An analytical model to estimate the performance of an indoor barrier at low-medium frequencies

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Applied Acoustics 69 (2008) 1343–1349 www.elsevier.com/locate/apacoust

An analytical model to estimate the performance of an indoor barrier at low-medium frequencies Y.J. Chu

a,b

, C.M. Mak

a,*

, X.J. Qiu

b

a

b

Department of Building Services Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People’s Republic of China Key Laboratory of Modern Acoustics and Institute of Acoustics, Nanjing University, Nanjing 210093, People’s Republic of China Received 26 January 2007; received in revised form 21 June 2007; accepted 23 August 2007 Available online 24 October 2007

Abstract Indoor barriers are now widely used for sound insulation. This paper examines the performance of indoor barriers in the low-medium frequency range and analyses the interaction between different natural modes of a room-barrier-room system. Morse proposed a theoretical model to calculate the sound field in a coupled-room, but this model neglects the surface integral of the boundary values of sound pressure. To estimate the performance of a barrier in an indoor environment, an analytical model is proposed that modifies the Green’s function for a non-rigid boundary enclosure and approximates the surface integral by a pre-estimated sound pressure based on Morse’s model. An additional approximation has been made in the proposed model to neglect the coupling area in the calculation of the surface integral. The proposed model used to predict the insertion loss of the barrier is verified by the experimental results using a 1:5 scale model. The predicted results agree well with the measured results at lower frequencies.  2007 Elsevier Ltd. All rights reserved. Keywords: Low-medium frequency; Indoor barrier; Coupled-room

1. Introduction It has been almost 40 years since Maekawa [1] presented one of the most influential graphical charts for outdoor screen attenuation that has been widely used in barrier design. Tatge [2] and Kohei [3] later proposed approximations to Maekawa’s design chart so that acoustical engineers could easily apply it to computer aided design. In the study of indoor barriers, geometrical and statistical acoustics are widely used and are based on the assumption that the wavelength of sound is considerably smaller than the room scale [4–6]. The most widely used techniques are the image source method and the ray-tracing method. However, these methods are not reliable in predicting the sound insulation at lower frequencies, where more interest has been arisen due to the growing impact of lower fre*

Corresponding author. Tel.: +852 2766 5856; fax: +852 2765 7198. E-mail address: [email protected] (C.M. Mak).

0003-682X/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2007.08.009

quency noise sources [7,8]. Osipov et al. [9], has shown that the sound insulation in this range depends on several parameters, i.e., the dimension of the room and the position of sound source. In this paper, the sound field response in a 1:5 scale model of a room-barrier-room system is analyzed. The geometry of the system and the scale model are shown in Fig. 1a and b, respectively. It is well-known that the methods based on wave acoustics, such as FEM [10] and BEM [11], are usually applied to low frequency problems and that the necessary computations are time-consuming. The present work focuses on the low-medium frequency range and the computations of the proposed method do not take up so much time as those of FEM and BEM. The modal coupling method has been used, in [9], to solve the problem of low-frequency airborne sound transmission through partitions in a room-wall-room system. This paper, however, proposes a coupled-room model as an analytical solution to the room-barrier-room problem

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Y.J. Chu et al. / Applied Acoustics 69 (2008) 1343–1349

a

passageway barrier z Room B

1.0 m

Room A

receiver

y

2.0 m

0.9 m

source 0

3.1 m

2.1 m

2.9 m

x

b

Fig. 1. (a) Geometry of the open-plan office and (b) the 1:5 scale model.

and is aimed at improving the understanding of the problem by revealing the interaction between the different natural modes of two coupled enclosures. The fundamental integral equation in terms of Green’s functions is used to describe such a system with reasonable approximations. This equation involves no new physics but modifies the calculation formulae originally proposed by Morse et al. [12] to provide predictions of the sound field in the room-barrier-room system and to obtain the insertion loss (IL) of the barrier. The proposed coupled-room analytical model is verified by the experimental results in the scale model. 2. Theory 2.1. The low-medium frequency range In the low-medium frequency range, the acoustic modes overlap and are difficult to distinguish from each other.

