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A note on instantaneous and time-averaged active and reactive sound intensity
Journal
ofSound and
A NOTE
Vibrafion (1991) 147(3), 489-496
ON
ACTIVE
INSTANTANEOUS AND
REACTIVE F.
AND SOUND
TIME-AVERAGED INTENSITY
JACOBSEN
The Acoustics Laboratory, Technical University of Denmark, Building 352, DK-2800 Lyngby, Denmark (Received 28 March 1990, and accepted 24 July 1990)
An overview is presented of the concepts and practical utilities of various “sound intensities”, related to the instantaneous flow of acoustical energy density as observed at a given location. The natures of these various quantities are illustrated and discussed. A principal conclusion is that although the time-averaged sound intensities to date have been found to have the greater usefulness in practice, insight can be gained into the physical significance of these by consideration of their instantaneous counterparts. 1. INTRODUCTION
Sound intensity measurement systems have now been on the market for about ten years, and today the sound intensity technique is a well-established supplement to more conventional methods. The principal quantity provided by this technique is the time-averaged active sound intensity. However, a number of other related field quantities have been proposed: e.g., the complex intensity [ 1,2], the reactive intensity [3,4], the instantaneous or time dependent intensity [5,6], the active and reactive instantaneous intensity components [7], “the semi-analytic intensity” [8], and “the complex instantaneous intensity” [9]. In view of the rapid development in this field it is not surprising that the terminology is sometimes ambiguous. The purpose of this note is to present an overview and to illustrate and discuss some of the various field quantities suggested in the literature. 2.
TIME-AVERAGED ACTIVE AND REACTIVE INTENSITY
It would seem that the concept of instantaneous intensity was more fundamental than the concept of time-averaged active and reactive intensity, but actually the time-averaged quantities are simpler to understand and rather more important. Obviously, time-averaged quantities are only meaningful in time-stationary sound fields. They are usually determined in frequency bands. The time-averaged active sound intensity is a vector defined as I=p(t)u(t).
(1)
The
usefulness of this quantity follows from the fact that it represents the time-averaged flow of sound energyt, or sound power flux density, from which a well-known fundamental
relation
can be derived, P, =
1. dS:
(2)
that is, the time-averaged net sound power generated within a given volume is identical with the integral over the enclosing surface of the normal component of the time-averaged t The
common
expression
“power flow” is a contradiction
in terms: power is flow of energy.
489 0022-460>
@ 1991 Academic
Press Limited
490
F. JACOBSEN
active intensity. The extraordinary interest in the sound intensity technique over the past decade is, of course, due to this relation. It is interesting, if less fundamentally important, that the time-averaged active intensity is proportional to the phase gradient of the sound pressure [lo]: in a pure-tone sound field one has
I = -(p*Wpc)(Vdk).
(3)
This leads to the conclusion that surfaces of constant phase (or “wavefronts”) at any position are orthogonal to the time-averaged active intensity [ 5,6]. However, the concept of such surfaces is only meaningful at discrete frequencies in coherent sound fields [ 111. The time-averaged reactive sound intensity? is a vector defined as
J=$(tMt),
(4)
where the symbol Adenotes the Hilbert transform (see, e.g., reference [ 121). This quantity is usually introduced as the imaginary part of the complex sound intensity I,, which, written in the usual complex notation, is defined as ’
1,=pu*:
(5)
that is, J = Im {pu*}.
(6)
However, complex notation is valid only for pure-tone sound fields. Equation (4) is valid in any stationary sound field. The time-averaged reactive intensity is less fundamental than the time-averaged active intensity, but the quantity is nevertheless quite useful: it indicates the presence of a reactive sound field, where the sound pressure and the particle velocity are in quadrature. Because of this property the normalized time-averaged reactive intensity is useful as a “field indicator” [13, 141. It is interesting that J is proportional to the gradient of the mean square pressure [15]; in a pure-tone sound field J = -V( p’(t))/(2pck). This means that surfaces of equal sound pressure are orthogonal the time-averaged reactive intensity [5,6]. 3. INSTANTANEOUS The instantaneous
(7) to the trajectories
of
INTENSITY
intensity is defined as I(r) = p(tMt),
(8)
and this quantity is meaningful also in a sound field generated by a transient source. Indeed, it is the physically fundamental sound intensity, as it represents the rate and direction of the total mechanical energy, kinetic plus potential, passing through any given point, per unit area. In a pure-tone sound field where the sound pressure at a given point is P(l) = POcm wof,
(9)
the particle velocity at the same point may, without loss of generality, be written as u(t) = u, cos war + u, sin w,r.
