Sddhand, Vol. 9, Part 4, December 1986, pp. 255-269. L~) Printed in India.
Noise reduction with perforated three-duct muffler components
K NARAYANA RAO* and M L MUNJAL t
Department of Mechanical Engineering, Indian Institute of Science,
Bangalore 560 012, India
::'Present address: Mechanical Engineering Department, University
College of Engineering, Osmania University, Hydcrabad 500 0~7 India
: Present address: Department of Mechanical Engineering, University of
Calgary,
25{)0 University
Drive
NW,
Calgary,
Alberta,
Canada T2N IN4
MS received 21 March 1986; revised 28 May 1986
Abstract. This paper describes the authors' dzstributed parameter
approach for derivation of closed-form expressions for the four-pole
parameters of the perforated three-duct muffler components. In this
method, three simultaneous second-order partial differential equations
are first reduced to a set 9f six first-order ordinary differential
equations. These equations are then uncoupled by means of a modal
matrix. The resulting 6 × 6 matrix is reduced to the 2 x 2 transfer matrix
using the relevant boundary conditions. This is combined with transfer
matrices of other elements (upstream and downstream of this perforated element) to predict muffler performance like noise reduction,
which is also measured. The correlation between experimental and
theoretical values of noise reduction is shown to be satisfactory.
Keywords.
Mufflers; noise control; transfer matrix method.
I. Introduction
The majority of the commercial mufflers used on internal combustion engines
contain one or more perforated tubes. These may interact with the surrounding
annular cavity as in a Helmholtz resonator, or may conduct the gases as in plug
mufflers and three-duct cross-flow or reverse-flow mufflers. Unfortunately,
however, analysis of such perforated-element mufflers was not known until a few
years ago, with the result that the design of commercial mufflers was based on trial
and error.
Recently, the authors developed and presented a generalized decoupling method
for the analysis of perforated-element mufflers with and without mean flow,
wherein the mean flow Mach numbers are allowed to have their actual (unequal)
values (Rao & Munjal 1984). But the method was demonstrated for two-duct
elements only. This method makes use of the distributed parameter approach in
256
K Narayana Rao and M L Munjal
contrast to Sullivan's segmentation approach (Sullivan 1979a, b), where a
perforated clement is notionally divided into a number of segments.
The present work extends the distributed parameter approach to three-duct
muffler element configurations by removing the ambiguities in evaluation of the
eigenvalues of the characteristic matrix. The three coupled partial differential
equations for wave propagation have been solved using the authors' generalized
decoupling approach with and without mean flow incorporating the relevant exact
boundary conditions. Explicit expressions have been derived for four-pole
parameters of the cross-flow expansion chamber as well as the reverse-flow
expansion chamber, shown in figure 1. The transfer matrices so derived have been
checked experimentally by comparing the predicted values of noise reduction with
those observed experimentally.
2. The coupled governing equations
Implicit in the following equations, and the solution thereof, are certain
assumptions that are typical of automotive muffler analysis, viz,
(i) working in the frequency domain, all acoustic variables are made to have
harmonic time-dependence everywhere;
(ii) at the frequencies of interest, only plane waves would propagate; higher order
modes could be ignored;
(iii) mean flow velocities are small enough; the flow is incompressible;
(iv) heat transfer and viscosity effects are neglected; all processes are isentropic;
mic rophone
dl~
!'
!2
a i2 [-
i: 'I ~h['~
M!
I
I
microphone
--.--.~M5
,
-I
it,)
microphone
MI
d1.[
micro.phone
-q0----M3
I
d3[
t3
Figure I. Three-duct muffler configurations: a. cross-flow expansion
b. reverse-flow expansion chamber.
chamber,
Noise reduction with perforated muffler components
257
(v) mean flow gradients along and across the duct are neglected. The same holds
for mean pressure and mean density;
(vi) velocity of cross flow is assumed to be uniform all over the perforate. This
allows us to assume a uniform acoustic impedance for the perforate.
