Transfer function method of measuring in-duct acoustic properties. I. Theory
J. Y. Chung and D. A. Blaser
Citation: The Journal of the Acoustical Society of America 68, 907 (1980); doi: 10.1121/1.384778
View online: http://dx.doi.org/10.1121/1.384778
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Transfer
function
method
of measuring in-duct
acoustic
properties. I. Theory
J. Y. Chungand D. A. Blaser
EngineeringMechanicsDepartment,GeneralMotorsResearchLaboratories,Warren,Michigan48090
(Received10 December1979;accepted
for publication5 June1980)
The theory of a transferfunctionmethodof measuringnormal incidentin-duct acousticpropertiesis
presented.In this method, a broadbandstationaryrandom acousticwave in a tube is mathematically
decomposed
into its incidentand reflectedcomponents
usinga simpletransfer-function
relationbetweentheacousticpressureat two locationson the tubewall. The wavedecomposition
leadsto the determinationof the
complexreflectioncoefficientfrom which the complexacousticimpedanceand the soundabsorption
coefficient
of a materialandthe transmission
lossof a silencerelementcanbe determined.Also presented
are
the theoriesof two techniques
for improvingtransferfunctionestimates:a sensor-switching
techniquefor
automaticsystemcalibrationand a coherence
functiontechniquefor signalenhancement.
PACS numbers: 43.50.Yw, 43.20.Mv, 43.55.Ev, 43.20.Bf
INTRODUCTION
In duct acoustics, the media through which the plane
wave propagates is characterized by the normal
acoustic impedance• which, along with other acoustic
properties
such as the transmission
loss of a silencer
andthesc•und
absorption
coefficient
ofa material,can
be determined by the well-known Standing-Wave-Ratio
and the sound absorption coefficient are evaluated. A
similar procedure for determining the transmission
loss of silencers was reported previously by Blaser
and Chung.s
I. MEASUREMENT
COEFFICIENT
single frequency operation, (b)a traveling microphone
inside the impedance tube to record the amplitudes of
the maximum and minimum acoustic pressures, and
(e) a means of accurately measuring the location of the
minimum acoustic pressure. The method is relatively
time consuming due to these requirements. In recent
years, measuring of reflection coefficients using the
rance between the microphones which are flush mounted
on the tube wall.
function
be-
acoustic pressures at the first and the second microphone locations, respectively.
Each pressure can be
written
all normal
incidence
as the sum of its incident
and reflected
com-
ponents, or
p• (t) =p•, (t) +P•r(t) ,
(1)
p2(t) =p2•(t) +p2r(t).
(2)
and
tween the acoustical pressure at two locations in a
tube to determine
The test sample is shown to be placed
at the end of the tube. Let p•(t) and p2(t) be the random
correlation functions and the cross-spectral density
function4 were reported. The new method presented
of the transfer
REFLECTION
With reference to Fig. 1, let s be the separation dis-
(SWR) method? The SWR method requires: (a)a
here uses a measurement
OF THE COMPLEX
acoustic
The following convolution integrals with impulsive re-
properties.
The theory underlying the new technique involves the
decomposition
ofa broadband
stationary
random
signal
sponsesr•, r•., hi, h•, and h•. are introduced to relate
their respective random acoustic pressures,
(generated by an acoustic driver) into its incident and
reflected components using a simple transfer-function
relation between the acoustic pressure at two locations
on the tube wall as depicted in Fig. 1. This wave decomposition leads to the determination of the complex
reflection coefficient from which acoustic properties
such as the acoustic impedance, the transmission loss,
p•(t) =
r•(r)p•i(t-
or
r)dr,
(3)
r•.(r)p•.i(t- r)dr,
(4)
h,(r)pxi(t- r)dr,
(5)
hr(,)p•r(t- r)dr,
(6)
,
p•,(t) =
P•.i(t)=
o
p•.•(t)=
o
ACOUSTIC
MICROPHONE
#1
MICROPHONE#2
and
DRIVER
p•.(t)=
h•..(r)p•(t- r)d,,
(7)
where the following definitions are used:
STATIONARY RANDOM S
r• and r•.: the impulsive responsescorrespondingto
ST MATERIAL
the reflected
FIG. 1. Testapparatus
for thet•'ansferfunctiontechnique
of
wave
evaluated
at the first
and the second
microphone locations, respectively.
measuring acoustic impedance and sound absorption coeffi-
hi and h•: the impulsive responses corresponding to
cient.