(10)
but it is apparently defined as a time t Usually the quantity J is simply termed “the reactive intensity”, average (it can also be defined in terms of the cross-spectrum between p and II [ 111). However, it is evidently a source of confusion that “the instantaneous reactive intensity” has zero mean.
PHYSICAL
AND
OTHER
SOUND
INTENSITIES
491
(It may readily be shown that u, is proportional to the phase gradient of the pressure, and that II, is proportional to the gradient of the mean square pressure [lo].) It now follows that I(t) = poua cos’ w,t +pOu, cos wOtsin o,t,
(11)
and since I = ;‘Doua,
(12,13)
J=;p+r,
it can be seen that I(t) = I( 1 + cos 2wot) + J sin 2~~2,
! 14)
which may be written in the form [7] I(t) = Re {I,(1 +e-jzwO’)}.
(15)
It can be concluded that the instantaneous intensity in a pure-tone jeld may be divided into two components, one associated with the active intensity and one associated with the reactive intensity [5-71. The time average of I(t) is I, of course, and the “reactive part” of the instantaneous intensity, i.e., the second term of the right side of equation (14), has zero mean. In the general case the sound field cannot be assumed to be monochromatic. Equation (8) is of general validity, of course, but since one cannot make an instantaneous separation of the particle velocity into components in phase and in quadrature with the pressure, as in equation (lo), there is no obvious, general analogue to equation (14). In fact, because of cross terms one cannot even decompose the instantaneous intensity in a sound field with two discrete frequency components-at least not in the simple manner suggested by equation (14). Nevertheless, several proposals can be found in the literature. Pettersen [16] has suggested the expression (16)
I,.(r) =p(t)(u(r)-j&(r)), while Uosukainen
[S] has suggested a similar expression, (17)
C(t) = (p(t)+j$(r))u(t),
termed “the semi-analytic intensity”. Yet another expression has been proposed by Heyser [91: I~(t)=~(p(r)+j~(t))(u(t)-j;(r)). This quantity is termed “the complex instantaneous p(tMt)
=p(tMt)
and
(18)
intensity”. Since
p(t)fi(t)
= -p^(tMr)
(19,X))
in a stationary sound field [12], it can be seen that the three expressions have the same time average: I,(t)=I:.(t)=I:‘(t)=I+jJ=I,. (21) It is also apparent that Re {I(t)} = Re {I:(r)}=I(r).
(22)
Moreover, when the pure-tone expressions (9) and (10) are inserted, equation (16) becomes I,.(t) = pOcos w,t(u, cos w,,t + u, sin w,,f - juu sin wOt+ ju, cos wof) = I,.( 1 + e-i’%‘),
(23)
492
F. JACOBSEN
which seems to be a natural extension of equation (15). Nevertheless, expressions the one suggested by Heyser is probably the most useful. It is well known that the signal r(t)=JxZ(t)+$2(t)
of the three
(24)
is the envelope of x( 1). A complex envelope w(t) can also be defined such that the analytic signal x(t) + j;( t) can be written in the form x(t)+jG(t)
= w(t) e+J;
(25)
see, e.g., reference [12]. Clearly, r(t) = Iw(t)l.
