Extension of frequency range of the sixteen-microphone method in normal-incidence sound transmission loss measurement Hiroshi Nakagawa 1 Nihon Onkyo Engineering Co., Ltd. 1–21–10 Midori, Sumida-ku, Tokyo, 130–0021 Japan Akira Sanada 2 Industrial Technology Center of Okayama Prefecture 5301 Haga, Kita-ku, Okayama, 701–1296 Japan ABSTRACT This study deals with development of a high-frequency measurement method of normal incidence sound transmission loss using acoustic impedance tube. On ordinary transfer-function method, meas- urable frequency of the method is limited by the diameter of the tube. Authors have developed eight microphone method to measure normal incidence absorption coefficient. This technique lays 4 mi- crophones on the same cross section and cancels rotational acoustic modes. This makes it possible to measure normal incidence absorption coefficient up to three times the upper frequency of the or- dinary two microphone method. In this study, authors apply the same technique to normal incidence transmission loss measurement using 16 microphones. 1. INTRODUCTION

• Normal-incidence sound absorption coefficient [1] and normal-incidence transmission loss [2] measurements using acoustic impedance tubes are widely used to evaluate the sound absorption and insulation properties of porous materials and to validate analytical models, because these measurements can be made with a smaller sample size and in a shorter time with simpler opera- tions than measurements using reverberation rooms. However, conventional measurements using acoustic impedance tubes assume a plane wave sound field, so the upper frequency limit depends on the inner diameter of the tube. Therefore, to measure higher frequencies, measurements must be made with smaller tubes, which requires smaller samples. This causes variations due to cutout areas caused by material inhomogeneity, variations in measurement results due to cutout accu- racy, and the effects of constraint conditions when setting the material in the tube. To make meas- urements at higher frequencies, Schultz et al. [3] have tried to use modal decomposition, but this is a complicated procedure. Therefore, Sanada [4-6] proposed an eight-microphone measurement method that can easily extract only the normal component, enabling measurement of normal in- cidence sound absorption coefficients up to frequencies almost three times higher than those of conventional methods. In this paper, the eight-microphone method applied by Sanada for sound

1 hiroshi_nakagawa@hibino.co.jp

2 akira_sanada@pref.okayama.lg.jp

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absorption coefficient measurements is extended and applied to transmission loss measurements with sixteen microphones. 2. THEORY

2.1. Sound Field in cylindrical Tube In this paper, we consider a cylindrical tube with radius R [m] (Figure 1). The sound field is ex- pressed by Equation (1):

Figure 1: Sound field in the tube.





sectionA _ Cross -section B Cross:

  , , ,   Ψ 

,   ,   

   , 

(1)



   

#   ,    

   ,  ! Ψ #

#   ,  !$  %&

,   ,   

  is the distance from the center of the cylindrical cross section, θ is the angle from x-axis and Ψ  , , Ψ # , are the functions representing acoustic modes in the cross section of the tube and are expressed by Equations (2) and (3):

,   ,  ' , (  ) *,   ! + (2)

Ψ 

,   ,  ' , (  ) *,   ! + (3)

Ψ #

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m and n are the orders of the acoustic modes in the cross section in the circumferential and radial directions, respectively; (  is the Bessel function of the first kind, and ' , is the normalization constant that satisfies the orthogonality of equations (4) and (5) among the acoustic modes:

⬚

 Ψ #∗

2 3 4, 5
, 6 7 (4)

, Ψ 

./

1

⬚

 Ψ #∗

0 3 9 4 : 5 9
: 6 9 7 (5)

, Ψ 

./

1

where S is the cross-sectional area of the cylindrical tube;    ,    ,   # , and   # are the amplitudes of the waves of the (m, n) mode; ) , is the wave number of the (m,n) mode in the

z-direction, expressed as ) , )  ; < ) *, ; ! = ; > , where )  = ω @  > ( @  : sound speed), and ) *, is the wave number in the cross-section of the cylindrical tube, which satisfies the bound- ary condition at the inner surface of the tube. ) *, A that satisfying the boundary condition is denoted as B , . The value of B , as shown in Figure 2. The (0,0) mode represents a plane wave propagating along the z-axis, and the other higher-order modes represent sound waves prop- agating obliquely to the z-axis. The obliquely propagating waves for which )  C ) *, cannot propagate in the tube because the wave number becomes purely imaginary and the wave becomes evanescent. Therefore, the minimum frequency at which the 5, 6 mode can propagate is:

" OCS8®e@ Aonn) =O — ann) = 1:84 Ramm) = 3.05 Amn) = 4.20 Amn) = 5:32 enn) = 3:83 Aomn) =5:33 Aum) = 6.71 ann) 8.02 Amn) = 9.28

F , G

D E,

;HI (6)

which is the so-called cut-on frequency of the mode. In the conventional standard transfer func- tion method, measurement is conducted under conditions where only the 0,0 mode can propagate in the tube. Thus, the upper frequency limit is less than the frequency below the (1, 0) mode. For example, for a 100 mm diameter tube, the theoretical upper measurement frequency is D E=, = 2013 [Hz] at temperature 20 °C.

