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In situ measurements of surface impedance and absorption coefficients of porous materials using two microphones and ambient noise
Applied Acoustics 66 (2005) 845–865 www.elsevier.com/locate/apacoust
In situ measurements of surface impedance and absorption coefficients of porous materials using two microphones and ambient noise Y. Takahashi a
a,*
, T. Otsuru b, R. Tomiku
c
Graduate School of Engineering, Oita University, 700 Dannoharu, Oita 870-1192, Japan b Faculty of Engineering, Oita University, 700 Dannoharu, Oita 870-1192, Japan c Venture Business Laboratory, Oita University, 700 Dannoharu, Oita 870-1192, Japan
Received 3 April 2003; received in revised form 10 November 2004; accepted 10 November 2004 Available online 20 January 2005
Abstract An in situ measurement method is proposed for obtaining the normal surface impedance and absorption coefficient of porous materials using two microphones located close to the material without a specific sound source such as a loudspeaker. Ambient environmental noise that does not excite distinct modes in the sound field is employed as the sound source. Measurements of the normal surface impedance of glass wool and rockwool have been made using this method in various sound fields. The repeatability and wide applicability of the method are demonstrated by comparing results of measurements in one room with different noise conditions and in three other environments (corridor, cafeteria and terrace). The assumed diffuse nature of the sound field on the material is validated by using absorption characteristics obtained experimentally at oblique incidence. This method allows simple and efficient in situ measurements of absorption characteristics of materials in a diffuse field. 2004 Elsevier Ltd. All rights reserved. Keywords: Impedance; In situ; Diffuse field; Two microphones; Ambient noise
*
Corresponding author. Tel.: +81 97 554 6406; fax: +81 97 554 7918. E-mail address: [email protected] (Y. Takahashi).
0003-682X/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2004.11.004
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1. Introduction Numerous methods have been proposed to measure material absorption characteristics. Garai [1] has classified the basic techniques as (i) reverberation room methods, (ii) tube methods, and (iii) reflection methods. The reverberation room method [2] allows measurement of the diffuse incidence absorption coefficient, which is used widely as a practical absorption index. However, creation of a diffuse sound field requires a large and costly reverberation room. A completely diffuse sound field can be achieved only rarely. Moreover, an accurate value of complex impedance cannot be derived from the absorption coefficient alone. The tube method has many variations [3–10]. Usually either the standing wave method or the transfer function method [4] is used. These methods can be used to measure accurate normal acoustic impedance at normal incidence with rather small equipment. However, Champoux et al. [11] and Garai [1] have pointed out some limitations of these tube methods. As well as these problems, the absorption characteristics obtained by the tube methods do not correspond to those in practical conditions. To predict and control acoustics in real environments, it is crucial to measure material absorption characteristics subject to various placement conditions in situ. Garai [1] has indicated that the reverberation room and tube methods are essentially laboratory methods so they are unsuitable for measuring absorption characteristics in situ. Various reflection methods [11–20] have been devised and developed to carry out in situ measurement in the expectation that they can solve the problems of other methods. In particular, the methods proposed by Allard et al. [11–14] or Garai [1] constitute efficient and useful reflection methods. Allard et al. describe reflection methods using two microphones. Their methods allow measurement of a small sample, area about 1 m2, in a free field. Moreover, they allow in situ measurement of a sample area of about 6 m2 at frequencies of 500 Hz or higher but the required sample size increases as the frequency decreases. GaraiÕs method for measuring the absorption coefficient in situ uses MLS as a test signal to improve background noise immunity. However, some problems remain in these methods, including (a) low-frequency limitation, (b) inaccuracy with weakly absorbing materials, (c) the need for complicated equipment when used in situ and (d) the practicality of the relative positioning of a sound source, microphones and sample. It is concluded that present methods do not allow in situ measurement of material absorption characteristics both simply and efficiently. The existing modes inside a room may cause measurement difficulties with the conventional reflection methods when they are excited by a loudspeaker source. The method presented here avoids the influence of room modes and enables simple and efficient in situ measurements of material absorption characteristics at low frequencies below 500 Hz, whereas such measurements are very difficult using conventional reflection methods. Background environmental noise is employed as the sound source in this measurement. We examine the basis and effectiveness of this measurement method and clarify details of measured absorption characteristics.
