- Email: [email protected]
The practical assessment of errors in sound intensity measurement
Journal
of Sound
THE
and Vibration (1986) 10!!(2),255-263
PRACTICAL
ASSESSMENT
INTENSITY
OF ERRORS
IN SOUND
MEASUREMENT
P. S. WATKINSON Plessey Marine, Wilkinthroop, Templecombe, BA8 ODH, England (Received 7 November 1984, and in revised form 27 March 1985)
Many sources of error in two transducer sound intensity measurements have been discussed in the literature. Many of these errors are difficult to assess in a practical situation, but two in particular can be calculated. These are the error associated with phase mis-match and random error. The use of the known phase mismatch is discussed as regards its use in evaluating the quality of a particular measurement, and the use of coherence is discussed as regards its use in calculating random error.
1.
INTRODUCTION
In previously published work [l-8] many sources of error associated with two transducer intensity measurements have been discussed. Despite these errors being quite well understood as presented, they can be far from applicable in practical situations: many practical fields do not approximate to plane wave behaviour (particularly where intensity measurements can be used to advantage) and practical sources rarely approximate to point sources at the close range necessary for the calculated near field errors to be significant [4]. Criteria are required whereby an intensity measurement can be made at a point in an unknown field and some estimate of its accuracy assessed. This is not yet possible when considering all the sources of error discussed in references [l-8], but is possible for two particular errors: phase error (if the phase mis-match is known) and random error. Pascal [9] has discussed two quality indicators which correspond to these two errors: the measured phase angle and the coherence between the two transducer signals. He also showed how the measured phase angle relates to the difference between the sound pressure level and the sound intensity level ( Lp - L,) which has been discussed as a criterion for measurement limitation by others [ 10-131. In this paper the use of phase and coherence criteria is discussed and also some analytically derived graphs of phase and finite separation errors versus frequency are presented for two hypothetical systems in a variety of fields to show some broad aspects of their behaviour. 2. PHASE MIS-MATCH
ERROR AND CRITERIA
If P1 and P2 are the two complex pressures which obtain at the two transducer locations (field points 1 and 2), then, with complete field generality, p1 = pA ej(+A++),
Pz = Ps ejlLe,
(192)
where PA and PB are the pressure amplitudes at points 1 and 2, (LAand I,G~are the pressure phases at points 1 and 2 (relative to a common arbitrary reference) and 4 is the difference 255
0022~460X/86/050255+09 $03.00/O
@ 1986Academic
Press Inc. (London)
Limited
256
P. S. WATKINSON
between the phase shifts introduced by the two transducers and their instrumentation. All these parameters are functions of frequency. The estimated intensity component (I,) from these two point measurements will be Z, = (ll~h)p,&
sin ($+ 9),
(3)
where h is the distance between points 1 and 2, w is the radian frequency, p is the density of the medium and 9 = $A - I/++The intensity estimate without the additional phase shift (I,,) is equation (3) with 4 = 0, so the extra error introduced by the phase shift will be: Ze/Z,,=sin($+4)/sin$.
(4)
To keep this equation a general function of frequency it cannot be simplified by the approximation sin Cc, = JI. The relationship between equations (3) and (4) to the true intensity (Z,) can be expressed as follows. If Z,,l Z, = E
(5)
(i.e., the error from all other sources except phase mis-match), then ZJZ, = (Z,/Z,l)(Z,,/ZJ = E sin (+++)lsin
(+).
(6)
If I,& is the measured phase angle ($I+ 4) then equation (4) becomes Z,lZ,, = sin (&J/sin
(4, - 4).
(7)
Equation (7) shows that, if the phase mis-match is known, the error in a raw measurement can be calculated and the measurement either accepted or rejected, or the error due to 4 can be compensated for. The advantage of simply calculating the error is knowing the error and limit of measurement with some certainty, the disadvantage is an unnecessarily limited range of measurement. The advantage of compensating for 4 is an increased range of measurement, the disadvantage is not knowing for certain what the limit of accurate measurement is. This error (given by equation (7)) can be discussed in terms of a limit on Lp - L,. This is particularly applicable in air because the reference levels for pressure and intensity are standardized such that Lp = LI in a plane wave. In water the reference levels are not so convenient, but similar criteria can apply with the appropriate reference levels taken into account. This criterion becomes less applicable as the finite separation error becomes increasingly dominant (i.e., with increasing frequency). The finite approximation error on intensity is different for that on pressure and therefore Lp and LI will differ even in the absence of phase mis-match. The two approaches to looking at the same error are useful under different circumstances. Use of the raw phases is useful in FFT style measurements where there is also some post-processing facility. The Lp -L, criterion is useful for sound level meter style measurements or FFT style measurements where post-processing is limited to displaying LI and Lp. If knowledge of phase mis-match is used to compensate phase mis-match error then some estimate must be made as to the accuracy with which 4 is determined compared with measurement conditions. This must include consideration of random errors and environmental conditions which may change the characteristics of the transducers and instrumentation. From this estimate of accuracy a maximum potential error can be calculated and used either to accept or reject a measurement. Any criteria for judging data with regard to phase mis-match or uncertainty must be based upon equation (7). Note that signs of +,,, and 4 must be preserved. In the case of a phase uncertainty then errors due to both t-4 and -4 must be considered, as one error may be significantly greater than the other: e.g., if )4\= ($1 then Z,/Z,, is either 0 or 2 cos 6.
