The measurement of complex acoustical properties of homogeneous materials by means of impulse response in a plane wave tube

Transcription

1 The measurement of complex acoustical properties of homogeneous materials by means of impulse response in a plane wave tube Paolo Bonfiglio, Francesco Pompoli, Nicola Prodi Dipartimento di Ingegneria, Università di Ferrara, Italia, {pbonfiglio, fpompoli, nprodi}@ing.unife.it The knowledge of the complex properties of fibrous and porous materials has become of great importance in the design of quieter environments. Several techniques were developed for measuring the transmission coicient, the characteristic impedance and the propagation constant (known also as complex wave number) of these materials. The current work focuses on a technique based on the transfer matrix approach. According to this approach, the sound field is firstly decomposed into an incident, a reflected and a transmitted wave, and then the complex pressure and particle velocity at each side of the sample material are calculated. From these quantities the transfer matrix is obtained and finally the complex acoustical properties are calculated. A two microphone system was used in order to measure the acoustic properties of the material and the sound field was sampled by impulse response measurements. This approach employs an exponential sine sweep signal which allows us to separate the non-linear components of the measurement system, and so to select the linear system component only. A further benefit of the test signal is the high S/N ratio it can achieve. Finally experimental results of characteristic impedance, propagation constant, complex reflection and transmission coicient for fibrous and porous materials are reported, and a comparison with results obtained by using traditional experimental methods and theoretical models is presented. Introduction At present the measurement of the characteristic acoustical properties for locally reacting materials is done by means of several techniques essentially based on two different test lay-outs in a standing wave tube: the samples are either placed on a rigid termination (two layer method [] and two cavity method[2]) or a single sample is placed in the middle of the tube receiving plane waves (transfer matrix method [3], two source method [4] and two load method [5]). In this paper a transfer matrix based measurement device is proposed. By means of impulse response measurements in four different microphone locations, the apparatus obtains the characteristic impedance, the complex wave number and the complex transmission coicient of materials. The results are compared and discussed with both traditional experimental methods and theoretical predictions. 2 Theoretical background 2. Determination of the complex acoustical properties The transfer matrix method, as described in [3], is essentially based on the decomposition of incident and reflected waves, upstream and downstream of a sample, in a standing wave tube. By considering the measurement set-p shown in Figure, it is possible to relate the acoustical pressures P and the particle velocities V at both surfaces of the tested sample by using a transfer matrix formulation. By using the decomposition technique, the contributions of the incident and reflected acoustical waves (A, B, and D in Figure ) at both sides of the material are calculated through the measured complex pressures at the different microphone positions, as follow: jkx2 jkx j( Pe 2 ) A = (.a) 2sinK x x B = = D = ( ) 2 jkx jkx2 ( 2 ) j Pe 2sinK x ( x ) 2 jkx4 jkx3 ( 3 4 ) j Pe 2sinK x ( x ) 2 jkx3 jkx4 ( 4 3 ) j Pe 2sinK x ( x ) 2 (.b) (.c) (.d) Then these quantities allow the determination of the pressure and of the particle on both sides of the porous sample through the following expressions: 2597

2 x d P = A + B (2.a) x= ( ) ρ V = A B c (2.b) x= jkd jkd P e De x d = = + (2.c) jkd jkd ( ) ρ V = e De c. (2.d) = Furthermore, for homogeneous, isotropic and locally reacting materials, the validity of symmetry and reciprocity conditions [6, 7] is proved; consequently the transfer matrix can be expressed in terms of characteristic impedance (Z c ) and complex wave number ( k c ) of the material as follows: cos kd jz sin kd P P = jsin k d V cos k d V x= x= d Z in which d is the thickness of the sample. After the acoustical pressures at the different microphone locations are measured, it is thus possible to determine the pressures and the particle velocities on the two surfaces of the sample and, by means of (3), the characteristic impedance and the complex wave number are calculated. (3) The solutions are: T ( R R e = ) (5.a) 2 jkd T 2 2 jkd ( R T ) e R where R=B/A, T=/A and R =D/. The transmission loss is given by: = R T R T (5.b) TL = 2 log [db] (6) T 3 Materials and methods 3. Measurement set-up The test set-up consists in a mm diameter plane wave tube equipped with four microphones positions and the material under test in the middle. On one end there is a loudspeaker and the other consists of a 25 cm sample of sound absorbing material placed on a rigid backing, its acoustic absorption lies between.6 and.9 for frequencies between Hz and 3 Hz, and for higher frequencies. In Figure 2 the measurement tube and some details are shown. Figure : Measurement set-up 2.2 omplex transmission coicient and transmission loss The measurement set-up shown in Figure allows to calculate the complex transmission coicient (R ) and the transmission loss (T ) of homogeneous materials too. In fact it can be proved that both complex reflection and transmission coicients are determined, regardless of the end termination, by solving the system: R = R T = T + T + R R T R T e 2 jkd (4) Figure 2: Measurement tube and details The sound pressures at the four positions are measured by impulse responses, obtained by exponential sine sweep method. The experimental set-up consist of 4 ¼ condenser microphones, a P and a 4 channels audio sound card. For the signal generation and acquisition the software Adobe Audition was used. The post-processing was carried out by using Matlab codes. 3.2 Amplitude and phase calibration of the microphones The amplitude and phase calibration procedures of the 4 microphones are obtained by a series of cross measurements, without any test sample, at two pre- 2598