(26)
It can now be seen that Heyser’s expression may be regarded as a generalized version of the pure-tone expression for the complex intensity (5), and it is a simple matter to show from equations (9), (10) and (18) that I:(r)=I, in a pure-tone sound field. In a narrow-band
(27)
sound field
I”(r)=Re{I~(r)}=~(p(r)u(r)+p^(r)G(r))
(28)
J”(r)=Im{I~(r)}=~(p*(r)u(r)-p(r)Q(r))
(29)
and
are low-pass signals which represent running short-time average values of, respectively, the active and the reactive intensity components p( r)u( r) and p^(r)u( r). 4. DISCUSSION The decomposition of the instantaneous intensity implied by equation (14) is interesting, since it shows that the actual flow of sound energy is much more complicated than suggested by the “energy streamlines” [ 171 (which follow the active intensity), as pointed out by Mann er al. [5,6] and by Uosukainen [8] (in the general case I and J have different directions). However, this decomposition is valid only in a pure-tone sound field and it can easily lead to erroneous conclusions. For example, one might well get the impression from equation (14) that the reactive intensity, which apparently is an amplitude [7], has no sign, or that the sign depends on the arbitrary time base. However, although the sign of the time-averaged reactive intensity is a mere convention it is certainly not arbitrary, as can be seen from equation (7). The reactive sound intensity has a definite physical significance; it points out of a source, for example [13, 181. One might also be led to the misapprehension that reactive intensity components associated with different frequency components do not simply add vectorially. However, they do indeed, as can easily be shown from equation (4). It follows that two reactive components of nearly the same magnitude and opposite sign lead to a small net reactive component. It can therefore be concluded that one cannot infer from a small wide-band indication of the reactive intensity that no oscillatory energy flux exists: this is the expense of analyzing in wide bands. (One cannot infer the absence of flow of sound energy from an insignificant band indication of the active intensity either.) These considerations can be illustrated by an example. Practically all the acoustic energy in a reverberant room driven with random noise is oscillatory, but the sound field is not reactive [ll]. In fact, contrary to usual belief, the sound field is no more reactive than active, as shown in a recent paper [ 191.
PHYSICAL
AND
OTHER
SOUND
INTENSITIES
493
Although the complex time dependent expressions (16), (17) and (18) seem to be of general validity, they are useful only for a sound field generated by a narrow-band source; time-domain representations of wide-band signals are of little use. The complex expressions (16) and (17), which are equivalent, can perhaps be justified with reference to equations (21) and (22), but the real and imaginary parts should not be interpreted as the instantaneous active and reactive components: a consistent definition would give the two terms of the right side of equation (14) in the single frequency case. Heyser’s expression is interesting since the two low-pass signals I”(f) and J”(t) illustrate the instantaneous transport of sound energy. It may have a potential in analyzing non-stationary sound fields. One might “re-modulate” the two signals by analogy with equation (15): the resulting signals would actually represent the instantaneous active and reactive components of the intensity. However, this would require determining the instantaneous frequency (this problem has been discussed by Uosukainen [8]), and the smooth signals are probably more useful. 5. EXPERIMENTAL
RESULTS
To illustrate some of the quantities described in section 3 a few experiments have been carried out. The sound pressure and a component of the particle velocity were measured with a sound intensity probe, Briiel & Kjaer (B & K) 35 19, in combination with an intensity analyzer, B 8z K4433, in various sound fields. Analogue signals proportional to sound pressure and particle velocity from the intensity analyzer were recorded and processed with a dual channel FFT analyzer, B & K 2032, in combination with a small computer, Hewlett Packard 9817. The FFT analyzer can produce Hilbert-transformed time signals; the computer was used in determining the instantaneous intensity I,(t) and the real and imaginary parts of Heyser’s complex instantaneous intensity; that is, the quantities I:‘(t) and J:(r) (the suffix r indicates a vector component in the r-direction). In Figure 1 is shown the result of a measurement about 30 cm from a small loudspeaker mounted in an enclosed cabinet and driven with noise in a one-third octave band with a centre frequency of 1 kHz. The result of a similar measurement very near the loudspeaker cone is shown in Figure 2; in this case the loudspeaker was driven with noise in a one-third octave band with a centre frequency of 250 Hz. Both measurements took place far from reflecting surfaces in a large, strongly damped room. In the first measurement the sound pressure and the normalized particle velocity (pcu,( 1)) are practically identical, and as a result the instantaneous intensity is always positive: this is an active sound field. By contrast, the sound pressure and the particle velocity are almost in quadrature in the near field measurement, and the normalized particle velocity is, as expected, larger than the pressure: this is a simple example of a strongly reactive sound field. Both measurements demonstrate the significance of the envelope signals I:(t) and J:‘(t). The next measurement was also made in the large, strongly damped room. This time the intensity probe was placed about 30 cm from one of the 3 mm sideplates of a steel box driven with noise in a one-third octave band with a centre frequency of 250 Hz by an electrodynamic exciter. This is a much more complicated sound field, partly active and partly reactive, and it is apparent that both the active and the reactive component can take negative values. It is not surprising, therefore, that the random errors associated with the use of a finite averaging time in intensity measurements can be very large in such circumstances [20]. In the measurement shown in Figure 4 the loudspeaker was placed in a large reverberation room with a reverberation time of about 5 s, and the intensity probe was placed at a random position far from the loudspeaker, which was driven with noise in a one-third
494
F. JACOBSEN
,:’
~
-&
_______.---
.Z?=======
E
‘.x.,_ ‘.__’
0
h+polah
apgdod
2
,.1’
______.-a-
cc=======
puoamssaJd
“d
c
PHYSICAL
AND
OTHER
SOUND
INTENSITIES
\ \., .,’ ,:’
._i..
