- Email: [email protected]

LETTERS MEASUREMENT

OF

PLANE

TO THE EDITOR

WAVE

ACOUSTIC

FIELDS

IN FLOW

DUCTS

Standing wave acoustic parameters, describing the acoustic field within flow ducts (e.g., wave components) can be conveniently derived by regarding a sufficiently long section of duct as an impedance tube. In principle, the analysis is straightforward, provided the waves remain plane. In practice, the classical method of measurement, which is based on an axial traverse to determine the position and relative magnitude of standing wave maxima and minima [l], suffers from a number of disadvantages. Flow and background noise can produce serious contamination of the data at pressure minima, while considerable experimental effort and time is required for data collection and analysis particularly when, as is often the case, the acoustic field is comprised of many discrete frequency components spread across several octave bands [l]. Furthermore, both the excitation and flow conditions must be maintained sufficiently constant throughout the time of the traverse, while the traverse rate must be carefully matched to the minimum bandwidth discrimination necessary to define each component. Finally the duct section chosen must be sufficiently accessible to perform the traverse. An alternative approach is described here, where the information is obtained by analysis of wall pressure records obtained simultaneously from a pair or an array of fixed points. This procedure minimizes most of the disadvantages just outlined, but requires facilities for fast digital data acquisition and processing. With one dimensional plane waves, letting the interface x = 0 represent a reflective termination, one can write for the incident and reflected waves, respectively, p+(x, t)=po+ expi([email protected]+x),

p-(x, t) = PO exp i(wt + p-x),

(1,2)

where pi and PO are the component (complex) wave amplitudes at x = 0, w is the radian frequency and p the appropriate complex wavenumber. Provided attenuation of the component waves remain small, then /? = w/c -icu = k -icu [2], where c is the local undisturbed sound speed relative to the gas. With sound transmission in pipes, the viscothermal attenuation coefficient (Y[Z], after Kirchhoff, can be expressed as cy = (l/ac)JvwTZ[l+(&-

l/Jy)J1Ipr],

(3)

where a is the pipe radius, v is the local kinematic viscosity, y the ratio of the specific heats and Pr the Prandtl number for the gas in the pipe. Finally, the appropriate wavenumbers in equations(1)and(2)aregivenbyPt=P/(1+M)andP-=P/(1-M),withMthevalueof the mean flow Mach number. Given two wall pressure records pi(r) and p2(f) obtained at positions x1 and x2 a distance 6 apart, one can easily show that the Fourier coefficients of the wave components at x1 are given by P2 - PI exp (ip-S) p_ _ P2-P1 exp (-$+a) P: = (4,5) exp (-i/3+6) - exp (ip-S) ’ ’ -exp (iP_6)-exp ([email protected]+a)’ In performing the evaluation of the components from equations (4) and (5) it is convenient to use Fourier methods and special programmes can be easily written to perform the required operations digitally. 539 0022-460X/80/200539