Figure 2: Acoustic mode in the cross-section of the tube.

2.2. 8 Microphone Method for Absorption Coefficient Measurement Sanada [4-5] showed that three circumferential modes of a cylindrical tube, (1,0), (2,0), and (3,0), can be canceled by placing microphones at four locations at 90-degree intervals on the same cross section and obtaining the sum of their sound pressure, as in Equation (7):

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J K
L M

N 
L M

O 
L M

P 
L M

Q 4    S       S  ! %& (7)

• Furthermore, it was shown that by placing the microphone not on the wall of the acoustic tube but at the node of the (0,1) mode in the radial direction, the (0,1) mode is no longer detected, so that only sound waves propagating in the z direction at frequencies below the (4,0) mode can be measured [6]. For exam- ple, for a 100 mm diameter acoustic tube, D ET, = 5820 [Hz], which is almost three times higher frequency than the conventional method. •

2.3. Sixteen-Microphone Method for Transmission Loss Measurement In this paper, the sound absorption coefficient measurement with eight microphones proposed by Sanada is extended to transmission loss measurement. An image of the microphone arrange- ment is shown in Figure 3. Microphones are placed at 90-degree intervals on four cross sections of the sample: two cross sections A and B on the surface side and two cross sections C and D on the back side. The microphones are positioned at the nodes of the (0,1) mode in the radial direc- tion so that the (0,1) mode in the radial direction is not detected, thus allowing us to measure only sound waves propagating normal direction in the tube at frequencies below the (4,0) mode. Let J UK , J VK , J GK , and J WK be the sound pressure signals obtained by summing the microphone signals at 90-degree intervals in the four cross sections A, B, C, and D as in Equation 8, then the sound pressure P 0 and particle velocity V 0 on the sample surface side, P d and V d on the sample back side placed inside the tube can be expressed as in Equations 9-12 [2].

Cross-sectionA Cross-sectionB Cross-sectionC — Cross-section D Termination

J UK
L M

XN 
L M

XO 
L M

XP 
L M

XQ J VK
L M

YN 
L M

YO 
L M

YP 
L M

YQ J GK
L M

ZN 
L M

ZO 
L M

ZP 
L M

ZQ J WK
L M

(8)

[N 
L M

[O 
L M

[P 
L M

[Q

Figure 3: Schematic view of sixteen-microphone transmission loss measurement.

J  J UK \]6)  ^ ; < J UK \]6)  ^ =

\]6)  \ = (9)

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_  `

a@\]6)  \ = (10)

J b J UK \]6)  ^ T < c ! < J UK \]6)  ^ d < c !

\]6)  \ ; (11)

_  `

a@\]6)  \ ; (12)

The relationship among the sound pressure P 0 and particle velocity V 0 at the sample surface and the sound pressure P d and particle velocity V d at the back of the sample can be expressed in a 2 × 2 transfer matrix as in Equation 13:

e J

_  f eg == g =; g ;= g ;; f eJ b

_ b f (13)

To obtain the four components of the 2 × 2 transfer matrix, measured data under two different termination conditions are required [2]. Let the two termination conditions be an absorptive ter- mination and a reflective termination, and let the sound pressure and particle velocity be P 0a , V 0a , P da , and V da at the absorptive termination and P 0r , V 0r , P dr , V dr at the reflective termination, they are given by equations (14) and (15):

e J h

_ h f eg == g =; g ;= g ;; f eJ bh

_ bh f (14)

e J *

_ * f eg == g =; g ;= g ;; f eJ b*

_ b* f (15)

From these two equations, the four components T 11 -T 22 of the transfer matrix can be ob- tained.

g == J U _ bI < J I _ bU

(16)

J U J I < J I _ U

g =; J I J bU < J U J bI

(17)

J U J I < J I _ U

g ;= _ U _ bI < _ I _ bU

(18)

J U J I < J I _ U

g ;; J bU _ I < J bI _ U

(19)

J U J I < J I _ U

The transmission coefficient τ and transmission loss TL are obtained from equations (20) and (21) using T 11 - T 22 .