Thus, the sound field behavior cannot be fully described by the modal properties of one or two individual modes even though the excitation frequency is at or very close to the natural frequencies of the modes. In addition, methods based on geometrical or statistical acoustics are not reliable when the sound field is not diffuse and the modal density of the sound field is not sufficiently high. In the present work, the frequency range from 100 Hz to 1000 Hz is chosen according to the Schroeder cutoff frequency, fc, and the lower frequency bound, fm, for medium-frequency range that is given by [13] pffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ fc ¼ 2000 T 60 =V ; ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c0 S 64c0 T 60 V 1þ 1 ; ð2Þ fm ¼ 16V 4pS 2 where T60 is the reverberation time in seconds; V is the volume of the enclosure in m3; S is the inner surface area in m2; and c0 is the speed of sound in air.

Y.J. Chu et al. / Applied Acoustics 69 (2008) 1343–1349

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op on

2.2. The coupled-room analytical model To predict the sound forced motion of the room-barrierroom system, an integral formula by means of the Green’s function is used pðrÞ ¼ f ðr0 ÞGðrjr0 Þdv0  ZZ  o o þ Gðrjr0 Þ pðr0 Þ  pðr0 Þ Gðrjr0 Þ dS 0 ; on0 on0 ð3Þ 2

where p is the sound pressure in Pa (N/m ) within the bounded volume and on the boundaries; f is the source function, which is the distribution of source pressure in Pa (N/m2); r is the coordinate of the receiver in meter while the subscript ‘‘0’’ indicates the coordinate of the source; G is the Green’s function; and o/on0 represents the normal derivative. The volume integral is over the bounded volume of the source function and the surface integral is over the boundary of the enclosure. In Morse’s model [12], all walls are assumed to be rigid enough that the surface integral over the boundary, except for the coupling area, can be neglected. In reality, however, the assumption may not be valid and such a surface integral should not be neglected. This work modifies Morse’s model by taking into account the boundary effect and making reasonable approximations to obtain satisfactory analytical solutions. Based on Eq. (3), Green’s functions (see Appendix for details) are chosen such that oG/on0 equals zero at the boundaries of each room. The integral equations for the pressure in the two rooms are ZZZ pa ðrÞ ¼ ikqc0 Sðr0 ÞGa ðrjr0 Þdv0 ZZ Ga ðrjr0 Þu0 ðr0 ÞdS 0 þ ikqc0 ZZ a Ga ðrjr0 Þp0a ðr0 ÞdS a þ ikb ZZ Ga ðrjr0 Þp0a ðr0 ÞdS 0 ;  ikba ð4Þ ZZ Gb ðrjr0 Þu0 ðr0 ÞdS 0 pb ðrÞ ¼ ikqc0 ZZ Gb ðrjr0 Þp0b ðr0 ÞdS b ikbb ZZ b Gb ðrjr0 Þp0b ðr0 ÞdS 0 :  ikb ð5Þ The wave number is k = x/c0, where x is the angular excitation frequency. The meanings of other symbols are as follows: q is the density of air in kg/m3; S(r0) is the strength of a simple source and is set to be unity for the calculation of IL; b = qc0u/p is the specific admittance of the inner surface, the superscripts indicating the particulate room; p0a and p0b are the sound pressure in the two rooms preestimated by Morse’s model; and u0(r0) is the normal velocity at the coupling area if the acoustical equation