3 determined positions (e.g. positions 3 and 4 in Figure ) indicated L and R, respectively. Let consider measurement with microphones in L and 2 in R. Measured complex sound pressures are P L and P 2R. Repeat the same measurement with microphones interchanged. Measured sound pressures are P R and P 2L. The same procedure is applied to the microphones 3 and 4; as result we measure the sound pressures P 3L, P 4R, P 3R, and P 4L. Then it is possible to define 4 complex calibration pressures, one for each microphone, as follows: P = P P i=, 2, 3 and 4 (7) ic il ir In order to determine the calibrated pressures it is enough to divide measured pressures by relative calibration pressure. 4 Experimental results In this section experimental results about characteristic acoustical properties are shown. In order to validate the proposed method some results for the same material measured with the method of different thicknesses are reported. The theoretical value of the surface impedance and of the sound absorption coicient are compared with the respective experimental data. Finally measurements of complex transmission coicient and transmission loss for different systems are presented and discussed. 4. Normalized characteristic impedance and complex wave number for homogeneous materials The measurements were performed on different types of fibrous and porous materials. The purpose is to check the validity of the results here obtained, thanks to a comparison with the two cavity technique [2] and the semi empirical model of Delany-Bazley [8] Figure 3 shows a comparison of the results of characteristic impedance and complex wave numbers obtained by using the transfer matrix method for a polyester fiber material (density is 4 kg/m3 and thickness is 4 mm) with measurements made in an impedance tube by means of the two cavity method (air gap: 3 mm e 3 mm). The experimental results are compared with those obtained using the Delany-Bazley model, with a flow resistivity of 46 N s/m 4. As it can be seen, an excellent agreement is achieved by the transfer matrix method, both for the two cavity method and for the theoretical Delany-Bazley model. In particular it has to be noted that the present technique leads to results closer to the theoretical ones for frequencies above 3 Hz. Real Zc / ρ c Imag Zc / ρ c Real kc Imag kc Figure 3: Normalized characteristic impedance and omplex wave number: comparison between Transfer matrix method, and Delany- Bazley prediction 2599

4 4.2 Measurements of samples with different thicknesses Figure 4 shows the acoustical characteristic impedance and the complex wave number of three different samples of a material with density (4 kg/m 3 ) but having different thicknesses of 2 mm, 25 mm and 4 mm respectively. Real Zc / ρ c Imag Zc / ρ c Real kc Imag kc mm 25 mm 4 mm mm 25 mm 4 mm 2 mm 25 mm 4 mm 2 mm 25 mm 4 mm Figure 4: Normalized char. imp. and com. wave number measured for different thicknesses onsidering the materials as locally reacting and neglecting the difference in density of the single samples, the experimental curves should perfectly coincide. The results are well satisfactory above nearly 3Hz. In the range below one can find some discrepancies probably due to the mounting conditions. 4.3 omparison of surface impedance and sound absorption coicients Based on the experimental measurements of characteristic impedance and complex wave number, the curves for surface impedance and sound absorption coicient for normal incidence were reconstructed, for a material with rigid backing. The following formulas were used: Z = Z cot ( k d) (8) S 4 Re[ Z S ] ρc α n = (9) 2 2 Z + 2ρ c Re[ Z ] + S S ( ρ c ) The comparison between the data obtained from the formulas (8) and (9) using the characteristic parameters measured with the transfer matrix technique, and the data measured directly with the transfer function method is shown in Figure 5 and 6. Real Zs / ρ c Imag Zs / ρ c Transfer function Transfer function Figure 5: omparison between the normalized surface impedance estimated by using the transfer matrix method and transfer function method 26

