INTER-NOISE 2006 3-6 DECEMBER 2006 HONOLULU, HAWAII, USA
Measurement of transmission loss of materials using a standing wave tube
Oliviero Olivieria J. Stuart Boltonb Brüel & Kjær Sound and Vibration Taewook Yooc Measurements A/S Ray W. Herrick Laboratories Skodsborgvej 307, DK-2850 Nærum, Denmark School of Mechanical Engineering Purdue University 140 S. Intramural Drive West Lafayette, IN 47907-2031, USA
ABSTRACT In this paper a measurement procedure for evaluating the normal incidence transmission loss of noise control materials using a four-microphone standing wave tube and an FFT analyzer is described. The mathematical formulation is based on the transfer matrix representation, which has been widely used in the past both to analyze and to measure the acoustical properties of flow system elements. The method described here differs from some earlier approaches that use a similar experimental setup in that knowledge of the tube termination condition is not required to ensure the accuracy of the result. Although the sound power transmitted through a sample in the tube depends in general both on its properties and on the tube termination conditions, the transfer matrix elements, which are used to calculate the transmission loss, are properties only of the sample.
1 INTRODUCTION In this paper a measurement procedure for evaluating the normal incidence transmission loss of a plug of noise control material placed in a four-microphone standing wave tube is described. A transfer matrix representation, which has been widely used in the past both to analyze and to measure the acoustical properties of flow system elements (e.g., automotive mufflers) [1-2], is used here to relate the sound pressures and the normal acoustic particle velocities on the two faces of the sample. After a brief section on the theory underlying the transfer matrix formulation, a procedure for determining the transfer matrix elements is described. The procedure is similar to those described in Refs. [3-4], however it differs from that of Ref. [4] in that knowledge of the tube termination condition is not required. Finally, it is shown how the normal incidence transmission loss may be related in closed form to the transfer matrix elements, and the advantage of using such an expression for the transmission loss is explained.
2 EXPERIMENTAL SETUP We consider a one-dimensional acoustical system consisting of a finite-length, rigid-walled tube with arbitrarily-shaped, uniform inner cross-section. At one end, the tube features a loudspeaker providing broadband, stationary random excitation to the system. At the other end, the tube can be fitted with arbitrary termination conditions (including, for example, an open-
a Email address: [email protected] b Email address: [email protected] c Email address: [email protected] ended termination). The internal volume of the tube is divided into two sections by a plug (a partition) of the material under investigation, extending across the tube cross-section, as shown in Figure 1. The two faces of the plug are planar and perpendicular to the tube walls. An example of suitable experimental apparatus is shown in Figure 2.
Figure 1. Idealized representation of the experimental setup.
Figure 2. The Brüel and Kjær Transmission Loss Kit Type 4206T is suitable for both low frequency (50 Hz to 1600 Hz) and high frequency (500 Hz to 6400 Hz) measurements since it includes both 10 cm (shown here) and 2.9 cm inner diameter tube sections.
3 THEORY
3.1 Background At frequencies below the cutoff frequency for the first dispersive mode, only plane waves can propagate in the tubed. In that frequency range the sound field in the two sections of the tube can be well approximated by a superposition of positive- and negative- going plane waves. By adopting a complex exponential representatione and a coordinate system with its origin at the surface of the sample terminating the upstream section of the tube (see Figure 1), the sound pressure and particle velocity in the up- and downstream tube sections can be written as: Re {(A(ω)e− jkx + B(ω)e jkx ) e jωt } x ≤ 0 p(x,t) Re{P(x, ) e jωt } = ω = − jkx jkx jωt Re{(C(ω)e + D(ω)e ) e } x ≥ l − jkx jkx A(ω)e − B(ω)e jωt Re e x ≤ 0 (1) jωt ρ0c v(x,t) = Re{V (x,ω) e } = C(ω)e− jkx − D(ω)e jkx Re e jωt x ≥ l ρ0c
