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Binaural source location
Journal of Sound and Vibration (1976) 44(2), 275-289
BINAURAL
SOURCE LOCATION R. KINNS ?
Department of Engineering, University of Cambridge, Cambridge CB2 lPZ, England (Received 14 January 1915, and in revisedform 8 July 1975)
It is shown how the coherence and phase spectra of signals from a closely spaced pair of microphones in the far-field can be used to compute the moments of a line distribution of arbitrarily correlated omni-directional sound radiators. This line source gives a far-field which is equivalent to that of model-scale and full-size turbojet engines in terms of measured power and cross-spectra of microphone signals. The necessary spectra can be computed rapidly on a small digital computer and the simplicity of the technique has meant that experiments could be performed in parallel with the usual far-field noise measurements. In this way, it has been possible to identify important properties of noise generation by turbojet engines at minimum cost and development and application of more sophisticated techniques has been accelerated. It is shown how the apparent properties of the source distribution may depend markedly on distance from the source to the microphones. Interpretation of results is guided by consideration of simple cases. 1. INTRODUCTION The requirement for a rational approach to the design of quiet turbojet engines requires a more detailed understanding of noise generation mechanisms than has hitherto been achieved. Although the theoretical framework provided by Lighthill [l, 21 and Ffowcs Williams [3] has led to a respectable correlation between noise measurements and the theory of aerodynamic sound generation, the need to introduce empirical constants in data reduction procedures could lead to erroneous conclusions about the relative importance of various mechanisms for the generation and propagation of sound. A useful survey of experimental data, and discussion of some of the unresolved anomalies, is given by Fisher, Lush and Harper Bourne [4]. In an effort to obtain a more definite understanding of the phenomena of sound generation associated with aero-engines, considerable effort is being devoted to the development of sound source location procedures. In particular, it is necessary to understand why jet engines generate more noise than would be expected from considerations of model-scale experiments. For instance, silencing policies will depend markedly on whether sound is radiated directly from a jet engine nozzle, or whether it is the result of temperature and velocity fluctuations in the emerging airstream. In addition, the influence of flight on practically important sources must be established. At model-scale, various attempts are being made to use acoustic mirrors, based on an optical analogy (see reference [5], for instance), but these become too unwieldy to be practicable for use on open sites with full-size engines. The use of arrays of microphones appears much more attractive (the theory and application of a 14 microphone “acoustic telescope” has been described by Kinns and Billingsley [6]). Another technique, utilizing the power and cross-spectra of the signals from a polar array of microphones, with respect to a fixed microphone, is being developed by Fisher and Harper Bourne [7]. These methods require corresponding complexity in the signal processing, although they can, in principle. provide detailed descriptions of an equivalent line source. i_Present address: Y-ARD Limited, Charing Cross Tower, Glasgow G2 4PP, Scotland 275
276
R. KINNS
However, the desire to acquire data from a large number of full-size engines and modelscale rigs, for a preliminary survey of the differences between them, led the author to develop a simpler technique using just two microphones. The simplicity of this arrangement has obvious attractions. With only one binuaral measurement and simple processing, the source centroid and scale are determined completely. These are the most fundamental characteristics of the source distribution. Once they are identified, the more active source regions can be probed in greater depth with the more elaborate multichannel techniques. 2. FORMULATION
OF A MATHEMATICAL
MODEL
OF THE SOURCE
REGION
It is known that noise from a turbojet is generated by the mixing of the emergent airstream with the surrounding atmosphere, by the entropy fluctuations arising from unsteady combustion in the engine, by the unsteady aerodynamic forces on turbomachinery blade rows and by a variety of other mechanisms such as the passage of eddies through the shock-cell system of an under-expanded exhaust stream. All these mechanisms may interact. For example, sound generated by the entropy fluctuations may be scattered by the emergent airstream and be diffracted at the nozzle exit plane. It may also trigger aerodynamic instabilities in the exhaust jet. Furthermore, the sound sources will have directivity associated with their motion with the multipole nature of turbulent eddies [l-3 ] and with the refraction and diffraction of sound as it propagates away from the source region. In this complex situation, it is inappropriate to assume any particular mechanism for sound generation in the design of source location experiments. However, there is no unique [8] specification for the origin of a given sound field and it is only possible to think in terms of an equivalent source region which shares the measurable far-field properties of the real sources. Source models are chosen for convenience in a tractable analogy. The chosen mathematical model should allow arbitrary directivity in the far-field and arbitrary correlation between measured microphone signals, with respect to frequency. If an array of microphones lies in a plane containing the axis of a jet engine and the power and cross-spectra of all the microphone signals are available, a convenient mathematical model for the source region is a line array of omni-directional radiators on the engine axis, which can have arbitrary correlation with respect to axial position. It is this model which has been developed for interpretation of signals from the microphone pair and the “acoustic
K
I I I -
__---
Yn
Polar origin ;
I& -Line
of sources
$
fK
Microphone
ot (r,JI)
Microphone
ot (r,8)
Figure 1. Notation for theory of line sources.
277
BINAURAL SOURCE LOCATION
telescope” [6]. If no correlation is allowed, the source region as a whole must be omnidirectional to an observer who is so far from the source region that variations due to the inverse square law can be neglected. In the following analysis, it is assumed that the pressure variations at any microphone are purely acoustic; that is, measurements are made sufficiently far from the source for other pressure fluctuations to be negligible. It is also assumed that sound propagates at constant speed outside the source region, so that atmospheric fluctuations are neglected in the first instance. Consider an equivalent source region which consists of the line of omni-directional radiators shown in Figure 1. Let the strength of these radiators be Q(y, t) with respect to position and time and consider the signals received by microphones at (r, 0) and (r, $) with respect to an origin on the line of sources. The signal received at (r, 0) is given by p(r, 0, t) = J’(l/r,)
-CC
(1)
Q(YI, t- rJc> dy,
while that received at (r,$) is
PG.,$9t) =
7V/r,) Q (~2, t - r,lc)dy,,
12)
where c is the speed of sound. The notation is shown in Figure 1. Thus OD (3) The cross-spectrum of microphone signals is
7Ex [p(r, 8, t) .p(r, II/, t + z)] eeznifrdr
= (1 ir’) C,, &(f, r),
say,
(4)
where Ex denotes expected value, and it is assumed that the source region is statistically stationary. Thus Lv
[Qh,t)Qb, t+ z-
G, ,Af,r) = jJj(rz/r1r2)Ex
c11 e-2nif(r-(rz_rl)lC) e2hf(rl -r2)/c drdy, Im(r2 - rI >I
dy,. 0)
If one now assumes that the microphones are sufficiently far from the source region that variations due to the inverse square law can be ignored, r2/rl r2 M
1,
(6)
so that
co,df,r)
= =
]r{ -m
Ex [Q(yl, t) Q(y2, t+ T’)e-2nifr’e2nif(rl-r2)/Cdzfdyl
m Q,,,,U> SI
eznif(rl-rz)lc
dy,
dy2,
dy2
(7)
(8)
-02
where the cross-spectrum of source components at y1 and y, is
Q,,,,U> = { Ex[Qh,t)Qb2, t+ 41e-2nifr dr.
(9)
By the cosine rule, rl and r2 are givln by @Jr)’ =
1 + (y,/r)2 - 2(y,/r)
cos 0
(10)
278
R. KINNS
and (rJr)2 = 1 + (yz/r>2 - 2(y,/r)cos
+.
(11)
Thus t1 so
-
r2
=
Wl(~l
+
r2)I
MY:
-
~:)PI
-
(Yl
cm
0 -
Y,
cm
$)I,
(12)
that, in the far-field in the vicinity of 0 = II/= n/2, r2
-
rl
=
(rl
cm
f3 -
y2
cm
ti)
+
0:
-
y3Pr)
+
rWy/r)4,
(y/r)’
cm2
(0.
W.
(13)
If the source region consisted of simple sources, such that Q,,,(f) decayed rapidly to zero with increasing 1y1 - y2), without oscillation with respect to (y, - vZ), the second term might be neglected. However, multipole sources consist of a nearly self-cancelling distribution of simple sources and the second term in expression (13) can remain significant up to a few hundred nozzle diameters from a jet source, even if the source region is restricted to ten diameters or so. Furthermore, disturbances in the jet exhaust may generate sound over several ma y remain significant for relatively large (vl - y2). With nozzle diameters, so that Q,,,(f) both terms retained in equation (13), equation (8) can be written as
CO,dL r) = jyQyl ,,(f> e(2ni/~)(Y:-Y:)/2re-(2~i/l)lv,cos
B--Y2COS ti) +,l
dy2,
-00
(14)
where I = c/f is the wavelength of sound at frequency f. If one now defines Q;,
,,V;
r)
=
Q,,
Y,cf)
e(2xill)(Yf-Y$)/2r,
05)
equation (14) can be written as Co, ,Jf, r) = fIQi,
y2(f, r) e-(2ni’A)(ylc0*e-y2cos*) dy, dy,.