907
J. Acoust. Soc. Am. 68(3), Sept. 1980
0001-4966/80/090907-07500.80
(D 1980 AcousticalSocietyof America
907
the incident and the reflected waves, respectively,
.= R•i,i(r/)
e'•"'½"
dr/ [h,,.(r)
-hi(r)]e' •"'½*dr
evaluated between the two microphone locations.
h,2: the impulsive response correspondingto the
combined
incident
and reflected
waves
evaluated
:
be-
tween the two microphone locations.
order of the microphone locations relative to the direction of propagation of the reflected wave). It can
or
S•,,•(f)f •[h•,.(r)
-h,(r)]e
-'"•'r
dr
s,,•,'(f):f;[h•(r)-h•,.(r)]e
-•"•dr '
(15)
whereSt• t•(f), the Fourier transform of R,• t•(r/) is the
be shown, however, that the impulsive response so
auto-spectral density of the incident pressure compo-
defined (i.e., hr) is related to that obtainableby re-
nent at the first microphone. Similarly, St• tr(f) is the
versing the system input and output through a simple
convolution integral i.e., the convolution of the two
cross spectral density between the incident and reflect-
ed pressure components at the same microphone location. The quantity at the left side of Eq. (15) is de-
is a delta function.
We find from Eqs. (1), (2), and (7) that
p•.•(t)+p•,(t) =
a,• ,•(r/)e'•"'•drl 0 [hr(r)-h,,.(r)]e'•'-•'dr,
(14)
It should be noted that for convenience, the system
input and output implicit in Eq. (6)was made opposite
to the actual physical condition (see Fig. I for the
quantities
oo
fined as the complex reflection coefficient* at the first
microphone location, or
h•.(r)[p•i(t - •) +p•r(t- •)]d•.
(8)
Using the relations in Eqs. (5) and (6), Eq. (8) be-
(16)
Using the relation in Eq. (3), one has
comes,
E{Pt,(t)'Pt•(t+r)}
= rt(rl)E{Pti(t)Pt,(t+r-rl)}drl.
(17)
foøøp•i(tr)[h•,.(r)-hi(r)]dr
It can be shown from Eq. (17) that
=fo•p,r(tr)[h•(r)-h,,.(r)]dr.
Multiplying both sides of Eq. {0) by p,•(t- a), then taking expectedvalues6 we have the following relation:
E
S•i•(f)= r,(rl)e.•,.,•,r
I'
(9)
S• •(f)
(18)
Thusfrom Eqs. (16) and (18), it is seenthatR,(f) is
the Fourier transform of r,(r/), or
Sti•_r(I)
R,(f)=f•or'(rl)e'•"'•ndrl=s,i
,•(I)'
P,i(t- r)P,i(t- a)[h,,.(r)-hi(r)]dr
o
(19)
Similarly, it can be shownfrom Eqs. (5) to (7) that
=E
p•r(t - r)p,i(t - a)[h•(r) - h,,.(r)]dr ,
or
o•E{P,i(t
- r)p,•(t
- a)}[h,,.(r)
-
Hi(f)= o h•(r)e'•'•*dr=S,•
•i(f)
$,• •i(f)'
(20)
St•,(f)
Hr(f
)= •o
hr(r)
e'•2'/*
dr=str
tr(f),
(21)
H,:(f): h,,.(r)
e'•"'•'dr:
Sz•.(f)
S,,(f) '
(22)
o
and
=fo
•E{P,r(t
- r)p,i(t
- a)}[h•{,)h,,.(r)]dr.