c---.-;,/ \“.., /’
F. JACOBSEN
496
octave band with a centre frequency of 500 Hz. In this case the sound pressure exceeds the normalized particle velocity [21], and the signals are very nearly uncorrelated [22]. The active and reactive intensity components fluctuate about zero: that is, the sound field is neither active nor reactive [19]. 6. CONCLUDING
REMARKS
Various instantaneous field quantities associated with sound intensity have been presented and demonstrated. Whereas the usefulness of the corresponding time-averaged quantities is fairly well established, the practical importance of the instantaneous quantities is less obvious. However, they demonstrate instructively the physical significance of the time-averaged quantities. REFERENCES 1. T. K. STANTON and R. T. BEVER 1979 Journal of the Acoustical Society of America 65,249-252.
Complex wattmeter measurements in a reactive acoustic field. 2. G. W. ELKO 1984 Ph.D. Thesis, Pennsylvania State University. Frequency domain estimation of the complex acoustic intensity and acoustic energy density. 3. J.-C. PASCAL 1981 Proceedings of the International Congress on Recent Development in Acoustic Intensity Measurements, 11-19. Mesure de l’intensit6 active et riactive dans differents champs acoustiques. 4. S. GADE 1982 Btief & Kjaer Technical Review 3, 3-39. Sound intensity (Part 1, Theory). 5. J. A. MANN III, J. TICHY and A. J. ROMANO 1987 Journal ofthe Acoustical Society ofAmerica 82, 17-30. Instantaneous and time-averaged energy transfer in acoustic fields. 6. J. A. MANN III 1988 Ph.D. Thesis, Pennsylvania State Uniuersity. Acoustic intensity: energy transfer, wave properties, and applications. 7. F. J. FAHY 1989 Sound Intensity. London: Elsevier Applied Science. See chapter 4. 8. S. UOSUKAINEN 1989 Ph.D. Thesis, Helsinki University of Technology, VTT Research Report No. 656. Properties of acoustic energy quantities. 9. R. C. HEYSER 1986 AES 81st Convention, Audio Engineering Society, Preprint 2399. Instantaneous intensity. 10. U. KURZE 1968 Acustica 20,308-310. Zur Entwicklung einges Geriites fiir komplexe Schallfeldmessungen. 11. F. JACOBSEN 1989 Journal of Sound and Vibration 130,493-507.Active and reactive, coherent and incoherent sound fields. 12. A. PAPOULIS 1984 Probability, Random Variables, and Stochastic Processes. New York: McGrawHill, second edition. See section 11-l. 13. J. A. MANN III and J. TICHY 1988Proceedings of Inter-Noise 88, 105-110. Extended intensity measurement technique for sound field identification in the noise source nearfield. 14. F. JACOBSEN 1990 Noise Control Engineering Journal 35,37-46. Sound field indicators: Useful Tools. 15. J.-C. PASCAL and C. CARLES 1982 Journal of Sound and Vibration 83, 53-65. Systematic measurement errors with two microphone sound intensity meters. 16. 0. K. 0. PETERSEN 1980 Ph.D. 7%esis, NTH Trondheim. MHling og anvendelse av akustisk intensitet for beskrivelse av avstriling og effekttransmisjon i kompliserte akustiske felt. 17. R. WATERHOUSE and D. FEIT 1986 Journal of the Acoustical Society of America 80,681-684. Equal-energy streamlines. 18. F. P. MECHEL 1988 Journal of Sound and Vibration 123, 537-572. Notes on the radiation impedance, especially of piston-like radiators. 19. F. JACOBSEN 1990Journal of Sound and Vibration 143, 231-240. Active and reactive sound intensity in a reverberant sound field. 20. F. JACOBSEN 1989 Journal of Sound and Vibration 131, 475-487. Random errors in sound power determination based on intensity measurement. 21. F. JACOBSEN 1979 The Acoustics Laboratory, Technical University of Denmark, Report No. 27. The diffuse sound field. 22. F. JACOBSEN and T. G. NIELSEN 1987Journal of Sound and Vibration 118, 175-180. Spatial correlation and coherence in a reverberant sound field.
ofSound and
A NOTE
Vibrafion (1991) 147(3), 489-496
ON
ACTIVE
INSTANTANEOUS AND
REACTIVE F.