(16)
-m As r increases, Q;, y,cf, r) tends to the limit :
,$JQL, y,ti 43 = Qy, y,cf)In practical noise measurements,
“far-field”
observations
are often conducted
(17) with
r/d < 50. Then, QilY,(f,r) may be markedly different from Qy,y2(f). The power spectrum of a microphone signal at (r, $) is given by
(18) This result is derived from expression (16) by setting 6 = $. When 8 = $ = 7112, C ,,.,,,cr;r)=~~Q:,,,ct;r)dy,dy,,
-co
(19)
while C,,,, *cf, r) = rj Q;, ,,cf, r) e(2ni/k)y+os* dy, dy,. -CO
(20)
In principle, the “acoustic telescope” [6] with linear signal processing allows Q,,,,(f) to be estimated to a spatial resolution of about half a wavelength. In the following section, it is shown how signals from closely spaced microphones can be used to estimate properties of the distribution of Qi, y,(J r).
BINAURAL
3. PROPERTIES
SOURCE
OF THE SIGNALS SMALL ANGULAR
LOCATIOA
FROM MICROPHONES SEPARATION
‘79 AT
If one defines
equation
(20) can be written
as
&,f, r) is complex, but its imaginary part has zero internal scale. since 06, ,,(,f: Y) =- Q;.; ,,>!,i. r) where * denotes the complex conjugate. This result also follows from equations (I 9) and ! 101, since power spectral density must be real. If o(y,,f. r) is separated into its real and imaginary parts. Q(y,f, r) = R(y,.f, equation
(22) can be written
r)
+ il(,v.f; r i.
:.I?)
as
(24) If $ is close to 7r/2 and it is assumed that the moments to _v, (24) can be expanded as a power series in k = (27r cos G);;..
of R and i are finite with respect
!,25)
it should be noted that k has dimensions of (length)- ‘, but I\ always associared with one of the length scales which specify the source distribution. Throughout the analysis. the explicit dependence of iT and ion v,fand r is to be understood. To third order in k,
where
1Rdv __“m and
Cc -aI
i dy
= 0.
(28)
_Cis the position of the source centroid, downstream ofthe origin. lt is entirely determmed by 8, that part of the source distribution which contributes to the sound intensity. The imaginary part 1, which is more an indication of source directivity. does not affect r.
280
R. KINNS
The cross-spectrum can be expressed in terms of magnitude and phase, j&, *cf, r), C,D. ti(f, r) = IC.,,, &(f, r)l eiSn’**(f,r),
(29)
so that Bn,n,ecf, r) = k[f - (k/2) vr - (~“/WPR + 3~,v31 (30) to third order in k. The coefficients are defined below. This equation shows how the source centroid is determined very simply from the single measurement of phase, it being k-l times the phase angle for small microphone separation where 0(k2) terms are negligible. m y,= myidy Rdy (31) 1 II-m -m represents the first moment of i divided by the intensity at (r, 7c/2). It is related to the first internal moment of correlated sources, which governs the directivity of the sound field. cc
1(y -y)‘Idy
VI =
-m
J Rdy I --m
(32)
is the second moment of i, divided by the sound intensity. It influences the manner in which the phase departs from a linear relationship with k as k increases. cc VR= j(y+Kdy PWdy (33) -m I -m is the second moment of R divided by the sound intensity. It indicates the extent of the source region. Its magnitude can be determined from the coherence spectrum of signals from microphones at small separation, together with the measured directivity of the sound field. m ,uR
=
(34)
s
-m
indicates the skewness of the distribution of R. It has an important influence on the behaviour of phase with respect to k, as k increases. If a is an even, and I an odd, function of (y -J), jn,2, *(Jr) will be proportional to k = (27ccos1j)/~ for all k. Otherwise, j3n,Z,Jr (f, r) will be proportional to kj for small k and the deviation from linearity will tend to be governed by the skewness of the distribution of i?, since lis self-cancelling (equation (28)) and its even moments tend to be small. 4. PROPERTIES OF THE COHERENCE SPECTRUM FOR SMALL MICROPHONE
SEPARATION
Although &, &Jr) is relatively easy to measure, as it is not dependent on the magnitudes of the microphone signals, accurate estimation of CX,2,*cf, r) for small (7c/2- $) is rendered difficult by its dependence on an accurate knowledge of the sensitivity of microphones and processing equipment. However, the cross-spectrum magnitude may be normalized to give the coherence spectrum, which is independent of gain factors. In addition, its statistical properties have been investigated in detail for Gaussian signals (see reference [9], for instance), and it is as easily computed as the cross-spectrum, by using modern data processing techniques [lo]. The properties of the coherence spectrum for small microphone separations can be deduced by using series expansion, as in the previous section. However, a more elegant deduction of its properties can be obtained via a Taylor series expansion, by using its derivatives with respect to $ at $ = n/2 and this approach is preferred.
BINAURAL
SOURCE LOCATION
181
The magnitude coherence spectrum, bounded by zero and unity, is given by Yn/2,ti(f, r) = ICWZ,ti(f,r)lI[C,,~, n,Z(L r) C,, a(.f,r)]1/2,
(35)
or Its first and second derivatives can be obtained in terms of derivatives of the power and crossspectra by differentiating equation (36) twice, after taking logarithms of both sides. Thus, (37) and a2 7n/2.
.&m21$42
=
w,:2,
+
.,2P(G,2.
u/c,,,,
n/2)
,>iw>wc,*,2,
32fcK,2,
e +
,>/wL~,2
c/2,
J -
c,,
+
,>!a
(38)
+2l$_n,2.
As before, the explicit dependence onfand r is to be understood. The first and second derivatives of C,,,, ti and C,, J, can be obtained as follows : (39) so that
= -(2ni/iJ fly2 Qi,Y2dYldY2.
a~,,~, #,&=n,2
(40)
-m
Similarly (41)
(42)
a2 C,*,,, ti/W21e=n,2
-(47r2ij.‘){j. Y: Q&h
=
dy2,
-co
(43)
(44) m
(45) By using equations (40), (42) and (44), it follows at once from equation (37) that (46)
aYn,2.ti,iW+nlz = 0 and that a2 Yn/2,
d~~21w2
=
(47r’in’) j-j Q;,&
-m
- j=j.Q;,,,dvl
-m
dy, dy, fl Ql:,+ -a
dy, fl Q;,,,y,My, -m
dy, dY2 dy2
flQ;,,,dp, -m
l2 dy,]. (47)
However, equation (47) can be greatly simplified by using definitions (3 1) and (33), together with :
v, = fj- Q:,,,(YI - y,)2duld~,/f -m
-m
Q;,,,dy,h.
(48)
282
R. KINNS
v, is a measure sound intensity Thus,
of the internal second moment of correlated sound sources, divided at (r, 7c/2). It is purely real, though it may have negative sign. a2~a,z, J&&=,,,
Comparison
of equations
= -(~~z~/J~~){v, - v,/2 - J$>,
comparison
(49)
(45) and (48) shows that (a* c,, ,iati*)ic,,,,
Similarly,
by the
of equations
n,Z1JI=n,2= -(4~*in*)
v,.
(50)
(31) and (44) gives the result
(a~,, ,ia*)ic,,,,
n,Z~JI_n,2= 47cY,1/1.
(51)
To second order, ($ - n/2) can be written as (-costi), so that the following relations from the analysis of this section, both being to second order in k = (2ncos$)/1: 1-
77~2,
G, ,(r,f)!C
&>.f)
=
(k2i2)(v,
,712, n,z(rJ->
=
-1 -
vi2
2ky,
-
(52)
~3, W/2>
result
(53)
v,.
Equations (52) and (53) show how the initial behaviour of the coherence with respect to increasing k is governed, not only by the external second moment of the effective source distribution but also by the directivity of the sound field, through the internal moments of the source distribution. If desired, higher order terms can be evaluated, by using the same approach. However, experiments at model-scale and on a full-size engine have shown that microphones can be sufficiently closely spaced for higher order terms to be ignored. A direct check can be obtained by using microphones at more than one spacing and some typical results are given by Kinns [ 111. The meaning of the various internal and external moments is clarified by considering two simple cases. The first is the trivial case of an array of uncorrelated omni-directional radiators, while the second is that of a stationary dipole.
5. ILLUSTRATIVE
SIMPLE
CASES
5.1. THE CASEOF A DISTRIBUTIONOF UNCORRELATEDOMNI-DIRECTIONALRADIATORS A line array of uncorrelated monopoles is actually of very little practical interest. Its distant sound field is totally devoid of directionality and its near-field behaviour is contrary to that of complex multipoles. Nonetheless, the model provides a simple illustration of how the binaural specification of source scales is determined. In conjunction with the following examples, it helps highlight the significance of correlation between components of the line source distribution which imparts directivity to the sound field. When sources are uncorrelated Q;,y2(f;r)=4y,
-~2)f',Jf)
=
Q,,,,cf>.