(lO)
But,
E{p,•(t - r)p•i(t- a)}=R,i•(a- r)
(11)
E{p•(t- r)P•i(t- a)}=R•i•(a- r),
(12)
and
where R,i,i and R,•r are the auto- and cross-correla-
tion functions, respectively. SubstitutingEqs. {11)
and (19.)into Eq. (10)andtakingFourier transforms,
we obtain,
f.•ofo•OR,i
,,(ar)[h,,.(r)-hi(r)]
e'•"•drda
=
It• ,•(a- r)[h•(r)- h•,.(r)]e'•"*• drda.
where Hi, Hr, and Hxaare the acoustical transfer functions corresponding to the impulsive responses hi, hr,
and hx,., respectively.
From Eqs. (15) to (22) the expression of the complex
reflection coefficient becomes,
R•.(f)= [H,:(f) - Hi(f) ]/[Hr(f) -H•.•.(/)].
(23)
Equation (23) is the basis of the new transfer function
technique for determining the acoustic impedance and
other acoustic parameters. From the definition in Eq.
(16), the magnitudeof Rt representsthe amplituderatio
of the reflected and incident pressure components at the
first microphone location.
The relation between R• and the reflection coefficient
at the second microphone, R,. can be found from the
Letting a- r =r/and da=dr/, then Eq. (13) can be rewritten as,
908
J.Acoust.
Soc.
Am.,Vol.68,No.3,September
1980
following relations:
From Eqs. (4)and (6), it is shown,
J.Y.Chung
andD.A.Blaser:
In-duct
acoustic
properties.
I
908
r•.(r)p•.i(t- r)dr=
(24)
hr(r)P•r(t- r)dr.
o•ø{p•i(t
- r)[hr(r)
-h•2(r)]
+P•,(t
- r)
x[h,(r)-h•2(r)]
Substituting Eqs. (3)and (5) into Eq. (24), one has
-p•(t- r)[h,O')-hi(r)]}dr=O,
fo
• fo•r2(r)hi(•)P•i(tr-•)d•dr
=
or
r•(•)hr(r)p•i(t- r- •)d• dr.
Multiplying both sides of Eq. (25) by p•i(t-a),
(25)
fo
•[P•i(tr)+p•,(tr)][h,(r)-h•2(r)]dr
and tak-
=
ing expected values, we obtain
fo•
fo•r2(r)hi(l•)R•i
•i(•-r-•)dl•
dr
=
(32)
r•(•)h•(r)R• •,(a- r - •)d• dr..
r)[hr(r)-hi(r)]dr.
o
(33)
Substituting the relation of Eq. (1) into Eq. (33), one
obtains
'P•(t
- r)[hr(r)-h•2(r)]dr
(26)
Taking the Fmrier transform of Eq. (26) and e•nging
the variable, a - r- • = •, da =d• we •ve•
=
p•(t - r)[h•(r) - h•(r)]ar.
(34)
Multiplying both sides of Eq. (34) by p•(t- a) and taking
R•i •i(•) e'•2'•rd•
expected values, we have, in terms of auto- and crosscorrelation
x
hi(•) e'j2'•td• =
x
rx(•)e'•2'S•d•
Rxi•(•7)e'•2'sc
h•(r)e'•2'•*dr.
functions,
o•R,,(e
- r)[h•(r)-h•2(r)]dr
(27)
Thus,
=
R•i(a-
r)[hr(r)-h•(r)]dr.
(35)
Taking the Fourier transform of Eq. (35), one shows,
Sxixi(f)'R2(f)'Hi(f)=Sx• x,(f)'Rx(f)'Hr(f)
Hr -H•2 =(S• •i/S•)(Hr-Hi).
A similar
or
Rx(f )/R2(f ) =Hi(f )/Hr(f ) .
leads to
Hx2-Hi = (Sxxr/Sn)(Hr-Hi).
(28)
Assuming plane-wave propagation, no mean flow,
and neglecting losses at the tube wall, we may express
H i and H r by
derivation
(36)
(37)
From Eqs. (23), (36), and (37), it is seen that the reflection coefficient Rx becomes indeterminate if Hr
=Hi. This conditionoccurs, accordingto Eqs. (29)
and (30), when
=e
ks=m•,
m=1,2,3,...
,
(38)
and
or
u,(f) =e
(30)
where k is the wavenumber and s is the microphone
spacing. Eqs. (28) to (30) show that the magnitude of
the reflection coefficient is independent of the location
at which it is measured. Physically, this should be
obvious since the magnitude of the reflection coefficient
represents the ratio of the reflected and the incident
sound power components in the tube. Provided the
losses at the tube wall are negligible, these components
are not functions of position along the tube.
s =m(X/2),
rn = 1, 2, 3, ....
