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Journal of the Acoustical Society of America, 69, 6, pp. 1710-1715, 1981-06
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Comments on the coherent and incoherent nature of a reverberant
sound field
Chu, W. T.
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National Research
Conseil national
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de recherches Canada
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COMMENTS ON THE COHERENT AND INCOHERENT
NATURE OF A REVERBERANT SOUND FLELD
by
W.
T. Chu
ANALYZED
Reprinted from
Journal of the Acoustical Society of America
Vol. 69, No. 6, June 1981
p. 1710
-
1715
DBR Paper No. 1020
Division of Building Research
I
N R C-
C l S T lBLDG.
RES.
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1
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NRCC 19921
L'auteur de cette note propose de
mieux
caractgriser les champs
de rEverb6ration en les classifiant selon leur nature
gr8ce
aux expressions "champ
coh6rent1' et "champ
incohgrent".
La nature des champs de rsverbgration est vErifi6e pour divers
types de signaux sonores et diverses salles acoustiques
pour
dgterminer
les
conditions
dans
lesquelles
les
!
Comments on the coherent and incoherent nature of a
reverberant sound field
W. T.Chu
Division of Building Research. Nationnl Research Council of Canada, Ottawa, Ontario. Canada KIA OR6 (Received 2 August 1978; accepted for publication 11 March 1981)
P
I& proposed that reverberant sound fields should be classified with betterdefined characteristics in t e r n of coherence and incoherence. Their exact nature created under different types of signal excitation and room condition are explored with the object of clarifying the conditions for which various results from the statistical theory apply.
''
PACS numbers: 43.55.Br. 43.45.Bk
AR
:"ts:"'/ - " ,INTRODUCTION $:(XI
-
I
B(x)1'
,
There i s little doubt that statistical room acoustics and
has made great advances since i t s inception in the mid-
P ( x )
-
IA(x)+
B(X)l a =
[A(x) + ~ b ) ] [ A * b ) + B*(x)I,fifties, following early work by Schroeder.' It has
yielded results for the cross-correlation 'function: the where
*
denotes complex conjugate and overbar definesinterference patterns at the boundaries of the room,3 time average.' Therefore
and the sampling distribution of the reverberant
field. ' s 4 e 5 It appears, however, that the general asso-
ciation of the theory of statistical room acoustics with random reverberant sound fields, together with the different possible interpretations of the definition of a diffuse sound field,' has created some misunderstand- ing in the practical application of the statistical theory.
For example, in the current ~ t a n d a r d s l * ~ for sound
power measurements in reverberation rooms the r e -
$ 2 ( ~ ) =R(x) +$:(XI
+
~ [ R ( X ) ~ ( X ) ] ~ / ~ cos6. (2)6 represents the phase difference of the arguments of
A and B. If this phase difference has a fixed value, the
nonadditivity of the mean-square pressure i s evident. The waves a r e said to be coperently related. If the phases of the original waves a r e random functions of time such that cos6 averages to zero, they a r e then referred to a s incoherently related.
quirement for microphone spacing and the boundary
Extending the above formulation t o the study of spatial interference corrections based on the results of Refs.
cross correlation at two points, 2 and 3 have been adopted indiscriminately for mea-
surement of random noise sources, as well as discrete p(x,)p(x,) =pl(xl)p,(x2) +p2(xl)pa(xa)
frequency sources. , The reverberant sound field
created by the two types of sources a r e , however,
+
[fi
( X ~ ~ ( X , ) I ' / ~ + [ ~ ( X , ) ~ ( X ~ ) ] ' / ~ C O S ~.
(3)rather different. If the waves a r e incoheregt, the last t e r m will average
This paper attempts to classify the reverberant sound to zero. In this case each wave contributes indepen-
fields with a better defined characteristics in terms of dently to the total c r o s s correlation.
coherence and incoherence and to clarify the conditions The above definitions of coherence and incoherence
f o r which various results from the statistical theory have to be modified slightly for the discussion of r e -
apply
-
verberant sound fields in an enclosure. In addition,there is need to discuss the behavior of the wave com-
I. DEFINITIONS ponents that make up the individual modes. If contribu- Morse and BoltQ initially considered the direct field
a s 'coherent" and the reverberant field as 'incoherent."