d Higher modes attenuate exponentially with distance, so they are not important at appreciable distances from the loudspeaker or sample [6-7]. e An e + jωt sign convention has been adopted. where Re{} means the real part, P is the complex pressure, V is the complex particle velocity, A to D are the complex amplitudes of the plane wave components, ρ 0 is the ambient fluid density, c is the ambient sound speed, ω is the angular frequency, k=(ω/c)-jα is the wave number, which is complex to account for the effects of viscous and thermal dissipation in the oscillatory, thermo-viscous boundary layer that forms on the inner surface of the ductf, and l is the thickness of the sample. From Eqs. (1), the coefficients A to D may easily be expressed in terms of complex pressures P1 to P4 at position x1 to x4, respectively, as shown in Figure 1. j(Pe jkx2 − P e jkx1 ) j(P e jkx4 − P e jkx3 ) A = 1 2 , C = 3 4 , 2sin k(x − x ) 2sin k(x − x ) 1 2 3 4 (2) j(P e− jkx1 − Pe− jkx2 ) j(P e− jkx3 − P e− jkx4 ) B = 2 1 , D = 4 3 . 2sin k(x1 − x2 ) 2sin k(x3 − x4 ) When a stationary (ergodic) random noise signal is used to excite the system, we can use the following equivalent formulae: j(H e jkx2 − H e jkx1 ) j(H e jkx4 − H e jkx3 ) A = G P1R P2 R , C = G P3R P4 R , RR 2sin k(x − x ) RR 2sin k(x − x ) 1 2 3 4 (3) j(H e − jkx1 − H e − jkx2 ) j(H e − jkx3 − H e − jkx4 ) P2 R P1R P4 R P3R B = GRR , D = GRR 2sin k(x1 − x2 ) 2sin k(x3 − x4 ) where the quantities A to D are now the estimated plane wave amplitudes whose phases are defined relative to a reference signal, R. Further, H is defined as Pi R H ( f ) = G ( f ) /G ( f ) : Pi R RPi RR (4) i.e., the frequency response functions between the complex sound pressures, Pi, and the complex reference signal, R. Finally, GRR is the averaged, one-sided auto-spectral density function of the reference signal, and G is the averaged, one-sided cross-spectral density function of the RPi reference signal and signal, Pi. In subsequent calculation of the transfer matrix elements, the quantities A to D always appear in ratios, with the result that GRR can be neglected.
4 TRANSFER MATRIX FORMULATION We assume that the partition of material under investigation can be described as a four-pole (two-port) passive, linear acoustic sub-system. Then, a transfer matrixg can be used to relate the exterior, complex pressures, P, and the exterior, complex normal acoustic particle velocities, V, on the two faces of the sample as follows:
P T11 T12 P = . (5) V x=0 T21 T22 V x=d
f A semi-empirical formula given by Temkin can be used to calculate the real and imaginary part of the wave number [8]. g Sometimes known as transmission matrix [6-7]. The quantities Tij, which are frequency-dependent quantities, may be directly related to the acoustical properties of the sample [3].
4.1 Two-load implementation Equation (5) represents two equations in four unknowns. In order to be able to solve for the transfer matrix elements, two additional, independent equations may be generated by introducing a second termination condition: i.e., (a) (b) (a) (b) P P T11 T12 P P = (6) (a) (b) T T (a) (b) V V x=0 21 22 V V x=d where superscripts (a) and (b) denote the two different termination conditions. The transfer matrix elements may then be easily determined by inverting the latter expression to obtain T T 1 11 12 = × (a) (b) (b) (a) T21 T22 P V − P V x=d x=d x=d x=d (7) P(a) V (b) − P(b) V (a) − P(a) P(b) + P(b) P(a) × x=0 x=d x=0 x=d x=0 x=d x=0 x=d . V (a) V (b) −V (b) V (a) − P(b) V (a) + P(a) V (b) x=0 x=d x=0 x=d x=d x=0 x=d x=0