ERRORS
IN
INTENSITY
257
MEASUREMENT
To illustrate the behaviour of phase error and its importance relative to finite separation error, the phase responses of two hypothetical intensity measuring systems have been created and the errors associated with the measurement of various idealized fields have been calculated. The error shown is the combination of phase and finite separation error and is expresssed as I,/ I, = 1 f E.
(8)
In all error graphs 10 loglo E is plotted versus frequency and all results are calculated from simple analytically derived equations from consideration of the field.
I
I
4
6 Frequency
8
(kHr)
Figure 1. Phase mis-match for two idealized sound intensity systems. 0.3 Hz-30 Hz with 10% mis-match (solid), 3 Hz-15 kHz with 20% mis-match (dashed).
Figure 1 shows plots of the phase mis-matches for the two systems. These phases are derived by assuming that each channel of the instrumentation is characterized by a first order high pass filter and a first order low pass filter, and that the cut-off frequencies of these filters are slightly different for each instrumentational channel. This model is quite realistic. For the first case (the solid line in Figure l), the nominal cut-off frequencies (i.e., 3 dB points, or phase response = 45”) are O-3 Hz and 30 kHz, where the actual values used are 10% different between the channels. The second case (the dashed line in Figure l), is where nominal lower and upper frequencies are 3 Hz and 15 kHz, but with a 20% mis-match in actual cut-off values. The transducer spacing used for calculating the subsequent graphs is 12 mm and the medium is assumed to be air. The phase mis-match is small at low frequency, but is significant compared to the performance required for accurate measurement. Figure 2 shows plots of the ratio of the phase mis-match to the phase angle expected to exist between the pressures at the two measurement points in a plane wave field, the direction of propagation being along the line joining the two measurement points. From the point of view of phase mis-match, the curves in this figure can be considered as figures of merit which can be interpreted with a view to intensity measurement better than by inspection of Figure 1, particularly as regards a low frequency limit and a maximum Lp -L, limit broadband. The flat portion of these plots correspond to the frequency range over which the phase mis-match is linear with frequency. Figure 3 shows plots of error versus frequency for a plane wave propagating in the direction of the axis joining the sensing points of the two transducers. At low frequency
258
P. S. WATKINSON
6
4 Frequency
(kHz)
Figure 2. Ratio of phase mis-match to plane wave phase angle for two idealized sound intensity instrumentation systems, for 12 mm transducer separation in air.
-201
0
2
6
4 Frequency
8
10
CkHz)
Figure 3. Error in measurement of intensity of a plane wave propagating in a direction parallel to the axis joining the pressure transducers of two idealized sound intensity instrumentation systems, for a 12 mm transducer separation in air.
0 I
I
I
I
I
I
I
I
4
6
8
Frequency
10
(kHz)
Figure 4. Error in measurement of component of intensity of a plane wave 6O’to the direction of propagation for the two idealized sound intensity instrumentation systems, for a 12 mm transducer separation in air.
ERRORS
IN INTENSITY
259
MEASUREMENT
the phase error dominates and the curves in this region are identical to those in Figure 2. With increasing frequency the error due to finite separation approximations increases until its effect is dominant and will be very nearly the same in both cases. LP and LI will be very nearly equal outside the range of significant finite separation error. The two graphs cross in the region where compensation between finite separation error and phase mis-match error becomes significant to distinguish the two curves. Figure 4 shows the error for the two systems when measuring the component of intensity in a direction 60”to the axis of propagation of the plane wave. The phase error is dominant over a much wider frequency range, the finite separation error being of a lesser magnitude at any particular frequency in the range shown than that in Figure 3. However, the overall error at lower frequency has increased by about 3 dB which is also the value of LP - L, and is also the difference between this graph and that of Figure 2.
1 2
4
6 Frequency
I 8
10
(Lb)
of component of intensity of a plane wave 84” to the direction of propagation Figure 5. Error in measurement for the two idealized sound intensity instrumentation systems, for a 12 mm transducer separation in air.
