Applied Acoustics 67 (2006) 106–111

www.elsevier.com/locate/apacoust

A note on the prediction method ofreverberation absorption coefficient of

double layer micro-perforated membrane

Jian Zou, Yong Shen *, Jianbin Yang, Xiaojun Qiu

State Key Laboratory of Modern Acoustics, Institute of Acoustics, Nanjing University,

Nanjing 210093, China

Received 14 December 2004; received in revised form 13 April 2005; accepted 6 May 2005

Available online 22 July 2005

Abstract

The absorption performance of micro-perforated absorber (MPA) has been usually esti-

mated by equivalent circuit (EC), however, it has been noted that the predicted absorption

coefficient by EC does not agree completely with the experiment in some frequency range.

Hence impedance transfer method (ITM) is adopted to predict the reverberation absorption

coefficient of a double layer micro-perforated membrane (MPM) structure. Experimental stud-

ies show that the prediction of ITM is better than that of EC.

� 2005 Elsevier Ltd. All rights reserved.

Keywords: Micro-perforated membrane; Absorption coefficient; Method of reverberation; Impedance

transfer method

1. Introduction

Theories and applications of micro-perforated absorber (MPA) [1–7] namely a

cavity backed plate or membrane with sub-millimeter apertures have been well devel-

oped. The predicted absorption curves in impedance tube and reverberation chamber

0003-682X/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.apacoust.2005.05.004

* Corresponding author. Tel./fax: +86 25 8359 3416.

E-mail address: yshen@nju.edu.cn (Y. Shen).

J. Zou et al. / Applied Acoustics 67 (2006) 106–111 107

were all calculated by equivalent circuit (EC). In this note, impedance transfer

method (ITM) is adopted to predict the reverberation absorption coefficient of a

double layer micro-perforated membrane (MPM) structure and difference between

ITM and EC is discussed.

2. Theory

When a plane sound wave propagates in direction h to the normal into an air gap

with a depth of D and arrives at the end of the air gap with a normalized (all the

specific acoustic impedance in this note are normalized by the characteristic imped-

ance of air, q0c0) normal specific acoustic impedance znD, it can be considered that

the sound wave propagates directly to a normalized normal acoustic load zn0 atthe opening position without the existence of the cavity. The relationship between

zn0 and znD can be determined from the impedance transfer method (ITM) [8]:

zn0 ¼1

cos h� jþ znD cos h � cotðxD cos h=c0ÞcotðxD cos h=c0Þ þ jznD cos h

; ð1Þ

where j is the complex number, x is the angular frequency, c0 (m/s) is the sound

speed in the air.

The membrane itself and the apertures are assumed to be locally reacting acousticelements [1,2]. That is, the acoustic impedance is independent of the angle of

incidence. Then the parallel normalized specific acoustic impedance z of MPM is

independent of the angle of incidence, which has been given in [7]

z ¼ zMzLzM þ zL

; ð2Þ

zM is the normalized specific acoustic impedance of membrane, zL is that of the aper-

tures. zM can be assumed to be that of a limp mass and calculated by

zM = rM + jxmM, where rM is the normalized specific acoustic resistance of the mem-

brane, which depends mainly on the mounting conditions, and mM = MM/q0c0,where MM is the surface density of the membrane (kg/m2). For zL, it has been given

in [2,7] as

zL ¼ r þ jxm ð3Þwith

r ¼ 0.147t

pd21þ k2MPP

32

� �12

þffiffiffi2

p

8

kMPPdt

" #;

m ¼ 0.294� 10�3tp

1þ 9þ k2MPP

2

� ��12

þ 0.85dt

" #.

Here d is the orifice diameter (mm), t is the membrane thickness (mm), p is the ratio

of apertures area to the total area of MPM (%).kMPP satisfies kMPP ¼ dffiffiffiffiffiffiffiffiffiffif =10

p, where

f is the frequency of sound wave.

108 J. Zou et al. / Applied Acoustics 67 (2006) 106–111

The total normalized normal specific impedance zT of the double layer MPM

structure (Fig. 1) under consideration for oblique incidence can be derived by Eq. (1)

zT ¼ z1 �j

cos hcot

xD1 cos hc0

� �

þ 1þ cot2ðxD1 cos h=c0Þcos h z2 cos h� j cotðxD1 cos h=c0Þ � j cotðxD2 cos h=c0Þ½ � ; ð4Þ

where z1 is the parallel normalized specific impedance of outer membrane and aper-

tures, z2 is that of inner MPM, and both of them can be calculated by Eq. (2). D1(m)

is the distance between the two layers and D2 (m) is the depth of inner cavity. In the

derivation, the acoustic load of the inner cavity is the rigid wall, and the acoustic

load of the outer air gap is the sum of z1 and the normalized normal acoustic imped-ance of the inner cavity.