For the three-duct model as shown in figure 2, the mass continuity equations may
be written as (Sullivan & Crocker 1978)
C]W I
C~Dl
4
P(1-- + Wo,
+--poUl,
~z
c)z
Opj
2 --
d~
(1)
c)t
for duct 1 of diameter d~,
014'2
,9p2
4dl
P o - - +Wo,
az
Oz
4d3
(d_~- d~ - d~)
poUl, 2 +
(d~ - d~ - d~)
ape
pou2, 3
(2)
Ot
for duct 2 of diameter d2, and
Ow3
P(1 - -
Oz
01)3
+ W,,, - -
4
Oz
d3
Po u2.3
Op3
-
(3)
3t
for duct 3 of diameter d3. The notation is given in appendix D.
The corresponding momentum equations are
Po
( ow, + wo,-c)%)
z O---t
Opj
az
"
j = 1''~ 3"
-"
(4), (5), (6)
The radial momentum at interfaces of duct 1 and duct 3 result in the equations
ul,z(z, t) = [p~ (z, t ) - p 2 ( z , t)]/(po co ~1),
(7a)
u2.3(z, t) = [p2(z, t ) - p 3 ( z , t)]/(p,, co ~z),
(7b)
and
respectively.
Assuming that the fluid is an ideal gas and the process involving wave
propagation is isentropic,
pj=pjc~,
j=
1,2,3.
(8)
For harmonic time dependence,
pj(z, t) = pi(z) e i't,
(9a)
j = 1, 2, 3,
and
ui, k(z, t) = Uj.k(Z) e i,o,, j = l , 2 ,
k=2,3.
(9b)
Substituting (7a), (7b), (8), (9a) and (9b) into (1) to (6) and eliminating Pl, P2,
f13, //1,2, /,/2,3, Wl, w2 and w3, yields the following three coupled differential
equations for ducts 1, 2 and 3, respectively (Munjal 1986),
258
K Narayana Rao and M L Munjal
iM,
[ d2
dz 2
"1-
(k ] +k2)
1 - M21
.
.
.
k
k2 ]
+ - p~(z)
dz
1 - M2
dz
a-M
(10)
.
I-M
I iM2
k~ - k 2
d
k 2 -- k 2
d
I pl(z)
1 - M--~ J
+
d
iM2
dz 2
1-M 2
(k~ + k 2)
k
dz
[ iM2 (kc2.-k 2 ) d
+ t 1-M 2 \ k
dz
iM____2_
3
k2 d
l_M2(
/~ k2) d z
+
dz 2
1-M 2
d
+
k b2 + k c 2- k ]
' l~M2~
J
p2(z)
k2c-k2 1
]--~222 j p3(z) = 0,
k2-k2
(11)
]
l_M2
k
2
p2(z)
dzz + - I - M
2
p3(z)=O,
(12)
where
k--
Wo,
O)
--,
Co
M1-
Wo
,
CO
M2-
:,
M3-
CO
i4k
i4kdl
, k2 = k2d, sr,
(d 2 - d 2 - d2) Srl '
i4kd3
k2 = k2(d 2 - d~ - - d 32) ~ " 2 '
i4k
k~ = k 2 - - d3~"2
,
Co
k2 = k2 _ _
and
(13)
3. Decoupling analysis
The analysis runs exactly as for the two-duct perforated element mufflers (Rao &
Munjal 1984). The three second-order differential equations (10), (11) and (12) are
first reduced to six first-order linear equatiolas. Treating the differential operator
as an algebraic variable, these six equations are decoupled through an eigenvalue
analysis and the use of principal coordinates. Finally acoustic state variables at
z = 0 are related to those at z -- l through a matrix, which is the required transfer
matrix.
259
Noise reduction with perforated muffler c o m p o n e n t s
Denoting d / d z by prime ('), and defining
Pl = Y | , P 2 = Y e , P 3 = Y3, Pl = Y4,
P2 = Y5 and
(14)
P3 = Y6,
(10),.(11) and (12) may be written in the form
--1
0
0 -1
D
0
0
Yl
0
0
D
0
Y2
0
0
0
D
Y3
0
0
Y4
0
0
D
0
0
0:lD+a2
0:3D + 0:.)