Figure 5 displays the error for measurement of the component of intensity in a ?rection 84” to the direction of propagation of the plane wave. The finite separation is not at all obviously in evidence and this graph is identical in form to that in Figure 2, except shifted up by 10 dB. Over the whole frequency range shown Lp - LI will be very close to 10 dB. It is clear that, over a particular frequency range, both systems can measure on-axis plane wave intensity with small error, but, for the case used to derive Figure 5, the less good system is beyond its range of accurate measurement (say, .e< -6 dB) whereas the better system will still only yield a small error. The field used to derive the error graph in Figure 6 consists of two plane waves travelling in opposite directions, one 0.8 of the pressure amplitude of the other. The phases of the plane waves are referenced to a point at a constant distance from the measurement point such that with varying frequency different parts of this “standing wave” field will obtain at the transducers: e.g., pressure nodes and maxima and particle velocity nodes and maxima. Although these field characteristics will vary, the intensity will always be constant and will be the difference between the individual intensities associated with each plane wave had it existed alone. It is in this type of field that one cannot assume I,$,,to be small such that sin +, = &,. Indeed, if a node in the pressure spatial waveform is situated between the two transducers then +,,, will equal T radians if the two plane waves are of equal magnitude. Criteria and errors based upon equation (4) will apply for all possible
260
P. S. WATKINSON C
-5 iI% s VI .g -10 _o 0 -15
-2c
1 0
I I I I Ill 2
I
I
I
4
6
8
Frequency (kHz)
Figure 6. Error in measurement of net intensity of two unequal plane waves (PB = 0,8P,) travelling in opposite directions for the two idealized sound intensity instrumentation systems, for a 12 mm transducer separation in air.
values of $,,,. Figure 6 shows that the system with the better phase matching is a consistently better performer, yet over limited frequency ranges compensation between phase error and finite separation error yields a small total error from the not so well matched system. This situation is fortuitous and therefore cannot be relied upon under other circumstances.
-201 0
2
4
6
8
10
Frequency (kHz)
Figure 7. Error in measurement of component of intensity at a point in a field comprising two equal plane waves, one travelling parallel to the measurement axis and one travelling at right angles to this, for the two idealized sound intensity instrumentation systems, and a 12 mm transducer separation in air.
The errors shown in Figure 7 are for a field comprising two equal amplitude plane waves travelling in directions at right angles to each other. The measurement is made in the direction of one. All previous fields discussed in this section have constant intensity through space; this field has an intensity which varies in space. The measurement position is stationary with respect to the phase reference position so with increasing frequency the field will change at this point. The curves show the superiority of the better matched system except over certain frequency ranges where the error is larger than the range of the graph axes. Over these ranges the error will peak at +oo dB and corresponds to a field point with zero intensity. This shows that where the error is very large the intensity is insignificantly small compared with the rest of the field anyway.
ERRORS
IN INTENSITY
261
MEASUREMENT
These worked examples should give the reader some insight into behaviour of phase errors in a typical measurement system which will assist in determining some form of criteria for accepting or rejecting a sound intensity measurement.
3. COHERENCE
AND
RANDOM
ERROR
CRITERIA
Seybert [6] has discussed statistical errors in acoustic intensity measurements and has related random error to coherence by E = (l/&)[(l/~*)+COt*
($){(l-
r’)/2r’]l”“,
where n is the number of raw estimates and y2 is the coherence function. Seybert discussed the application of this equation in reference [7] and Dyrlund [8] introduced Lp - LI (which reflects $) as a parameter which indicates an expected higher random error (due to its dependence upon I,-~). There are four specific problems with using coherence as a criterion for assessing the quality of a particular measurement and thereby accepting or rejecting it: bias errors, knowledge (or rather lack of knowledge) of $, resolution of y2 in a two channel Fourier analyzer, and the assumption that the signals are of a random character in the derivation of equation (9). Bias error for spectral estimates has been discussed quite thoroughly in reference [ 141. The problem is that equation (9) predicts only a random error if the reason for y* being less than unity is due only to random error. If there are bias errors then the random error can be grossly over-estimated. The bias error is said to be proportional to the square of the derivative of the phase spectrum between the two pressure signals and therefore would be particularly large for measurements in, say, a standing wave or some other field with large phase changes with frequency. Estimates of cross spectrum and estimates of coherence are subject to this bias error and it is not clear how to evaluate an actual error from these estimates. The presence of such an error is characterized by an improvement in coherence for finer spectral resolution. It is good in many ways that a parameter such as coherence responds to bias error, but it is not good that it is not easy to distinguish random and bias errors so that one could use equation (9). Equation (9) contains the phase angle Q. In the general case the only way this is known is by measurement. If this