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Applied Acoustics 65 (2004) 985996 www.elsevier.com/locate/apacoust
A chart for estimating the distance attenuation of anking sound passing through open windows in the exterior wall of adjoining rooms and its experimental verication
, Tadao Fukuyama b, Yuzo Tsuchiya
Department of Architecture, Faculty of Engineering, Kansai University, Japan b Technical Research Institute, Toda Corporation, Japan
Received 22 November 2003; received in revised form 9 March 2004; accepted 2 April 2004 Available online 19 June 2004
Abstract Sound transmission between adjoining rooms is often inuenced by a anking sound passing through open windows placed in the exterior wall of the two rooms. It is important to predict such anking sound propagation when considering the sound insulation between the adjoining rooms. In this paper, a chart for estimating the distance attenuation of the anking sound, which is obtained from analysis using the boundary integral equation method, is rst provided. Next, 1:10 scale model experiments are carried out. The experimental results are in good agreement with the chart, the eectiveness of the chart thus being veried. 2004 Elsevier Ltd. All rights reserved.
Keywords: Flanking sound; Open window; Adjoining room; Transmission rate; Boundary integral equation; BEM; Image method
1. Introduction Sound transmission between adjoining rooms is often inuenced by anking sound passing through open windows in the shared exterior wall of the rooms. It is thought that such anking sound might be more inuential than sound transmitted through
Corresponding author. E-mail address: firstname.lastname@example.org (Y. Kawai).
0003-682X/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2004.04.002
Y. Kawai et al. / Applied Acoustics 65 (2004) 985996
the partition wall, especially in the summer when the windows are frequently open. In order to predict anking sound propagation, one of the authors constracted a chart based on two-dimensional analysis using boundary integral equations . It is more important, however, to predict three-dimensional propagation. In this paper, a chart for predicting anking sound obtained from three-dimensional analysis using boundary integral equations is presented and its eectiveness veried by 1:10 scale model experiments.
2. Formulation Let a source room, which includes an omnidirectional point sound source, be adjoined to a receiving room, as shown in Fig. 1. Each room has an aperture, A1 or A2 , in its exterior wall, S1 or S2. The thickness of the exterior wall is considered to be acoustically rigid and thin enough for the wavelength. The anking sound means here the sound which is emitted from the aperture A1 propagated in the exterior space X0 and entering through the aperture A2 into the receiving room. For simplicitys sake, in this analysis, the source room and the receiving room are assumed to be semi-innite. Though the two rooms are illustrated as quarter-innite spaces in Fig. 1, this assumption means that the surfaces of the partition wall between them are both perfectly absorbent. This assumption also presumes a one-directional sound incidence on the aperture A1 and no reradiation from the aperture A2. With this assumption, the anking sound, which is derived from the energy emitted from aperture A1 and that entering through the aperture A2 , can be precisely estimated. Let us consider an innitely large rigid at surface, in two parts of which are apertures A1 and A2. Also, let a semi-innite space X0 bounded by the innitely large semi-sphere R, the rigid surfaces S1 and S2 , and the apertures A1 and A2 include a receiving point P , and let Pi be the image point of P with respect to the surfaces S1 , S2 , A1 and A2 (see Fig. 2).
Fig. 1. Propagation of anking sound passing through open windows.
Fig. 2. Derivation of integral formula with respect to exterior space: X0 semi-innite space bounded by rigid walls S1 , S2 , apertures A1 , A2 , and innitely large semi-sphere R; n normal vector; P receiving point; Pi the image of P with respect to S1 ,S2 , A1 , A2 ; r small sphere of center P and radius.
In order to derive an integral formula with respect to the space X0 , we will use GP ; Q expikr expikri ; 4pr 4pri 1
as a fundamental solution, in which the image point Pi is taken into account. Here, r PQ and ri Pi Q. The time factor expixt is omitted throughout this paper. We
Fig. 3. Transmission rate TR in decibels of anking sound varying with distance between the midpoints of the apertures of the source and receiving rooms.
apply Greens theorem to the space X0 r, where r is a small sphere of center P with radius. Taking into consideration (1) the normal component of particle velocity vanishes throughout S1 and S2 , (2) oG=on 0 throughout S1 , S2 , A1 and A2 , and (3) Sommerfelds radiation condition , we can obtain Z 1 oUQ expikr UP 2 dS; P 2 X0 ; A1 ; A2 ; S1 ; S2 ; 2p A1 A2 on r where UP denotes velocity potential at P and n the inward drawn normal. Eq. (2) is valid R when P is located on S1 , S2 , A1 or A2 (i.e., P Pi ), since R lim or=2 dS UP ; P 2 S1 ; S2 ; A1 ; A2 . !0 As for the space X1 , considering the point source is located within it and Eq. (1) is used as the fundamental solution, we have
Fig. 4. Outline gure of the 1:10 scale model used in experiments.