¼ q ou ¼ ikqc0 u is used. The surface integral dS0 is ot over the coupling area, while dSa and dSb are over the inner surface of the two rooms. Notice that, for convenience, u0 is assumed to be positive for flow from room A to room B. The last two terms in both Eqs. (4) and (5) are neglected by Morse’s model. The further processing is based on the assumption that the pressure on both sides of the coupling area is practically constant [12,14] and equal. Thus, according to Eqs. (4) and (5), the following can be obtained: Z  a  P ðrÞ þ ikqc0 G ðrjr0 Þ þ Gb ðrjr0 Þ u0 ðr0 ÞdS 0 ¼ 0; ð6Þ R where RP ðrÞ ¼ ikqc0 Sðr0 ÞGa ðrjr 0 þ P a  P b , with P a R 0 Þdv a a a a 0 0 ¼ ikb ðrjr Þp ðr ÞdS  ikb ðrjr G G 0 0 a 0 Þp a ðr 0 ÞdS 0 , PRb ¼ a R b R b b b 0 0 ikb G ðrjr0 Þpb ðr0 ÞdS b  ikb G ðrjr0 Þpb ðr0 ÞdS 0 ; and is an integral expression. In order to use the variation method, a variational expression for the integral equation, Eq. (6), is set up as Z T  u0 ðr0 ÞP ðr0 ÞdS 0 : ð7Þ Multiplying Eq. (6) by u0, integrating over S0, and then adding Eq. (7), the function to be varied is Z Z T ¼ 2 u0 ðr0 ÞP ðr0 ÞdS 0 þ ikqc0 u0 ðrÞdS Z  a   G ðrjr0 Þ þ Gb ðrjr0 Þ u0 ðr0 ÞdS 0 ; ð8Þ resulting in the optimal solution for the trial value of the velocity A: R  P ðrÞdS R R u0 ffi A ¼ ; ð9Þ ikqc0 dS ½Ga ðrjr0 Þ þ Gb ðrjr0 ÞdS 0 and the corresponding solutions for the steady-state pressure in the two rooms, which can be obtained from Eqs. (4) and (5). To calculate Eqs. (9), (4), (5) excited by the source at frequency fj = kjc0/2p, Green’s functions will involve only N eigenfunctions of the cavities, and the pressure amplitudes of the two rooms are expressed in vector forms as follows: T

Aj ¼ paj ¼ pbj

¼

½P N þ P aN  P bN  ½S 0a N  2 T a ik j qc0 ð½ðS 0a N Þ  ½1=E N =va T a ½P N þ P aN þ P 0a N  ½UN ; T b ½P bN  P 0b N  ½UN ;

2 T b þ ½ðS 0b N Þ  ½1=EN =vb Þ

;

ð10Þ ð11Þ ð12Þ

where the nth elements of the vectors are (n = 1, 2, . . . N), R Sðr0 ÞUan dv0 ik j qc0 Pn ¼   ; ð13Þ va Ean R R ik j bt ½P Nt pre ½UtN Utn dS t  ½P Nt pre ½UtN Utn dS 0  ; ð14Þ P tn ¼ vt Etn P 0t n ¼

ik j qc0 Aj S 0t  tn ; vt En

ð15Þ

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S 0t n ¼

Y.J. Chu et al. / Applied Acoustics 69 (2008) 1343–1349

Z

Utn dS 0 ;

ð16Þ

2

2

Etn ¼ Ktn ½ðK tn Þ  ðk j Þ :

ð17Þ

½P Nt pre 

Here, is the pre-estimated sound pressure vector of Morse’s model and superscript t = a, b is an indication of the particulate space. It is difficult to calculate the surface integral in Eq. (14) if no approximation is made. Owing to the small value of the specific admittance as well as the small area of the coupling aperture compared with the total surface area of the cavity, the surface integral over the coupling area is comparatively small. Regarding the small surface integral over the coupling area, the second term of the surface integral in Eq. (14) can be ignored. Thus, the integral covers all the inner surfaces where the cross terms are cancelled out and the squares are left. Eq. (14) is thus reduced to

(0, 0, 0), where the most modes of the space may be excited. Behind the barrier in room B, there is a receiver, located at a height of 1.1 m. The barrier is located from the position (0, 0) to (0, 3.1) in the x–y coordinate system shown in the figure. The geometric representation used assumes that the inner surface of the model, including the floor, ceiling, and walls, is thick enough that the waves being transmitted to the exterior walls are not reflected back by the outer surfaces of these elements. Since the transmission loss of the barrier is generally high enough, the sound transmitted through the barrier is negligible. Each boundary surface has a specific admittance, which is assumed to be as small as 0.001 + 0.001i. In particular, the floor is usually covered with thin carpet, the specific admittance of which is estimated to be as small as required by Morse to make the first-order approximation in the Green’s function shown

h

i  t    t t t pre K et lx ly b þ b þ etnx ly lz btx0 þ btx1 þ etny lz lx bty0 þ bty1 z0 z1 n nz ik P j n P tn ¼  : Etn vt