(54)
and the distribution is unique in its independence of observation distance (apart from terms arising from the inverse square law). 6 is Dirac’s delta function and J’,,,(.f) is an intensity measure. All internal moments of the source distribution are zero (equations (31) and (48)) and the loss of coherence for small k is given by equation (52) with v, = y, = 0. Thus, the source scale, 6, is given directly by the expression
v%=(Wh’2(1 Furthermore,
consideration
of equation
-~n/z.df,rN.
(24) shows that in this trivial
(55) case where 1 is
BINAURAL
283
SOURCE LOCATION
identically zero, the polar cross-spectrum Cn12,@cfr) h as real and imaginary parts which are even and odd functions of (7c/2- $), This is not the case in the examples that follow. 5.2.
ANALYSIS FOR THE CASE OF A DIPOLE
Consider the case of two point sources situated at y = a and y = b. Let these sources have equal intensity Z’(f) and unity coherence at all frequencies and let the source at y = b lag that aty=abyT. Then the cross-spectrum of source components is given by
Q,,,,(f)
= KY, - 48~~
- VU-)
+
d(y, - b)&y,
- 4W)
+
+ StyI - 4 &y2 - b) p(f) e- 2nifT + 6(yl - b) 6(y2 - a)P(f)eZn*fT.
(56)
Thus
Q;, ,,(_Ar) =
W)@(Y,
- 4
W2
-
+ a(y, _ a) &y, &y,
_
b)
a(
4
+
&YI
_ 6) e-27MT
y2
_
a)
e2nifT
- b) @y, - 4 +
e(2WU(a2-b2)/2r
+
e-(2nili)(a2-b212r)},
(57) from equation (15). It follows from equation (16) that the cross-spectrum of signals from microphones at (r, 0) and (r, $) is proportional to C,, *(f, r) = p(f) {e-(2ni/~)(aZC‘35 0--lrCOS ,b) + e-(2nill)(bCos 8-b Co5 J) + +
e-2nifT
e(2n1/A)(.*-b2)/2r
+
e2n,fT
e-(2ni/l)(o~-b~)/2r
e-(2”‘/L)(~
e-(2WMbcos
cos O-b
B-0
cos IJS) +
C‘W)}~ (58)
Therefore C, *(f, r) = 2P(f) e(2ni/l)(a+b/2)(cos~-c0se) {cos (lc/l.(b - a) (cos 0 - cos t,h))+ + cos (cc+ rt/J.(b - a)(cos 6 + cos I,+))},
(59)
where a(f, r) is given by cc(f, r) = 27c[-fr + (a2 - b2)/2rl].
(60)
The dependence of the cross-spectrum on r indicates how the latter can have a marked influence on apparent source distributions through the directivity of individual parts of the source region. It can be seen from equation (59) that the phase PO.Jf,r) is Be,+(.f,r) = (244{(a + b)/2)(cos Ic/- cos o),
(61)
which is the result which would be obtained for an omni-directional point source at y = (a + b)/2 for any $ and 8. It is easily shown by direct analysis that the coherence Ye,&r) is unity, so that the phase and coherence spectra are identical to those for the point source. It is useful to consider the moments 1, y,, vR, vi and v, defined previously, in order to demonstrate their importance for a compact source. From equations (21), (57) and (60), the complex source distribution is &v,f, r) = P(f) (XV - a)11 + vial f S(y - b)[l + eta]}.
(62)
Therefore %~,.f, r) = WMY I(y,f, r) = W){S(y
- a) + S(y - b)}(l + cos a),
(63)
- b) - S(y - a)} sin tl.
(64)
Thus, the centroid position is ~7= (a + b)/2, which is the point midway between the dipole elements.
R. KINNS
284
The internal first moment is y, = {(b - a)/2) tan(a/2) and is seen to give a measure of the separation of the dipole elements. The external second moment is vR = {(b - ~r)/2}~,while vi = 0, since I is antisymmetric about jj. The internal second moment is v, = 2{(b - a)/2}‘{ 1 - tan2(a/2)}, which is a third measure of the separation between elements. It can be seen that the sign of v, depends on ~1,but that (va - v,/2 - y,Z)is identically zero. It will be noted that CIin equation (60) is dependent on the polar radius, r, while j is always the point midway between the dipole elements. In principle, yj? and v, can be evaluated from the first and second derivatives of the power spectral density of a microphone signal with respect to $, but the need for evaluation of microphone sensitivities hinders exact measurement. The above results indicate how v, and yf can both be larger than vRfor a multipole source. This is a simple example of a case in which i can be much larger than i? and where the directivity of the sound field is critical, even where correlated sources are not widely separated so that it is tempting to discard the second term in equation (13). In order to illustrate this, consider the case of a dipole whose elements are separated by A/4 and where one element emits at time 31/8c after the other. It is easily shown that the far-field is given by c,, JI= (l/2)0 + cos [(n/2)(3/2 -
and c,.
COST
(6%
tbis shown as a function of $ in Figure 2.
Figure 2. Theoretical other by 31i/8c.
field shape for a dipole with elements separated by A/4 where one element lags the
It can also be shown that if the dipole elements are positioned at y. - A/8 and y. + A/8 from the assumed origin and observations of the source are made at distance r, then the magnitudes of the delta functions representing R and I are proportional to : 8, = 1 + cos [(n/2)(3/2 + ye/r)]
(66)
II= sin [(n/2)(3/2 + yo/r>l.
(67)
and w, and 1, are shown in Figure 3 for -0.2 < y,/r < 0.2. In this case, the directivity of the source leads to significant variation in apparent source distribution with respect to observation distance. The variation in R reflects the change in intensity of the sound pressure level
BINAURAL
SOURCE LOCATION
185
Figure 3. Variation of the magnitudes of the impulse functions at yO - A/8 and yO+ 118, which represent the complex source distribution of the dipole whose field shape is shown in Figure 2, with respect to dipole
position and observationdistance. as the source is moved, while the variation in i reflects the change in the local directivity with respect to the assumed origin. It is only necessary for a source region to have substantial directivity for large variations in apparent source distributions with respect to observation distance to occur. All the moments of the apparent source distribution may be affected and the difficulty of making “far-field” noise measurements in a situation where there are substantial variations in directivity with respect to source position will be obvious. However, the apparent centroid position is clearly useful in providing an origin for the extrapolation of noise spectral density measurements, made at modest distances from the source region, to the far-field. In this example, such extrapolation would be ideal, because the centroid is independent of far-field observation position. 6. CALIBRATION AND A TYPICAL RESULT Detailed considerations of experimental technique and interpretation of results are given by Kinns [l 11.The intention here is to illustrate the method of calibration of a microphone pair with a practical example and to show a typical result for jet noise. 6.1.
CALIBRATION
OF A MICROPHONE
PAIR
It is clearly unrealistic to measure polar radius r to very high accuracy, but it is relatively easy to measure the distance between a pair of microphones which are rigidly mounted on the same support. Thus, the angular separation of microphones can be accurately determined, but small differences in polar radius must be expected. If the microphones are positioned at (r, 6) and (r + E,I/I), where 6 z $ z n/2 and (t/r(cos$ - cos@l < 1, then
rz - rl = E + 1~~cos 0 - y, cos $ + (yz - yf)/2r,
(68)
to the order of accuracy of equation (13). Equation (16) becomes co
Co. &f, r) = e-(znia’A)
II Q:, ,,(_A r) e-m
(hi /l)(Y,coss-y,Cos~‘)d,~~ dy,.
(69)
Thus, the influence of the small error Ein the polar radius is just to introduce a phase factor in the cross-spectrum. It follows that providing a microphone pair is calibrated by using a known source, the small error E can be evaluated without difficulty. In order to demonstrate the excellent agreement between predicted and measured results when using a known source, and the method for eliminating E, consider the experimental arrangement shown in Figure 4. In this case, $ in microphones were positioned at 90” and
286
R. KINNS
Grourdcoveredby acousticfoam
wedges
Axis of jet
(a)
(b)
Figure 4. Experimental
configuration.
(a) Top view; (b) side view.
89.5” with respect to the exit of a nozzle and its axis of symmetry, for tests on jet noise. They were mounted 12 ft 7 in from the nozzle and the microphone arrangement was approximately symmetric with respect to its support stand and the nozzle exit. The spacing of microphones was only 1.32 in and was much smaller than that normally used for either model-scale or full-size engine studies. Experiments were conducted with acoustic foam wedges on the ground, in flat-calm conditions. A Tannoy loudspeaker with 7 in exit diameter was positioned 5 in and 27 in downstream of the nozzle exit and was driven by pink noise in each case.