(39)
Equation (39) indicates that the reflection coefficient
cannot be determined from Eq. (23) at discrete frequency points for which the microphone spacing is an
integer multiple of the half-wavelength of sound. In
order to avoid these points up to a frequency fro, the
microphone spacing s must be chosen such that
[accordingto Eq. (39)],
s •<c/2f.,,
(40)
The expression of the reflection coefficient in Eq. (23)
becomes indeterminate when H,-H•2 = 0. This singular
where c is the speed of sound. In order to determine
condition can be further investigated as follows:
material
the reflection
coefficient
which is not at the microphone location,
'
fo•ø{p•(tr)[h•2(r)h•
(r)]-p•,(tr)
(r)-
at=0,
of a test
'
(41)
,
where
(31)
H• =e'•
,
(42)
(43)
or
909
Eq.
(28) can be applied, or
From Eq. (9), it is seen,
x
R on the surface
J. Acoust.Soc.Am.,Vol.68, No.3, September
1980
J.Y. ChungandD. A. Blaser:In-ductacoustic
properties.I
909
and I is the distance from the first
TL = 101Oglo(Wa/Wt)
microphone to the
'surfaceof thetest material'.Usingtheaboverelations
we have,
Ht[
+101Oglo(•)
=20loglo
1+R.I_201Ogxol
[I+R•.
(44)
=20loglo
H,-//•2- 201Oglol
Ht[+10loglo ,
II.
THE COMPLEX
ACOUSTIC
IMPEDANCE
(51)
Using Eqs. (23) and (44) and the standard relation
between the complex acoustic impedance and the reflection coefficient, one can obtain the complex acoustic
impedance on the cross-sectional area of the tube at a
distance l from the first microphone, or
also
is the transfer function of the acoustic element, which
z/pc = (1 +R)/(1-R)
canbe determinedindependently
from H •2andH •.,
Hi•sin(k/) sin[k(/-s)]
(45)
=Jcos[k(/-s•
- H12
cos(kli"
with its real and imaginary
sin(2kl)})/Ha,
within the range of linear acoustic propagation.
It
shouldbe noted that the final form for TL in Eq. (51) is
valid only if the nondimensional
wavenumberks [see
Eqs. (29) and (30)] is the same for both upstream and
parts being,
x/pc=(Re(HI•.)
sin[k(2/- s)] - 1/2{sin[2k(l- s)]
+I
where H u
12and H a12are the transfer functions measured
at the upstream and downstream locations, respectively,
downstream'
measurements.
(46)
and
r/pc = [- Im(H12)sin(ks)]/Ha,
(47)
respectively, where pc is the characteristic impedance
of air
and
x cos[/,(/-s)] + I
parts, respectively.
(48)
the real and imaginary
All quantities on the right side of
The sound absorption coefficient of a material is
defined as the fraction of the total sound power that
absorption coefficient
function between the two microphone signals.
a is related to the incident-wave
auto-spectrum S1•1, and the reflected-wave auto-spectrum S•, lr (bothbeing evaluated at the first microphonelocation) by the following expression;
Eqs. (45) to (48) except for HI•. are knownconstantsfor
a given measurement configuration. Thus, evaluation
of the acoustic 'impedance of a given material becomes
a relatively simple problem of determining HI•., the
transfer
COEFFICIENT
is absorbed by the material in the tube (see Fig. 1 for
the test configuration). It can be shown easily that the
Ha= cos2[k(l-s)] - 2Re(H12)
cos(k/)
with Re and Im representing
IV. THE ABSORPTION
ot=
= 1-Sl,, 1,,/S•i1•'
(53)
But from Eq. (3), it is seen that
E{px,(t).pl,(t +r)} =
III.