ð18Þ

Based on the calculation above, IL can be obtained to assess the performance of the barrier. The 1/3 octave band insertion loss can be obtained as follows: , ! M M X X 0 2 b 2 IL ¼ 10 log jpi j jpi j ; ð19Þ i¼1

i¼1

where M is the number of frequencies involved within a certain octave band and p0i is the sound pressure at the receiver position without the screen barrier. 3. Materials and methods Measurements were made on the 1:5 scale model to validate the proposed predicted model. The dimension of the test model is 5.0 · 4.0 · 3.0 m3 (L · W · H), as shown in Fig. 1a. A barrier of 2 m high and 3.1 m wide separates the room into two rectangular spaces; thus, it constitutes a room-barrier-room system. Room A contains a noise source set in one of the corners of the room; that is, position

Room A

Room B 3

6

9

2

5

8

1

4

7

Fig. 2. Plan of the receiver points.

Fig. 3. Predicted IL in room B at 100 Hz (a) and 1000 Hz (b).

Y.J. Chu et al. / Applied Acoustics 69 (2008) 1343–1349

in the Appendix. That is the real part of the admittance ranges from 0.03 to 0.09 and the imaginary part from 0.03 to 0.015 as the frequency increases. K tn is the first-order approximation of the eigen-wave-number, which is reasonable for the small specific admittance. The predicted IL by the proposed method in the receiver room is measured. The sound pressure levels were measured in 1/3 octave bands (500–5000 Hz range) at nine fixed positions at a height of 1.1 m of the office, as shown in Fig. 2. IL values were calculated according to ISO 11821 [15].

4. Results and discussion The analytical model for determining the sound field and IL values of a room-barrier-room system was described in Section 2. In this section, the predicted results based on the proposed model are compared with those based on Morse’s model and with measured results in the scale model. With regard to the distribution, the predicted IL values of the proposed model shown in Fig. 3 exhibit a behavior dependent on the frequency. It is found that, at lower

12

16 Measured results Proposed model Morse's model

14

10 8 IL at Point 2 (dB)

IL at Point 1 (dB)

12 10 8 6

4

Measured results Proposed model Morse's model

0

2 0

-2 100

200

400 Frequency (Hz)

100

1000

200

400 Frequency (Hz)

1000

16

16 Measured results Proposed model Morse's model

14

Measured results Proposed model Morse's model

14 12 IL at Point 5 (dB)

12 IL at Point 3 (dB)

6

2

4

10 8 6

10 8 6 4

4

2 0

2 100

200

400 Frequency (Hz)

1000

100

14

8 6 4

400 Frequency (Hz)

100 0

Measured results Proposed model Morse's model

12 IL at Point 9 (dB)

10

200

14

Measured results Proposed model Morse's model

12 IL at Point 7 (dB)

1347

10 8 6 4

2

2

100

200

400 Frequency (Hz)

1000

100

200

400 Frequency (Hz)

Fig. 4. The measured and predicted IL at chosen receiver points.

100 0

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Y.J. Chu et al. / Applied Acoustics 69 (2008) 1343–1349

frequencies, in particular 100 Hz, IL values are much more smoothly distributed. It can be seen that there are some areas where the IL values are higher than those of the rest of the receiver room; that is, regions near the corner behind the barrier and regions near the back wall. This is because the sound field at a lower frequency is determined mainly by several dominant natural modes whose wave length is comparable to the size of the model and whose wave peaks are distinguishable. As the frequency increases, it can be foreseen that more modes will become excited and overlap, resulting in indistinguishable peaks of sound pressure level as well as IL values. It can be seen in Fig. 3a and b that predicted IL values at a frequency of 1 kHz are more evenly distributed than those at 100 Hz. Fig. 4 shows the measured and predicted IL values based on the proposed model and Morse’s model at six receiver points. The predicted IL values at six receiver points obtained from the office are the average values over nine points of 0.05 m · 0.05 m at a height of 1.1 m. These six receiver points are chosen to give their representative values over the room-barrier-room system. It can be seen that the IL values predicted by Morse’s model are overestimated at all frequencies at each receiver point. This is because the model neglects the surface integral of the boundary values of sound pressure. Such integrals, P tn , as shown in Eq. (18), can be proved to be coincident, although not exactly, in phase with the integrals over the 0b b coupling area P 0t n . Thus, subtracting P n from P n in Eq. (12) leads to a smaller absolute value. It can be seen that the results predicted by the proposed model agree better with the measured results at all frequencies than those predicted by Morse’s model. However, better agreement for the proposed model is obtained at lower frequency bands; for example, at position nine. This is because it is increasingly difficult to predict the natural modes as the frequency becomes higher due to the complexity and accuracy of the necessary computations. Fig. 5 shows the comparison of the measured IL values at different positions at 100 Hz and 1 kHz. It can be seen that the IL values are more position-sensitive at 100 Hz

