed in Ref. 2 was a
speaker driven by a sine wave generator at 1560 Hz. The 1560 Hz driving
frequency assured that only plane waves would propagate in the straight por-
attached to the upstream end of the test section. The horn was filled with
an acoustic foam wedge to approximate a L = pd C 0 impedance termination
where the pressure reelection factor is k = 0.0 (Eq. (18)) which minimizes
retlect„ons anu simulates an infinitely long duct.
Further information on the apparatus including a description of the
inicrophone installation and the measurement procedure ;s given in Ref. t.
DISCUSSION OF RESULTS
The measured and theoretical normaiized pressure magnitude profiles arc
compared in fig. Z and the measured and theoretical phase angle profiles are
compared in Fig. 3.
	
The theoretical curves calculated using as an exit
boundary condition a duct exit pressure reflection tactor of k = U.O which
corresponds to an impedance termination of c = p0
 C0 are a near perfect
match upstream of -A a area change. however, due to design and material
limitations the pressure retlection tactor is not exactly zero. Lonsequent-
ly, the pressure magnitude is not constant downstream of the area change and
the experimental and theoretical curves are not in good agreement in this
region.
Impedance tube theory indicates that one unique value of the exit re-
tlection factor will generate the stanoing wave pattern shown in Fig. k.
In order to determine this value, small chances in the exit boundary condi-
tier wire made so that the theoretical curves were in better agreement with
the data downstream of the area change. The resulting boundary condition
was R - -0.030 + j 0.010 (Z = n 0C 0 (0.94 + j 0.003G)). This boundary
condition as shown in Figs. 7 and 3 yields a more perfect match not only for
the downstream data paints but at all data points.
9
the data discussed in Ret. 2 wes used to verify the "steady" state
finite element theory of Astley and Eversman discussed in Ref. 4 and the
transient finite difference theory of White discussed in Ref. 5 for the
limitin g
 case of plane wave sound propa g ation ;.ithc)ut flow. Both of these
numerical methods yield a two dimension„1 solution. 	 I t'is shown in Ref. 2
that both of these methods predict the proper treads of the data and are in
excellent agreement with the measurements. In both cases, howe er, the cal-
culations were made assuminq that the impedance was 7 - r oc o . Con-
seauently, the numerical calculations resembled the clashed curve in the
jownstrPam ,)ortion of duct (x greater than unity).
CONCUPING REIiARhS
the var i able area transmission matrix theory (Ref. 1) produces good
a g reement with the axial data both in terms of the pressure magnitude and
ph2^ , . Similar results can be obtained by numerical methods (Refs. 4 and
5). However, these latter methocs are more complex than nccessary if only
plane wave propagation is of interest. Th, variable area transmission
wetrix approach is simple to use by comparison with two dimensional numeri-
cal methods. Consequently, the acoustic transmission matrix analysis of
sound propa g ation in a variable area duct is an ideal procedure to use for
plane wave propa g ation of sound in a hard will duct with large area
variation.
REFERENCES
1. J. H. Miles. "Acoustic transmission maxtrix of a veriabie area duct or
no77le carryin g
 a compressible subsonic flow," J. Acoust. Soc. Am. n9.
1577 -1(1 86 (1981).
10
2. K. J. Baumeister, W. Eversman, R. J. Astley, and W. White, "Application
of "steady" state finite element and transient finite difference theory
to sound propagation in a variable area duct: a comparison with
experiment," A.IAA Paper No. 81-2016 (October 1981).
'z . G. P. Succi, K. J. Baumeister, and K. U. Ingard, "Interaction of a
turbulent jet noise source with transverse modes in a rectangular duct,"
NASA TP-1?48, June 1978.
4. R. J. Astley and W. Eversmen, "Acoustic transmission in nonuniform ducts
with mean flow, part 11 the finite element method. J. Sound Vib. 74,
103-121 (1981).
5. J. W. White, "A general mcppinq procedure for variable area duct
acoustics," A1AA Paper No. 81-0094 (January 1961).
(7.62 c m) i-{
Y	 H 1^Cm)
1	 _^
ORIGINAL PAGE IS
OF POOR QUA LITY
1.42 m	 --•^
TES! PI RTES
	
10 cm,	 MICROPHONE A
(16 TOTAL)	 ; '16.6 cm MICROPHONE S
EXPONENTIAL HORN
POWE R
 
TER	
```SPEAKER	 SOUND ABSORBINGI'%
ACOUSTIC WEDGE -
S wE WAVE
GENERATOR
(a) No flow acot , Aic duct test section and instrumentation.
(bl Variable area test piece with 5 microphone locations (microphone plugs not shown).
Figure 1. - Apparatus.
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Verification of an acoustic transmission matrix analysis of sound propagation in a variable area duct without flow

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