UP UD P UD Pi
oUQ expikr dS; on r
P 2 X1 ; A1 ; S1 ;
where UD denotes a direct sound and n the outward drawn normal. The point Pi also denotes the image point of P 2 X1 with respect to S1 and A1. Considering that no source is located, we can also obtain an integral formula for the space X2 in the same way: i.e. Z 1 oUQ expikr dS; P 2 X2 ; A2 ; S2 : UP 4 2p A2 on r When P is located on A1 or A2 , then Eqs. (2)(4) are integral equations with an unknown function oU=on on A1 and A2. When P is located on A1 , subtracting the dierence between Eqs. (2) and (3) yields Z Z 1 oUQ expikr 1 oUQ expikr dS dS 2UD P ; P 2 A1 : p A1 on r 2p A2 on r 5
Table 1 Distances between the midpoints of windows in the scale model Window width (mm) 400 Distance between the midpoints of windows (mm) Extreme approach 500 Two times 1000 Three times 1500 Four times
Fig. 5. Arrangement for model experiments. Sound pressure levels at ve points denoted by dots in each room are measured (see also side view of Fig. 4).
Also, when P is located on A2 , subtracting the dierence between Eqs. (2) and (4) yields Z Z 1 oUQ expikr 1 oUQ expikr dS dS 0; P 2 A2 : 6 2p A1 on r p A2 on r Solving the simultaneous integral Eqs. (5) and (6), we can obtain oU=on on A1 and A2 , which gives the velocity potential U in X0 , X1 and X2 by substituting it into Eqs. (2)(4), respectively. The energy I1 emitted from the aperture A1 and the energy I2 entering through the aperture A2 can be obtained by the potential U and oU=on on A1 and A2 using the following equations: p ixqU; 7 v oU ; on and
Fig. 6. Comparison of the radiation directional characteristics: (a) when diusion objects are suspended from the ceiling (see side view of Fig. 4), and (b) when in addition to the characteristics of case (a), columns made of wood and absorbent expanded polystyrene are installed near three side walls (see Fig. 5). The results are converted to that of real scale.
1 pv p v dS; pv p v dS; 2
where p and v are complex conjugates of sound pressure p and particle velocity v, respectively . The transmission rate TR (in dB) of the anking sound is obtained by TR 10 log10 I2 : I1 10
3. A chart of distance attenuation For purposes of noise control, it is useful to know the transmission rate of anking sound under a condition of 1 octave band noise random incidence. In order to simulate the above condition, numerical calculations are carried out for 825 plane waves that are incident on the aperture A1 from all directions at regular solid angle intervals and for six frequencies taken in the octave band. The energies of both the emitted wave from A1 and of the incoming one are calculated for each condition and summed up separately. The transmission rate under conditions of random incidence can thus be obtained by using the resultant values. Fig. 3 shows a chart of the distance attenuation of anking sound calculated using the above-mentioned method when the dimension of both apertures is m2. The abscissa in this chart denotes the distance between the midpoints of the apertures.
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Fig. 7. Block diagram of measurement apparatus.
We can see that the transmission rate reduces by %6 dB for every doubling of distance. If the apertures have dimensions Sa1 and Sa2 , on the basis of the diuse sound eld assumption, we have L1 L2 TR 10 log10 A ; Sa1 Sa2 11
50 Sound pressure level difference (dB) Sound pressure level difference (dB)
20 200mm open 300mm distance 200mm open 600mm distance 200mm open 900mm distance 200mm open 1200mm distance
20 300mm open 400mm distance 300mm open 800mm distance 300mm open 1200mm distance 300mm open1600mm distance
500 1k 2k 4k
500 1k 2k 4k 1/3 Octave band center frequency(Hz)
1/3 Octave band center frequency(Hz)
Sound pressure level difference (dB)
20 400mm open 500mm distance 400mm open 1000mm distance 10 400mm open 1500mm distance
Fig. 8. Dierences between sound pressure levels in the source and receiving rooms with: (a) width of apertures 200 mm, (b) width of apertures 300 mm, and (c) width of apertures 400 mm. The averaged level for ve points in each room is used to obtain the level dierences.
where A denotes the absorption power of the receiving room, and L1 and L2 are energy density levels in the source room and the receiving room, respectively.