This definition, however, i s too general for present use. Instead, the original definition from the classical wave theory will be adopted in this paper.
tions to the mean-square pressure o r the spatial c r o s s correlation from the different modes cannot be con- sidered a s independent, the field is labeled 'coherentw;
otherwise, it i s 'incoherent." If, in addition, the con-
tributions from the wave components within each mode can be shown to be independent, the field i s considered
Consider two harmonic waves of the same frequency to be 'incoherent in the-strict senbe." . -
pl(x, t ) = Re{~(x)e-'~'), The cross-product t e r m s between modes o r those in
p2(x, t) = Re(B(x)e-'*'), Eqs. (2) and (3) can sometimes be eliminated by some
kind of spatial averaging o r ensemble averaging.
where A(x) and B(x) a r e complex quantities and Re( ) These special cases will be termed 'equivalent inco-
denotes U r e a l part of." The total excitation at a point x herent." -
i s given by
fib, t ) =pl +p, =Re(A(x)
+
~(x))e-'*'.
II. EIGENMODE ANALYSIS OF STEADY-STATESOUND
It follows for the mean-square pressure that
To illustrate the coherent o r incoherent nature of the
f i ( x )
-
/A(X)l 2
,
sound field in a room, the space-time cross-correla-tion function will be used. As the sound field i s not necessarily homogeneous in space, the proper nor- malization of the cross-correlation function by the lo- cal root-mean-square values is required. For sim- plicity, a rectangular room of sides L,, L,, L, and volume V = L&,L, was chosen. The sound field inside such a room can readily be analyzed by means of the
eigenmode theory. The sound field at the point
r
= (x, y, z) due to a simple source ~ e " ~ ' at r, =
Gc,,
y,,z,)has been written as''
k = w/c i s the wavenumber, p i s the density of the
medium, and
0
i s the specific normal admittance of thechamber surface assumed to be real and small. Equa- tion (4) can be rewritten a s
where
Where the origin of coordinates i s at one corner of the
chamber, N stands for the trio of numbers 1, m, n, and The space-time cross-correlation function of the acous-
tic at two points r, and r2 i s given by
A N = 1 / ~ , ~ , ~ . , E , = I , and c i = 2 , for i + 0 ,
JIN(x, y, Z ) = cosk, x cosk, y cosk. z
,
p(r,)p(r2, 7) =i
~e[p(r,)p*(r,, r) ].
In r n ~ n R
k,=- k
--,
k#=LT, k;=g+k;+k:,L,' , - L Y Thus, using Eq. (51,
Define
Equation (6) can be written, with the cosine term expanded, as
In addition, if
tanq =
C
B, sin.pN BN C O S + ~,
N
IF
=cos(q
-
0.-wr). (10)Equation (10) shows that R varies between 4, corre- sponding to a coherent field. This i s in agreement with the definition set out in Sec. I, since contributions from different modes have not been considered to be
D=[( ~ ~ ~ s i n @ ~ r + ( ~ B ~ c o s ~ ~ ] ~ ' ~ , independent here. Thus the sound field created under
N pure -tone excitation i s definitelv coherent.
On the other hand, if the field were incoherent, the cross-correlation coefficient would have a different form. To illustrate this point, an equivalent incoherent Eq. (8) can be written a s
field obtained by averaging the previous result over many source positions will be used. The cross-product p(rl)P(r2, 7) =+DE[(cosq cosa
+
sinq sina)coswr-(cosq s i n a
-
sinq cosa)sinwr],
=+DE C O S ( ~
-
a-
WT).
terms between modes were eliminated by the ortho- gonality of the characteristic functions associated with the source. lo Thus
where [
1,
signifies averaging over all possible sourcepositions in the room. Chien and Sorokal' have shown
the normalized cross-correlation coefficient is
that for high frequencies, by replacing the summation by integration, Eq. (11) can be,written a s
where S i s the surface a r e a of the chamber. Chien and Soroka heuristically assumed the "other termsn to be negligible. A better argument for dropping them, to- gether with observations concerning these terms, will be given in Appendix A. As indicated there, the 'other terms" can be ignored in the central region of the room where the equivalent incoherent field i s nearly homo- geneous. Thus the cross -correlation coefficient be
-
comesIn contrast with Eq. (101, [ R ] , varies between *sinkr/ kr, a magnitude that depends on k and r. Chien and ~ o r o k a " derived a result similar to Eq. (10) for r = 0 by carrying out the actual summation over the modes. Their analysis was for the nearfield of the source. The present experimental results, however, indicate that it i s also true in the reverberant field.