measurement is greatly in error then circumstances could be such that the estimated error is predicted to be erroneously small. This circumstance should correspond to a particularly low coherence, but would require independent inspection of coherence. An automated system may not be designed with this facility and may unwisely rely upon inspection of the estimated random error. For measurements in field regions where + is particularly small then, for a given maximum error and number of averages, a coherence very close to unity may be sought. The existence of 1 - y2 in equation (9) means that there are significant differences in the estimated value of E for values of the coherence of, say, O-9, 0.99, O-999 and 0.9999, etc., each extra 9 after the decimal point decreasing the error by a factor of 10. Coherences extremely close to unity can physically exist between pressures at two closely spaced points in a sound field and will probably be limited by electrical noise in the instrumentation. There is also a limit due to the resolution of the value of the coherence due to the discrete arithmetic in whatever processor and software is in use, particularly in some two channel Fourier analyzers where resolution is sacrificed for speed and the assumption is made that resolution sufficient for a plot of coherence on a scale of 0 to 1 is acceptable. If the coherence is limited to confidence values less than 0.99999 then the random error
P. S. WATKINSON
262
may be over-estimated for + < 1”. This situation is not improved by increasing the number of averages used to estimate coherence. Lastly, equation (9) has been derived by assuming Gaussian random signal types. It is not clear if this equation is applicable to, say, deterministic signals, or a mixture of deterministic and random signals. Random error formulae for auto spectra and cross spectra are clearly in error for deterministic signals. If y2 = 1.0 is substituted into equation (9) then E = l/&i, which is wrong for a purely deterministic signal (E = 0). An alternative approach for the determination of random error is to calculate directly the mean and variance for a set of raw spectra. This will then yield random error whatever the signal type. To store all the raw spectra and then operate upon them would generally require an unreasonable amount of computer storage and time and would also preset the maximum number of averages. A compromise is to compute a running mean and variance, an option not available on current 2 channel FFT analyzers. The benefit is freedom from the uncertainties introduced by employing coherence to calculate random error. 4. CONCLUSION In this paper criteria for the assessment of errors in two transducer intensity measurements when the nature of the sound field in the region of the measurement points is not known have been discussed. The first criterion discussed is based upon phase mis-match. Present criteria usage is related to the effect of phase mis-match upon general field measurements and a number of specific fields have been analyzed here to show error behaviour. The second criterion is based upon coherence, a very commonly used error assessment parameter. Its relevance to error assessment for sound intensity measurements has been discussed and four uncertainties identified: the influence of bias errors, the dependence upon knowing the measured phase angle, which may be in error, the problem of resolving a coherence very close to unity, and the applicability of random error formulae to signals of type other than Gaussian random. It is proposed that instead of using coherence to calculate random error, this latter parameter can be calculated from the directly determined mean and variance of successive raw spectral estimates of intensity. ACKNOWLEDGMENT The work described in this paper was supported by the Procurement Ministry of Defence.
Executive of the
REFERENCES
ofSound and Vibration 75, 229-238. Finite difference approximation errors in acoustic intensity measurements. 2. S. J. ELLIOTT 1981 Journal of Sound and Vibration 78, 439-443. Errors in acoustic intensity measurement. 3. J. K. THOMPSON 1981 Journal of Sound and Vibration 78, 444-445. Author’s reply. 4. J. POPE and J. Y. CHUNG 1982 Journal of Sound and Vibration 82, 459-462. Comments on 1. J. K. THOMPSON and D. R. TREE 1981 Journal
“Finite difference approximation errors in acoustic intensity measurements”.
5. J. K. THOMPSON 1982 Journal of Sound and Vibration 82,463-464. Author’s reply. 6. A. F. SEYBERT 1981 Journal ofSound and Vibration 75, 519-526. Statistical errors in acoustic intensity measurements. 7. A. F. SEYBERT1982 Proceedings of Inter-Noise 82,119-122. Some observations on the application of the acoustic intensity method.
8. 0. DYRLUND 1983 Journal of Sound and Vibration 90, 585-589. A note on statistical errors in acoustic
intensity
measurements.
ERRORS
11.
12.
13. 14.
263
of Inter-Noise 83, 1071-1074. FFT intensity meters used for sound sources and acoustical environment characterisation. J. ROLAND 1982 fioceedings ofInter-Noise 82, 715-718. What are the limitations of intensity technique in a hemi-difhtse field. G. HUBNER 1983 Z+oceedings oflnter-hroise 83, 1043-1046. Determination of sound power of sound sources under in-situ conditions using intensity method-field of application, suppression of parasitic noise, reflection effect. S. GADE, K. B. GINN, 0. ROTH and M. BROCK 1983 Proceedings oflnter-Noise 83, 1047-1050. Sound power determination in highly reactive environments using sound intensity measurements. G. RASMUSSEN and M. BROCK 1983Proceedings ofthe 11 th Zntemational Congress on Acousrics 6, 177-180. Transducers for intensity measurements. J. S. BENDAT and A. G. PIERSOL 1980 Engineering Applications of Correlation and Spectral Analysis. New York: John Wiley and Sons.
9. J. C. PASCAL 1983 Proceedings
10.