+ 04 $02.00/O

@ 1980 Academic Press Inc. (London) Limited

540

LETTERS

TO

THE

EDITOR

The magnitude of the denominator of equations (4) and (5) varies cyclically with frequency between zero and two. Furthermore, the minima occur when pS = rzn( 1 -M’), n = 0, 1,2, etc. Ideally this should not introduce any difficulties in the analysis since the ratio of numerator to denominator always remains finite, by definition. However, in practice, the pressure records and their Fourier coefficients are always contaminated in various ways. These factors include transducer calibration and response, signal transmission channel bias, electronic and flow noise and digitization noise, etc. The effects of all these factors on the reliability of the results should be minimized by careful calibration, but some residual errors will always remain. It can be shown [3,4], that any such small errors will be magnified by a factor inversely proportional to the magnitude of the denominator in equations (4) and (5). If confidence in the estimates obtained of P: and PT is to be maintained at the same level as for the relative values of the Fourier coefficients of the original signal records pr(t) and pz(t), reliable decomposition into the wave components P: and P; can only be obtained when sin pS > 0.5. Thus a single pair of records from fixed points, separated by an axial distance S, can be expected to give reliable estimates of the wave components with n + l/6 < kS/n < (n + 1) -l/6, n = 0, 1,2, etc. Experience suggests that the first continuous band provides better estimates than the later ones, as might be expected. This means that one fixed transducer pair provides reliable values for the wave components for a band covering just over two octaves. If the bandwidth to be analyzed exceeds this, a further transducer spaced at S/4 from x1, and sampled simultaneously, will extend the continuous bandwidth to more than four octaves, and so on. Measurement of the characteristics of an acoustic field comprising many discrete frequencies also requires some care and consideration [3,4], in the choice of signal sampling rate and of the sample record length that is processed digitally. Failure to do so can result in the generation of sets of additional spurious components in the resultant wave component spectra, while contamination of all the components recovered is also possible. As well as the characteristics of the signals representing the acoustic field in the duct, the precautions necessary, with the constraints on the appropriate choices, depend on the characteristics of the computer system used for the digital processing. Sufficient information exists in the literature to highlight many of the problems that can arise, as well as the precautions that will be necessary to avoid their occurrence. Since the details of the precautions required will depend both on the nature of the acoustic field and on the facilities offered by the machine used for processing, there seems little value in pursuing this subject further here. Direct comparison with standing wave traverses [3], for discrete frequency excitation in a standing wave tube, showed that both methods of measurement gave similar results at low flow Mach numbers so long as the standing wave ratio was less than about 28 dB. Even so, it was necessary to use a narrow band filter (6%) during the traverse to reduce flow noise contamination near a standing wave minimum. Such precautions do not seem to be necessary with the signal decomposition approach, described here. This is illustrated in Figure 1, which shows such a comparison with a standing wave ratio of about 30 dB and with one wall microphone (No. 1) situated near a standing wave minimum. The results also indicate the level of uncertainty normally found in the measurements at the standing wave minima, during traverses made when the standing wave ratio was higher than 28 dB. A further check on the reliability of the new technique is provided by the comparison, shown in Figure 2, between the measured wall pressure at position 3 of Figure 1 and that predicted by translating the components found at position 1 by using equations (1) and (2). There are some slight discrepancies between the two waveforms which arise mainly from noise in the measured pressure signal. This result shows that a measurement with a third

LETTERS

TO THE Microphone

positions

2

3

I

, I

6

I

-0.6

It

I

541

EDITOR

I

11

-0.4

I

I

I1

-0.2

Axial

distance

X0

(m)

Figure 1. Standing wave pattern at 1.4 kHz, M = 0.06, tube diameter standing wave traverse; filtered background noise 94 dB.

37 mm. -,

Wave decomposition;

a,

transducer can always be employed, if desired, to provide a sensitive check on the reliability of the decomposition into wave components. The decomposition technique has also been applied with success [5] to acoustic field measurements in automotive engine exhaust pipes, where many harmonics exist simultaneously [l]. Here the difficulty in maintaining experimental conditions constant for long enough reduces the feasibility of the standing wave technique, while the experimental effort necessary renders it impractical. One should note, however, that a standing wave measurement gives the wavelength of the sound waves and, if the traverse extends over several wavelengths, permits a fair estimate to be made of the attenuation coefficient. The wave decomposition technique

Time

Figure 2. Predicted

signal at position

3 of Figure

1, compared

with measured

signal. +, Predicted;

x, measured.

542

LETTERS TO THE EDITOR

requires the value of the complex wavenumber /3, so that the local sound speed and attenuation constant must be determined in advance. In a practical situation however, the additional analytical complexity and necessity for computer facilities are more than adequately compensated by the substantial reduction in experimental effort compared with a standing wave traverse. Institute of Sound and Vibration Research, University of Southampton, Southampton SO9 5NH, England

P.O. A. L.DAVIES M. BHATTACHARYA J.L. BENTOCOELHO

(Received 7 May 1980)

REFERENCES 1. R. J. ALFREDSON and P.0. A. L.DAVIES 1970Journal of Sound and Vibration 13,389-408. The radiation of sound from an engine exhaust. 2. LORD RAYLEIGH 1894 The Theoryof Sound. London: Macmillan, second edition. See Articles 346-350. 3. M. BHATTACHARYA 1980 Ph.D.Thesis, University of Southampton. Engine noise source characterisation-an analysis and evaluation of experimental techniques. 4. J. L. BENTO COELHO 1980M.Sc. Thesis, University of Southampton. Study of the characteristics of acoustic elements in flow ducts. 5. J. E. TEMPLE 1980 M.Sc. Thesis, University of Southampton. An investigation into the source region characteristics of internal combustion engine exhaust systems.