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Firstly, we describe a procedure for measuring absorption characteristics of a material. Secondly, glass wool and rockwool measurements are made in a room to verify the methodÕs reliability. Then, we compare results measured in different environmental noise conditions to verify the repeatability of measurements. We also demonstrate the wide applicability of measurements through comparison of results measured in various sound fields. Finally, we discuss details of the measured absorption characteristics and clarify the incidence condition of the signals used in the measurement.
2. Description of the method 2.1. The measurement setup The authors recently proposed a new simple and efficient method for in situ measurement of material absorption characteristics using two microphones and ambient noise [21]. The method is based on measurement of the transfer function between two microphones using a spectrum analyzer. Fig. 1 shows a schematic diagram of the measurement setup. The two microphones are located at Ma and Mb. The distance d from the material surface to Mb is 10 mm and the space l between the two microphones is 13 mm. Two identical 1/2-in. (1.27 cm) microphones are employed to measure the sound pressure. When the microphones have different sensitivities, they are calibrated relative to each other using the standard switching technique. The FFT resolution is set to 1.0 Hz and a Hamming window is employed. Measured data are averaged 30 times. The material to be measured is backed by a rigid wall. Ambient noise around the material, so-called background noise, is employed as the sound source instead of a specific sound source such as a loudspeaker. Previously [21], the authors have called such noise ‘‘environmental anonymous noise’’ (EA-Noise) to indicate that the noise does not excite distinct modes. It is assumed that EA-Noise represents ambient noise with no strong
Fig. 1. A schematic diagram of the measurement setup.
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directivity. The required characteristics of the noise will be clarified in following sections. 2.2. Impedance measurement Let pi(x) and pr(x) represent the incident and reflected sound pressure in the frequency domain at the material surface. Assuming plane waves (implied time dependence is ejxt), with oblique incidence at angle h, the sound pressures pa(h,x) and pb(h,x) measured at Ma and Mb (see Fig. 2) are given by pa ðh; xÞ ¼ pi ðxÞejkðlþdÞ cos h þ pr ðxÞejkðlþdÞ cos h
ð1aÞ
pb ðh; xÞ ¼ pi ðxÞejkd cos h þ pr ðxÞejkd cos h :
ð1bÞ
and
Then, the reflection coefficient r(h) and the normal surface impedance of material ZN(h,x) are calculated from rðhÞ ¼
pr ðxÞ p ðh; xÞ pb ðh; xÞejkl cos h 2jkd cos h ¼ a e pi ðxÞ pb ðh; xÞejkl cos h pa ðh; xÞ
ð2Þ
and Z N ðh; xÞ ¼ ¼
qc 1 þ rðhÞ cos h 1 rðhÞ qc H ab ðh; xÞð1 e2jkðlþdÞ cos h Þ ejkl cos h ð1 e2jkd cos h Þ ; cos h H ab ðh; xÞð1 þ e2jkðlþdÞ cos h Þ ejkl cos h ð1 þ e2jkd cos h Þ
ð3Þ
where k is the wavelength constant, q is the density of air, c is the velocity of sound, and Hab(h,x) is the transfer function pb(h,x)/pa(h,x) in the direction defined by microphones at the angle of incidence h. The absorption coefficient aN(h,x) is calculated from the normal surface impedance ZN(h,x). When EA-Noise is utilized as the sound source, the sound pressure paEA(x) and pbEA(x) measured at Ma and Mb are values averaged over time and incident angle. Consequently, using averaged sound pressures, the transfer function between two
Fig. 2. A schematic of the impedance measurement at oblique incidence of angle h.