4.2 One-load implementation When the plane wave reflection and transmission coefficients from the two surfaces of the sample are the same, it is possible to take advantage of the reciprocal nature of the layer to generate two additional equations instead of making a second set of measurementsh: T = T 11 22 (8) T11T22 − T12T21 = 1 . By combining Eqs. (5) and (8), the transfer matrix elements for a sample satisfying the above conditions can be expressed directly for one termination condition: i.e., P V + P V P 2 − P 2 T T 1 x=d x=d x=0 x=0 x=d 11 12 x=0 = . (9) 2 2 T21 T22 P V + P V V −V P V + P V x=0 x=d x=d x=0 x=0 x=d x=d x=d x=0 x=0
4.3 Determination of the transfer matrix elements From Eq. (1) the pressures and particle velocities at the two surfaces of the sample may easily be expressed in terms of coefficients A to D: P(s) = A(s) + B(s) , P(s) = C (s)e− jkd + D(s)e jkd , x=0 x=d A(s) − B(s) C (s)e− jkd − D(s)e jkd (10) V (s) = , V (s) = x=0 x=d ρ0c ρ0c
h Pierce noted that reciprocity requires that the determinant of the transfer matrix be unity [6]. Ingard has noted that the latter constraint is a general property of passive, linear four-pole networks [9]. Allard has also shown that this condition follows directly from the requirement that the transmission coefficient of a planar, arbitrarily layered acoustical system be the same in both directions [10]. Further, Pierce notes that for symmetrical systems, T11 = T22 . where the superscript (s) denotes that the quantities are determined for an arbitrary termination condition s. After substituting Eqs. (10) into Eqs. (7) or (9), the transfer matrix elements are then explicitly determined in terms of coefficients A(s) to D(s). Additionally, note that by using Eq. (10) and Eq. (5), and after some algebraic manipulations (omitted here), it can be shown that: 1 T12 − jkd 1 T12 + jkd T11 + + ρ0cT21 + T22 e T11 − + ρ0cT21 − T22 e (s) (s) A 2 ρ0c 2 ρ0c C = . (11) B(s) 1 T 1 T D(s) 12 − jkd 12 + jkd T11 + − ρ0cT21 − T22 e T11 − − ρ0cT21 + T22 e 2 ρ0c 2 ρ0c It should be noted that coefficients A(s) to D(s) are dependent on the termination condition s. On the contrary, the transfer matrix elements Tij are independent of the termination condition.
5 DETERMINATION OF TRANSMISSION LOSS The sound transmission loss of a partition is defined as: W i TL(ω) ≡ 10log10 (12) Wt where Wi and Wt are the airborne sound power incident on the partition and the sound power transmitted by the partition and radiated from the other side, respectively. Typically, the sound transmission loss is determined according to standard procedures [11-13], where the specimen is exposed to a diffuse sound field. In the case of the experimental setup described in Section 2, with a perfectly anechoic termination (that is D=0), the sound transmission loss is: 2 A(anechoic) TL (ω) = 10log (13) n 10 C (anechoic) where the subscript n indicates that the sample is exposed to a normally incident, plane wave field. When Eqs. (10) with the condition D=0 are substituted into Eq. (5), the normal incidence sound transmission loss is found to be: 2 1 T TL (ω) = 10log T + 12 + ρ cT + T . (14) n 4 11 ρ c 0 21 22 0 Note that the sound power, Wt transmitted by a sample in a tube depends in general both on the properties of the sample and on the termination conditions. For example, in the extreme case of a perfectly rigid termination, the plane wave coefficients C(rigid) and D(rigid) are equal in magnitude and the sound power transmitted by the sample is, in principle, zero, thus causing the transmission loss to be apparently infinite. Even when a tube termination is “nearly-anechoic”, small reflections from the termination may have a noticeable impact on the transmission loss if it (nearly−anechoic) (nearly−anechoic) 2 is calculated simply as 10log10 ( A / C ) : see Ref [4]. On the contrary, the normal incidence transmission loss given in Eq. (14) is expressed in terms of the transfer matrix elements, which are properties only of the sample and not of the measurement environment (e.g., the tube termination), thus providing an expression that is unambiguous. It should be noted that in Eq. (14) there are no terms that refer explicitly to the sample thickness or position.