It i s worth noting that although the analysis i s for a rectangular room, the result should be applicable to nonrectangular rooms o r rooms with fixed diffusers. For these cases the only difference i s that D, E, q ,
and a! of Eq. (9) become unknown functions of the room
characteristics.
Ill. EXPERIMENTAL RESULTS
The experiment was carried out in one of the reverb- eration chambers at DBR/NRC. It i s equipped with fixed diffusers and a rotating vane. For this particular experiment the rotating vane was not in operation and the room was excited with pure tone using a single loudspeaker at the corner. Space -time correlation coefficients were obtained for six pairs of microphones scattered throughout the middle of the room. Their space separations ranged from 1.65 to 3.95 m. The nearest microphone was 3 m from the source, a dis- tance greater than the nearfield criteria set out in Ref. 11. Figure 1 shows that the correlation coefficients are in agreement with Eq. (10) rather than with Eq. (13). The reverberant field is definitely coherent. At 1.6-kHz excitation frequency the modal overlap for the room i s about 90. Similar results a r e obtained for 1 kHz.
To provide further experimental evidence about the nature of the reverberant field, spatial cross-correla- tion coefficients between a fixed pair of microphones have been obtained under various conditions. In these. experiments the two *-in. B & K microphones were fixed 10 cm apart, located near the center of the room. The separation vector between them was not parallel to any boundary of the room and different frequencies were used to generate various kt- values. Three room and signal conditions were investigated.
(1) The room was excited with a pure tone using a
1712 J. Acwst. Soc. Am., Vol. 69, No. 6, June 1981
TIME D E L A Y , ms
FIG. 1. Space-time correlatioq coefficients of six pairs of microphones in a reverberation chamber equipped with fixed diffusers. Single frequency excitation at 1.6 kHz.
single loudspeaker at the corner and the rotating vane not in operation.
(2) The test was repeated for 12 fixed orientations of the rotating vane and the results from the 12 tests were averaged to simulate the effect of averaging over source positions. The assumption i s that modifying the room geometry and thus the arrangement of the image sources i s equivalent to changing the source position.
(3) Third-octave band random noise was used to ex- cite the room under the first condition. Some pre- liminary results a r e presented in Figs. 2 and 3, to- gether with the theoretical prediction (sinkr/kt-) for an incoherent field.
Although these experimental results appear to be rather limited, they a r e sufficient to indicate the dif- ferent nature of the reverberant sound field. Results represented by the open circles in Fig. 2 support the conclusion drawn from Fig. 1 that the sound field cre- ated by the pure -tone excitation i s coherent. An equiv
-
alent incoherent field can be approached, however, if averaging over source positions i s taken a s indicated by the results depicted a s solid triangles in Fig. 2. When the exciting signal is band-limited random noise, the sound field i s definitely incoherent, a s shown by the results presented in Fig. 3. Discrepancy at large val- .ues of kt- i s probably caused by diffraction effects (asexplained by Cook et al?) that have not been corrected in the present case. Better agreement with theory was obtained by Balachandran12 in the DBR facility some twenty years ago.
IV. DISCUSSION
FIG. 2. Spatial correlation coefficient between a pair of fixed microphones under different conditions.
-
0- pure-tone ex-citation, rotating vane not in operation, r pure-tone excitation, averaged result over twelve fixed rotating vane orientations.
-
S i n ( k r l l k rFIG. 3. Spatial correlation coefficient between a pair of fixed microphones under 4-octave-band noise excitation. rotating vane not in operation.
1713 J. Acoust. Soc. Am., Vol. 69, No. 6, June 1981
It has been established that the reverberant field .
created under pure -tone excitation is coherent. Recent studies, experimentallS and computer simulation,14 have confirmed the sampling distribution derived by
Schroeder' and waterhouse4 for such a coherent field in
rooms with a certain finite boundary absorption. Their
assumptions that the modes o r wave components were statistically independent should not, however, be taken to imply an incoherent field. It is interesting that the
mean-square pressure can be written as
f r o m the present eigenmode analysis of Sec. 11. This formulation i s similar to those used by both Schroeder and Waterhouse. The reconciliation of the two models f o r the sampling distributions will be the subject of another study.
When the room is excited by band-limited random noise, the reverberant field will become incoherent be
-
cause a large number of modes will be excited and
resonant a t their own characteristic frequencies. As harmonic waves a t different frequencies a r e uncorre-
lated, the modes become noninterfering
.