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microphones becomes HabEA(x) = pbEA(x)/paEA(x). So for measurements using EANoise, substituting HabEA(x) for Hab(h,x) and an average value of h in Eq. (3), the impedance can be calculated. However, it is difficult to ascertain an average value of incident angles of sound in measurements because of the difficulty in obtaining the instantaneous incident angle of sound for each FFT used in the measurements. Moreover, the first object of this study is to carry out simple and stable measurements for absorption characteristics of materials in situ. Therefore, the first stage of this study considers normal incidence, i.e., cosh = 1 in Eq. (3). In this case, the impedance can be expressed as Z EA ðxÞ ¼ qc
H abEA ðxÞð1 e2jkðlþdÞ Þ ejkl ð1 e2jkd Þ : H abEA ðxÞð1 þ e2jkðlþdÞ Þ ejkl ð1 þ e2jkd Þ
ð4Þ
The absorption coefficient aEA(x) is calculated from the impedance ZEA(x). The relationship between ZN(h,x) and ZEA(x) is shown in Appendix A.
3. Method reliability This section discusses the repeatability and applicability of the method. Using the proposed method, the absorption characteristics of glass wool (density = 32 kg/m3) and rockwool (density = 150 kg/m3) have been measured in four rooms during different ambient noise conditions, with various types of incident sound fields, and the results have been compared. The sound pressure levels (Lp) of noise during measurement were monitored at one-second intervals. The area and thickness of each material was 0.3 · 0.6 m2 and 0.05 m, respectively. Table 1 shows details of the environments, i.e., volumes and reverberation times (500 Hz, one octave band). Measurements were ceased when any of the one-second Lp exceeded 10 dB above the mean during the measurements. In following sections, measured values are averaged by frequency range and plotted in 100 Hz steps to show any differences distinctly. 3.1. Measurements in a room Measurements of two materials were carried out in a room with three different ambient noise conditions (Symbol: R-L) to investigate repeatability and robustness. Fig. 3 shows a plan view of the room with furniture layout and sample location. In
Table 1 Outlines and reverberation time (one octave band, center frequency = 500 Hz) of sound fields used in measurement Symbol
Sound field
Volume (m3)
Reverberation time (s)
R-L Cr Caf Tr
Laboratory Corridor Cafeteria Terrace
70 177 2062 (outdoor)
0.17 1.57 1.63
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Fig. 3. Plan view of a room (R-L) with furniture layout (dimensions in mm).
condition (i), nine personal computers were turned on, including the one used in measurement, all windows and a door were open, and two people operating measurement instruments were in the room. In condition (ii), the computers and people were the same as in condition (i) but all windows and the door were closed. In condition (iii), the computers and people were the same as in condition (i), but six additional people were present either performing light tasks using computers or chatting quietly around the material. Fig. 4 shows the Lp monitored during the measurement p Þ and variances period for each room condition. Table 2 lists the averaged values ðL (r2(Lp)). The mounting and locations of the material were identical for each measurement. In each case the ambient noise can be regarded as EA-Noise. Figs. 5 and 6 show the measured absorption coefficients aEA (x) and complex impedance values ZEA(x). For glass wool, mean deviations (MDs) of absorption coefficients are less than 0.015 at frequencies above 200 Hz. MDs for real parts of impedances are about 100 Pa s/m at 200 Hz and less than 42 Pa s/m at frequencies above 300 Hz, whereas those for imaginary parts are about 125 Pa s/m at 200 Hz and less than 35 Pa s/m at frequencies above 300 Hz. As for rockwool, MDs of absorption coefficients are about 0.02 at 200 Hz and less than 0.02 at frequencies
Fig. 4. Monitored Lp at three room conditions in R-L: (a) glass wool measurement; (b) rockwool measurement.
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Table 2 Averaged values and variances of the monitored Lps at three room conditions in R-L r2(Lp) (dB)
(a) Glass wool measurement Condition (i) Condition (ii) Condition (iii)
58.2 54.2 60.3
6.8 4.1 10.9
(b) Rockwool measurement Condition (i) Condition (ii) Condition (iii)
57.9 55.5 59.6
3.4 3.0 6.3
Glass wool
1.2 1.0 0.8 0.6
Condition (i)
0.4 Condition (ii)
0.2
Condition (iii)
0.0 0
500 1000 Frequency [Hz]
(a)
Absorption coefficient αEA
Absorption coefficient α EA
p (dB) L
Rock wool
1.2 1.0 0.8 0.6
Condition (i)
0.4 Condition (ii)
0.2
Condition (iii)
0.0 0
1500
500 1000 Frequency [Hz]
(b)
1500
Fig. 5. Comparisons of the absorption coefficients measured at three room conditions in R-L: (a) glass wool; (b) rockwool.
