error propagation analysis of four sound-power measurement techniques

Error propagation analysis of four sound techniques

-power measurement

G. A. Russell

Department of Mechanical Engineering, University of Massachusetts, Amherst, Massachusetts 01003 (Received 7 August 1979; accepted for publication 28 August 1979)

Error propagation equations are derived for four different sound-power measurement techniques: hemispherical freefield, reference source, reverberation time, and two surface. The analysis shows the hemispherical freefield and two-surface methods to be governed by the same error equation. Of the four methods examined, the reference-source method is shown to exhibit the strongest dependence on errors in measuring the space averaged sound pressure. The results obtained can be used in planning and executing more accurate sound-power measurements.

PACS numbers: 43.85.Fm, 43.50. Cb

I NT RO DUCTI ON ' whe re

The measurement of the sound-power emitted by an acoustic source can be accomplished in several different ways. Whatever technique may be used--freefield, reverberation room, reference source, two-surface method, etc.---the sound power must ultimately be computed from other observed parameters. As such, the computed value of the sound power will have an uncer- tainW which is a function of the uncertainties associated with each of the variables actually measured (space- averaged sound pressure, distance, etc.). The manner in which these individual measurement errors influence

the error in the computed sound power can be examined by an error propagation analysis applied to the equation used to compute the sound power.

Since the different techniques used to find the sound power are based on different equations relating sound power to the experimentally observed quantities, the error propagation will depend on which measurement technique is used. In particular, the propagation of errors associated with the space-averaged sound-pressure level is of special interest as this variable is (1) required for all of the sound-power measurement techniques and (2) subject to non-neglibible errors due to practical considerations.

The purpose here is to derive the error propagation equations for four different sound-power measurement techniques: the hemispherical freefield method, the reference-source method, the reverberation time method, and the two-surface method; and to briefly discuss how these equations can be used to estimate the uncer- tainW in the computed sound-power level.

I. ERROR ANALYSIS

A. Hemispherical freefield method ,

In this method the source is placed on a reflecting surface in an otherwise freefield and the spatially averaged root-mean-squared sound pressure is measured over a hypothetical hemisphere surrounding the source. The emitted power, in the frequency band of interest, is computed from

w =2rYVp, (z)

W= emitted power, watts

r = radius of hemisphere, m

p = spatially averaged root-mean-squared sound pressure, N/m •'

pc= characteristic air resistance, N-s/m s .

Treating the characteristic air resistance, pc, as a single variable, Eq. (1) can be expanded in partial derivative form as

/x W_ 27rr22p 27r2rp 2 27rr •-p 2 pc (ap)+ (at) -• (apc) (2) pc (pc) •- '

where the differential quantities Ap, At, and apc repre- sent the variations, or estimated errors, associated with these measured variables. The a W variation is the ex-

pected error in the calculated power level due to the individual Ap, At, and Apc errors. Combing Eq. (1) with Eq. (2) gives the fractional error as

aw_ 2 a__r_r _ af. (3) W p r pc ß

Because of the stochastic nature of the individual mea-

surement errors, a root-mean-square summation gives a more valid estimate • of the expected error in W. That is, •,W/W is more appropriately expressed as

aW-[4(••+ 4 (•r--)•'+ (4) W \pc /j

Equation (4)defines the fractional error to be expected in the calculated acoustic power in terms of the frae- tional errors in each of the measured variables. This

result thus provides a convenient means for predicting the expected error in the calculated power level when the hemispherical freefield measurement method has been used.

B. Reference sound-source method

When it is not practical to locate the source in a hemispherical freefield, a reference sound source (of known acoustic power) may be employed in a comparison procedure to measure the power emiffed by the source. To do this the source to be measured is located in a rever-

663 J. Acoust. Soc. Am. 67(2), Feb. 1980 0001-4966/80/020663-03500.80 ¸ 1980 Acoustical Society of America 663

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berant room and the space-averaged root-mean-squared- sound pressure p is measured over some appropriate surfaceø The source being tested is then replaced with the reference source and the space-averaged root-mean squared pressure p, is measured over the same spatial surface. The sound power of the source being tested (in a given frequency band) is then calculated from

where

W,=sound power of reference source, watts,

p = space-averaged root-mean-squared sound pressure for test source, N/m 2,

p,= space-averaged root-mean-squared sound pressure for reference source, N/m •'.

Expanding Eq. (5) in partial derivative form leads to

aw aw r ,• at,, = + 2--- 2 . (6) w w, • •,

It is resortable [o assume that [he fractional error in

measuring p is equal [o that associa[ed wi[h p• such [ha[, when expressed in a roo[-mean-square formal, Eq. (6) becomes

w = ••/ +8 . (•) This last result defines the fractional error to be expected in the calculated power from the source when the reference sound-source measurement technique is employed.

C. Reverberation time me•od

In this methodolo• the reverberation time (in each frequency band of interest) of the test room, with the source in place in the room, is measured directly. It is also necessary to measure the space-averted root- mean-squared pressure produced by the source in the room. The sound-power level is then computed from •

VWL = SVL+ •0 •ogV/T[i + (SS/8V)]- 13.5 dS, (8)

where

PWL=sound-power level, dB re 10 '•a W,

SPL = space-averaged root-mean-square sound pressure level, dB re 2 x 10 -s N/m z,

V= test room volume (less source volume), m 3,

T= test room (with source in place) reverberation time, s,

S= surface area of test room, m z,

X= wavelength at center frequency of test band, m.