THE TRANSMISSION
r 1(i)r 1(•)E{px,(t- •)
xpli(t+ r- •)}d•d<,
LOSS
or
The
transmission
loss
of an acoustic
element
in a
duct system can be determined using the complex refleetion coefficients measured at two locations upstream
and downstream of the acoustic element. Let Ru and
Rd be the upstream and downstream complex reflection
coefficients, respectively, then the sound power incident
on the acoustic
element
can be shown
to be
Wi =S.,. A./(pc[ 1+R.12),
sectional area A d and the reflection coefficient as
910
(50)
loss can then be defined using Eqs.
(49) and (50) as,
J. Acoust.Soc.Am., Vol. 68, No. 3, September
1980
(54)
Taking the Fourier transform of Eq. (54), it can be
shown that,
12
sectional area respectively at the upstream measure-,
ment location. Similarly the sound power transmitted
through the element can be determined at the downstream location using the auto-spectrum Sad, the cross-
The transmission
rl(•).r•(•)Rl•l•(r+•-•)d•d•.
r•(• ) ' r•(•) e'•'•( •'•)d•d•
(49)
where S.u and A. are the auto-spectrum s and the cross-
w, =s,, .A,/(pcl 1 +Ra12).
Ri, 1,(r)=
=S• •/S• • .
(•)
Using Eq. (55) in Eq. (53), the absorption coefficient
can be expressed in terms of the reflection coefficient
a = 1 - [R•I .
(56)
Thus usingRt in Eq. (23), the soundabsorptioncoefficient shown in Eq. (56) can be rewritten as,
a = 1-
-
.
J.Y. ChungandD. A. Blaser:In-ductacoustic
properties.I
(sv)
910
V.
THEORETICAL
TECHNIQUE
tained from the geometric mean of H•. and H•., or
BASIS OF A SENSOR-SWITCHING
FOR AUTOMATIC
SYSTEM
H•. =(//•. ß•s
CALIBRATION
As mentioned earlier,
=(s ).
the determination of the
acousticimpedance
[Eq.(45)]thetransmission
loss
[Eq.(51)]or thesound
absorption
coefficient
[Eq.(57)]
using the new technique amounts to measuring the
transfer function between acoustic pressures at two
closely spaced microphones. Using a modern spectral
analyzer, the measurement of transfer functions is a
routine
matter.
An accurate
determinationof
the trans-
fer function, however, requires a careful calibration
of not only the gain factor, but also the phase factor of
the entire measurement system. This calibration procedure is generally quite difficult in practice, especially when it becomes necessary to use microphone
adaptors to measure pressures remotely from the system in question.
A sensor switching procedure was therefore developed to eliminate the difficult task of calibrating both
the gain factor and the phase factor of the measure-
ment system. In this procedure, the measurement of
the transfer function is made with an initial microphone
configuration and a second measurement is made with
the microphone sensing locations switched or interchanged. The final result of the measurement is then
obtained from the geometric mean of the original and
the
switched
results.
The theory behind a related switching technique for
estimating a cross-spectral density was presented pre-
viously by Chung.9 It was demonstrated in Ref. 9 that
the technique can eliminate the phase-mismatch error
in estimating the cross spectral density. When the
switching technique is applied to estimating the transfer function, not only is the measurement error due
to phase-mismatch between the two microphone channels eliminated, but also the result becomes independent of the gain factors of the two measurement
channels.
The theoretical basis of the sensor switching technique for measuring the transfer function is shown in
the following: Let Hx, and Hx•.be the complex instrument frequency responses associated with the first and
the second microphone channels. From linear theory,
the transfer function measured from two microphone
signals with the original microphone configuration can
be written
as
,, ) . i-i?,
(58)
where
H• = (H,*t ß
.
(59)
and where * denotes the complex conjugate. If the two
microphone channels are interchanged or switched, the
measured transfer function becomes,
H•. = (St•./Sn ) ' HI ,
(60)
H• =(H•*•-H•t)/I H,,•.
i•'o
(61)
•here
Since/•o H• = 1, the transfer function,H•. canbe ob=
911
J. Acoust.Soc.Am.,Vol.68, No.3, September
1980
'
=S•/S n .
(62)
It is obviousfrom Eq. (62) t•t
the quantityH• is the
desired ratio of the cross spectral densi• S• and the
auto-spectral density Sn without the effect of the instmment gain and p•se factors (i.e., independent of
H•x and H•).