Glass wool Real part
1000 0
Condition (i) -1000 Imaginary part
0
(a)
Condition (ii) Condition (iii)
-2000
Rockwool
2000
500 1000 Frequency [ Hz]
Impedance ZEA [Pa s/m]
Impedance ZEA [Pa s/m]
2000
1000
Real part
0 Condition (i) -1000 Imaginary part
Condition (iii)
-2000
1500
0
(b)
Condition (ii)
500 1000 Frequency [ Hz]
1500
Fig. 6. Comparisons of the impedance measured at three room conditions in R-L: (a) glass wool; (b) rockwool.
above 300 Hz. At frequencies above 200 Hz, MDs for the real part of impedance are less than 85 Pa s/m, whereas those for the imaginary part are less than 90 Pa s/m. Thus, although the averaged values and variance of the monitored Lp differ in each measurement, as shown in Table 2, the overall tendencies of three results agree well at frequencies above 200 Hz.
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Furthermore, Fig. 7 compares the absorption coefficients aEA(x) measured four times in condition (i) and shows the averaged values and variance of the monitored Lp in each measurement. MDs of absorption coefficients for glass wool are less than 0.02 within measurement frequency range. For rockwool, they are less than 0.025 within the frequency range. Though discrepancies among measured results are observed because of slight differences of noise conditions in each measurement, the four results agree well within the frequency range. We have confirmed also that MDs of 15 results over the three conditions have the same values as the four for room condition (i). Thus, the proposed measurement method has been shown to give repeatable results for different ambient noise conditions in a given room. 3.2. Measurements in various sound fields
Glass wool
1.2 1.0 0.8
L p [dB] σ (L p ) [dB] 58.2 6.8 57.1 2.5 57.8 3.5 58.2 2.7 2
0.6 0.4 0.2 0.0 0
(a)
500 1000 Frequency [Hz]
Absorption coefficient α EA
Absorption coefficient αEA
To investigate the general applicability of the method, measurements of the two materials have been carried out in three other environments [a corridor (Symbol: Cr), a cafeteria (Symbol: Caf) and a terrace (Symbol: Tr)], volumes and reverberation times of which are shown in Table 1. Plan views of furniture layouts and material locations in the corridor and cafeteria are shown in Figs. 8 and 9. Figs. 10 and 11 show the measurement conditions in the cafeteria and the Terrace. In the corridor, windows 1 and 2, shown in Fig. 8, were opened and there were two people inside. In the cafeteria, there were many people chatting, as shown in Fig. 10. On the terrace, material samples were located at 2 m distance from a building. Fig. 12 shows the impedance ZEA(x) measured in the three sound fields. Results for R-L given in the previous section are shown also. The monitored Lp values in each sound field are shown in Fig. 13 and Table 3 shows their averaged values and variances. For glass wool, MDs of impedance are less than 60 Pa s/m at frequencies above 200 Hz except for a MD of about 145 Pa s/m at 200 Hz for the imaginary part. At frequencies above 200 Hz, MDs of real parts are less than 70 Pa s/m whereas those of imaginary parts are less than 100 Pa s/m for rockwool. Moreover, we have found
1500
Rock wool
1.2 1.0 0.8
L p [dB] σ (L p ) [dB] 57.9 3.4 57.8 2.3 58.7 4.9 57.1 2.9 2
0.6 0.4 0.2 0.0 0
(b)
500 1000 Frequency [Hz]
1500
Fig. 7. Comparisons of the absorption coefficients among four times measurement in R-L with condition (i) and the averaged values and variances of monitored Lp: (a) glass wool; (b) rockwool.
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Fig. 8. Plan view of the corridor (Cr) with sample location (dimensions in mm).
Fig. 9. Plan view of the cafeteria (Caf) with furniture layout and sample location (dimensions in mm).