In a diffuse sound field, the angle-averaged absorption coefficient can be calcu-

lated by Eq. (5) with ah ¼ 1� jzT cos h�1zT cos hþ1

j2

as ¼Z p

2

0

ah sin 2h dh. ð5Þ

Alternatively, the equivalent circuit of Fig. 1 is shown in Fig. 2, where cotðxD1 cos h=c0Þj cos h is

the normalized normal specific acoustic impedance of outer cavity and cotðxD2 cos h=c0Þj cos h is

that of inner cavity. The total normalized normal specific acoustic impedance z0Th ofthis structure can then be derived to calculate the reverberation absorption coeffi-

cient with

z0Th ¼ z1 � jcotðxD1 cos h=c0Þ

cos h

þ cot2ðxD1 cos h=c0Þcos h z2 cos h� j cotðxD1 cos h=c0Þ � j cotðxD2 cos h=c0Þ½ � . ð6Þ

Fig. 1. Schematic of double-layer micro-perforated membrane (MPM) of oblique incidence.

2cot( cos )

cos

j D cω θ

θ

1z 2z

1 0 0cot( cos )

cos

j D cω θ

θ

– –

Fig. 2. Equivalent circuit (EC) of Fig. 1.

J. Zou et al. / Applied Acoustics 67 (2006) 106–111 109

3. Numerical calculations and measurements

The parameters ofMPMare t = 0.105mm, d = 0.21mm, p = 1%,Mm = 0.14 kg/m2.Two depth combinations of the outer and inner cavity are chosen: D1 = 30 mm,

D2 = 50 mm; D1 = 30 mm, D2 = 100 mm. In the calculation, rM is chosen as 1.5 since

it has been experimentally shown that a value around 1 is appropriate for themounting

condition [7]. The random incidence absorption coefficient of ITM and EC can then be

calculated separately by Eqs. (4) and (6) through Eq. (5).

Measurements were carried out in the reverberation room of Nanjing University,

according to GBJ47-83 (compatible to ISO 354). The membranes are installed on a

frame and the available surface area of the MPM is 9.1 m2. This fairly large surfacearea was used in the measurements to minimize edge effects.

The results of measurements and predictions are shown in Fig. 3. Single-frequency

values are at one-third octave intervals. It can be seen that the prediction result of

ITM is better than EC. The difference is much more obvious from about 800 to

4000 Hz.

In Eq. (1), when znDcosh is infinite, the acoustic impedance at the opening position

is cotðxD cos h=c0Þj cos h . That is just the normalized normal specific impedance of the cavity in

EC. It is correct for the inner cavity because its acoustic load is the rigid wall, but it isnot quite appropriate for the outer cavity because its acoustic load is not infinite.

Fig. 3. Calculated and measured diffuse field absorption coefficients of the double MPM.

Fig. 4. Contour plot of cot(xD1/c0) = 30 mm.

110 J. Zou et al. / Applied Acoustics 67 (2006) 106–111

However, when cot2(xD1cosh/c0) � 1, Eq. (4) can be simplified to Eq. (6) and EC

can still be used.

Fig. 4 shows the contour plot of cot2(xD1cosh/c0) for D1 = 30 mm. The curves

with number labels divide the f–h plane into regions marked as I and II. The labelsare the value of cot2(xD1cosh/c0). In region I, cot2(xD1cosh/c0) � 1 and EC results

are similar to those of ITM, while in region II, cot2(xD1cosh/c0) � 1 and significant

differences appear.

It can be seen from Fig. 4 that EC differs significantly from ITM in frequency

regions defined by a lower frequency fiL and an upper frequency fiH. For example,

let cot2(xD1/c0) = 1, then fiL and fiH can be estimated by Eq. (7) as

fiL ¼ ð4iþ 1Þc08D1

; f iH ¼ ð4iþ 3Þc08D1

; i ¼ 0; 1; 2 . . . ð7Þ

It can be seen that if D1 is sufficiently small then fiL exceeds 5000 Hz, and the two

methods will have little differences on absorption coefficient curves below 5000Hz. It can also be seen that if D1 is large enough, more than one fiL and fiH can

be obtained by Eq. (7) at frequencies below 5000 Hz and the two methods will differ

significantly between each pair of fiL and fiH. This is confirmed in Fig. 5. In Fig. 5(a),

the two methods have little differences, since the smallest fiL = 8500 > 5000 Hz.

When D1 = 60 mm, it can be estimated by Eq. (7) that, in the frequency regions

708–2125 and 3541–4958 Hz, the two methods have significant differences, as shown

in Fig. 5(b).

Fig. 5. Calculated diffuse field absorption coefficients of two methods: ITM and EC.

J. Zou et al. / Applied Acoustics 67 (2006) 106–111 111

4. Conclusion

In this note, the impedance transfer method (ITM) as well as the equivalent circuit

(EC) has been adopted to predict the reverberation absorption coefficient of a

double-layer micro-perforated membrane. An equation for calculating the absorp-

tion coefficient is presented. Experiment results show better agreement with ITM

than with EC. Further discussions show that EC can be obtained by simplifying

ITM in case that the distance between the two layers is less than c08f h

, where fh (Hz)is the highest frequency that EC remains in agreement with ITM.

Acknowledgments

The work is supported by the National Natural Science Foundation of China

(10474046). The authors are indebted to associate Professor Qixue Wu, Mr. Xiaox-

iang Shen for their help in the experiments, and the authors also want to give theirthanks to Beijing Di-Ma High Technology Ltd., for its MPM samples.

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