6 REMARKS The signal from microphone 1 may be conveniently used as the reference to calculate H , P2R H , and H as defined in Eq. (4). Obviously, H shall be set equal to unity in that case. P3R P4R P1R However, when that approach is followed, the signal-to-noise ratio may not be satisfactory at certain frequencies, in particular when the sample under investigation is highly reflective. To overcome this problem, either a fifth microphone positioned close to the loudspeaker or the signal provided to the loudspeaker may be used instead. It should be noted that, in the latter case, it must be assumed that the loudspeaker is perfectly linear, which may be confirmed by an inspection of the coherence of the transfer function estimates as a function of source level. The method does not require simultaneous measurements of H , although this would Pi R considerably speed up the measurement process. Measurements can be conducted by the use of a dual-channel FFT analyzer and a single microphone, provided that the signal to the loudspeaker is used as reference. Consider Eq. (11), which relates the complex amplitudes of the plane wave components for both the upstream and downstream tube sections. To simplify the notation, we write: (s) (s) A a11 a12 C (s) = (s) . (15) B a21 a22 D (anec) (anec) It should be noted first that a11 = A / C in the case for D=0. Equation. (15) represents two equations in four unknowns. In order to be able to solve for the matrix element a11, two additional equations may be generated by introducing a second termination condition: i.e., (a) (b) (a) (b) A A a11 a12 C C (a) (b) = (a) (b) (16) B B a21 a22 D D where superscripts (a) and (b) denote the two different termination conditions. After some algebraic manipulations (omitted here), it can be shown that: A(a) D(b) − A(b) D(a) T (a) R(b) − T (b) R(a) a = = (17) 11 C (a) D(b) − C (b) D(a) R(b) − R(a) where R(s) ≡ D(s) / C (s) is the reflection coefficient created by the non-ideal termination and T (s) ≡ A(s) / C (s) . Clearly, a problem with singularity occurs if the two measured terminations produce almost identical reflection coefficients at one or more frequencies. The presence of a nearly-anechoic termination in at least one measurement condition causes the sound field in the downstream section to be almost purely propagational in that case, thus maximizing the phase difference between the sound pressures at the two downstream microphone locations and thus minimizing the effect of microphone phase-mismatch on the downstream transfer function estimates. Note that a microphone switching procedure may be performed at a calibration stage to reduce the effects of microphone and measurement channel mismatch, as recommended, for example, in Refs. [14-15]. 7 CONCLUSIONS In the present article, we have described a quick and convenient method for determining the normal incidence transmission loss of samples placed in a standing wave tube, based on the well- known transfer matrix representation. Although the sound power transmitted through the sample depends in general both on its properties and on the tube termination conditions, the procedure provides the normal incidence transmission loss as if the sample were backed by a perfectly anechoic termination independent of the actual tube termination conditions used during the measurements. Moreover, knowledge of such termination conditions is not required. Both one- and two-load implementations have been described, and the conditions under which the one-load method may be used have been specified.
8 REFERENCES [1] M. L. Munjal and A. G. Doige, “Theory of two source-location method for direct experimental evaluation of the four-pole parameters of an aeroacoustic element,” J. Sound Vib. 141, 323-333 (1990). [2] Z. Tao and A. F. Seybert, “A Review of Current Techniques for Measuring Muffler Transmission Loss,” Proceedings of the SAE Noise & Vibration Conference and Exhibition (2003). [3] B. H. Song and 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., 107 (3), 1131-1152 (2000). [4] J. S. Bolton, R. J. Yun, J. Pope, and D. Apfel, “Development of a new sound transmission test for automotive sealant materials,” SAE Technical Papers, Doc. Nr. 971896 (1997). [5] J.S. Bendat and A.G. Piersol, Engineering applications of correlation and spectral analysis (John Wiley & Sons, New York, 1980). [6] A. D. Pierce, Acoustics: An Introduction To Its Physical Principles And Applications (Woodbury, New York, Acoustical Society of America, 1991). [7] L.E. Kinsler, A.R. Frey, A.B. Coppens and J.V. Sanders, Fundamentals of Acoustics, 4th Ed. (John Wiley & Sons, New York, 2000). [8] S. Temkin, Elements of Acoustics (Wiley, New York, 1981). [9] K. U. Ingard, Notes on Sound Absorption Technology (Noise Control Foundation, Poughkeepsie, NY, 1994). [10] J. F. Allard, Propagation of Sound in Porous Media (Elsevier Applied Science, London and New York, 1993). [11] Standard terminology relating to environmental acoustics, ASTM C 634-02 (ASTM International, West Conshohocken, PA, USA, 2002). [12] Standard test method for the laboratory measurement of airborne sound transmission loss of building partitions and elements, ASTM E-90-02 (ASTM International, West Conshohocken, PA, USA, 2002). [13] Acoustics – Measurement of sound insulation in buildings and of building elements – Part3: Laboratory measurements of airborne sound insulation of building elements, ISO 140-3:1995 (International Organization for Standardization, Geneva, Switzerland, 1995). [14] Standard test method for impedance and absorption of acoustical materials using a tube, two microphones and a digital frequency analysis system, ASTM E-1050-98 (ASTM International, West Conshohocken, PA, USA, 1998). [15] Acoustics—Determination of Sound Absorption Coefficient and Impedance in Impedance tubes-Part 2: Transfer-Function Method, ISO 10534–2:1998 (International Organization for Standardization, Geneva, Switzerland, 1998).