IN INTENSITY MEASUREMENT
of Sound
THE
and Vibration (1986) 10!!(2),255-263
PRACTICAL
ASSESSMENT
INTENSITY
OF ERRORS
IN SOUND
MEASUREMENT
P. S. WATKINSON Plessey Marine, Wilkinthroop, Templecombe, BA8 ODH, England (Received 7 November 1984, and in revised form 27 March 1985)
Many sources of error in two transducer sound intensity measurements have been discussed in the literature. Many of these errors are difficult to assess in a practical situation, but two in particular can be calculated. These are the error associated with phase mis-match and random error. The use of the known phase mismatch is discussed as regards its use in evaluating the quality of a particular measurement, and the use of coherence is discussed as regards its use in calculating random error.
1.
INTRODUCTION
In previously published work [l-8] many sources of error associated with two transducer intensity measurements have been discussed. Despite these errors being quite well understood as presented, they can be far from applicable in practical situations: many practical fields do not approximate to plane wave behaviour (particularly where intensity measurements can be used to advantage) and practical sources rarely approximate to point sources at the close range necessary for the calculated near field errors to be significant [4]. Criteria are required whereby an intensity measurement can be made at a point in an unknown field and some estimate of its accuracy assessed. This is not yet possible when considering all the sources of error discussed in references [l-8], but is possible for two particular errors: phase error (if the phase mis-match is known) and random error. Pascal [9] has discussed two quality indicators which correspond to these two errors: the measured phase angle and the coherence between the two transducer signals. He also showed how the measured phase angle relates to the difference between the sound pressure level and the sound intensity level ( Lp - L,) which has been discussed as a criterion for measurement limitation by others [ 10-131. In this paper the use of phase and coherence criteria is discussed and also some analytically derived graphs of phase and finite separation errors versus frequency are presented for two hypothetical systems in a variety of fields to show some broad aspects of their behaviour. 2. PHASE MIS-MATCH
ERROR AND CRITERIA
If P1 and P2 are the two complex pressures which obtain at the two transducer locations (field points 1 and 2), then, with complete field generality, p1 = pA ej(+A++),
Pz = Ps ejlLe,
(192)
where PA and PB are the pressure amplitudes at points 1 and 2, (LAand I,G~are the pressure phases at points 1 and 2 (relative to a common arbitrary reference) and 4 is the difference 255
0022~460X/86/050255+09 $03.00/O
@ 1986Academic
Press Inc. (London)
Limited
256
P. S. WATKINSON
between the phase shifts introduced by the two transducers and their instrumentation. All these parameters are functions of frequency. The estimated intensity component (I,) from these two point measurements will be Z, = (ll~h)p,&
sin ($+ 9),
(3)
where h is the distance between points 1 and 2, w is the radian frequency, p is the density of the medium and 9 = $A - I/++The intensity estimate without the additional phase shift (I,,) is equation (3) with 4 = 0, so the extra error introduced by the phase shift will be: Ze/Z,,=sin($+4)/sin$.
(4)
To keep this equation a general function of frequency it cannot be simplified by the approximation sin Cc, = JI. The relationship between equations (3) and (4) to the true intensity (Z,) can be expressed as follows. If Z,,l Z, = E
(5)
(i.e., the error from all other sources except phase mis-match), then ZJZ, = (Z,/Z,l)(Z,,/ZJ = E sin (+++)lsin
(+).
(6)
If I,& is the measured phase angle ($I+ 4) then equation (4) becomes Z,lZ,, = sin (&J/sin
(4, - 4).
(7)
Equation (7) shows that, if the phase mis-match is known, the error in a raw measurement can be calculated and the measurement either accepted or rejected, or the error due to 4 can be compensated for. The advantage of simply calculating the error is knowing the error and limit of measurement with some certainty, the disadvantage is an unnecessarily limited range of measurement. The advantage of compensating for 4 is an increased range of measurement, the disadvantage is not knowing for certain what the limit of accurate measurement is. This error (given by equation (7)) can be discussed in terms of a limit on Lp - L,. This is particularly applicable in air because the reference levels for pressure and intensity are standardized such that Lp = LI in a plane wave. In water the reference levels are not so convenient, but similar criteria can apply with the appropriate reference levels taken into account. This criterion becomes less applicable as the finite separation error becomes increasingly dominant (i.e., with increasing frequency). The finite approximation error on intensity is different for that on pressure and therefore Lp and LI will differ even in the absence of phase mis-match. The two approaches to looking at the same error are useful under different circumstances. Use of the raw phases is useful in FFT style measurements where there is also some post-processing facility. The Lp -L, criterion is useful for sound level meter style measurements or FFT style measurements where post-processing is limited to displaying LI and Lp. If knowledge of phase mis-match is used to compensate phase mis-match error then some estimate must be made as to the accuracy with which 4 is determined compared with measurement conditions. This must include consideration of random errors and environmental conditions which may change the characteristics of the transducers and instrumentation. From this estimate of accuracy a maximum potential error can be calculated and used either to accept or reject a measurement. Any criteria for judging data with regard to phase mis-match or uncertainty must be based upon equation (7). Note that signs of +,,, and 4 must be preserved. In the case of a phase uncertainty then errors due to both t-4 and -4 must be considered, as one error may be significantly greater than the other: e.g., if )4\= ($1 then Z,/Z,, is either 0 or 2 cos 6.