0
0
D
0
asD+0:6
0:7D+0:~
agD+al(i
0
-1
0
0]
0
D
0
0:11D+0:)2
Y5
I
0:13D+0:14
,. Y%
1
(15a)
0
0
or
[~l{y} = {0},
(15b)
where
0:1 - -
~
k
k]-k
"0:4 - -
2
.~ ,
I - Mi
-
iM9
0:5 - -
1 - M?~
0:l"7-
1
, 0:-, - -
1 - Mi
, 0:3-
I - M~
k
7
- (kc - k 2)
--
~
-iM3
a13-
iM 3
, 0:11 --
1-M~
l_M2
k2 + k 2 ~ ,
k
]
a,4 -
0:6 --
' 0:~
I - M~
-
M~
[ k~ - k2 ~
1 - M ~ "~
k
( k ~ , - k 2)
-) '
" 0:~ -
,
1 - M~
[ k~--k2]
-
1 -
0:to
----
1 - M~
k
k2
k
,
)
k,et - k 2
, 0:12-
f-M 2
,
(16)
1-M~
and
d
D - --.
dz
(17)
Making use of principal coordinates, these equations may be decoupled.
The new variables, i.e., the principal co-ordinates F~, Fe, [`3, [`4, r'5 and I"6 are
related to variables yj, Y2, Y3, Y4, Y5 and y¢,, through the modal matrix [~] as
260
K Narayana Rao and M L Munjal
where
~ l , j = 1.0
't'2,j
(say),
= - (~
'q-t3, j =
+ 0/l & + 0/2)/(0/3& + 0/4),
(/3~ + 0/,/3j +0/~) (t~ +,~t3j +,~s)- (0/.,t~j +0/~)(0/5/3j +0/~)
(0/3~j +0/4) (0/9~j + O~10)
~4,j
= 1"0//3i,
~II5,j = ~It2.j/ [~j ,
and
qr64 = ~3,j/[3j,
(19)
and subscript j takes the values 1, 2 . . . .
polynomial
]al = 0,
6. /3's are roots of the sixth degree
(20)
to be found numerically on a computer by means of one of the standard
subroutines.
The general solutions to first order equations (20) can be obtained as
rj(z) = Cje ~'z,
(21)
where subscript j takes values 1, 2 . . . . 6.
Next, (4), (5) and (6) may be used to obtain expressions for wl(z), w2(z) and
w3(z). Finally, we may write
p1(z)
p2(z)
p3(z)
pocowl(z)
C2
C~
=D]
poCoW2(Z)
G
poCow3(z)
C6
(22)
where elements Ai, j involving only M, k and l are given by expressions noted in
appendix A.
Now the pressure and velocity at z = 0 can be related to the pressure and
velocity at z = 1; that is,
p,(0)
p,q)
p2(O)
p2(l)
p3(0)
meow, (0)
-- [TI
p~(t)
p, :owl (1)
poCow2 (0)
poco w2 (1)
poCoW3(O)
poCoW3(l)
(23)
where the transfer matrix is given by
[T] = [.4(0)1 [Aft)]-'.
(24)
Noise
reduction
with perforated
muffler
components
26l
4. Transfer matrix for cross-flow expansion chamber
To eliminate four of the six state variables in the foregoing general solution, (23),
we make use of four boundary conditions. In the case of cross-flow expansion
chamber with rigid end termination, the boundary conditions are (Thawani &
Jayaraman 1983)
z = 0 : pocow2(O) = - i
tan(k l,) p2(0)
z = 0 " pocow3(O) = - i
tanik 1,,) p.~(O)
z = 1 : pocowl(1)
= -i
tan(k ll,) p L ( l )
z = l : pocow2(l)
= -i
tan(k lb) p 2 ( l ) .
(25)
Skipping the elimination details, the final expressions for the upstream state
variables and the downstream variables can be written as (Rap 1984)
poCoWl (0)
T~.
T,I
pocow3(l)
The expressions for T are given in appendix B.
5. Transfer matrix for reverse flow expansion chamber
The following changes have to be noted in this element:
(a) mean flow in duct three is negative with respect to the reference direction;
(b) the boundary conditions are different; these are
z = 0 : pocow2(O) = - i
tan(k t.) p2(O)
z = l : poc~jwl(l)
= -i
tan(k lt,) p l ( l )
z = I : pocow2(1)
= -i
tan(k lh) p2(l)
z = l : pocow3(I) = - i tan(k 11,) p 3 ( l ) .
(27)
Skipping the algebraic details of the elimination process, the transfer matrix
relation for this element may be written as (Rap 1984)
[ o~c~w~(0)
L,
Tt,
-
'3'°' I
oocow3(O)
and the resulting expressions for T,,, Tb, Tc and T,t are given in appendix C (Rap
1984).