AND SOUND
TIME-AVERAGED INTENSITY
JACOBSEN
The Acoustics Laboratory, Technical University of Denmark, Building 352, DK-2800 Lyngby, Denmark (Received 28 March 1990, and accepted 24 July 1990)
An overview is presented of the concepts and practical utilities of various “sound intensities”, related to the instantaneous flow of acoustical energy density as observed at a given location. The natures of these various quantities are illustrated and discussed. A principal conclusion is that although the time-averaged sound intensities to date have been found to have the greater usefulness in practice, insight can be gained into the physical significance of these by consideration of their instantaneous counterparts. 1. INTRODUCTION
Sound intensity measurement systems have now been on the market for about ten years, and today the sound intensity technique is a well-established supplement to more conventional methods. The principal quantity provided by this technique is the time-averaged active sound intensity. However, a number of other related field quantities have been proposed: e.g., the complex intensity [ 1,2], the reactive intensity [3,4], the instantaneous or time dependent intensity [5,6], the active and reactive instantaneous intensity components [7], “the semi-analytic intensity” [8], and “the complex instantaneous intensity” [9]. In view of the rapid development in this field it is not surprising that the terminology is sometimes ambiguous. The purpose of this note is to present an overview and to illustrate and discuss some of the various field quantities suggested in the literature. 2.
TIME-AVERAGED ACTIVE AND REACTIVE INTENSITY
It would seem that the concept of instantaneous intensity was more fundamental than the concept of time-averaged active and reactive intensity, but actually the time-averaged quantities are simpler to understand and rather more important. Obviously, time-averaged quantities are only meaningful in time-stationary sound fields. They are usually determined in frequency bands. The time-averaged active sound intensity is a vector defined as I=p(t)u(t).
(1)
The
usefulness of this quantity follows from the fact that it represents the time-averaged flow of sound energyt, or sound power flux density, from which a well-known fundamental
relation
can be derived, P, =
1. dS:
(2)
that is, the time-averaged net sound power generated within a given volume is identical with the integral over the enclosing surface of the normal component of the time-averaged t The
common
expression
“power flow” is a contradiction
in terms: power is flow of energy.
489 0022-460>
@ 1991 Academic
Press Limited
490
F. JACOBSEN
active intensity. The extraordinary interest in the sound intensity technique over the past decade is, of course, due to this relation. It is interesting, if less fundamentally important, that the time-averaged active intensity is proportional to the phase gradient of the sound pressure [lo]: in a pure-tone sound field one has
I = -(p*Wpc)(Vdk).
(3)
This leads to the conclusion that surfaces of constant phase (or “wavefronts”) at any position are orthogonal to the time-averaged active intensity [ 5,6]. However, the concept of such surfaces is only meaningful at discrete frequencies in coherent sound fields [ 111. The time-averaged reactive sound intensity? is a vector defined as
J=$(tMt),
(4)
where the symbol Adenotes the Hilbert transform (see, e.g., reference [ 121). This quantity is usually introduced as the imaginary part of the complex sound intensity I,, which, written in the usual complex notation, is defined as ’
1,=pu*:
(5)
that is, J = Im {pu*}.
(6)
However, complex notation is valid only for pure-tone sound fields. Equation (4) is valid in any stationary sound field. The time-averaged reactive intensity is less fundamental than the time-averaged active intensity, but the quantity is nevertheless quite useful: it indicates the presence of a reactive sound field, where the sound pressure and the particle velocity are in quadrature. Because of this property the normalized time-averaged reactive intensity is useful as a “field indicator” [13, 141. It is interesting that J is proportional to the gradient of the mean square pressure [15]; in a pure-tone sound field J = -V( p’(t))/(2pck). This means that surfaces of equal sound pressure are orthogonal the time-averaged reactive intensity [5,6]. 3. INSTANTANEOUS The instantaneous
(7) to the trajectories
of
INTENSITY
intensity is defined as I(r) = p(tMt),
(8)
and this quantity is meaningful also in a sound field generated by a transient source. Indeed, it is the physically fundamental sound intensity, as it represents the rate and direction of the total mechanical energy, kinetic plus potential, passing through any given point, per unit area. In a pure-tone sound field where the sound pressure at a given point is P(l) = POcm wof,
(9)
the particle velocity at the same point may, without loss of generality, be written as u(t) = u, cos war + u, sin w,r.