ERRORS
IN
INTENSITY
257
MEASUREMENT
To illustrate the behaviour of phase error and its importance relative to finite separation error, the phase responses of two hypothetical intensity measuring systems have been created and the errors associated with the measurement of various idealized fields have been calculated. The error shown is the combination of phase and finite separation error and is expresssed as I,/ I, = 1 f E.
(8)
In all error graphs 10 loglo E is plotted versus frequency and all results are calculated from simple analytically derived equations from consideration of the field.
I
I
4
6 Frequency
8
(kHr)
Figure 1. Phase mis-match for two idealized sound intensity systems. 0.3 Hz-30 Hz with 10% mis-match (solid), 3 Hz-15 kHz with 20% mis-match (dashed).
Figure 1 shows plots of the phase mis-matches for the two systems. These phases are derived by assuming that each channel of the instrumentation is characterized by a first order high pass filter and a first order low pass filter, and that the cut-off frequencies of these filters are slightly different for each instrumentational channel. This model is quite realistic. For the first case (the solid line in Figure l), the nominal cut-off frequencies (i.e., 3 dB points, or phase response = 45”) are O-3 Hz and 30 kHz, where the actual values used are 10% different between the channels. The second case (the dashed line in Figure l), is where nominal lower and upper frequencies are 3 Hz and 15 kHz, but with a 20% mis-match in actual cut-off values. The transducer spacing used for calculating the subsequent graphs is 12 mm and the medium is assumed to be air. The phase mis-match is small at low frequency, but is significant compared to the performance required for accurate measurement. Figure 2 shows plots of the ratio of the phase mis-match to the phase angle expected to exist between the pressures at the two measurement points in a plane wave field, the direction of propagation being along the line joining the two measurement points. From the point of view of phase mis-match, the curves in this figure can be considered as figures of merit which can be interpreted with a view to intensity measurement better than by inspection of Figure 1, particularly as regards a low frequency limit and a maximum Lp -L, limit broadband. The flat portion of these plots correspond to the frequency range over which the phase mis-match is linear with frequency. Figure 3 shows plots of error versus frequency for a plane wave propagating in the direction of the axis joining the sensing points of the two transducers. At low frequency
258
P. S. WATKINSON
6
4 Frequency
(kHz)
Figure 2. Ratio of phase mis-match to plane wave phase angle for two idealized sound intensity instrumentation systems, for 12 mm transducer separation in air.
-201
0
2
6
4 Frequency
8
10
CkHz)
Figure 3. Error in measurement of intensity of a plane wave propagating in a direction parallel to the axis joining the pressure transducers of two idealized sound intensity instrumentation systems, for a 12 mm transducer separation in air.
0 I
I
I
I
I
I
I
I
4
6
8
Frequency
10
(kHz)
Figure 4. Error in measurement of component of intensity of a plane wave 6O’to the direction of propagation for the two idealized sound intensity instrumentation systems, for a 12 mm transducer separation in air.
ERRORS
IN INTENSITY
259
MEASUREMENT
the phase error dominates and the curves in this region are identical to those in Figure 2. With increasing frequency the error due to finite separation approximations increases until its effect is dominant and will be very nearly the same in both cases. LP and LI will be very nearly equal outside the range of significant finite separation error. The two graphs cross in the region where compensation between finite separation error and phase mis-match error becomes significant to distinguish the two curves. Figure 4 shows the error for the two systems when measuring the component of intensity in a direction 60”to the axis of propagation of the plane wave. The phase error is dominant over a much wider frequency range, the finite separation error being of a lesser magnitude at any particular frequency in the range shown than that in Figure 3. However, the overall error at lower frequency has increased by about 3 dB which is also the value of LP - L, and is also the difference between this graph and that of Figure 2.
1 2
4
6 Frequency
I 8
10
(Lb)
of component of intensity of a plane wave 84” to the direction of propagation Figure 5. Error in measurement for the two idealized sound intensity instrumentation systems, for a 12 mm transducer separation in air.