Fig. 10. Condition of the cafeteria (Caf) used for measurements.
that MDs of absorption coefficients at frequencies above 200 Hz are less than 0.025 for glass wool and less than 0.02 for rockwool, but these absorption coefficient values are not shown here. The overall tendency is that the impedances measured in the four sound fields agree well with each other at frequencies above 200 Hz.
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Fig. 11. Condition of the terrace (Tr) used for measurements.
Glass wool Real part
1000 0 -1000 Imaginaryp art
-2000 0
(a)
2000
R-L
Caf
Cr
Tr
500 1000 Frequency [Hz]
Impedance ZEA [Pa s/m]
Impedance ZEA [Pa s/m]
2000
Rock wool
1000
Real part
0 Imaginary part
-1000 -2000
1500
0
(b)
R-L
Caf
Cr
Tr
500 1000 Frequency [Hz]
1500
Fig. 12. Comparisons of the impedances measured in four sound fields: (a) glass wool; (b) rockwool.
Fig. 13. Monitored Lp in four sound fields: (a) glass wool measurement; (b) rockwool measurement.
These results indicate that certain types of material absorption characteristics can be measured universally using the method even if the incident sound fields are different. Furthermore, it is noteworthy that results measured in the cafeteria are
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Table 3 Averaged values and variances of the monitored Lps in four sound fields p (dB) L
r2(Lp) (dB)
(a) Glass wool measurement R-L Cr Caf Tr
58.2 55.9 71.6 59.3
6.8 1.8 4.7 6.4
(b) Rockwool measurement R-L Cr Caf Tr
57.9 55.0 71.2 59.1
3.4 2.3 0.6 1.1
indistinguishable from the others despite great differences in the averaged values of one-second Lp. Consequently, the method may be considered to have sufficient reliability as an in situ method for measuring absorption characteristics of porous materials. In following sections, values obtained in the cafeteria are chosen as representative values of absorption characteristics measured by the method; they are referred to as ‘‘EA-Caf’’.
4. Clarification of measured absorption characteristics What kind of absorption characteristics are measured by this method? In this section, incident condition of signals to materials in measurement is investigated. A particular incident condition is assumed and the assumption is verified experimentally. 4.1. Investigation on incident condition of signals 4.1.1. Comparison with values obtained by the Miki model Fig. 14 shows comparisons between EA-Caf and absorption characteristics obtained using the regression model by Miki [22] (MikiÕs-eq) for glass wool. For absorption coefficients, distinct differences can be found between EA-Caf and MikiÕs-eq within measurement frequency range. The values of the imaginary part of impedance are close to MikiÕs-eq, while real part values are not. Clearly, the incident condition in the present method differs from normal incidence because EA-Noise is utilized as the sound source. Therefore, the differences between EA-Caf and MikiÕs-eq are likely to result from differences in the incident conditions. 4.1.2. Impedance averaged on angle of incidence Hereafter, the authors assume that the incident condition in the measurements is diffuse. Then, we consider that the impedance at an incident condition corresponding to a diffuse field can be obtained according to the calculation procedure for the random incidence absorption coefficient.
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1.0 0.8 0.6 0.4
EA-Caf
0.2 Miki's-eq
0.0 0
(a)
Impedance
2000
Impedance [Pa s/m]
Absorption coefficient
1.2
500 1000 Frequency [Hz]
Real part
1000 0 -1000
Imaginary part
EA-Caf Miki's-eq
-2000 1500
0
(b)
500 1000 Frequency [Hz]
1500
Fig. 14. Comparisons between EA-Caf and MikiÕs-eq for glass wool: (a) absorption coefficient; (b) impedance.