This means any c•nge in the amplinde
and p•se
response characteristics
tem
not affect
would
of the sensor sys-
the mea•rement
result.
This
is
very impor•nt,
because in practice both gain and p•se
error in transfer-function
estimates could induce significant inaccuracy in the subsequent compu•tion of '
quantities such as the absorption coefficient.
It should be noted that the switching technique described here is more effective than the calibration procedure
described
procedure,
with
earlier
in Ref.
5.
In the
earlier
the microphone responses were calibrated
the relative
transfer
function
between
the two
microphone systems, which was pre=determined by
exposing the two microphones to the same sound field.
The pre-determined transfer function is then used to
correct the error in estimating the phase factor of a
transfer function or a cross-spectral
density. The
switching procedure, however, eliminates the requirement of exposing the two microphones to the same
sound field. This latter requirement can be difficult
to fulfill for a broadband analysis.
Another calibration procedure without the need of
exposing the two microphones to the same sound field
is to make corrections using the instrument frequency
response obtained by a variation of the sensor-switching
procedure. In this procedure, the relative phase factor
of the two instrument systems is obtained from the
square root of the ratio of the original and the switched
cross-spectral densities. The relative gain factor in
addition to the phase factor is obtainable from the same
procedure, if the cross-spectral densities are replaced.
by the transfer functions. The above correction methods, however, are not as convenient to apply as the
geometric-mean switching technique.
Vl.
THEORETICAL
ENHANCEMENT
=
'
BASIS OF A SIGNAL
TECHNIQUE
USING
COHERENCE
FUNCTIONS
In the present method, the acoustic impedance and the
absorption coefficient can be evaluated most convenient-
ly by a programmable digital spectral analyzer. In
digital spectral computation, however, the signal-tonoise ratio is one of the most important factors affecting computational accuracy. In general, the compu-
tation of a transfer function based on Eq. (62) will be
erroneous if signal interference is present. The. interference is usually greater within frequency regions
where the signal amplitude is relatively low. In an
impedance tube system, the signal dynamic range is
relatively wide, due to the longitudinal resonances of
J.Y. ChungandD. A. Blaser:In-ductacoustic
properties.
I
911
the tube, thus signal enhancement is generally useful
in this
and
case.
a= 1- I(C '•,•-Hi)/(H,.-
A signal enhancement technique similar to the one
signal-to-noise ratio in the experimental evaluation of
the transfer function. This new procedure is based
on the following analysis.
It has been shown in Ref. 10, that using three pressure sensors, the auto-spectral densities of the sig-
rials, Su, S22, and Saawithout noise contaminations can
be expressed in terms of the contaminated auto spectral
densities S•,, S•.•.,
½ and Saaby the following relations:
ß
(64)
- ½(rs,' Y•.s)/Y,•.
$as-$ss
,
(65)
where e.g. u,•. is the positive square root of the oror
=[is.iV(s11*S22)F
(66)
ß
It can be shown from Eqs. (62) to (65) that
H,2 --H,2 ßC, ,
(67)
=
ß
(68)
is the coherencefactor and H•. is the measuredtransfer function with contamination.u Equations (67)and
(68) are the basis of the signal enhancement technique
using coherence function relations of a three-sensor
system.
The three sensors must be placed in the coherent
sound field. Also it is necessary to place the microphones close to each other relative to the wave length
to satisfy the linearity requirement between the microphone signals. 'ø It should be noted that in a noise-free
condition, all three coherence functions become unity;
thus it is seen from Eqs. (67) and (68) that the signal
enhancement
scheme
remains
NEW FORMULAE
MEASUREMENT
AND
ßu
s W•',
H•.=(H•.
--•2,
(73)
and where superscripts o and s represent quantities
associated with the original and the switched micro-
phoneconfigurations, respectivelyß Also H",9. and H,2
and downstream locations with a switching procedure
shownin Eq. (73). It shouldbe notedthat {Ht{ canbe
determinedby means of auto spectra [Eq. (52)] when
the microphone systems have matched amplitude responses. To eliminate the calibration for the gain
may be used to measure both the upstream and the
downstream auto spectra in two measurements provided a stationary sound field is maintained during the
entire measurement period.