Their con-tribution to both mean-square pressure and spatial cor- relation can be considered independent. This argument is supported by the more quantitative analysis given in Ref. 10. In addition, it is shown in Appendix A that in the central region of the room most of the coherent contributions from the wave components of the individ- ual independent modes a r e also very small and can be neglected. This region of the sound field can be called "incoherent in the strict sensem; only here will Cook's theory of spatial c r o s s correlation2 be applicable. In spaces close to the boundaries these coherent contribu- tions from the wave components cannot be ignored. In fact, they a r e responsible f o r the interference pattern and build up of the mean-square pressure near the boundaries described by Waterhouse.
To explain this situation using the free-wave model,. consider the free waves coming from different direc- tions as being generated by the images of the real source in the room. When the source emits band-
limited random noise, it will have a finite coherence
time (correlation time length) that is inversely propor- tional to the bandwidth. Thus the'free-wave packets coming from different image sources will be incoherent when the time delays between them, in reaching a c e r - tain region inside the room, a r e longer than the co- herence time of the real source. This will be true for the majority of the wave packets crisscrossing the cen- t r a l region of the room, but not for the space close to boundaries. Here, because of the symmetrical dispo- sition of the images with respect to these surfaces, many of the differences in time delay become small o r
zero. On the other hand, if the source emits only a
single frequency, all the waves coming from the image sources will be coherent in the sense that they a r e only shifted in phase and will still interfere with each other. It is therefore not possible t o neglect c r o s s t e r m s be- W. T. Chu: Coherent and incoherent reverberant sound fields 1713
tween any two of them in forming spatial cross-corre- lations o r interference patterns near the boundaries. If the bandwidth of the random noise used is very narrow,
a partially coherent situation will result. This particu-
lar case has not been discussed.
As indicated here, however, and in Ref. 10, an equiv-
alent incoherent condition for even a coherent field can
be approached by applying some kind of averaging tech- nique over frequency o r space. Averaging over source positions has been chosen for illustration because of i t s practical application, but another spatial averaging technique has been used in Ref. 10.
So far, the word 'random" has been avoided in de- scribing the sound field because of i t s ambiguity. For example, the t e r m urandom sound field" used in Ref. 2 is to be interpreted as a 'field containing many uncor- related plane waves," whereas the t e r m 'randomphase" used in Ref. 4 in describing the plane waves should be interpreted a s 'the phase relations of the plane waves constitute a fixed set of random variables and they a r e not uncorrelated." One must be very specific when us- ing the word 'random" to describe the reverberant sound field.
ACKNOWLEDGMENTS
This paper is a contribution from the Division of Building Research, National Research Council of Canada, and is published with the approval of the Di- rector of the Division.'
APPENDIX A:
Let the two position vectors be rl= ( ~ , , ~ , , z , ) and r,
= (x,, y,,~,). According to definition
JIN(rl)JIN(r2)= (coskXx1 C O S ~ ~ Y , C O S ~ , ~ , )
x ( C O S R ~ , cosk, y, cask, 2,)
.
(Al) Using the trigonometric relationcoskXx1 C O S ~ , X , =
4
C O S [ ~ ~ ( X ~ -x,)]+
COS[~,(X~ +x,)],and the definitions
Cx(Dx) = C O S ~ , ( X ~ -x2)
,
Cx(S,) = coskX(x1 + x,),
Cy(Dy)=cosky(yl -y,), etc.,
Eq. (All can be rewritten as
Chien and Sorokall assumed that the first t e r m is the c,(S,)C,(D,)C,(D,) = cos(2kNx, sinBsin@)
dominant term, following an argument presented origi-
x
cos(k,r sin0 sin@ sinpsiny)nally by Morrow.15 A more quantitative assessment of
the importance of these t e r m s will be given here. A better x cos(kNr cos0 cos0)
.