If the errors associated with measuring S, V, and X are assumed to be negligible, Eq. (8) can be rewritten as

w= •x (p /•),

which leads directly to

__ + . (9) w

D. Two-surface method

In many situations it is necessary to make measurements of the emitted power with the source located in a less than ideal acoustic environment. The two-surface method may be employed to advantage in these situations. It is based on the equation (assuming the source to be located on a hard floor)

p2 =pcW(1/2vr 2 + 4/R ) , (10) where

r= radius of hemisphere over which p is space averaged, m,

R = room constant, m 2.

This equation is employed over two different test hem- ispheres of radii r• and r•. to give two versions of Eq. (10). The two equations are combined to eliminate the room constant R, giving

w_ 2• (P• -P•) (ll) pc (1/• - 1/•) '

where p• is the space-averaged pressure over the r• hemisphere and p•. the pressure obtained over the r•. hemisphere. Taking the partial derivative of Eq. (11) gives

W 2W (.2Ap• AP2) aW= pc (a•)+ (p•-pz) p,•-P•

2W (1 •r, 1 •) (12) -(•/• •/•) -•W•+.• • ß Now, if the fractional error in measuring p• (&p•/P•) is assumed to be equal to the fractional error in measuring P2 (&P2/P2) and the &r•/r• and &r2/r 2 errors are also assumed to be equal, Eq. (12) r•uces to

&W &pc --= _ +2 aP+ 2 --, W pc p r

which, when expressed in root-mean-square format becomes

aw= a• +4 +4 . w

Equation (13) defines the fractional error in the sound- power level when the two-surface measurement technique is used. It should be noted that Eq. (13) is idential to Eq. (4), the equivalent error expression for the hemispherical freefield methodology.

II. DISCUSSION

The four error analyses outlined above pertain only to the propagation of measurement errors in the calcula- tion of the sound-power level. While the analysis car- ried out for the four different test procedures can shed no light on the magnitude of the individual measurement errors, it does illustrate the relative importance of the various measurement errors. For example, in the

hemispherical freefield method, Eq. (4), and the two- surface method, Eq. (13), the fractional errors in p and r are twice as important as the pc' error contribution.

It is interesting to note how the error associated with

664 J. Acoust. Soc. Am., Vol. 67, No. 2, February 1980 G.A. Russell' Error propagation analysis 664


the space-averaged pressure influences the A W/W error in the different measurement techniques. With the hemispherical freefield and the two-surface methods the zip/p error is multiplied by a factor of 4; whereas, in the reference source method, Eq. (7), the zip/p mul- tiplier is 8. And in the reverberation time method, Eq. (9), the zip/p coefficient is unity. This observation suggests that if the other measurement errors were neglibible and if any of the four measurement procedures could be employed with the same zip/p error, then the reverberation time method would give the least fractional error in the computed sound power, W.

The error equations derived above illustrate the importance of the ap/p error term tn setting a lower bound on the ziW/W error. In particular, the ziW/W error can never be less than the ap/p error with the reverberation time procedure; tt can never be less than 2zip/p with the hemispherical freefield or two-surface procedures; and tt can never be less than • zip/p with the reference-source procedure. Beranek •' discusses some of the difficulties tn obtaining accurate spattally averaged sound-pressure measurements and suggests that an uncertainty of 1 or,2 dB tn the spattally averaged sound-pressure level can be expectedø

The following example illustrates how this uncertainty in sound-pressure level influences the ziW/W error. Assume that the two-surface method is used and that

the measured space-averaged sound pressure levels (over both surfaces) are accurate to within + 1.0 dB. This 1.0-dB variation implies that the sound pressure, p, can be between 0.79 and 1.26 times its nominal measured value such that Ap/p =0.26. If r and pc are both measured with an uncertainty of 1% (zipc/pc = zir/r =0.01), then Eq. (13) gives

ziW/W=[ (0.01)2+4 (0.20)2+4 (0.01)2]'/•' = 0.52,

for the fractional error in the computed sound power. A 52% error in sound power represents a deviation of approximately 1.8 dB in the sound-power level.

The same magnitude of zip/p error in the reference- source method, together with a + 1.0 dB uncertainty in the reference-source power level, will combine accord- ing to Eq. (7) to generate

ziW/W= [ (0.26) 2 + 8 (0.26)2] '/2 = 0.78,

or a deviation of about 2.5 dB in the computed sound- power level.

III. CONCLUSIONS

The error propagation equations which have been derived for four different sound-power measurement methodologies show how the individual measurement errors contribute to the total uncertainty in the calculated sound power. The magnitude of the sound-power error can be readily estimated for any of the measurement procedures considered, as illustrated by the examples given above. More importantly perhaps, the results of the error propagation analysis described here can be put to better use in planning experiments such that excessive errors in the computed sound power may be avoided.

1j. p. Holman, Experimental Methods for Engineers (McGraw- Hill, New York, 1971), p. 38.

2L. L. Beranek, "The Measurement of Power Levels and Di- rectivity Patterns of Noise Sources," in Noise and Vibration Control, edited by L. L. Beranek (McGraw-Hill, New York, 1971), pp. 138-163.

665 J. Acoust. Soc. Am., Vol. 67, No. 2, February 1980 G.A. Russell' Error propagation analysis 665


error propagation analysis of four sound-power measurement techniques

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