When normal surface impedance at oblique incidence ZN(h,x) is given, the impedance in a diffuse field Zfield(x) can be expressed as following equations by averaging admittance bN(h,x) = 1/ZN(h,x) over the angle of incidence h (h = 0–78: field incidence): R 78 bN ðh; xÞ sin h cos h dh bfield ðxÞ ¼ 0 R 78 ; ð5Þ sin h cos h dh 0 and Z field ðxÞ ¼
1 ; bfield ðxÞ
ð6Þ
where bfield(x) is the admittance in a diffuse field. 4.2. Experimental verification 4.2.1. Experimental procedure of obtaining absorption characteristics at an incident condition of field incidence This section explains an experimental procedure to derive the impedance ZEA(x) measured by the method. The previous section showed that the impedance at field incidence, Zfield(x), is obtainable from the impedance ZN(h,x) calculated by Eq. (3). However, it is considered that the impedance Zfield(x) differs from impedance ZEA(x) because the impedance ZN(h,x) is different from the component of the impedance ZEA(x) at incident angle h. If this component is denoted by ZEAcom(h,x), it is considered that Z EAcom ðh; xÞ ¼
H ab ðh; xÞð1 e2jkðlþdÞ Þ ejkl ð1 e2jkd Þ qc: H ab ðh; xÞð1 þ e2jkðlþdÞ Þ ejkl ð1 þ e2jkd Þ
ð7Þ
The transfer function Hab(h,x) in Eq. (7) is equal to that in Eq. (3). Using the impedance ZEAcom(h,x) and Eq. (5), the impedance in a diffuse field ZEAfield(x) is obtained by averaging admittance bEAcom(h,x) = 1/ZEAcom(h,x) over the angle of incidence h (h = 0–78). Hence
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R 78 bEAfield ðxÞ ¼
0
bEAcom ðh; xÞ sin h cos h dh ; R 78 sin h cos h dh 0
857
ð8Þ
and Z EAfield ðxÞ ¼
1 ; bEAfield ðxÞ
ð9Þ
bEAfield(x) is the admittance in a diffuse field obtained using ZEAcom(h,x). We consider that the impedance ZEA(x) is equal to impedance ZEAfield(x) when the incident condition in the method is diffuse. For that reason, we infer that comparisons between ZEA(x) and ZEAfield(x), the latter of which is obtained experimentally, should verify the type of the measured absorption characteristics. Measurements of absorption characteristics at oblique incidence have been carried out in an anechoic room using the technique described by Allard et al. [12] (AllardÕs method). A sound source was placed 2.5 m distant from the material surface as shown in Fig. 15. The incident angle hi was varied in 15 steps from 0 to 75 with respect to the normal to the material surface. The receiving system settings were the same as those described in Section 2. The impedance at each angle of incidence was calculated using Eq. (7). Thereby, six discrete impedance spectra were obtained. A discrete approximation of equation of Eq. (8) is P5 b ðhi ; xÞm sinðhi Þ cosðhi ÞDh bEAfield ðxÞ ¼ i¼0 EAcom ; ð10Þ P5 i¼0 m sinðhi Þ cosðhi ÞDh where hi ¼ i Dh;
Dh ¼ 15;
m ¼ 1=2 ði ¼ 0; 5Þ; m ¼ 1 ði ¼ 1; 2; 3; 4Þ;
and
bEAcom ðhi ; xÞ ¼ 1=Z EAcom ðhi ; xÞ: Using the six measured discrete impedances and Eqs. (10) and (9), the impedance at field incidence ZEAfield(x) and hence the absorption coefficient at field incidence
Fig. 15. Experimental setup of the normal surface impedance measurement at oblique incidence of angle hi.
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aEAfield(x) have been obtained. The resulting absorption characteristics are denoted hereafter as ‘‘Field-EA’’. Likewise, a discrete approximate of Eq. (5) can be defined. bfield and Zfield(x) can be obtained from that equation. Those absorption characteristics are denoted by ‘‘Field-Avg’’. 4.2.2. Comparisons with experimental averaged absorption characteristics Field-EA on glass wool and rockwool are obtained experimentally with the procedure described in the previous section. Figs. 16 and 17 show comparisons between EA-Caf and Field-EA. In the experiments at oblique incidence, the accuracy of values obtained at low frequencies below 500 Hz is insufficient because of the small sample areas and the influence of the positioning of the sound source, microphones and sample. However, despite the slight differences between EA-Caf and Field-EA, the values of EA-Caf correspond well with those of Field-EA at frequencies above 400 Hz for both glass wool and rockwool. Thus, the incident condition assumption of the method described in Section 4.1 is confirmed to be sufficiently appropriate. More specifically, the method enables us to measure impedance of materials in a diffuse field, ZEAfield(x). Absorption coefficient
1.0 0.8 0.6 0.4 EA-Caf
0.2
Field-EA
0.0 0
(a)
500 1000 Frequency [Hz]
Impedance
2000
Impedance [Pa s/m]
Absorption coefficient
1.2
Real part
1000 0 EA-Caf
-1000
Imaginary part
Field-EA
-2000 0
1500
(b)
500 1000 Frequency [Hz]
1500
Fig. 16. Comparisons between EA-Caf and Field-EA for glass wool: (a) absorption coefficient; (b) impedance.