6. Experimental evidence
In order to test the validity of the mathematical model at various mean flow
conditions, noise reduction measurements were made on typical three-duct
perforated element muffler configurations of figure 1.
262
K Narayana Rao and M L Munjal
Noise reduction, NR, is the difference in sound pressure levels at two arbitrarily
selected points in the exhaust pipe and the tail pipe, located upstream and
downstream of the muffler proper, respectively. Referring to figure 1,
NR
=
SeLl- SPL3
20 logm
=
IP~/P3[,
(29)
where p denotes the root mean square value of the acoustic pressure perturbation.
For the purpose of calculation, since p~ is not known directly, one can write
P,/P3 = (Pl/Po) "(Po/P3),
(30)
where P0 is the acoustic pressure a t the radiation end. Applying the wave
relationships for the pipe section of length le, one obtains (Munjal 1986)
(P3/Po) =
kole+ ( ig3/zo)
cos
where Zo is the radiation impedance,
The other pressure ratio
kole,
po/vo.
sin
P~/Pomay be obtained
[ PTl l l v l= [
(31)
(32)
from the transfer matrix relation
T12]
[ P ° I T z vo
2 '
T:I
(33)
where v denotes acoustic mass velocity, poAu, and Tii (i, j = 1, 2) are the fourpole parameters of the overall transfer matrix relating the state variables at points
1 and 0. This matrix is a product of the transfer matrices of the upstream pipe,
perforated element, and downstream pipe.
From (32) and (33) we obtain
(Pl/Po) =
Tll @ T 1 2 1 Z o
(34)
•
Equations (29), (30), (31) and (34) may be combined to obtain
NR = 20 log10
r . + r,2/ Zo
cos
[
kole+ i Y3/Zo'sin kole I
.
(35)
7. Results and discussion
General Fortran programs have been developed for computation (on D E C 10
computer) of noise reduction for three-duct element configurations shown in
figure 1 over the frequency range covered by pure plane wave propagation, in steps
of 20 Hz. The following perforated impedances were incorporated in the noise
reduction prediction (Sullivan 1979),
~= (6×10-3+i4.8×10-5f)/o -, for
~r = (0-514 dM/lo-+i4.8× 10-5 f)/~r,
M = 0,
for M >I 0.05.
Figure 2 shows the typical curves of noise reduction versus frequency of three-duct
cross-flow muffler represented in figure la for a stationary medium (M = 0
condition). Neglecting the temperature gradient and the tube wall thickness, there
is good agreement between the predictions of the present investigation and those of
experimental results. In figures 3 and 4 a similar exercise is repeated for inlet flow,
Mach number M = 0.10 and M = 0.2. Though there are small discrepancies, yet
the agreement between theory and experiment is generally good.
Noise reduction with perforated muffler components
dB
120
I
1
•
I
•
•
I
!
mtosurcmeNt~s
p l"edlcttons
100
80
60
A
40
20 ¸
-20
m
0
Figure 2.
0.5
i
1.0
I
1.5
frequency
I
I
2.0
2.5
3.0 kHz
Noise reduction of three-duct cross-flow expansion chamber of figure la for
M~ = 0
dB
120 j
,
/
i
,
• • •
meosurement$
predictions
100 L
80
tO
60
40
m
o
e-.
20
-2C
0
Figure 3.
M1 = 0-1.
I
0.5
I
lO
1
1-5
frequency
I
2.0
t
2.5
3.0 kHz
Noise reduction of three-duct cross-flow expansion chamber of figure la for
263
264
K Narayana Rao and M L Munjal
d8
120
I
I
IOO
I
•
•
_
_
•
I
I
measurefneNtE
predictions
8o
°_
6C
~q
" 20
-20
i
0-5
i
I
1.0
1.5
I
I
2-0
2.5
3.0kHz
frequency
Figure 4. Noise reduction of three-duct cross-flow expansion chamber of figure ia for
M1 = 0-2.
Noise reduction at zero mean flow of the reverse-flow expansion chamber of
figure lb is shown in figure 5. The agreement between theoretical predictions and
experimental observations is good in terms of peak locations as well as magnitudes.
Figures 6 and 7 present computed and experimental noise reduction curves for the
same muffler with M = 0.1 and M = 0-2 respectively. Here too, the theoretical
results match the experimental results well.