VIII.
DISCUSSION
The formulas shown in Eqs. (69) to (73) conclude the
where
VII.
and
factor in determiningIH, I, however,the samesensor
dinary coherence function between the first and the
sensors
(72)
in Eq. (70) are determined respectively at the upstream
(63)
S,.,.=S•2(v2a.
u,•.)/va,,
second
(71)
where
previously reported'ø can be used to improve the
=sh
C 'Ex2)],
SOUND
OF THE
ACOUSTIC
of the instrumentation.
1For the definition of normal acoustic impedance and trans-
valid.
mission loss, see, e.g., Noise and Vibration Control, edited
by L. L. Beranek (McGraw-Hill, New York, 1971).
2For the test procedureof SWHmethod, see ANSI/ASTM
C384-77, "Standard Test Method for Impedanceand Absorption of Acoustical Materials by the Tube Method," revised
FOR PRACTICAL
ABSORPTION
theoretical development of the transfer function method
of determining in-duct acoustic properties. Both the
coherence-function method for signal enhancement and
the microphone switching procedure for automatic
system calibration are implemented in the formulae.
When signal interference may be considered to be insignificant, the signal enhancement procedure may be
omitted, since it requires an additional microphone
channel in the method. The microphone switching procedure, however, should always be employed, since it
is easy to use and it effectively eliminates all the difficult tasks of correcting for the gain and phase factors
IMPEDANCE
COEFFICIENT
When the sensor switchingprocedure [Eq. (62)] and
the coherencefunction technique [Eqs. (67) and (68)]
are employed in the transfer function estimate, a new
expression for the measurement of the acoustic imperlance, transmission loss and sound absorption coef-
ficient can be obtained. From Eqs. (45), (51), (57),
(62), and (67), an expression for the acoustic impe-
as,
C '•,a' sin(kl,)-,,sin,[k(1-s)]
Reflections from Obstructions in a Pipe with Flow," NSF Report PD-20 (1975).
4A. F. SeybertandD. F. Ross, "ExperimentalDetermination
of Acoustic Properties Using a Two-Microphone RandomExcitation Technique," J. Acoust. Soc. Am. 61, 1362-1370
dance z, the transmission loss TL and the sound absorption coefficient • can be rewritten respectively
cos[k(/-s)]
- C.•,2 cos(k/)
'
1977.
3W. E. Schmidt and J.P. Johnston, "Measurement of Acoustic
(69)
H r - C.
(1977).
5D. A. Blaser and J. Y. Chung, "A Transfer-Function Technique for Measuring the Acoustic Characteristics of Duct
Systemswith Flow," Proc. Inter-Noise '78, 901-908 (1978).
6For the definition of expectedvalue, see, e.g., A. Papoulis,
Probability, Random Variable and Stochastic Processes
(McGraw-Hill,
New York, 1965).
VThereflection coefficient so defined applies to both stationaryrandom and deterministic signals. In the case of deterministic signals, it becomes identical to the classical definition
of the complex reflection coefficient.
IHr- C •/•.
-"1- 201Og,
TL=20logxo
ol
HtI+101Og,o(A•)
i7o)
912
J. Acoust.
Soc.Am.,Vol.68, No.3, September
1980
J.Y. Chung
andD.A. Blaser:In-duct
acoustic
properties.
I
912
SHerethe auto-spectrumis definedas the auto spectral density
multiplied by the frequency resolution bandwidth.
sJ.Y. Chung,"Cross-SpectralMethodof MeasuringAcoustic
Intensity Without Error Caused by Instrument Phase Mismatch," J. Acoust. Soc. Am. 64, 1613-1616 (1978).
ence Function Method," J. Acoust. Soc. Am. 62, 388-395
(1977).
l•Thecross spectraldensityS•2in Eq. (66)is notcontaminated.
Also the contaminations of the auto-spectral
assumed to be mutually uncorrelated.
densities are
10j. y. Chung, "The Rejectionof Flow Noise Usinga Coher-
913
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Soc.Am.,Vol.68, No.3, September
1980
J.Y. ChungandD. A. Blaser:In-ductacoustic
properties.
I
913