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4 2  S b

(20)

g ==  j g =; a@  > k  a@  g ;=  g ;;

gl 10n:o p 1

|4| ; r (21)

3. EXPERIMENT

To validate the proposed method, measurements were conducted on two different samples.

3.1. Experimental Setup Figure 4 shows an overview of the experimental setup. Figure 5 shows a photograph of the acoustic impedance tube. To compare the proposed method with the conventional method [2], measurements were conducted on the tube with an inner diameter of 40 mm and 15 mm.

Sch. Microphone Amplifier - [8ch. Microphone TOMI Amplifier Tom Power Amplifier l8ch. Audio Interface lech. Audio Interface

Figure 4: Measurement Setup of transmission loss measurement

Figure 5: Photograph of the tube for transmission loss measurement.

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3.2. Measurement Result The list of samples measured is shown in Table 1. Two types of materials were measured: natu- ral rubber as a typical impervious material and melamine foam as an open-cell porous material. Figure 6 shows the result of natural rubber and figure 7 shows the result of melamine foam. The mass law under normal incidence conditions is appended to the Natural rubber graph for refer- ence. Natural rubber, an impervious material, has a significant difference in transmission loss depending on the sample size due to its stiffness. The conventional method [2], which uses a smaller tube, a sharp dip due to the first-order resonance of the bending mode of the sample and the effect of stiffness at lower frequencies appear strongly within the measurement frequency band. The proposed method, on the other hand, allows measurement of a larger sample and shows no sharp peaks or dips in the transmission loss in the measured frequency range, which is almost in agreement with the mass law above 1500 Hz. Melamine foam, although a porous ma- terial, also shows a steep dip due to the natural vibration caused by being constrained in the tube. The dip appears from 2000 to 4000 Hz when measured with thin acoustic tubes of 40 mm and 15 mm inner diameters, whereas the dip can be reduced to around 500 Hz with this method, which allows measurement with a larger inner diameter tube, so the transmission loss measure- ment results can be evaluated more effectively. Table 1. Measured sample list

Normal Incience T Vv 20 16 mic. @100 —4 mic. @40 10 —4 mic. p15 —Mass Law oO 0 2,000 4,000 Frequency [Hz] 6,000

Sample name Thickness [mm] Area density [g/m^2] Bulk density [kg/m^3]

Natural rubber 3.5 4760 1360

Melamine foam 25 242 9.7

60 2 2 ° B $ & [ap] 5507 uorssiwisue.

Figure 6: Measurement result of natural rubber.

Red line: proposed method in 100 mm diameter, blue line: conventional method in 40 mm diameter, green line: conventional method in 15 mm diameter, black line: mass law in normal incidence.

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Normal Incience T 16 mic. @100 4 mic. 940 —4 mic. p15 2000 Frequency [Hz] 4000 6000

Figure 7: Measurement result of melamine foam.

Red line: proposed method in 100 mm diameter, blue line: conventional method in 40 mm diameter, green line: conventional method in 15 mm diameter.

4. CONCLUSIONS

A new method using 16 microphones was proposed for measuring transmission loss in acoustic tubes. We showed that this method can measure transmission loss up to three times higher fre- quencies than the conventional method. The validity of the proposed method was confirmed by the measurement results of two samples, natural rubber and melamine foam, as actual examples. This method is expected to be useful for the verification of flexible materials with low permea- bility and Biot models, which have been difficult to measure with conventional acoustic tubes. 5. REFERENCES

10 cy © [ap] sso7 uolssiusues

1. ISO 10534-2, Acoustic–Determination of sound absorption coefficient and impedance in imped- ance tubes — Part 2: Transfer-function method (1998). 2. ASTM E2611, Standard Test Method for Normal Incidence Determination of Porous Material Acoustical Properties Based on the Transfer Matrix Method 3. Schultz, T. Cattafesta III L. N. Sheplak, M. Modal decomposition method for acoustic impedance testing in square ducts. Journal of Acoustic Society of America, 120(6) , 3750-3758 (2006). 4. Sanada, A. Extension of the frequency range of normal incidence sound absorption coefficient measurement using four or eight microphones. Acoustical Science and Technology , 38(5) , 261– 263 (2017). 5. Sanada, A. Iwata, K. Nakagawa, H. Extension of the frequency range of normal-incidence sound absorption coefficient measurement in impedance tube using four or eight microphones. Acous- tical Science and Technology , 39(5) , 335–342 (2018). 6. Sanada, A. Iwata, K. Nakagawa, H. High-frequency measurement of normal-incidence sound ab- sorption coefficient using eight microphones. Proceedings of ICSV 25 , pp. 3225-3232. Hiro- shima, Japan, August 2018.

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