12 100 Hz 1000 Hz

10

IL (dB)

8 6 4 2 0 -2 1

2

3

4 5 6 7 Receiver point number

8

Fig. 5. Distribution of IL at two frequency bands.

9

than those at 1 kHz. For example, IL values at 100 Hz are higher at points 1, 4, and 7 than at other points, while the IL values at 1 kHz fluctuate at all positions slightly at around 4 dB. These measured results agree with the predicted results in Fig. 3 that there are higher IL values at 100 Hz and that the sound fields are more evenly distributed at 1 kHz. 5. Conclusion In this paper, an analytical model has been proposed to modify Morse’s model and used to predict the insertion loss of a barrier in a room-barrier-room system where the boundary of the enclosure cannot be assumed to be rigid. The predicted results agree well, particularly at lower frequencies, with the measured results in the scale model of the office. The proposed predicted model can be applied to predict the insertion loss of an indoor barrier in the low-medium frequency range. Appendix. Green’s functions The Green’s function for a rectangular enclosure with a specific acoustic admittance bij(x) on each of the boundaries is 1 X Utn ðrÞUtn ðr0 Þ Gt ¼ ; ðA:1Þ vt n Ktn ½ðK tn Þ2  k 2  where Utn is the shape functions of the nth rigid-walled modal in a rectangular room, which can be written as

    Utn ¼ cos pntx x=Ltx cos pnty y=Lty cos pntz z=Ltz ; ðA:2Þ with the corresponding eigenvalue,  t 2  t t 2 t t 2  t t 2 gn ¼ pnx =Lx þ pny =Ly þ pnz =Lz ;

ðA:3Þ

where Ltx , Lty , and Ltz are the room dimensions and ntx , nty , and ntz are non-negative integers. Definitions of the symbols that need to be declared: i ¼ fx; y; zg Indication of the j ¼ f0; lg boundaries of the rectangular enclosure; t = {a, b} Indication of the particulate space; the default value indicates an empty office; vt The volume of the particular  enclosure; 0; ni ¼ 0 Ktn ¼ enx e1ny enz ; eni ¼ Eigenvalue; 2; ni 6¼ 0

Peni bi0Lþbi i1 K tn ¼ gtn  2gikt Eigen-wavei¼x;y;z n number.

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[9] Osipov A, Mees P, Vermeir G. Low-frequency airborne sound transmission through single partitions in buildings. Appl Acoust 1997;52:273–88. [10] Papadopoulos CI. Redistribution of the low frequency acoustic modes of a room: a finite element-based optimization method. Appl Acoust 2001;62:1267–85. [11] Tadeu A, Santos P. Assessing the effect of a barrier between two rooms subjected to low frequency sound using the boundary element method. Appl Acoust 2003;64:287–310. [12] Morse PM, Ingard KU. Theoretical acoustics. New York: McGrawHill; 1968. [13] Sum KS, Pan J. A study of the medium frequency response of sound field in a panel – cavity system. J Acoust Soc Am 1998;103(3):1510–9. [14] Harris CM. On the acoustics of coupled rooms. J Acoust Soc Am 1950;22(5):572–8. [15] ISO 18221:1997. Acoustics – Measurement of the in situ sound attenuation of a removable screen. 1997.