(10)
but it is apparently defined as a time t Usually the quantity J is simply termed “the reactive intensity”, average (it can also be defined in terms of the cross-spectrum between p and II [ 111). However, it is evidently a source of confusion that “the instantaneous reactive intensity” has zero mean.
PHYSICAL
AND
OTHER
SOUND
INTENSITIES
491
(It may readily be shown that u, is proportional to the phase gradient of the pressure, and that II, is proportional to the gradient of the mean square pressure [lo].) It now follows that I(t) = poua cos’ w,t +pOu, cos wOtsin o,t,
(11)
and since I = ;‘Doua,
(12,13)
J=;p+r,
it can be seen that I(t) = I( 1 + cos 2wot) + J sin 2~~2,
! 14)
which may be written in the form [7] I(t) = Re {I,(1 +e-jzwO’)}.
(15)
It can be concluded that the instantaneous intensity in a pure-tone jeld may be divided into two components, one associated with the active intensity and one associated with the reactive intensity [5-71. The time average of I(t) is I, of course, and the “reactive part” of the instantaneous intensity, i.e., the second term of the right side of equation (14), has zero mean. In the general case the sound field cannot be assumed to be monochromatic. Equation (8) is of general validity, of course, but since one cannot make an instantaneous separation of the particle velocity into components in phase and in quadrature with the pressure, as in equation (lo), there is no obvious, general analogue to equation (14). In fact, because of cross terms one cannot even decompose the instantaneous intensity in a sound field with two discrete frequency components-at least not in the simple manner suggested by equation (14). Nevertheless, several proposals can be found in the literature. Pettersen [16] has suggested the expression (16)
I,.(r) =p(t)(u(r)-j&(r)), while Uosukainen
[S] has suggested a similar expression, (17)
C(t) = (p(t)+j$(r))u(t),
termed “the semi-analytic intensity”. Yet another expression has been proposed by Heyser [91: I~(t)=~(p(r)+j~(t))(u(t)-j;(r)). This quantity is termed “the complex instantaneous p(tMt)
=p(tMt)
and
(18)
intensity”. Since
p(t)fi(t)
= -p^(tMr)
(19,X))
in a stationary sound field [12], it can be seen that the three expressions have the same time average: I,(t)=I:.(t)=I:‘(t)=I+jJ=I,. (21) It is also apparent that Re {I(t)} = Re {I:(r)}=I(r).
(22)
Moreover, when the pure-tone expressions (9) and (10) are inserted, equation (16) becomes I,.(t) = pOcos w,t(u, cos w,,t + u, sin w,,f - juu sin wOt+ ju, cos wof) = I,.( 1 + e-i’%‘),
(23)
492
F. JACOBSEN
which seems to be a natural extension of equation (15). Nevertheless, expressions the one suggested by Heyser is probably the most useful. It is well known that the signal r(t)=JxZ(t)+$2(t)
of the three
(24)
is the envelope of x( 1). A complex envelope w(t) can also be defined such that the analytic signal x(t) + j;( t) can be written in the form x(t)+jG(t)
= w(t) e+J;
(25)
see, e.g., reference [12]. Clearly, r(t) = Iw(t)l.