T IOdB
1
0
I
I
I.6
3.2 Freqwncy
I
46
I
e
6.4
(kHr)
Figure 5. Power spectra of signals from microphone at B = 90” for the loudspeaker calibration and for an experiment on jet noise. -, Power spectral density for loudspeaker in first position (Figure 4); -, power spectral density for jet noise. Jet Mach number 0.85 with ambient stagnation temperature at exit from a 3 in conical nozzle.
BINAURAL SOURCE LOCATION
287
Signals from the microphones were recorded on a Racal Store-4 F.M. tape recorder at 60 in/s. Tapes were replayed at 15/ 16 in/s for analysis on an IBM I 130 computer [lo] to give the required power, coherence and phase spectra in the frequency range O-16 kHz with about 77 Hz resolution. Results were obtained from 3.0 second data samples. Slight misalignment of tape recorder heads and differences in electronics between channels have the same effect as the spatial error E, but it is unnecessary to evaluate these separately if the same analysis procedure is used for calibration and jet noise signals.
O~Sl
0
I
I
I
I.6
3.2
4.6
Frequency
(kHz)
/
6.4
IU 8.0
Figure 6. Magnitude coherence spectra for loudspeaker calibration and jet noise. -, Coherence spectrum for loudspeaker calibration; -, coherence spectrum for jet noise. Conditions given in caption to Figure 5. 0 = 90”; v/ = 89.5.
The power spectrum of one of the microphone signals and the coherence spectrum for the first loudspeaker position are shown in Figures 5 and 6. The phase spectra for both loudspeaker positions are shown in Figure 7. It can be seen from Figure 5 that the loudspeaker had a highly non-uniform gain with respect to frequency and that the variation in spectral density has a pronounced effect on the coherence magnitude (Figure 6). These variations are easily explained in terms of extraneous noise from a variety of sources and statistical errors [1 11, but the phase spectra show near perfect linearity with respect to frequency in regions where the coherence is close to unity. It should be noted that the power spectral density at frequencies above about 2 kHz was much lower than in studies of jet noise and results are therefore more susceptible to electronic noise. It is for this reason that spectra are not shown for frequencies greater than 8 kHz, while estimates below 800 Hz are not shown because they are appreciably influenced by high-pass filters. The effective value of E can be deduced from the phase spectrum for the first loudspeaker position. The dotted line extension of the phase spectrum for the 27 in loudspeaker location
A
Phase spectrum ---pfedlcted frov
ISiresult
Phase spectrum for I*’ loudspeaker pcxit~o~, 3.2
Frequency
4.8
(kHz)
Figure 7. Phase spectra for two loudspeaker positions and for jet noise. -, Phase spectra for two loudspeaker positions; ----, phase spectrum for second position predicted from result for first position; -, phase spectrum forjet noise. Conditions given in caption to Figure 5.
288
R. KINNS
was computed from that for the 5 in position, by using the known microphone spacing, loudspeaker postions and speed of sound. Excellent agreement between predicted and measured results will be noted. Often, diffraction from microphone supports causes oscillation of the phase with respect to frequency, owing to lack of symmetry of the microphone arrangement with respect to the source, but the agreement shown in Figure 7 is typical of that found in extensive calibration experiments. Of course, a loudspeaker is not an omni-directional radiator and has marked directivity at high frequencies. Nevertheless, it has the property that (vR - v,/2 - y,“) is close to zero, as for the dipole example described earlier, and is therefore appropriate for calibration. 6.2. A TYPICAL RESULT FOR JET NOISE The power, coherence and phase spectra for an airjet with ambient stagnation temperature and exit Mach number of 0.85 are shown in Figures 56 and 7 to the same scales as the calibration results discussed in section 6.1. The exit diameter of the conical nozzle was 3 in. An instrumentation rake was positioned upstream of the nozzle exit plane and this caused an appreciable increase in spectral density levels at frequencies beyond 10 kHz. Figure 5 shows significant oscillation in the power spectral density due to diffraction from the microphone supports at frequencies in the vicinity of 5 kHz, but this has little effect en the phase or coherence spectra, owing to the symmetry of the microphone support with respect to the source region. It can be seen from Figure 6 that the scale of sources vR - v,/2 - JJ~increases on a wavelength scale as the frequency is increased, while Figure 7 shows that the centroid moves closer to the nozzle exit as frequency increases. In this case, v, and y,’are negligible in relation to vs, since the directivity of the source region is weak at the low jet velocity U, in relation to the atmospheric speed of sound.
4-
33 0.5
I.5
I.0
2.0
2.5
Strouhol number g
Figure 8. Approximate distributions of 7 and fi for jet operating conditions of 0.85 math number and ambient stagnation temperature. -, Centroid position f/d; -, scale of source region vR/d.
Finally, Figure 8 shows the approximate dependence of jj and 6
on jet Strouhal number
fd/U, in the range 0.5 to 2.5, for this jet condition. In this frequency range the instrumentation
rake might be expected to lead to an apparent increase in source scale and a shift of centroid position towards the nozzle exit. A series of experiments to establish the dependence of centroid position on jet pressure ratio and temperature for pure jet noise is described by Kinns [ Ill.
7.
CONCLUSIONS
The phase and coherence spectra of signals from a pair of closely spaced microphones can be used to determine source position and scale. In the paper, it has been shown how such
BINAURALSOURCELOCATION
189
binaural signals? can be analysed to determine the characteristics of a line distribution of arbitrarily correlated omni-directional sound radiators which gives the same acoustic field as a real turbojet engine. The extent of the source region and its directivity have an important influence on the measurements and there is a subtle interplay between the internal and external moments of the source distribution. When a pair of microphones is sufficiently closely spaced, the phase at any frequency is proportional to the position of the source centroid, relative to an origin determined by calibration. The loss of coherence is governed by the root mean square extent of sources about the centroid and by the internal moments of the equivalent source distribution, which govern the directivity of the sound field. When used in conjunction with ordinary far-field measurements, both the internal and external first and second moments of the distribution can be determined. It is these which characterize a source region. It is also shown that these characteristics may depend significantly on the distance from which sources are viewed, through the directivity resulting from self-cancellation. This makes “far-field” measurements so difficult to obtain in practice. However, the apparent centroid position provides an origin for the extrapolation of noise spectral density estimates, made at modest distances from the source region, to the “far-field” and good results are likely to be obtained whenever the centroid is independent of observation position. ACKNOWLEDGMENTS The work was supported by a grant from Rolls-Royce (1971) Limited and was carried out while the author was Maudslay Research Fellow of Pembroke College, Cambridge. The enthusiastic support of Professor J. E. Ffowcs Williams and of the staff of the Noise Department at the Bristol Engine Division of Rolls-Royce (1971) Limited, is gratefully acknowledged. REFERENCES 1. M. J. LIGHTHILL 1952 Proceedings of the Royal Society A 211, 564-587. On sound generated aerodynamically 1. General theory. 2. M. J. LIGHTHILL1954 Proceedings of the Royal Society A 222, l-32. On sound generated aerodynamically 2. Turbulence as a source of sound. 3. J. E. FFOWCS WILLIAMS1963 Philosophical Transactions of the Royal Society A 255, 4699503. The noise from turbulence convected at high speed. 4. M. J. FISHER, P. A. LUSH and M. HARPER BOURNE 1973 Journal of Sound and Vibration 28, 563-585. Jet noise. 5. F. R. GROSCHE 1973 AGARD Conference CP-131 on Noise Mechanisms. Distributions of sound source intensities in subsonic and supersonic jets. 6. R. KINNS and J. BILLINGSLEY 1974 (to be published as an Aeronautical Research Council Current Paper). Application of the Acoustic Telescope to noise measurements on a Viper engine, with
considerations on the theory and development of the Telescope. 7. M. J. FISHERand M. HARPERBOURNE1974 Aeronautical Research Council Report ARC 35 353 N 910. Source location in jet flows. 8. J. E. FFOWCSWILLIAMS1974 AGARD Technical Evaluation Report AR-66. In Noise mechanisms. 9. G. C. CARTER,C. H. KNAPP~~~A. H. NUTTAL1973 ZnstituteofElectronicandElectricalEngineers Transactionson AudioandElectroacoustics AU-21,388. Statisticsoftheestimateofthemagnitudecoherence function. 10. R. KINNS 1973 International Journal for Numerical Methods in Engineering 6, 395-411. Spectral analysis on a small computer. 11. R. KINNS 1975 (to be published in Journal of Sound and Vibration). Experiments using binaural source location. t Note added in proof: The use of two receivers for sound source location has a long history. It is discussed at the end of Lamb’s book Hydrodynamics and the title of this paper reflects the physiological inspiration of early acousticians.