Figure 5 displays the error for measurement of the component of intensity in a ?rection 84” to the direction of propagation of the plane wave. The finite separation is not at all obviously in evidence and this graph is identical in form to that in Figure 2, except shifted up by 10 dB. Over the whole frequency range shown Lp - LI will be very close to 10 dB. It is clear that, over a particular frequency range, both systems can measure on-axis plane wave intensity with small error, but, for the case used to derive Figure 5, the less good system is beyond its range of accurate measurement (say, .e< -6 dB) whereas the better system will still only yield a small error. The field used to derive the error graph in Figure 6 consists of two plane waves travelling in opposite directions, one 0.8 of the pressure amplitude of the other. The phases of the plane waves are referenced to a point at a constant distance from the measurement point such that with varying frequency different parts of this “standing wave” field will obtain at the transducers: e.g., pressure nodes and maxima and particle velocity nodes and maxima. Although these field characteristics will vary, the intensity will always be constant and will be the difference between the individual intensities associated with each plane wave had it existed alone. It is in this type of field that one cannot assume I,$,,to be small such that sin +, = &,. Indeed, if a node in the pressure spatial waveform is situated between the two transducers then +,,, will equal T radians if the two plane waves are of equal magnitude. Criteria and errors based upon equation (4) will apply for all possible
260
P. S. WATKINSON C
-5 iI% s VI .g -10 _o 0 -15
-2c
1 0
I I I I Ill 2
I
I
I
4
6
8
Frequency (kHz)
Figure 6. Error in measurement of net intensity of two unequal plane waves (PB = 0,8P,) travelling in opposite directions for the two idealized sound intensity instrumentation systems, for a 12 mm transducer separation in air.
values of $,,,. Figure 6 shows that the system with the better phase matching is a consistently better performer, yet over limited frequency ranges compensation between phase error and finite separation error yields a small total error from the not so well matched system. This situation is fortuitous and therefore cannot be relied upon under other circumstances.
-201 0
2
4
6
8
10
Frequency (kHz)
Figure 7. Error in measurement of component of intensity at a point in a field comprising two equal plane waves, one travelling parallel to the measurement axis and one travelling at right angles to this, for the two idealized sound intensity instrumentation systems, and a 12 mm transducer separation in air.
The errors shown in Figure 7 are for a field comprising two equal amplitude plane waves travelling in directions at right angles to each other. The measurement is made in the direction of one. All previous fields discussed in this section have constant intensity through space; this field has an intensity which varies in space. The measurement position is stationary with respect to the phase reference position so with increasing frequency the field will change at this point. The curves show the superiority of the better matched system except over certain frequency ranges where the error is larger than the range of the graph axes. Over these ranges the error will peak at +oo dB and corresponds to a field point with zero intensity. This shows that where the error is very large the intensity is insignificantly small compared with the rest of the field anyway.
ERRORS
IN INTENSITY
261
MEASUREMENT
These worked examples should give the reader some insight into behaviour of phase errors in a typical measurement system which will assist in determining some form of criteria for accepting or rejecting a sound intensity measurement.
3. COHERENCE
AND
RANDOM
ERROR
CRITERIA
Seybert [6] has discussed statistical errors in acoustic intensity measurements and has related random error to coherence by E = (l/&)[(l/~*)+COt*
($){(l-
r’)/2r’]l”“,
where n is the number of raw estimates and y2 is the coherence function. Seybert discussed the application of this equation in reference [7] and Dyrlund [8] introduced Lp - LI (which reflects $) as a parameter which indicates an expected higher random error (due to its dependence upon I,-~). There are four specific problems with using coherence as a criterion for assessing the quality of a particular measurement and thereby accepting or rejecting it: bias errors, knowledge (or rather lack of knowledge) of $, resolution of y2 in a two channel Fourier analyzer, and the assumption that the signals are of a random character in the derivation of equation (9). Bias error for spectral estimates has been discussed quite thoroughly in reference [ 141. The problem is that equation (9) predicts only a random error if the reason for y* being less than unity is due only to random error. If there are bias errors then the random error can be grossly over-estimated. The bias error is said to be proportional to the square of the derivative of the phase spectrum between the two pressure signals and therefore would be particularly large for measurements in, say, a standing wave or some other field with large phase changes with frequency. Estimates of cross spectrum and estimates of coherence are subject to this bias error and it is not clear how to evaluate an actual error from these estimates. The presence of such an error is characterized by an improvement in coherence for finer spectral resolution. It is good in many ways that a parameter such as coherence responds to bias error, but it is not good that it is not easy to distinguish random and bias errors so that one could use equation (9). Equation (9) contains the phase angle Q. In the general case the only way this is known is by measurement. If this measurement is