In general, experimental values of noise reduction are in good agreement with
the predicted results, thereby substantiating the validity of the transfer matrices
derived and used in the predictions.
These transfer matrices may be used along with those for other non-perforated
elements like a tube, sudden contraction, sudden expansion, extended inlet,
extended outlet, flow-reversal contraction and flow-reversal expansion (Munjai
1975; Panicker & Munjal 1981), to predict the overall performance (irt terms of
insertion loss, transmission loss, level difference or noise reduction) of a given
muffler. These performance curves for typical dimensions may be used for
table-design of an efficient muffler, keeping in mind other non-acoustical
considerations like overall size, weight, cost of fabrication, back pressure on the
engine, durability etc. (Munjal 1986).
The first author would like to thank the authorities of the Osmania University for
the study leave granted. The second author acknowledges with thanks the financial
support received from the Stiftung Volkswagenwerk, Hannover, Germany, for a
project in collaboration with Professor M Heckl and Professor M Hubert of the
Technische Universitiit Berlin.
Noise reduction with perforated muffler components
dB
1213
I
I
I
•
•
•
I
I
rneosuremen[s
p redictaons
10C
80
~o 6O
~- ~0
1
3
C
20
--20
Figure 5.
M] = 0.
t
°
o o
L
I
0.5
1.0
L
1.5
frequency
I
I
2.0
2.5
3-0 kHz
Noise reduction of three-duct reverse-flow expansion chamber of figure lb for
d8
1"20
I
[
l
•
-
100
•
-
•
I
I
meosuremen~s
p redlcQons
80
=
o
6O
40
o
c
•
20
•
• •
•
0
-20
F i g u r e 6.
Mi = 0.1.
I
i
0.5
I-0
J
1-5
frequency
I
L
2.0
2.5
3.0 kHz
Noise reduction of three-duct reverse-flow expansion chamber of figure lb for
265
266
K Narayana Rao and M L Munjal
dB
120
l
I
I
•
•
-
-
•
l
meosur@rnc~nts
predicU0ns
100
80
tO
t)
60
/.0
tn
o
c
20
-20
0
I
0.5
I
1.0
1
1.5
I
2.0
t
2.5
3.0kHz
frequency
Figure 7.
Noise reduction of three-duct reverse-flow expansion chamber of figure l b for M1 = 0-2.
Appendix A
The following relations hold for both the three-duct cross-flow expansion chamber
and the three-duct reverse flow expansion chamber.
A 1 j = ~bn,je~jz,
A2,i = ~5,jet3j~,
A3, j = dd6,/eOjz,
Z4,j = -e~jZ/(ik + Ml/3j),
A s j = -~2deOjZ/(ik+M2~y),
A6, j =
-~3,je t~jz/(ik + M3flj ).
Appendix B
The following relations hold for three-duct perforated element cross-flow
expansion chamber:
T, = TTI,2+A3C3;
Tb
7",. = T T 3 , 2 + A 3 D 3 ;
Td = TT3,4+B3D3;
= TTI,4+B3C3;
A3 = (TT2,2X2- TT4,2)/~;
B3 = (TT2.4X2- TT4,4)/F2;
Noise reduction with perforated muffler components
C3 = T T I , I + X I T T I , 3 ;
267
D3 = TT3,1+X1TT3,3;
F2 = TT4.I "I-X1TT4.3 -- X2(TT2.1 + X I T T2.3).
[ T T ] is an intermediate 4 x 4
matrix defined as
m
p2(O)
poCowl (0)
= [ T T]4x4
pocow2 (0)
P3(/)
poCow2(l)
O~cow2(l) _
with
TTl.1 = A1A2+ TI,2;
TT1,2 = BIA2 + T~.3;
TT1,3 = C1A2+ T1,5;
TTI.4 = DIA2+ Tl,6;
TTe.1 = A1Be+ ire,2 ;
TT2.2 = B1B2+ T2,3;
TT2.3 = C1B2+ T2.5;
TT2,4 = D1B2 + T2,6;
TT3.1 = A~C2+ T4",;
TT3.2 = B1C2 + T4.3;
TT3.3 = C1C2 -t- T4.5;
TT3,4 = D1C2+ T4.6;
TT4.1 = AID2 + Ts.2;
TT4.2 = BID2+ T5.3;
T T4.3 = C1D2 + T5.5 ;
TT4. 4 = D1D2+ T5.6;
A1 = (T3,2X2-T6,2)/F1;
B1 = (T3,3X2 - T6,3)/F1;
C1 = (T3.5X2- T6,5)/FI;
D1 = (T3.6X2- T6.6)/FI;
A2 = TI.I+XIT1.4;
B2 = T2.1+XIT2.4;
C2 = T4.1+XIT4.4; D2 = Ts.l+X1T5.4;
F1 = T6,1 + X I T 6 , 4 - X 2 ( T 3 , 1 +X1T3,4);
x1 = i tan(klb);
and
X2 = - i tan
(kla).