BINAURAL
SOURCE LOCATION R. KINNS ?
Department of Engineering, University of Cambridge, Cambridge CB2 lPZ, England (Received 14 January 1915, and in revisedform 8 July 1975)
It is shown how the coherence and phase spectra of signals from a closely spaced pair of microphones in the far-field can be used to compute the moments of a line distribution of arbitrarily correlated omni-directional sound radiators. This line source gives a far-field which is equivalent to that of model-scale and full-size turbojet engines in terms of measured power and cross-spectra of microphone signals. The necessary spectra can be computed rapidly on a small digital computer and the simplicity of the technique has meant that experiments could be performed in parallel with the usual far-field noise measurements. In this way, it has been possible to identify important properties of noise generation by turbojet engines at minimum cost and development and application of more sophisticated techniques has been accelerated. It is shown how the apparent properties of the source distribution may depend markedly on distance from the source to the microphones. Interpretation of results is guided by consideration of simple cases. 1. INTRODUCTION The requirement for a rational approach to the design of quiet turbojet engines requires a more detailed understanding of noise generation mechanisms than has hitherto been achieved. Although the theoretical framework provided by Lighthill [l, 21 and Ffowcs Williams [3] has led to a respectable correlation between noise measurements and the theory of aerodynamic sound generation, the need to introduce empirical constants in data reduction procedures could lead to erroneous conclusions about the relative importance of various mechanisms for the generation and propagation of sound. A useful survey of experimental data, and discussion of some of the unresolved anomalies, is given by Fisher, Lush and Harper Bourne [4]. In an effort to obtain a more definite understanding of the phenomena of sound generation associated with aero-engines, considerable effort is being devoted to the development of sound source location procedures. In particular, it is necessary to understand why jet engines generate more noise than would be expected from considerations of model-scale experiments. For instance, silencing policies will depend markedly on whether sound is radiated directly from a jet engine nozzle, or whether it is the result of temperature and velocity fluctuations in the emerging airstream. In addition, the influence of flight on practically important sources must be established. At model-scale, various attempts are being made to use acoustic mirrors, based on an optical analogy (see reference [5], for instance), but these become too unwieldy to be practicable for use on open sites with full-size engines. The use of arrays of microphones appears much more attractive (the theory and application of a 14 microphone “acoustic telescope” has been described by Kinns and Billingsley [6]). Another technique, utilizing the power and cross-spectra of the signals from a polar array of microphones, with respect to a fixed microphone, is being developed by Fisher and Harper Bourne [7]. These methods require corresponding complexity in the signal processing, although they can, in principle. provide detailed descriptions of an equivalent line source. i_Present address: Y-ARD Limited, Charing Cross Tower, Glasgow G2 4PP, Scotland 275
276
R. KINNS
However, the desire to acquire data from a large number of full-size engines and modelscale rigs, for a preliminary survey of the differences between them, led the author to develop a simpler technique using just two microphones. The simplicity of this arrangement has obvious attractions. With only one binuaral measurement and simple processing, the source centroid and scale are determined completely. These are the most fundamental characteristics of the source distribution. Once they are identified, the more active source regions can be probed in greater depth with the more elaborate multichannel techniques. 2. FORMULATION
OF A MATHEMATICAL
MODEL
OF THE SOURCE
REGION
It is known that noise from a turbojet is generated by the mixing of the emergent airstream with the surrounding atmosphere, by the entropy fluctuations arising from unsteady combustion in the engine, by the unsteady aerodynamic forces on turbomachinery blade rows and by a variety of other mechanisms such as the passage of eddies through the shock-cell system of an under-expanded exhaust stream. All these mechanisms may interact. For example, sound generated by the entropy fluctuations may be scattered by the emergent airstream and be diffracted at the nozzle exit plane. It may also trigger aerodynamic instabilities in the exhaust jet. Furthermore, the sound sources will have directivity associated with their motion with the multipole nature of turbulent eddies [l-3 ] and with the refraction and diffraction of sound as it propagates away from the source region. In this complex situation, it is inappropriate to assume any particular mechanism for sound generation in the design of source location experiments. However, there is no unique [8] specification for the origin of a given sound field and it is only possible to think in terms of an equivalent source region which shares the measurable far-field properties of the real sources. Source models are chosen for convenience in a tractable analogy. The chosen mathematical model should allow arbitrary directivity in the far-field and arbitrary correlation between measured microphone signals, with respect to frequency. If an array of microphones lies in a plane containing the axis of a jet engine and the power and cross-spectra of all the microphone signals are available, a convenient mathematical model for the source region is a line array of omni-directional radiators on the engine axis, which can have arbitrary correlation with respect to axial position. It is this model which has been developed for interpretation of signals from the microphone pair and the “acoustic
K
I I I -
__---
Yn
Polar origin ;
I& -Line
of sources
$
fK
Microphone
ot (r,JI)
Microphone
ot (r,8)
Figure 1. Notation for theory of line sources.
277
BINAURAL SOURCE LOCATION
telescope” [6]. If no correlation is allowed, the source region as a whole must be omnidirectional to an observer who is so far from the source region that variations due to the inverse square law can be neglected. In the following analysis, it is assumed that the pressure variations at any microphone are purely acoustic; that is, measurements are made sufficiently far from the source for other pressure fluctuations to be negligible. It is also assumed that sound propagates at constant speed outside the source region, so that atmospheric fluctuations are neglected in the first instance. Consider an equivalent source region which consists of the line of omni-directional radiators shown in Figure 1. Let the strength of these radiators be Q(y, t) with respect to position and time and consider the signals received by microphones at (r, 0) and (r, $) with respect to an origin on the line of sources. The signal received at (r, 0) is given by p(r, 0, t) = J’(l/r,)
-CC
(1)
Q(YI, t- rJc> dy,
while that received at (r,$) is
PG.,$9t) =
7V/r,) Q (~2, t - r,lc)dy,,
12)
where c is the speed of sound. The notation is shown in Figure 1. Thus OD (3) The cross-spectrum of microphone signals is
7Ex [p(r, 8, t) .p(r, II/, t + z)] eeznifrdr
= (1 ir’) C,, &(f, r),
say,
(4)
where Ex denotes expected value, and it is assumed that the source region is statistically stationary. Thus Lv
[Qh,t)Qb, t+ z-
G, ,Af,r) = jJj(rz/r1r2)Ex
c11 e-2nif(r-(rz_rl)lC) e2hf(rl -r2)/c drdy, Im(r2 - rI >I
dy,. 0)
If one now assumes that the microphones are sufficiently far from the source region that variations due to the inverse square law can be ignored, r2/rl r2 M
1,
(6)
so that
co,df,r)
= =
]r{ -m
Ex [Q(yl, t) Q(y2, t+ T’)e-2nifr’e2nif(rl-r2)/Cdzfdyl
m Q,,,,U> SI
eznif(rl-rz)lc
dy,
dy2,
dy2
(7)
(8)
-02
where the cross-spectrum of source components at y1 and y, is
Q,,,,U> = { Ex[Qh,t)Qb2, t+ 41e-2nifr dr.
(9)
By the cosine rule, rl and r2 are givln by @Jr)’ =
1 + (y,/r)2 - 2(y,/r)
cos 0
(10)
278
R. KINNS
and (rJr)2 = 1 + (yz/r>2 - 2(y,/r)cos
+.
(11)
Thus t1 so
-
r2
=
Wl(~l
+
r2)I
MY:
-
~:)PI
-
(Yl
cm
0 -
Y,
cm
$)I,
(12)
that, in the far-field in the vicinity of 0 = II/= n/2, r2
-
rl
=
(rl
cm
f3 -
y2
cm
ti)
+
0:
-
y3Pr)
+
rWy/r)4,
(y/r)’
cm2
(0.
W.
(13)
If the source region consisted of simple sources, such that Q,,,(f) decayed rapidly to zero with increasing 1y1 - y2), without oscillation with respect to (y, - vZ), the second term might be neglected. However, multipole sources consist of a nearly self-cancelling distribution of simple sources and the second term in expression (13) can remain significant up to a few hundred nozzle diameters from a jet source, even if the source region is restricted to ten diameters or so. Furthermore, disturbances in the jet exhaust may generate sound over several ma y remain significant for relatively large (vl - y2). With nozzle diameters, so that Q,,,(f) both terms retained in equation (13), equation (8) can be written as
CO,dL r) = jyQyl ,,(f> e(2ni/~)(Y:-Y:)/2re-(2~i/l)lv,cos
B--Y2COS ti) +,l
dy2,
-00
(14)
where I = c/f is the wavelength of sound at frequency f. If one now defines Q;,
,,V;
r)
=
Q,,
Y,cf)
e(2xill)(Yf-Y$)/2r,
05)
equation (14) can be written as Co, ,Jf, r) = fIQi,
y2(f, r) e-(2ni’A)(ylc0*e-y2cos*) dy, dy,.
