Calculation of spatial sound intensity distribution based on synchronised measurement of acoustic pressure

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1 Calculation of spatial sound intensity distribution based on synchronised measurement of acoustic pressure Witold Mickiewicz, Michał Jakub Jabłoński West Pomeranian University of Technology Szczecin Faculty of Electrical Engineering Szczecin, ul. Sikorskiego 3 witold.mickiewicz@zut.edu.pl Michał Pyła West Pomeranian University of Technology Szczecin Faculty of Maritime Technology and Transport Szczecin, Al. Piastów 4 mpyla@zut.edu.pl Abstract The paper describes a new approach in sound intensity measurement. Nowadays sound intensity has become a popular parameter which describes the sound field. Current methods which lead to sound intensity are generally based on two microphone probes or hot wire anemometry. Using the double microphone approach involves some problems with matching of the used transducers. Where the measured phenomena are excited with a known signal which is repeatable there exists a possibility of applying one microphone method. That method eliminates the necessity of matching two microphones. This goal is achieved by using resampling of measured data and synchronisation of pressure data in all measurement points of our volume of interest. Some notes on that proposed method has been presented in that paper. I. INTRODUCTION Imaging of spatial distribution of sound field parameters is useful in many branches of technology. In case of socalled regions of far field parameter that describes sufficiently properties of the field is acoustic pressure measured in several points by a measurement class microphone. Generally measurement in each point is independent and on that basis we calculate some time averaged parameters like effective value of acoustic pressure or its level. In case of a detailed analysis, especially of acoustic sources in near field or waves that propagate in waveguides, due to its strong impact of interference phenomena that influence the propagation of acoustic signal, imaging of sound intensity vector seems to be a better approach [7][6]. Measurement of that parameter is far more complicated. It requires simultaneous measurement of two physical parameters: acoustic pressure and acoustic velocity, moreover also requires applying some computational algorithms. Over last 4 years many measurement methods have been introduced and proven but still the most common from the industrial point of view seems to be the method based on pressure-pressure probe (p-p probe) [2]. That type of probe consists of two matched microphones placed in some distance and it allows to measure the normal component of the sound intensity vector using the simultaneous observation of time dependent pressure waveform in two point of space. Imaging of the spatial distribution of intensity vector requires also scanning of the volume of interest point-by-point and performing a measurement in each of those points. That kind of process currently is automatised and the probe is displaced by industrial robot. Application of numerically controlled robot in that type of measurement enables high accuracy of positioning and high accuracy in following the scanning trajectory. The topic of that paper is proposal of a new measurement procedure of sound field characterisation excited by a electroacoustic source using only one microphone (or miniature microphone probe) that scans with high accuracy the volume of interest together with synchronous acquisition of microphone signal and the exciting signal. Proposed method eliminates the problem of precise microphone pairing and guarantees possibility of measurement of high levels of sound pressure limited only by maximum parameters of the microphone which permits levels up to 68 db (microphone probe BK47). Proposed method can be also a complement to visualisation of sound field by particle image velocimetry. From the previous work authors have got some experience in measurements of excited sound fields in waveguides measured with other type of transducer (pressure velocity probe) [9]. Avoiding getting in details of principles of operation of that type of probes, they have one significant advantage comparing to p-p probes, it is very small (5mm x 5mm x 5mm) and the three components of acoustic particle velocity are measured with application of hot-wire anemometry. That approach has one fundamental disadvantage, it is not suitable for measurement of sound field where sound pressure exceeds level of db and velocity 3dB. In case of visualisation of non-linear acoustic phenomena is a significant limitation. II. BACKGROUND OF SOUND INTENSITY MEASUREMENT USING PRESSURE GRADIENT METHOD Measuring the sound intensity is more difficult and complex that measuring sound pressure. Every imperfection of the measuring equipment will have a significant impact on the results. Instantaneous sound intensity is defined in equation 2, where p is acoustic pressure and u is acoustic velocity /3/$3. 23 IEEE 696