Appendix C
The following relations hold only for three-duct perforated element reverse-flow
expansion c h a m b e r
7",, = Bl. IDl.l + BI.2D2.1+ BI.3D3.1,
Tt, = B~,IDI.2+ B~.2D2,2+ BI,3D3. 2,
7",. = B4,1Dt,I + B4,2D2,1 + Ba.3D3.1,
Td = B4,1DI.2 + B4,2D2,~_+ B4,3D3.2,
268
K Narayana Rao and M L Munjal
Bil,i 2 = Til,i 2 + X 1 T/1,(i2+3),
i1=1,2 .....
i2 = 1 , 2 , 3 ,
6,
)(1 = i tan(k/b).
Dl,1
= Cl.lD2.1+Cl.3D3.1,
DI,z = C1,1D2,2 + C1,2D3,2,
1)2,1 = C3,2/F4;
D2,2 = - C2,2/F4,
D3,1 = _C3,1/F4;
D3,2 = C2,1/F4,
F 4 = C2,1C3,2-C2,2C3,1,
CI, 1 = ( B 5 , 2 - X 2 B 2 , i ) / F 3 ;
C1,2 = ( B s , 3 - X 2 B 2 , 3 ) / F 3 ,
C2,1 = B3,z+CI,IB3,1;
C2,2 = B3,3+Cl,2B3,~,
C3,1 -~ B6,2+Cl,lB6,1;
C3,2 = B6,3+CI,zB6,1,
F3 = B2,1X2-B5,1,
X2
=
- i
tan (k la).
Appendix D. Notation
A1, Ao
Co
d
f
i
k
/
M
Mi
go
Po
P
t
temp
UI,2, U2,3
Wo
w
Y
z
po
O
o)
~1, ~2
internal areas of inlet and exit pipes, respectively,
velocity of wave propagation
internal diameter of pipe
frequency,
iota, X/~-1,
wave number, (tO~Co),
length of pipe,
Mach number, (Wo/co),
inlet Mach number,
exit Mach number,
pressure of the undisturbed fluid,
fluctuating pressure,
time co-ordinate,
temperature,
radial fluctuating velocities at 1, 2 and 2, 3 interfaces of the control volume, respectively,
velocity of the undisturbed fluid,
fluctuating velocity,
characteristic impedance, co~A,
axial co-ordinate,
density of undisturbed fluid,
fluctuation in density,
circular frequency,
acoustical impedances at 1, 2 and 2, 3 interfaces, respectively.
Noise reduction with perforated muffler components
269
References
Munjal M L 1975 J. S o , n d Vib. 39: 1115-I19
Munjal M L 1986 Acousttcs o f duct,s attd mufflers (New York: John Wiley) (in print)
Panickcr V B, Munjal M L 1981a J. India, ht~'t. SoL A63:1-19
Panicker V B, Munjal M L 1981b J. hulian hzst. Sci. A63:21-38
Rao K N 1984 Predictton and verlification ~f the aero-acoustic performance o f perfi)rated element
;m(l~flers. Ph.D. thesis, Indian Institute of Science, Bangalore
Rao K N, Munjal M L 1984 A gencralized decoupling method for analysing perforated element
mufflers, Proceedings of the Nelson Acoustics Conference, Madison, USA
Sullivan J W 1979a J. Acoust. Soc. Am. 66:772-778
Sullivan J W 1979b J. A~oust. Soc. Am. 66:779-788
Sullivan J W, Crocker M J 1978 J. Acoust, Soc. Am. 64:2(17-215
Thawani P T, Jayaraman K 1983 J. Acous7. Soc Am. 73:1387-1389