(16)
-m As r increases, Q;, y,cf, r) tends to the limit :
,$JQL, y,ti 43 = Qy, y,cf)In practical noise measurements,
“far-field”
observations
are often conducted
(17) with
r/d < 50. Then, QilY,(f,r) may be markedly different from Qy,y2(f). The power spectrum of a microphone signal at (r, $) is given by
(18) This result is derived from expression (16) by setting 6 = $. When 8 = $ = 7112, C ,,.,,,cr;r)=~~Q:,,,ct;r)dy,dy,,
-co
(19)
while C,,,, *cf, r) = rj Q;, ,,cf, r) e(2ni/k)y+os* dy, dy,. -CO
(20)
In principle, the “acoustic telescope” [6] with linear signal processing allows Q,,,,(f) to be estimated to a spatial resolution of about half a wavelength. In the following section, it is shown how signals from closely spaced microphones can be used to estimate properties of the distribution of Qi, y,(J r).
BINAURAL
3. PROPERTIES
SOURCE
OF THE SIGNALS SMALL ANGULAR
LOCATIOA
FROM MICROPHONES SEPARATION
‘79 AT
If one defines
equation
(20) can be written
as
&,f, r) is complex, but its imaginary part has zero internal scale. since 06, ,,(,f: Y) =- Q;.; ,,>!,i. r) where * denotes the complex conjugate. This result also follows from equations (I 9) and ! 101, since power spectral density must be real. If o(y,,f. r) is separated into its real and imaginary parts. Q(y,f, r) = R(y,.f, equation
(22) can be written
r)
+ il(,v.f; r i.
:.I?)
as
(24) If $ is close to 7r/2 and it is assumed that the moments to _v, (24) can be expanded as a power series in k = (27r cos G);;..
of R and i are finite with respect
!,25)
it should be noted that k has dimensions of (length)- ‘, but I\ always associared with one of the length scales which specify the source distribution. Throughout the analysis. the explicit dependence of iT and ion v,fand r is to be understood. To third order in k,
where
1Rdv __“m and
Cc -aI
i dy
= 0.
(28)
_Cis the position of the source centroid, downstream ofthe origin. lt is entirely determmed by 8, that part of the source distribution which contributes to the sound intensity. The imaginary part 1, which is more an indication of source directivity. does not affect r.
280
R. KINNS
The cross-spectrum can be expressed in terms of magnitude and phase, j&, *cf, r), C,D. ti(f, r) = IC.,,, &(f, r)l eiSn’**(f,r),
(29)
so that Bn,n,ecf, r) = k[f - (k/2) vr - (~“/WPR + 3~,v31 (30) to third order in k. The coefficients are defined below. This equation shows how the source centroid is determined very simply from the single measurement of phase, it being k-l times the phase angle for small microphone separation where 0(k2) terms are negligible. m y,= myidy Rdy (31) 1 II-m -m represents the first moment of i divided by the intensity at (r, 7c/2). It is related to the first internal moment of correlated sources, which governs the directivity of the sound field. cc
1(y -y)‘Idy
VI =
-m
J Rdy I --m
(32)
is the second moment of i, divided by the sound intensity. It influences the manner in which the phase departs from a linear relationship with k as k increases. cc VR= j(y+Kdy PWdy (33) -m I -m is the second moment of R divided by the sound intensity. It indicates the extent of the source region. Its magnitude can be determined from the coherence spectrum of signals from microphones at small separation, together with the measured directivity of the sound field. m ,uR
=
(34)
s
-m
indicates the skewness of the distribution of R. It has an important influence on the behaviour of phase with respect to k, as k increases. If a is an even, and I an odd, function of (y -J), jn,2, *(Jr) will be proportional to k = (27ccos1j)/~ for all k. Otherwise, j3n,Z,Jr (f, r) will be proportional to kj for small k and the deviation from linearity will tend to be governed by the skewness of the distribution of i?, since lis self-cancelling (equation (28)) and its even moments tend to be small. 4. PROPERTIES OF THE COHERENCE SPECTRUM FOR SMALL MICROPHONE
SEPARATION
Although &, &Jr) is relatively easy to measure, as it is not dependent on the magnitudes of the microphone signals, accurate estimation of CX,2,*cf, r) for small (7c/2- $) is rendered difficult by its dependence on an accurate knowledge of the sensitivity of microphones and processing equipment. However, the cross-spectrum magnitude may be normalized to give the coherence spectrum, which is independent of gain factors. In addition, its statistical properties have been investigated in detail for Gaussian signals (see reference [9], for instance), and it is as easily computed as the cross-spectrum, by using modern data processing techniques [lo]. The properties of the coherence spectrum for small microphone separations can be deduced by using series expansion, as in the previous section. However, a more elegant deduction of its properties can be obtained via a Taylor series expansion, by using its derivatives with respect to $ at $ = n/2 and this approach is preferred.
BINAURAL
SOURCE LOCATION
181
The magnitude coherence spectrum, bounded by zero and unity, is given by Yn/2,ti(f, r) = ICWZ,ti(f,r)lI[C,,~, n,Z(L r) C,, a(.f,r)]1/2,
(35)
or Its first and second derivatives can be obtained in terms of derivatives of the power and crossspectra by differentiating equation (36) twice, after taking logarithms of both sides. Thus, (37) and a2 7n/2.
.&m21$42
=
w,:2,
+
.,2P(G,2.
u/c,,,,
n/2)
,>iw>wc,*,2,
32fcK,2,
e +
,>/wL~,2
c/2,
J -
c,,
+
,>!a
(38)
+2l$_n,2.
As before, the explicit dependence onfand r is to be understood. The first and second derivatives of C,,,, ti and C,, J, can be obtained as follows : (39) so that
= -(2ni/iJ fly2 Qi,Y2dYldY2.
a~,,~, #,&=n,2
(40)
-m
Similarly (41)
(42)
a2 C,*,,, ti/W21e=n,2
-(47r2ij.‘){j. Y: Q&h
=
dy2,
-co
(43)
(44) m
(45) By using equations (40), (42) and (44), it follows at once from equation (37) that (46)
aYn,2.ti,iW+nlz = 0 and that a2 Yn/2,
d~~21w2
=
(47r’in’) j-j Q;,&
-m
- j=j.Q;,,,dvl
-m
dy, dy, fl Ql:,+ -a
dy, fl Q;,,,y,My, -m
dy, dY2 dy2
flQ;,,,dp, -m
l2 dy,]. (47)
However, equation (47) can be greatly simplified by using definitions (3 1) and (33), together with :
v, = fj- Q:,,,(YI - y,)2duld~,/f -m
-m
Q;,,,dy,h.
(48)
282
R. KINNS
v, is a measure sound intensity Thus,
of the internal second moment of correlated sound sources, divided at (r, 7c/2). It is purely real, though it may have negative sign. a2~a,z, J&&=,,,
Comparison
of equations
= -(~~z~/J~~){v, - v,/2 - J$>,
comparison
(49)
(45) and (48) shows that (a* c,, ,iati*)ic,,,,
Similarly,
by the
of equations
n,Z1JI=n,2= -(4~*in*)
v,.
(50)
(31) and (44) gives the result
(a~,, ,ia*)ic,,,,
n,Z~JI_n,2= 47cY,1/1.
(51)
To second order, ($ - n/2) can be written as (-costi), so that the following relations from the analysis of this section, both being to second order in k = (2ncos$)/1: 1-
77~2,
G, ,(r,f)!C
&>.f)
=
(k2i2)(v,
,712, n,z(rJ->
=
-1 -
vi2
2ky,
-
(52)
~3, W/2>
result
(53)
v,.
Equations (52) and (53) show how the initial behaviour of the coherence with respect to increasing k is governed, not only by the external second moment of the effective source distribution but also by the directivity of the sound field, through the internal moments of the source distribution. If desired, higher order terms can be evaluated, by using the same approach. However, experiments at model-scale and on a full-size engine have shown that microphones can be sufficiently closely spaced for higher order terms to be ignored. A direct check can be obtained by using microphones at more than one spacing and some typical results are given by Kinns [ 111. The meaning of the various internal and external moments is clarified by considering two simple cases. The first is the trivial case of an array of uncorrelated omni-directional radiators, while the second is that of a stationary dipole.
5. ILLUSTRATIVE
SIMPLE
CASES
5.1. THE CASEOF A DISTRIBUTIONOF UNCORRELATEDOMNI-DIRECTIONALRADIATORS A line array of uncorrelated monopoles is actually of very little practical interest. Its distant sound field is totally devoid of directionality and its near-field behaviour is contrary to that of complex multipoles. Nonetheless, the model provides a simple illustration of how the binaural specification of source scales is determined. In conjunction with the following examples, it helps highlight the significance of correlation between components of the line source distribution which imparts directivity to the sound field. When sources are uncorrelated Q;,y2(f;r)=4y,
-~2)f',Jf)
=
Q,,,,cf>.