2 I(t) = p(t) u(t) () In most cases the active part of the sound intesity is measured which is defined as [5]: I a = T T p(t) u(t)dt (2) Since the velocity measurement in implicit form is quite complicated, the gradient pressure two microphone technique is used in most of the cases. The p-p method uses two microphones, which are closely spaced and separated with a spacer of a given length (fig.). Every microphone measures sound pressure and using Euler s relation we are able to calculate the acoustic velocity (eq.3). p = ρ o u n t That leads to eq 4 where p and p 2 are the sound pressure signals from two separated microphones by r on n direction. û n (t) = t (3) p 2 (τ) p (τ) dτ (4) ρ r Pressure between the two microphones in the middle between two probes is estimated as ˆp = p + p 2 2 And that leads to the final equation (eq. 7) passing through an intermediate step eq. 6. (5) Î n = ˆp(t)û n (t) (6) I n (t) t 2ρ r [p (t) + p 2 (t)] [p (τ) p 2 (τ)] dτ (7) Fig.. Schematic of p-p probe, where microphone A provides measurement of p and B provides p 2 In some sound intensity analysers principle of crossspectrum is used to measure the intensity in selected frequencies (ω). Î n (ω) = ω ρ r Im {S 2(ω)} (8) where ρ is air density and S 2 is cross-spectrum of p (t) and p 2 (t) [][3]. As we see in formulas 7-8 in order to obtain intensity values simultaneously measured time evolution of instantaneous value of pressure in both points is needed. III. REQUIREMENTS FOR P-P PROBES FOR SOUND INTENSITY MEASUREMENT One of the main factors that influences the actual accuracy of the measurement done by a p-p probe is the level of sameness of applied microphones. Ideal fulfilment of that criteria in case of real devices in never possible. The main source of error resulting from the difference between the sensors is the unequal sensitivity and phase shift characteristics. In order to estimate the quality of matching of the microphones used in p-p probe an objective parameter pressure-intensity index (PI index) has been introduced [8]. There exist many ways of compensation of the errors caused by the nonequality of the sensors. Generally it is made by performing a series of consecutive measurements with the probe in that the microphones had been exchanged or by applying some complex calibration procedures that permit to estimate the error. In the next step the mentioned error is taken into account during the calculation phase [4]. Nowadays the calculation of sound intensity process is made using a method based on two channel analyser and cross spectrum method. Independently of all mentioned procedures always the microphones are separated by a suitable spacer which has to be chosen in accordance to the measured frequency band. The BK company guarantees for their pair of microphones for pp 497 phase matching of inch microphone pair Type 497 is better than.5 between 2Hz and 25 Hz and is better than (f /5) at higher frequencies where f is the frequency. Such phase matching is possible as a result of the integral microphone phase-corrector units. The normalised microphone frequency responses differ by less than.2 db up to khz and by less than.4 db up to 7.kHz. In datasheet of matched pair of microphones for sound intensity 5AI we can see that the GRAS company guarantees achievement of the PI coefficient about 29dB, which means that the phase mismatch error at 7Hz frequency around.2 []. IV. MEASUREMENT OF SYNCHRONIZED ACOUSTIC PRESSURE DISTRIBUTION In many cases of acoustic measurements, the averaged distribution of sound intensity is calculated for known excitation source. It is assumed that the energy propagation is in given conditions a stationary process. In that sort of cases there is a possibility to calculate the sound intensity by only one microphone which is sequentially displaced (fig. 2). In order to perform such measurement in each of desired points of the volume of interest the same excitation signal is emitted and the signal from single channel together with the synchronisation signal is recorded. The data recorded that way is now synchronised and basing on the synchronised pairs of signals that represent two points in the volume of interest calculate sound intensity in the exactly the same way as in case of real p-p probe. The main problem in that type of procedure is the synchronisation accuracy, which is related to the sampling rate which offer current digital systems suitable for audio measurements. Generally the sampling rates are in 697

3 range from 4 ks/s to 2 ks/s. In case of synchronisation with a tone of frequency of 7Hz and standard sampling rate 44.kS/s it means that the phase synchronisation resolution is 5.7 which is a too small value to provide high accuracy and precision of sound intensity component. Below results of some research have been presented which generally consist of an increasing of synchronisation accuracy without sampling rate change during the measurement but, a suitable post processing which involves some resampling and smoothing procedures and statistic computation of the shift is applied. In order to estimate the accuracy of used algorithms an experiment was done, which aim was to obtain normal component of the averaged vector of sound intensity in function of the distance between the end of a square waveguide (intersection 7cm x 7 cm). The measurement was limited to 88 measurement points which are on the symmetry axis of the waveguide. At the opposite end of the waveguide a dynamic loudspeaker was placed which emits a tone of frequency of 7Hz. To measure a double microphone probe p-p 5AI manufactured by GRAS was used []. During the measurement a simultaneous acquisition of microphone signals, synchronisation signal and the value of sound intensity calculated by the Norsonic RTA-84 analyser were recorded. Time of registration in each point was 2s. Fig. 2. Displacement of measurement microphone. Upper part - measurement microphone in position of microphone a in p-p probe. Lower part - measurement microphone displaced to position of microphone B in p-p probe. Distribution of the value of sound intensity along the symmetry axis is shown in figure 3. As we can see, the sound intensity vector does not change the direction, and its value oscillates around 2 db. We can suppose that the change of normal value of sound intensity vector in that region is caused by the interaction between the wave that propagates from the source and the wave reflected in the region of jump of impedance related to the end of the waveguide. V. SYNCHRONIZATION REFINEMENT ALGORITHM The required accuracy of synchronisation can be estimated on the base of parameters given in the previous section. That means that in case of presented microphone the phase accordance has to be at least f/5 = 7/5 =.4 Thus the required sampling rate should be at least /.4 =.8MS/s [db] Fig Sound Intensity Measurement points Profile of the sound intensity level in the measured line which is a high frequency taking into account that we have to use some high resolution converters which can provide sufficiently dynamic measurement. The required accuracy of synchronisation can be also estimated from PI index (GRAS calibration procedure) and for its value of 29dB the phase mismatch error at 7Hz frequency should be around.2 [8] and []. In comparison to the previous value we see that we have to increase the sampling rate more 7 times. If we consider both conditions in the category of oversampling related to 44.kHz sampling rate we see that the results are highly divergent and fluctuates between 4 times and even 285 times oversampling. In presented experiment we have set the task of examining the relation between error of one microphone measurement compared to two microphone measurement in function of the level of resampling of data used in the synchronisation of data recorded from many measurement points. Of course, we have to be conscious that the error of synchronisation it not the only source of the measurement error. One of important issues related to the summary error is the change of the placement of the probe, which induces other disturbances of acoustic field. The relation which we are looking for will asymptotically converge to the error resulting from other phenomena than the phase error. VI. OPTIMISATION AND ERROR ANALYSIS In order to compute the relationship between the measurement error and the level of oversampling the authors propose the following method of procedure. In the first step we calculate the vector of phase displacement taking as a base the synchronising signal of the measurements in the consecutive points. For that purpose the measured synchronising signal is converted into ideal square wave by applying a threshold operation. In the square signal we search for the first rising slope. In order to increase the accuracy of estimation of the shift we use a priori knowledge and divide the synchronising signal into sections of 2 periods and for each section we calculate the number of sample that describes the exact time moment of the rising slope. Doing like that we obtain a set of non-ideally equal values which we obtain the searched value of 698

