Bendat J. S.

Finn Jacobsen

Gade S.

Kuttruff H.

Morse P. M.

Polack J.-D.

Yu Luan

The acoustic Green's function can be measured using a device with two matched microphones mounted in a tube driven by a loudspeaker combined with another microphone that represents the observation point. Good agreement is obtained between the measured and theoretical Green's function in a rectangular room below 320 Hz. At higher frequencies the agreement is less good because of the imperfect geometry of the room

Luan, Yu

Jacobsen, Finn

Online Research Database In Technology

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Downloaded from orbit.dtu.dk on: Dec 17, 2017A method of measuring the Green’s function in an enclosureLuan, Yu; Jacobsen, FinnPublished in:Journal of the Acoustical Society of AmericaLink to article, DOI:10.1121/1.2917804Publication date:2008Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Luan, Y., & Jacobsen, F. (2008). A method of measuring the Green’s function in an enclosure. Journal of theAcoustical Society of America, 123(6), 4044-4046. DOI: 10.1121/1.2917804A method of measuring the Green’s function in an enclosure (L)Yu Luana and Finn JacobsenAcoustic Technology, Department of Electrical Engineering, Technical University of Denmark,Building 352, Ørsteds Plads, DK-2800 Kgs. Lyngby, DenmarkReceived 10 January 2008; revised 8 April 2008; accepted 8 April 2008The acoustic Green’s function can be measured using a device with two matched microphonesmounted in a tube driven by a loudspeaker combined with another microphone that represents theobservation point. Good agreement is obtained between the measured and theoretical Green’sfunction in a rectangular room below 320 Hz. At higher frequencies the agreement is less goodbecause of the imperfect geometry of the room.© 2008 Acoustical Society of America. DOI: 10.1121/1.2917804PACS numbers: 43.55.Mc, 43.20.Rz, 43.20.Ks, 43.58.Fm AJZ Pages: 4044–4046I. INTRODUCTIONThere is a growing need, e.g., in the automotive industryfor experimental determination of acoustic transfer functionsin connection with analysis of complicated sources. Thetransfer function of concern, henceforth called the Green’sfunction, is the complex ratio of the sound pressure at agiven position to the volume acceleration of a monopole atanother position. The purpose of this letter is to examine asimple technique for experimental determination of theGreen’s function based on pressure microphones mounted intube driven by a loudspeaker. The technology is commer-cially available and has been validated in an anechoic room.1In this letter the performance of the device is evaluated in alightly damped room.II. OUTLINE OF THEORYThe acoustic Green’s function is the solution to the in-homogeneous wave equation with a mass injection sourceterm and given boundary conditions,2Gr,r0 =prjQ , 1where pr is the sound pressure at r, Q is the volume ve-locity of a harmonic monopole at r0,  is the radian fre-quency, and  is the density of air. Experimental determina-tion of this quantity obviously involves measuring thefrequency response between the volume velocity or accel-eration of a small source and the signal from a pressuremicrophone. However, ordinary loudspeakers do not providea signal proportional to their volume velocity.Figure 1 shows an arrangement for determining the vol-ume velocity at the opening of the tube, Q, driven by aloudspeaker at the other end. The sound pressure signals pAand pB are measured at a distance of l+l and l from theopening with matched quarter-inch microphones. The tubesection has a diameter of 4 cm, from which it follows thatthe nonaxisymmetric 1, 0 and 2, 0 modes can propagateabove 5.0 and 8.3 kHz, respectively.2 However, since the mi-crophones measure the sound pressure on the axis of the tubethese modes, which do not contribute to the volume velocityat the opening, are not detected. The cut-on frequency of thefirst axisymmetric mode is 10.5 kHz.In the frequency range where it can be assumed thatonly plane waves are measured, the entire sound field can bedetermined if the sound field is sampled at two positionsunless they are spaced a multiple of half a wavelength,3 andthe volume velocity at the opening of the tube can easily beshown to beQ = Sc·pA cos kl − pB coskl + lj sin kl , 2where S is the cross-sectional area of the tube, c is the speedof sound, and k is the wave number.1 It now follows that theGreen’s function isGr,r0 =pCjQ =Q*pCjQ2 3where pC= pr. Expressed in terms of frequency responsesbetween the three pressure signals the Green’s function be-comesGr,r0 =sin klkSHAC cos kl − HAB2HBC coskl + lcos2kl − 2 ReHABcos kl coskl + l + HAB2 cos2kl + l,4where the frequency responses HAB, HAC, and HBC are esti-mated in the usual manner from cross and auto spectra,4 e.g.,the frequency response between microphone signals A and Bis the ratio of the cross spectrum SAB to the auto spectrumSAA,HAB =SABSAA. 