greatly in error then circumstances could be such that the estimated error is predicted to be erroneously small. This circumstance should correspond to a particularly low coherence, but would require independent inspection of coherence. An automated system may not be designed with this facility and may unwisely rely upon inspection of the estimated random error. For measurements in field regions where + is particularly small then, for a given maximum error and number of averages, a coherence very close to unity may be sought. The existence of 1 - y2 in equation (9) means that there are significant differences in the estimated value of E for values of the coherence of, say, O-9, 0.99, O-999 and 0.9999, etc., each extra 9 after the decimal point decreasing the error by a factor of 10. Coherences extremely close to unity can physically exist between pressures at two closely spaced points in a sound field and will probably be limited by electrical noise in the instrumentation. There is also a limit due to the resolution of the value of the coherence due to the discrete arithmetic in whatever processor and software is in use, particularly in some two channel Fourier analyzers where resolution is sacrificed for speed and the assumption is made that resolution sufficient for a plot of coherence on a scale of 0 to 1 is acceptable. If the coherence is limited to confidence values less than 0.99999 then the random error
P. S. WATKINSON
262
may be over-estimated for + < 1”. This situation is not improved by increasing the number of averages used to estimate coherence. Lastly, equation (9) has been derived by assuming Gaussian random signal types. It is not clear if this equation is applicable to, say, deterministic signals, or a mixture of deterministic and random signals. Random error formulae for auto spectra and cross spectra are clearly in error for deterministic signals. If y2 = 1.0 is substituted into equation (9) then E = l/&i, which is wrong for a purely deterministic signal (E = 0). An alternative approach for the determination of random error is to calculate directly the mean and variance for a set of raw spectra. This will then yield random error whatever the signal type. To store all the raw spectra and then operate upon them would generally require an unreasonable amount of computer storage and time and would also preset the maximum number of averages. A compromise is to compute a running mean and variance, an option not available on current 2 channel FFT analyzers. The benefit is freedom from the uncertainties introduced by employing coherence to calculate random error. 4. CONCLUSION In this paper criteria for the assessment of errors in two transducer intensity measurements when the nature of the sound field in the region of the measurement points is not known have been discussed. The first criterion discussed is based upon phase mis-match. Present criteria usage is related to the effect of phase mis-match upon general field measurements and a number of specific fields have been analyzed here to show error behaviour. The second criterion is based upon coherence, a very commonly used error assessment parameter. Its relevance to error assessment for sound intensity measurements has been discussed and four uncertainties identified: the influence of bias errors, the dependence upon knowing the measured phase angle, which may be in error, the problem of resolving a coherence very close to unity, and the applicability of random error formulae to signals of type other than Gaussian random. It is proposed that instead of using coherence to calculate random error, this latter parameter can be calculated from the directly determined mean and variance of successive raw spectral estimates of intensity. ACKNOWLEDGMENT The work described in this paper was supported by the Procurement Ministry of Defence.
Executive of the
REFERENCES
ofSound and Vibration 75, 229-238. Finite difference approximation errors in acoustic intensity measurements. 2. S. J. ELLIOTT 1981 Journal of Sound and Vibration 78, 439-443. Errors in acoustic intensity measurement. 3. J. K. THOMPSON 1981 Journal of Sound and Vibration 78, 444-445. Author’s reply. 4. J. POPE and J. Y. CHUNG 1982 Journal of Sound and Vibration 82, 459-462. Comments on 1. J. K. THOMPSON and D. R. TREE 1981 Journal
“Finite difference approximation errors in acoustic intensity measurements”.
5. J. K. THOMPSON 1982 Journal of Sound and Vibration 82,463-464. Author’s reply. 6. A. F. SEYBERT 1981 Journal ofSound and Vibration 75, 519-526. Statistical errors in acoustic intensity measurements. 7. A. F. SEYBERT1982 Proceedings of Inter-Noise 82,119-122. Some observations on the application of the acoustic intensity method.
8. 0. DYRLUND 1983 Journal of Sound and Vibration 90, 585-589. A note on statistical errors in acoustic
intensity
measurements.
ERRORS
11.
12.
13. 14.
263
of Inter-Noise 83, 1071-1074. FFT intensity meters used for sound sources and acoustical environment characterisation. J. ROLAND 1982 fioceedings ofInter-Noise 82, 715-718. What are the limitations of intensity technique in a hemi-difhtse field. G. HUBNER 1983 Z+oceedings oflnter-hroise 83, 1043-1046. Determination of sound power of sound sources under in-situ conditions using intensity method-field of application, suppression of parasitic noise, reflection effect. S. GADE, K. B. GINN, 0. ROTH and M. BROCK 1983 Proceedings oflnter-Noise 83, 1047-1050. Sound power determination in highly reactive environments using sound intensity measurements. G. RASMUSSEN and M. BROCK 1983Proceedings ofthe 11 th Zntemational Congress on Acousrics 6, 177-180. Transducers for intensity measurements. J. S. BENDAT and A. G. PIERSOL 1980 Engineering Applications of Correlation and Spectral Analysis. New York: John Wiley and Sons.
9. J. C. PASCAL 1983 Proceedings
10.
IN INTENSITY MEASUREMENT