(54)
and the distribution is unique in its independence of observation distance (apart from terms arising from the inverse square law). 6 is Dirac’s delta function and J’,,,(.f) is an intensity measure. All internal moments of the source distribution are zero (equations (31) and (48)) and the loss of coherence for small k is given by equation (52) with v, = y, = 0. Thus, the source scale, 6, is given directly by the expression
v%=(Wh’2(1 Furthermore,
consideration
of equation
-~n/z.df,rN.
(24) shows that in this trivial
(55) case where 1 is
BINAURAL
283
SOURCE LOCATION
identically zero, the polar cross-spectrum Cn12,@cfr) h as real and imaginary parts which are even and odd functions of (7c/2- $), This is not the case in the examples that follow. 5.2.
ANALYSIS FOR THE CASE OF A DIPOLE
Consider the case of two point sources situated at y = a and y = b. Let these sources have equal intensity Z’(f) and unity coherence at all frequencies and let the source at y = b lag that aty=abyT. Then the cross-spectrum of source components is given by
Q,,,,(f)
= KY, - 48~~
- VU-)
+
d(y, - b)&y,
- 4W)
+
+ StyI - 4 &y2 - b) p(f) e- 2nifT + 6(yl - b) 6(y2 - a)P(f)eZn*fT.
(56)
Thus
Q;, ,,(_Ar) =
W)@(Y,
- 4
W2
-
+ a(y, _ a) &y, &y,
_
b)
a(
4
+
&YI
_ 6) e-27MT
y2
_
a)
e2nifT
- b) @y, - 4 +
e(2WU(a2-b2)/2r
+
e-(2nili)(a2-b212r)},
(57) from equation (15). It follows from equation (16) that the cross-spectrum of signals from microphones at (r, 0) and (r, $) is proportional to C,, *(f, r) = p(f) {e-(2ni/~)(aZC‘35 0--lrCOS ,b) + e-(2nill)(bCos 8-b Co5 J) + +
e-2nifT
e(2n1/A)(.*-b2)/2r
+
e2n,fT
e-(2ni/l)(o~-b~)/2r
e-(2”‘/L)(~
e-(2WMbcos
cos O-b
B-0
cos IJS) +
C‘W)}~ (58)
Therefore C, *(f, r) = 2P(f) e(2ni/l)(a+b/2)(cos~-c0se) {cos (lc/l.(b - a) (cos 0 - cos t,h))+ + cos (cc+ rt/J.(b - a)(cos 6 + cos I,+))},
(59)
where a(f, r) is given by cc(f, r) = 27c[-fr + (a2 - b2)/2rl].
(60)
The dependence of the cross-spectrum on r indicates how the latter can have a marked influence on apparent source distributions through the directivity of individual parts of the source region. It can be seen from equation (59) that the phase PO.Jf,r) is Be,+(.f,r) = (244{(a + b)/2)(cos Ic/- cos o),
(61)
which is the result which would be obtained for an omni-directional point source at y = (a + b)/2 for any $ and 8. It is easily shown by direct analysis that the coherence Ye,&r) is unity, so that the phase and coherence spectra are identical to those for the point source. It is useful to consider the moments 1, y,, vR, vi and v, defined previously, in order to demonstrate their importance for a compact source. From equations (21), (57) and (60), the complex source distribution is &v,f, r) = P(f) (XV - a)11 + vial f S(y - b)[l + eta]}.
(62)
Therefore %~,.f, r) = WMY I(y,f, r) = W){S(y
- a) + S(y - b)}(l + cos a),
(63)
- b) - S(y - a)} sin tl.
(64)
Thus, the centroid position is ~7= (a + b)/2, which is the point midway between the dipole elements.
R. KINNS
284
The internal first moment is y, = {(b - a)/2) tan(a/2) and is seen to give a measure of the separation of the dipole elements. The external second moment is vR = {(b - ~r)/2}~,while vi = 0, since I is antisymmetric about jj. The internal second moment is v, = 2{(b - a)/2}‘{ 1 - tan2(a/2)}, which is a third measure of the separation between elements. It can be seen that the sign of v, depends on ~1,but that (va - v,/2 - y,Z)is identically zero. It will be noted that CIin equation (60) is dependent on the polar radius, r, while j is always the point midway between the dipole elements. In principle, yj? and v, can be evaluated from the first and second derivatives of the power spectral density of a microphone signal with respect to $, but the need for evaluation of microphone sensitivities hinders exact measurement. The above results indicate how v, and yf can both be larger than vRfor a multipole source. This is a simple example of a case in which i can be much larger than i? and where the directivity of the sound field is critical, even where correlated sources are not widely separated so that it is tempting to discard the second term in equation (13). In order to illustrate this, consider the case of a dipole whose elements are separated by A/4 and where one element emits at time 31/8c after the other. It is easily shown that the far-field is given by c,, JI= (l/2)0 + cos [(n/2)(3/2 -
and c,.
COST
(6%
tbis shown as a function of $ in Figure 2.
Figure 2. Theoretical other by 31i/8c.
field shape for a dipole with elements separated by A/4 where one element lags the
It can also be shown that if the dipole elements are positioned at y. - A/8 and y. + A/8 from the assumed origin and observations of the source are made at distance r, then the magnitudes of the delta functions representing R and I are proportional to : 8, = 1 + cos [(n/2)(3/2 + ye/r)]
(66)
II= sin [(n/2)(3/2 + yo/r>l.
(67)
and w, and 1, are shown in Figure 3 for -0.2 < y,/r < 0.2. In this case, the directivity of the source leads to significant variation in apparent source distribution with respect to observation distance. The variation in R reflects the change in intensity of the sound pressure level
BINAURAL
SOURCE LOCATION
185
Figure 3. Variation of the magnitudes of the impulse functions at yO - A/8 and yO+ 118, which represent the complex source distribution of the dipole whose field shape is shown in Figure 2, with respect to dipole
position and observationdistance. as the source is moved, while the variation in i reflects the change in the local directivity with respect to the assumed origin. It is only necessary for a source region to have substantial directivity for large variations in apparent source distributions with respect to observation distance to occur. All the moments of the apparent source distribution may be affected and the difficulty of making “far-field” noise measurements in a situation where there are substantial variations in directivity with respect to source position will be obvious. However, the apparent centroid position is clearly useful in providing an origin for the extrapolation of noise spectral density measurements, made at modest distances from the source region, to the far-field. In this example, such extrapolation would be ideal, because the centroid is independent of far-field observation position. 6. CALIBRATION AND A TYPICAL RESULT Detailed considerations of experimental technique and interpretation of results are given by Kinns [l 11.The intention here is to illustrate the method of calibration of a microphone pair with a practical example and to show a typical result for jet noise. 6.1.
CALIBRATION
OF A MICROPHONE
PAIR
It is clearly unrealistic to measure polar radius r to very high accuracy, but it is relatively easy to measure the distance between a pair of microphones which are rigidly mounted on the same support. Thus, the angular separation of microphones can be accurately determined, but small differences in polar radius must be expected. If the microphones are positioned at (r, 6) and (r + E,I/I), where 6 z $ z n/2 and (t/r(cos$ - cos@l < 1, then
rz - rl = E + 1~~cos 0 - y, cos $ + (yz - yf)/2r,
(68)
to the order of accuracy of equation (13). Equation (16) becomes co
Co. &f, r) = e-(znia’A)
II Q:, ,,(_A r) e-m
(hi /l)(Y,coss-y,Cos~‘)d,~~ dy,.
(69)
Thus, the influence of the small error Ein the polar radius is just to introduce a phase factor in the cross-spectrum. It follows that providing a microphone pair is calibrated by using a known source, the small error E can be evaluated without difficulty. In order to demonstrate the excellent agreement between predicted and measured results when using a known source, and the method for eliminating E, consider the experimental arrangement shown in Figure 4. In this case, $ in microphones were positioned at 90” and
286
R. KINNS
Grourdcoveredby acousticfoam
wedges
Axis of jet
(a)
(b)
Figure 4. Experimental
configuration.
(a) Top view; (b) side view.
89.5” with respect to the exit of a nozzle and its axis of symmetry, for tests on jet noise. They were mounted 12 ft 7 in from the nozzle and the microphone arrangement was approximately symmetric with respect to its support stand and the nozzle exit. The spacing of microphones was only 1.32 in and was much smaller than that normally used for either model-scale or full-size engine studies. Experiments were conducted with acoustic foam wedges on the ground, in flat-calm conditions. A Tannoy loudspeaker with 7 in exit diameter was positioned 5 in and 27 in downstream of the nozzle exit and was driven by pink noise in each case.