4 .5 Raw pressure data and presented in figure 6. ɛ(l) = n I pl (9) U [V] time [samples] Fig. 4. Raw pressure data, before synchronisation process where ɛ(l) is the error for each resampling coefficient l and is the reference value obtained from p-p procedure, while I pl is result from each resampling all calculated in n points of space. Analysing the figure 6 we can assume that the the sufficient resampling level is and for that value we have processed all date set. In figure 7 we can see sound intensity obtained from p-p measurement and also from one-microphone method. As we can see the error has not exceeded the range of 2dB. The error has its peaks in the places of the highest change of value..5 Synchonised pressure data.48 error fit.46 U [V] error [db] time [samples] resampling Fig. 5. Synchronised waveforms of pressure in measurement points Fig. 6. Total error values and its approximation between p-p two-microphone and one-microphone method in function of applied resampling level shift by applying procedure of linear regression. The obtained value we round to integer number of samples in oversampled signal. The vector of shifts which we have calculated then is used to synchronise the measurement data and to calculate the sound intensity value in each point. To achieve that we upsample with a given oversampling coefficient, then we apply smoothing and shift the data according to the recently calculated vector of shift. The next step includes single downsampling procedure. Data prepared in that way is used to calculate the sound intensity using the standard algorithm for two microphone probe. For given measurement point the signal from microphone a represents synchronised signal truly originating from a microphone, while the signal from B microphone represents a synchronised signal from a microphone but for the position which corresponds to shift which is in accordance to the spacer used before in the p-p probe. Figure 4 and figure 5 show examples of data before and after synchronisation procedure. The described algorithm was run for resampling parameter from to 25. The total error was calculated using equation 9 [db] Sound Intensity Measurement points Fig. 7. Comparison of sound intensity levels obtained from p-p () method and from one-microphone method (I p) I p 699

5 VII. CONCLUSION In the paper a one-microphone sound intensity method was presented. The method is based on synchronisation of pressure signal with the external excitation. The new method might be used in situations where we measure sound intensity and we use one defined signal to excite the object of interest. The presented method eliminates problem of matching and differences between microphones. Proper phase accuracy is obtained in postprocessing so there is no need to use higher sampling rate during the measurement. Increment of measurement error might be eliminated by virtual spacer adjustment. This will be covered in the next stage of our research. REFERENCES [] F.J. Fahy, Measurement of acoustic intensity using the crossspectral density of two microphone signals, J. Acoust. Soc. Am., 62(L), 5759 (977). [2] F.J. Fahy, Sound Intensity, 2nd edition London, England: E&FN Spon, 995. [3] J.Y. Chung, Cross-spectral method of measuring acoustic intensity, Research Publication, General Motors Research Laboratory, GMR-267, Warren, Michigan (977). [4] J.Y. Chung, Cross spectral method of measuring acoustic intensity without error caused by instrument phase mismatch, 2nd edition London, J. Acoust. Soc. Am. Volume 64, Issue 6, 978. [5] F. Jacobsen, A note on instantaneous and time-averaged active and reactive sound intensity, J. Sound Vib., 47, (99). [6] G. Rasmussen, Measurement of vector sound fields, Proc. 2nd Int. Congr. Acoustic Intensity, pp (985). [7] S. Weyna, An Image of the Energetic Acoustic Field in a Parallelepipeded Room Models, Acta Acustica Intern. Journal on Acoustics, Vol. 82, 996, 72-8 [8] F. Jacobsen, Sound Intensity and its measurement applications, Lyngby, Denmark: B&K 2. [9] W. Mickiewicz, M. Jablonski, M. Pyla Automatized system for 3D sound intensity field measurement, MMAR Conference: Miedzyzdroje, 2. [] Sound-intensity Probe Type 5AI - datasheet, G.R.A.S 26. [] Sound Intensity Probe Kit - Type datasheet, B&K 26. 7