(A5)understanding of the sound field can be obtained by care- When the summation over all possible modes i s carried
fully studying the other terms. Rewrite the c r o s s cor-
relation as a function of the separation distance, Y, be- out by means of integration over the wavenumber space
tween the two points and the position vector, re, of the f o r high frequencies, as f o r the procedure used in ~ e f . midpoint between them. Referring to Fig. A l ,
x1 =x,
+
(~/2)sinfl cosy, x, = x,-
(r/2)sinB cosy, y =ye-
(~/2)sinflsiny, y, = y,+
(r/2)sinp siny, (A31 z l = z ,-
(r/2)cosfl, z, =z,+ (r/2)cosp.Changing the wavenumber space from rectangular to spherical-coordinate system it follows
kx = kN sine cos@
,
ky = kN sinOsin+, k,=kN cos0.Thus the first t e r m of Eq. (A2) becomes
x cos(kNr C O S ~ cosp)
,
(A4)/
which is the same as that used by Chien and Soroka in
obtaining the sinkr/kr t e r m in Eq. (12) of this paper.
Similarly, expanding the second t e r m of Eq. ( ~ 2 ) , gives FIG. A l . Coordinate system.
10, the following expression i s obtained t e r m to the total cross-correlation fuhction. A similar
I
(A6) derivation can be performed f o r the other terms. Thusthe seven expressions that constitute the "other termsn
I
where jo(A) = sinA/A for the contribution of the second in Eq. (12) a r e
I
One may observe that the "other termsn become negligible when the c r o s s correlation i s considered in the central portion of the room, but not close to the boundaries. In fact, these t e r m s a r e responsible for the interference pattern near the boundaries, as derived in Ref. 10. The same result can be obtained by putting r = 0 in Eq. (A71 and in the sin k r / k r term.
Another observation may be made by decomposing the modes into their wave components. As was demonstrated
by Lubman,ls an oblique mode can be written
as
the summation of eight traveling waves. That isI
$N(r)e'wt = cosk,x cosky y cosk, ~ e ' ~ '
= ${exp[i(k,x+kyy
+
k,z)]+exp[-i(k,x+kyy +k,z)]+exp[i(k,x-k,y -k,z)]+exp[-i(k,x-&,,y -k,z)]The spatial c r o s s correlation due to the f i r s t wave component can be written as
I
If the eight wave components a r e considered as incoherent, s o that their contribution to the c r o s s correlation can
be summed independently,
I
This is the f i r s t t e r m of Eq. (A21 that gives rise to sin kr/kr except for a factor of
i
which comes from the additional time averaging applied to Eq. (A8). The "other terms" then'must be due to the coherent inter- ference of the wave components within each individual mode. The interesting fact is that in an incoherent r e - verberant sound field t h e w i s a region where the wave components of each individual mode can also be treatedas independent o r incoherent. This region i s "incoher- ent in the strict sense."
'M. R. Schroeder, Acustica 4, 594-600 (1954).
'R. K. Cook, R. V. Waterhouse, R. D. Berendt. S. Edelman, and M. C. Thompson. Jr., J. Acoust. Soc. Am. 27, 1072-
1077 (1955).
3 ~ . V. Waterhouse, J. Acoust. Soc. Am. 27, 247-258 (1955).
'R. V. Waterhouse, J. Acoust. Soc. Am. 43, 1436-1444 (1968).
5 ~ . Lubman. J. Acoust. Soc. Am. 44, 1491-1502 (1968). %. J. Schultz, J. Sound Vib. 16, 17-28 (1971).
'ANSI Standard S1.21-1972. American National Standard In- stitute. 1430 Broadway, New York, NY 10017.
'ISO Standard 3742-1 975 (E), International Organization for Standardization, Geneva, Switzerland.
'P. M. Morse and R. H. Bolt, Rev. Mod. Phys. 16. 69-150
(1944).
'%. T. Chu, "Eigenmode analysis of the interference patterns in reverberant sound fields,'' J. Acoust. Soc. Am. 68, 184-
190 (1980).
"c.
F. Chien and W. W. Soroka, J. Sound Vib. 48. 235-242(1976).
"c.
G. Balachandran, J. Acoust. Soc. Am. 31, 1319-1321 (1959).13w.
T. Chu, J, Acoust. Soc. Am. Suppl. 1 64, S77 (1978).14w. T. Chu, J. Acoust. Soc. Am. Suppl. 1 65, S52 (1979).
T. Morrow, J. Sound Vib. 16, 29-42 (1971).
1 6 ~ . Lubman, J. Acoust. Soc. Am. Suppl. 1 60, S59 (1976).
This publication is being d i s t r i b u t e d b y the Division of
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.
Stamps a r e not acceptable.A l i s t of a l l publications of the Division i s available and m a y b e obtained f r o m the Publications Section. Division of Building R e s e a r c h , National R e s e a r c h Council of Canada, Ottawa. KIAOR6.