T IOdB
1
0
I
I
I.6
3.2 Freqwncy
I
46
I
e
6.4
(kHr)
Figure 5. Power spectra of signals from microphone at B = 90” for the loudspeaker calibration and for an experiment on jet noise. -, Power spectral density for loudspeaker in first position (Figure 4); -, power spectral density for jet noise. Jet Mach number 0.85 with ambient stagnation temperature at exit from a 3 in conical nozzle.
BINAURAL SOURCE LOCATION
287
Signals from the microphones were recorded on a Racal Store-4 F.M. tape recorder at 60 in/s. Tapes were replayed at 15/ 16 in/s for analysis on an IBM I 130 computer [lo] to give the required power, coherence and phase spectra in the frequency range O-16 kHz with about 77 Hz resolution. Results were obtained from 3.0 second data samples. Slight misalignment of tape recorder heads and differences in electronics between channels have the same effect as the spatial error E, but it is unnecessary to evaluate these separately if the same analysis procedure is used for calibration and jet noise signals.
O~Sl
0
I
I
I
I.6
3.2
4.6
Frequency
(kHz)
/
6.4
IU 8.0
Figure 6. Magnitude coherence spectra for loudspeaker calibration and jet noise. -, Coherence spectrum for loudspeaker calibration; -, coherence spectrum for jet noise. Conditions given in caption to Figure 5. 0 = 90”; v/ = 89.5.
The power spectrum of one of the microphone signals and the coherence spectrum for the first loudspeaker position are shown in Figures 5 and 6. The phase spectra for both loudspeaker positions are shown in Figure 7. It can be seen from Figure 5 that the loudspeaker had a highly non-uniform gain with respect to frequency and that the variation in spectral density has a pronounced effect on the coherence magnitude (Figure 6). These variations are easily explained in terms of extraneous noise from a variety of sources and statistical errors [1 11, but the phase spectra show near perfect linearity with respect to frequency in regions where the coherence is close to unity. It should be noted that the power spectral density at frequencies above about 2 kHz was much lower than in studies of jet noise and results are therefore more susceptible to electronic noise. It is for this reason that spectra are not shown for frequencies greater than 8 kHz, while estimates below 800 Hz are not shown because they are appreciably influenced by high-pass filters. The effective value of E can be deduced from the phase spectrum for the first loudspeaker position. The dotted line extension of the phase spectrum for the 27 in loudspeaker location
A
Phase spectrum ---pfedlcted frov
ISiresult
Phase spectrum for I*’ loudspeaker pcxit~o~, 3.2
Frequency
4.8
(kHz)
Figure 7. Phase spectra for two loudspeaker positions and for jet noise. -, Phase spectra for two loudspeaker positions; ----, phase spectrum for second position predicted from result for first position; -, phase spectrum forjet noise. Conditions given in caption to Figure 5.
288
R. KINNS
was computed from that for the 5 in position, by using the known microphone spacing, loudspeaker postions and speed of sound. Excellent agreement between predicted and measured results will be noted. Often, diffraction from microphone supports causes oscillation of the phase with respect to frequency, owing to lack of symmetry of the microphone arrangement with respect to the source, but the agreement shown in Figure 7 is typical of that found in extensive calibration experiments. Of course, a loudspeaker is not an omni-directional radiator and has marked directivity at high frequencies. Nevertheless, it has the property that (vR - v,/2 - y,“) is close to zero, as for the dipole example described earlier, and is therefore appropriate for calibration. 6.2. A TYPICAL RESULT FOR JET NOISE The power, coherence and phase spectra for an airjet with ambient stagnation temperature and exit Mach number of 0.85 are shown in Figures 56 and 7 to the same scales as the calibration results discussed in section 6.1. The exit diameter of the conical nozzle was 3 in. An instrumentation rake was positioned upstream of the nozzle exit plane and this caused an appreciable increase in spectral density levels at frequencies beyond 10 kHz. Figure 5 shows significant oscillation in the power spectral density due to diffraction from the microphone supports at frequencies in the vicinity of 5 kHz, but this has little effect en the phase or coherence spectra, owing to the symmetry of the microphone support with respect to the source region. It can be seen from Figure 6 that the scale of sources vR - v,/2 - JJ~increases on a wavelength scale as the frequency is increased, while Figure 7 shows that the centroid moves closer to the nozzle exit as frequency increases. In this case, v, and y,’are negligible in relation to vs, since the directivity of the source region is weak at the low jet velocity U, in relation to the atmospheric speed of sound.
4-
33 0.5
I.5
I.0
2.0
2.5
Strouhol number g
Figure 8. Approximate distributions of 7 and fi for jet operating conditions of 0.85 math number and ambient stagnation temperature. -, Centroid position f/d; -, scale of source region vR/d.
Finally, Figure 8 shows the approximate dependence of jj and 6
on jet Strouhal number
fd/U, in the range 0.5 to 2.5, for this jet condition. In this frequency range the instrumentation
rake might be expected to lead to an apparent increase in source scale and a shift of centroid position towards the nozzle exit. A series of experiments to establish the dependence of centroid position on jet pressure ratio and temperature for pure jet noise is described by Kinns [ Ill.
7.
CONCLUSIONS
The phase and coherence spectra of signals from a pair of closely spaced microphones can be used to determine source position and scale. In the paper, it has been shown how such
BINAURALSOURCELOCATION
189
binaural signals? can be analysed to determine the characteristics of a line distribution of arbitrarily correlated omni-directional sound radiators which gives the same acoustic field as a real turbojet engine. The extent of the source region and its directivity have an important influence on the measurements and there is a subtle interplay between the internal and external moments of the source distribution. When a pair of microphones is sufficiently closely spaced, the phase at any frequency is proportional to the position of the source centroid, relative to an origin determined by calibration. The loss of coherence is governed by the root mean square extent of sources about the centroid and by the internal moments of the equivalent source distribution, which govern the directivity of the sound field. When used in conjunction with ordinary far-field measurements, both the internal and external first and second moments of the distribution can be determined. It is these which characterize a source region. It is also shown that these characteristics may depend significantly on the distance from which sources are viewed, through the directivity resulting from self-cancellation. This makes “far-field” measurements so difficult to obtain in practice. However, the apparent centroid position provides an origin for the extrapolation of noise spectral density estimates, made at modest distances from the source region, to the “far-field” and good results are likely to be obtained whenever the centroid is independent of observation position. ACKNOWLEDGMENTS The work was supported by a grant from Rolls-Royce (1971) Limited and was carried out while the author was Maudslay Research Fellow of Pembroke College, Cambridge. The enthusiastic support of Professor J. E. Ffowcs Williams and of the staff of the Noise Department at the Bristol Engine Division of Rolls-Royce (1971) Limited, is gratefully acknowledged. REFERENCES 1. M. J. LIGHTHILL 1952 Proceedings of the Royal Society A 211, 564-587. On sound generated aerodynamically 1. General theory. 2. M. J. LIGHTHILL1954 Proceedings of the Royal Society A 222, l-32. On sound generated aerodynamically 2. Turbulence as a source of sound. 3. J. E. FFOWCS WILLIAMS1963 Philosophical Transactions of the Royal Society A 255, 4699503. The noise from turbulence convected at high speed. 4. M. J. FISHER, P. A. LUSH and M. HARPER BOURNE 1973 Journal of Sound and Vibration 28, 563-585. Jet noise. 5. F. R. GROSCHE 1973 AGARD Conference CP-131 on Noise Mechanisms. Distributions of sound source intensities in subsonic and supersonic jets. 6. R. KINNS and J. BILLINGSLEY 1974 (to be published as an Aeronautical Research Council Current Paper). Application of the Acoustic Telescope to noise measurements on a Viper engine, with
considerations on the theory and development of the Telescope. 7. M. J. FISHERand M. HARPERBOURNE1974 Aeronautical Research Council Report ARC 35 353 N 910. Source location in jet flows. 8. J. E. FFOWCSWILLIAMS1974 AGARD Technical Evaluation Report AR-66. In Noise mechanisms. 9. G. C. CARTER,C. H. KNAPP~~~A. H. NUTTAL1973 ZnstituteofElectronicandElectricalEngineers Transactionson AudioandElectroacoustics AU-21,388. Statisticsoftheestimateofthemagnitudecoherence function. 10. R. KINNS 1973 International Journal for Numerical Methods in Engineering 6, 395-411. Spectral analysis on a small computer. 11. R. KINNS 1975 (to be published in Journal of Sound and Vibration). Experiments using binaural source location. t Note added in proof: The use of two receivers for sound source location has a long history. It is discussed at the end of Lamb’s book Hydrodynamics and the title of this paper reflects the physiological inspiration of early acousticians.