4. Scale model experiment 4.1. Outline of experiments As shown in Fig. 4, 1:10 scale model rooms are made from vinyl chloride plate 10 mm thick. Each room has an aperture whose dimensions can be changed between 200 200, and mm2. The distances between the midpoints of the apertures in the above three cases are shown in Table 1. The receiving room is
cal - 125 cal - 250 cal - 500 cal - 1k cal - 2k exp- 125 exp- 250 exp- 500 exp- 1k exp-2 k
-Distance [m] 100 1
10 Distance [m]
Fig. 9. Comparison of experimental with theoretical values.
isolated from the source room in order not to be excited. In the experiment for distance attenuation, the receiving room is moved, keeping the source room xed. Model experiments are carried out in a chamber whose peripheral walls are covered with triple absorbent layers. The measured inverse square law of sound pressure level in this chamber coming from the point source is shown to be satisfactory.
Fig. 10. Measurement of the radiation directional characteristics when the setting of tweeters is changed. Dots in front of the aperture denote measuring points.
Fig. 11. Comparison of the radiation directional characteristics of (a) the arrangement shown in Fig. 5, and (b) the arrangement shown in Fig. 10 (250 Hz).
In order to simulate the diuse sound eld in the model source room, the following two cases are investigated: (1) diusion objects made from plastic board are suspended (see side view of Fig. 4), and (2) in addition, columns made of wood and absorbent expanded polystyrene are installed near three side walls as shown in Fig. 5. The radiation directional characteristics from the source room for the two cases are indicated in Fig. 6. As the result for case (2) is better than that for case (1), we adopt the case (2) setting. Columns are also installed to the model receiving room (see also Fig. 5). The measurements also show that the deviation of the sound pressure levels in the source room is relatively small under this arrangement. A block diagram of our measurement apparatus is given in Fig. 7. Noise source signals of 1 octave band from 500 to 40 kHz are generated from four tweeters. Three 1/4-in. microphones are used for receivers and 1/3 octave band levels are measured using an FFT spectrum analyzer. 4.2. Results of experiments All the results obtained from model experiments are converted to those of real scale. Fig. 8(a)(c) show the dierences between sound pressure levels (measured for 1/3 octave bands) in the source room and the receiving room when both the aperture widths are 200, 300 and 400 mm, respectively. Though sound pressure levels are expected to increase monotonically with increasing frequency, they rise at about 250 Hz in all cases. After the results shown in Fig. 8(a)(c) are converted to those of 1 octave band, the transmission rates are calculated using Eq. (11) and compared with
cal-250Hz SP-one side
-10 Distance [m]
Fig. 12. Comparison of experimental and theoretical values at the 250 Hz band for the dierent arrangements.
the distance attenuation chart, as shown in Fig. 9(a)(c). As uneven sound pressure level distribution is expected in the source or the receiving room mainly due to the existence of the aperture, the averaged level for ve points in each room are used for energy density level L1 or L2 (see Fig. 5). We can see that the experimental values are in good agreement with the theoretical values, within 2 dB for all octave bands except one band at the center frequency 250 Hz. The reason of the discrepancy at the 250 Hz band is probably due to bad radiation directional characteristics from the aperture of the source room as shown in Fig. 11(a). If the setting of tweeters is changed as shown in Fig. 10, better radiation directional characteristics is obtained (see Fig. 11(b)). Under this setting the discrepancy between the experimental and theoretical values at the 250 Hz band is improved as shown in Fig. 12.
5. Conclusion The eectiveness of the chart to predict anking sound obtained from analysis using boundary integral equations was veried experimentally. As the chart was derived under a condition of 1 octave band noise random incidence, it should prove useful when diuse sound elds in the source and receiving rooms are approximated.
 Kawai Y. J Acoust Soc Jpn 2001;57(2):13943 (in Japanese).  Baker BB, Copson ET. The mathematical theory of Huygens principle. New York: Chelsea; 1987. p. 25.  Kawai Y. J Acoust Soc Jpn 2000;56(3):1437 (in Japanese).  Morse PM, Ingard KU. Theoretical acoustics. New York: McGraw-Hill; 1968. p. 258.
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