5Equation 4 can be simplified if the three signals can beassumed to be perfectly coherent. However, in measurementsin enclosures a poor signal-to-noise ratio of the signal frommicrophone C may occur at antiresonance frequencies.aAuthor to whom correspondence should be addressed; electronic mail:yl@oersted.dtu.dk.4044 J. Acoust. Soc. Am. 123 6, June 2008 © 2008 Acoustical Society of America0001-4966/2008/1236/4044/3/$23.00III. EXPERIMENTAL RESULTSTo test the method with a complicated Green’s functiona simple experiment has been carried out in a lightly dampedrectangular room with dimensions 3.294.383.29 m anda reverberation time of 5 s in the frequency range of concern,corresponding to a Schroeder frequency5 of 650 Hz. Thetube, and thus the room, was driven by a Brüel & Kjær“OmniSource” B&K 4295, a loudspeaker mounted in an“inverted horn” of hard plastic, and radiating through a smallopening.6 The volume of the inverted horn is about 2.3 l. Onthe source a “Volume velocity adaptor” of type B&K 4299was mounted. This device is a 10-cm-long tube of hard plas-tic that fits the OmniSource and makes it possible to measurethe sound pressure at two positions in the tube with amatched set of B&K 4178 1 /4 in. microphones micro-phones A and B in the foregoing. The dimensions l and lare 3 and 2 cm, respectively. Finally, the sound pressure atthe observation point was measured with a 1 /2 in. micro-phone of type B&K 4192 microphone C in the foregoing.The three frequency responses were measured using a B&K“PULSE” analyzer using a frequency span of 200 Hz, centerfrequencies between 160 and 600 Hz, and 3200 spectral linescorresponding to a resolution of 62.5 mHz. The PULSE ana-lyzer was also used for measuring the reverberation time ofthe room using the conventional interrupted noise method.No correction for possible phase and amplitude mismatchbetween the matched microphones was attempted, but theagreement between their responses was ascertained by mea-suring one time with the microphones interchanged.In what follows measured Green’s functions are com-pared with the theoretical function for an enclosure2,5:Gr,r0 = −1V m=0mrmr0k2 − km2− jk/mc, 6where V is the volume of the room, m is the mode shape asimple product of three cosines in a rectangular room, km isthe wave number corresponding to the natural frequency ofthe mth mode, and m is the time constant of the mode. Thecorresponding reverberation time is 13.8 times larger.5Figure 2 shows a typical example of a measured andpredicted Green’s function in the frequency range between60 and 260 Hz for a source position of 1.65, 2.19, 0.01 mand a receiver position of 1.65, 2.19, 1.04 m. As can beseen the agreement is generally quite good, although devia-tions occur around 160 Hz. It can also be seen that somepeaks are underestimated and some troughs are overesti-mated; and some peaks and troughs are shifted a little. Theunderestimation of a few peaks and overestimation of a fewtroughs are probably due to the fact that the broadband re-verberation time has been measured in the frequency rangeof concern, whereas the actual modal time constants of dif-ferent modes may vary. To determine the individual timeconstants would require measuring, say, 3 dB bandwidths ofall the modes; and this is only possible when the modes arewell separated. The small frequency shifts that occur, e.g., at175 Hz are in all probability due to the fact that the roomwas not perfectly rectangular.Figure 3 shows an example of a measured and predictedGreen’s function in the frequency range between 260 and460 Hz. The agreement is clearly less good, in particularbetween 320 and 350 Hz. This is undoubtedly due to theirregular geometry of the room the floor of the room was, infact, neither completely horizontal nor completely flat.Above 460 Hz the agreement not shown is in general nolonger acceptable, but there is no reason to expect this to bedue to a failure of the experimental arrangement.FIG. 1. Device for measuring the output volume velocity.FIG. 2. Measured and predicted Green’s function in a rectangular enclosureof 47 m3.FIG. 3. Measured and predicted Green’s function in a rectangular enclosureof 47 m3.J. Acoust. Soc. Am., Vol. 123, No. 6, June 2008 Y. Luan and F. Jacobsen: Letters to the Editor 4045IV. CONCLUSIONSA method of measuring the Green’s function has beenexamined in a rectangular enclosure and found to give resultsin good agreement with theoretical predictions in the fre-quency range below 320 Hz. The poor agreement at higherfrequencies is due to deviations between the assumed per-fectly rectangular geometry and the actual, slightly irregulargeometry of the room.1S. Gade, N. Møller, J. Hald, and L. Alkestrup, “The use of volume veloc-ity source in tranfer measurements,” in Proceedings of Inter-Noise 2004,Prague, Czech Republic, 2004.2P. M. Morse and K. U Ingard, Theoretical Acoustics McGraw–Hill, NewYork, 1968.3J. Y. Chung and D. A. Blaser, “Transfer function method of measuringin-duct acoustic properties. I. Theory,” J. Acoust. Soc. Am. 68, 907–9131980.4J. S. Bendat and A. G. Piersol, Engineering Applications of Correlationand Spectral Processing, 2nd ed. Wiley, New York, 1993.5H. Kuttruff, Room Acoustics, 4th ed. E & FN Spon, London, 2000.6J.-D. Polack, L. S. Christensen, and P. M. Juhl, “An innovative design foromnidirectional sound sources,” Acta. Acust. Acust. 87, 505–512 2001.4046 J. Acoust. Soc. Am., Vol. 123, No. 6, June 2008 Y. Luan and F. Jacobsen: Letters to the Editor

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A method of measuring the Green’s function in an enclosure

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