5 α n Transfer function Figure 6: omparison between the absorption coicient estimated by using the transfer matrix method and transfer function method 4.4 Transmission loss measurements In Figure 7 an example of the complex transmission coicient (amplitude and phase respectively) for a polyester fiber sample (thickness 4mm and density 65 Kg/m 3 ) is reported. The figure includes the prediction of the complex transmission coicient obtained by the semi-empirical model of Delany-Bazley [8]. mod(t) phase (T) [rad] Experimental Delany-Bazley Experimental Delany-Bazley thickness mm) is shown. In the graphic the theoretical curve obtained by using the normal incidence mass law is also reported. It should be observed that the comparison between the plaster sample and theoretical prediction is good, although the boundary condition of the tested sample. A layer of rubber was applied along the perimeter of the sample in order to ensure an adequate lateral constrain. In the frequency range around 2 Hz lower values of TL were found, probably because of the resonance of the entire system (i.e. plaster and rubber). TL [db] TL Mass law Figure 8: Transmission loss for a plaster sample: comparison between experimental results and mass law prediction. It has to be noted that for frequencies above 7 Hz the TL curve shows an irregular behaviour. This is due to the low S/N ratio at both the sides of the sample. In fact, in that range the sound reduction is more than 4 db and consequently an higher dynamic range is necessary. In order to achieve the suitable S/N ratio, a different pre-amplification of the microphone should be used. In Figure 9 the results of TL for a non-homogeneous system, made up by rigid aluminium membrane (7 µm thick) and a polyester fiber layer (5mm thick) are shown. The measurements were carried both with the membrane and with the polyester facing the loudspeaker. This was done in order to verify the validity of the (3) for highly non-homogeneous systems. The results are satisfactory over all the frequency range. It has to be remarked that TL can be measured with a higher precision with the two load method [5]. Figure 7: omplex transmission coicient (amplitude and phase) for a polyester fiber sample. In Figure 8 the curve of transmission loss (TL) of a plaster sample (whose density is 23 Kg/m 3 and 26

6 TL [db] M - F F - M Figure 9: Transmission loss for a non-homogeneus system 5 onclusions The present work has shown an apparatus for the determination of the characteristic impedance and of the complex wave number in the case of homogeneous materials placed in a plane wave tube. The system is based on established theory but the approach here presented gives substantial advantages. In particular the robustness of the signal processing is achieved by the swept sine technique and the time-saving in the execution of measurements is due to a multichannel acquisition. The comparisons with the experimental results obtained by means of the two-cavity method and with the theoretical values calculated by using Delany-Bazley model are quite satisfactory. Furthermore results for samples of the same material having different thicknesses were considered. The experimental and theoretical values of surface impedance and sound absorption coicient have been compared and the validity of the apparatus was further confirmed. In addition, the measurement system allows the determination of the complex transmission coicient and of the transmission loss for homogeneous materials and multilayer systems. For the tested materials the results are in satisfactory agreement with theoretical predictions and the sample reversibility was also considered as a limit application to the above approach. Further results in this directions are expected in the future. References [] M. A. Ferrero, G. G. Sacerdote, Parameters of Sound Propagation in Granular Absorption Materials, Acustica,, pp (95) [2] H. Utsuno, T. Tanaka, T. Fujikawa, A. F. Seybert, Transfer Function Method for Measuring haracteristic Impedance and Propagation onstant of Porous Materials, J. Acoust. Soc. Am., Vol. 86 (2), pp (989) [3] B. H. Song, J. S. Bolton, A transfer matrix approach for estimating the characteristic impedance and wave numbers of limp and rigid porous materials, J.Acoust. Soc. Am., Vol.7 (3), pp (2) [4] M. L. Munjal, A. G. Doige, Theory of a Two Source-location Method for Direct Experimental Evaluation of the Four-pole Parameters of an Aeroacoustic Element, Journal of Sound and Vibration, Vol. 4 (2), pp (99) [5] T. Y. Lung, A. G. Doige, A Time-averaging Transient Testing Method for Acoustic Properties of Piping Systems and Mufflers, J. Acoust. Soc. Am., Vol. 73, pp (983) [6] J.F. Allard, Propagation of Sound in Porous Media, Elsevier, Applied Science, London and New York (993) [7] A.D. Pierce, Acoustics: An Introduction to its Physical Principles and Applications, McGraw- Hill, New York (98) [8] M. E. Delany and E. N. Bazley, Acoustics properties of fibrous absorbent materials, Applied Acoustics, Vol. 3, pp (97). Acknowledgments G. Amadasi of S..S ontrolli e Sistemi s.r.l and P. Zambusi of AVINTEH s.r.l are acknowledged for the support to the project. The system is now on the market as SS92B/TL. 262