(26)
It can now be seen that Heyser’s expression may be regarded as a generalized version of the pure-tone expression for the complex intensity (5), and it is a simple matter to show from equations (9), (10) and (18) that I:(r)=I, in a pure-tone sound field. In a narrow-band
(27)
sound field
I”(r)=Re{I~(r)}=~(p(r)u(r)+p^(r)G(r))
(28)
J”(r)=Im{I~(r)}=~(p*(r)u(r)-p(r)Q(r))
(29)
and
are low-pass signals which represent running short-time average values of, respectively, the active and the reactive intensity components p( r)u( r) and p^(r)u( r). 4. DISCUSSION The decomposition of the instantaneous intensity implied by equation (14) is interesting, since it shows that the actual flow of sound energy is much more complicated than suggested by the “energy streamlines” [ 171 (which follow the active intensity), as pointed out by Mann er al. [5,6] and by Uosukainen [8] (in the general case I and J have different directions). However, this decomposition is valid only in a pure-tone sound field and it can easily lead to erroneous conclusions. For example, one might well get the impression from equation (14) that the reactive intensity, which apparently is an amplitude [7], has no sign, or that the sign depends on the arbitrary time base. However, although the sign of the time-averaged reactive intensity is a mere convention it is certainly not arbitrary, as can be seen from equation (7). The reactive sound intensity has a definite physical significance; it points out of a source, for example [13, 181. One might also be led to the misapprehension that reactive intensity components associated with different frequency components do not simply add vectorially. However, they do indeed, as can easily be shown from equation (4). It follows that two reactive components of nearly the same magnitude and opposite sign lead to a small net reactive component. It can therefore be concluded that one cannot infer from a small wide-band indication of the reactive intensity that no oscillatory energy flux exists: this is the expense of analyzing in wide bands. (One cannot infer the absence of flow of sound energy from an insignificant band indication of the active intensity either.) These considerations can be illustrated by an example. Practically all the acoustic energy in a reverberant room driven with random noise is oscillatory, but the sound field is not reactive [ll]. In fact, contrary to usual belief, the sound field is no more reactive than active, as shown in a recent paper [ 191.
PHYSICAL
AND
OTHER
SOUND
INTENSITIES
493
Although the complex time dependent expressions (16), (17) and (18) seem to be of general validity, they are useful only for a sound field generated by a narrow-band source; time-domain representations of wide-band signals are of little use. The complex expressions (16) and (17), which are equivalent, can perhaps be justified with reference to equations (21) and (22), but the real and imaginary parts should not be interpreted as the instantaneous active and reactive components: a consistent definition would give the two terms of the right side of equation (14) in the single frequency case. Heyser’s expression is interesting since the two low-pass signals I”(f) and J”(t) illustrate the instantaneous transport of sound energy. It may have a potential in analyzing non-stationary sound fields. One might “re-modulate” the two signals by analogy with equation (15): the resulting signals would actually represent the instantaneous active and reactive components of the intensity. However, this would require determining the instantaneous frequency (this problem has been discussed by Uosukainen [8]), and the smooth signals are probably more useful. 5. EXPERIMENTAL
RESULTS
To illustrate some of the quantities described in section 3 a few experiments have been carried out. The sound pressure and a component of the particle velocity were measured with a sound intensity probe, Briiel & Kjaer (B & K) 35 19, in combination with an intensity analyzer, B 8z K4433, in various sound fields. Analogue signals proportional to sound pressure and particle velocity from the intensity analyzer were recorded and processed with a dual channel FFT analyzer, B & K 2032, in combination with a small computer, Hewlett Packard 9817. The FFT analyzer can produce Hilbert-transformed time signals; the computer was used in determining the instantaneous intensity I,(t) and the real and imaginary parts of Heyser’s complex instantaneous intensity; that is, the quantities I:‘(t) and J:(r) (the suffix r indicates a vector component in the r-direction). In Figure 1 is shown the result of a measurement about 30 cm from a small loudspeaker mounted in an enclosed cabinet and driven with noise in a one-third octave band with a centre frequency of 1 kHz. The result of a similar measurement very near the loudspeaker cone is shown in Figure 2; in this case the loudspeaker was driven with noise in a one-third octave band with a centre frequency of 250 Hz. Both measurements took place far from reflecting surfaces in a large, strongly damped room. In the first measurement the sound pressure and the normalized particle velocity (pcu,( 1)) are practically identical, and as a result the instantaneous intensity is always positive: this is an active sound field. By contrast, the sound pressure and the particle velocity are almost in quadrature in the near field measurement, and the normalized particle velocity is, as expected, larger than the pressure: this is a simple example of a strongly reactive sound field. Both measurements demonstrate the significance of the envelope signals I:(t) and J:‘(t). The next measurement was also made in the large, strongly damped room. This time the intensity probe was placed about 30 cm from one of the 3 mm sideplates of a steel box driven with noise in a one-third octave band with a centre frequency of 250 Hz by an electrodynamic exciter. This is a much more complicated sound field, partly active and partly reactive, and it is apparent that both the active and the reactive component can take negative values. It is not surprising, therefore, that the random errors associated with the use of a finite averaging time in intensity measurements can be very large in such circumstances [20]. In the measurement shown in Figure 4 the loudspeaker was placed in a large reverberation room with a reverberation time of about 5 s, and the intensity probe was placed at a random position far from the loudspeaker, which was driven with noise in a one-third
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octave band with a centre frequency of 500 Hz. In this case the sound pressure exceeds the normalized particle velocity [21], and the signals are very nearly uncorrelated [22]. The active and reactive intensity components fluctuate about zero: that is, the sound field is neither active nor reactive [19]. 6. CONCLUDING
REMARKS
Various instantaneous field quantities associated with sound intensity have been presented and demonstrated. Whereas the usefulness of the corresponding time-averaged quantities is fairly well established, the practical importance of the instantaneous quantities is less obvious. However, they demonstrate instructively the physical significance of the time-averaged quantities. REFERENCES 1. T. K. STANTON and R. T. BEVER 1979 Journal of the Acoustical Society of America 65,249-252.
Complex wattmeter measurements in a reactive acoustic field. 2. G. W. ELKO 1984 Ph.D. Thesis, Pennsylvania State University. Frequency domain estimation of the complex acoustic intensity and acoustic energy density. 3. J.-C. PASCAL 1981 Proceedings of the International Congress on Recent Development in Acoustic Intensity Measurements, 11-19. Mesure de l’intensit6 active et riactive dans differents champs acoustiques. 4. S. GADE 1982 Btief & Kjaer Technical Review 3, 3-39. Sound intensity (Part 1, Theory). 5. J. A. MANN III, J. TICHY and A. J. ROMANO 1987 Journal ofthe Acoustical Society ofAmerica 82, 17-30. Instantaneous and time-averaged energy transfer in acoustic fields. 6. J. A. MANN III 1988 Ph.D. Thesis, Pennsylvania State Uniuersity. Acoustic intensity: energy transfer, wave properties, and applications. 7. F. J. FAHY 1989 Sound Intensity. London: Elsevier Applied Science. See chapter 4. 8. S. UOSUKAINEN 1989 Ph.D. Thesis, Helsinki University of Technology, VTT Research Report No. 656. Properties of acoustic energy quantities. 9. R. C. HEYSER 1986 AES 81st Convention, Audio Engineering Society, Preprint 2399. Instantaneous intensity. 10. U. KURZE 1968 Acustica 20,308-310. Zur Entwicklung einges Geriites fiir komplexe Schallfeldmessungen. 11. F. JACOBSEN 1989 Journal of Sound and Vibration 130,493-507.Active and reactive, coherent and incoherent sound fields. 12. A. PAPOULIS 1984 Probability, Random Variables, and Stochastic Processes. New York: McGrawHill, second edition. See section 11-l. 13. J. A. MANN III and J. TICHY 1988Proceedings of Inter-Noise 88, 105-110. Extended intensity measurement technique for sound field identification in the noise source nearfield. 14. F. JACOBSEN 1990 Noise Control Engineering Journal 35,37-46. Sound field indicators: Useful Tools. 15. J.-C. PASCAL and C. CARLES 1982 Journal of Sound and Vibration 83, 53-65. Systematic measurement errors with two microphone sound intensity meters. 16. 0. K. 0. PETERSEN 1980 Ph.D. 7%esis, NTH Trondheim. MHling og anvendelse av akustisk intensitet for beskrivelse av avstriling og effekttransmisjon i kompliserte akustiske felt. 17. R. WATERHOUSE and D. FEIT 1986 Journal of the Acoustical Society of America 80,681-684. Equal-energy streamlines. 18. F. P. MECHEL 1988 Journal of Sound and Vibration 123, 537-572. Notes on the radiation impedance, especially of piston-like radiators. 19. F. JACOBSEN 1990Journal of Sound and Vibration 143, 231-240. Active and reactive sound intensity in a reverberant sound field. 20. F. JACOBSEN 1989 Journal of Sound and Vibration 131, 475-487. Random errors in sound power determination based on intensity measurement. 21. F. JACOBSEN 1979 The Acoustics Laboratory, Technical University of Denmark, Report No. 27. The diffuse sound field. 22. F. JACOBSEN and T. G. NIELSEN 1987Journal of Sound and Vibration 118, 175-180. Spatial correlation and coherence in a reverberant sound field.