Fig. 17. Comparisons between EA-Caf and Field-EA for rockwool: (a) absorption coefficient; (b) impedance.
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Fig. 18. Comparisons between the absorption characteristics obtained by Eq. (3) [ZN(h,x)] and those obtained by Eq. (7) [ZEAcom(h,x)] at angle of incidence h = 15 for glass wool: (a) absorption coefficient; (b) impedance.
Fig. 19. Comparisons between the absorption characteristics obtained by Eq. (3) [ZN(h,x)] and those obtained by Eq. (7) [ZEAcom(h,x)] at angle of incidence h = 45 for glass wool: (a) absorption coefficient; (b) impedance.
4.2.3. Comparisons between two absorption characteristics at oblique incidence The impedance ZEAcom(h,x) differs from the normal surface impedance at oblique incidence ZN(h,x). To discern differences between ZEAcom(h,x) and ZN(h,x) at every incident angle h, the absorption characteristics for glass wool obtained as described at incident angles of 15, 45, and 75 are compared with those at same angles calculated by Eq. (3) using the same measured transfer functions. These results are averaged by frequency range and are shown in Figs. 18–20 at 100 Hz steps. Averaged residuals of results are shown with the figures. Although some differences are observed for impedance at angle of incidence h = 75, averaged residuals of absorption coefficients are less than 0.03 at all incident angles and differences between aEAcom(h,x) and aN(h,x) are relatively small. Comparisons between Field-EA and Field-Avg are shown with their averaged residuals in Fig. 21. For both real and imaginary parts of impedance, averaged residuals are less than 75 Pa s/m. They are less than 0.02 for absorption coefficients. It seems
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Fig. 20. Comparisons between the absorption characteristics obtained by Eq. (3) [ZN(h,x)] and those obtained by Eq. (7) [ZEAcom(h,x)] at angle of incidence h = 75 for glass wool: (a) absorption coefficient; (b) impedance.
Fig. 21. Comparisons between Field-EA and Field-Avg for glass wool: (a) absorption coefficient; (b) impedance.
that the differences between absorption characteristics obtained by Eq. (3) and those obtained by Eq. (7) become larger as the angle of incidence h increases. Nevertheless, differences between Field-EA and Field-Avg are relatively small compared to those of Fig. 20. Hence, it is concluded that the impedance obtained by this method ZEA(x) is close to Zfield(x) which represents the average of ZN(h,x) over the angle of incidence h (h = 0–78) where ZN(h,x) is the normal surface impedance at oblique incidence.
5. Conclusions A new in situ measurement method for obtaining absorption characteristics of porous materials at field incidence using EA-Noise has been presented. The effectiveness and validity of the method have been examined. The measurement method offers good repeatability and robustness: MDs of absorption coefficients are less than
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0.025 and those of impedance are less than 100 Pa s/m at frequencies above 200 Hz. Also, it is applicable in different sound fields: MDs of absorption coefficients are less than 0.025 and those of impedance are less than 145 Pa s/m at frequencies above 200 Hz. It is interesting to note that the method can measure absorption characteristics of porous materials at frequencies below 500 Hz even when measurements are carried out on small samples in non-free field conditions. The nature of the incident condition of signals to materials in this measurement has been examined experimentally. It has been shown that the incident condition in this measurement can be regarded as field or random incidence. We conclude that this method offers a simple and efficient in situ measurement method for obtaining the impedance of porous materials in a diffuse field, Zfield(x), and eliminates the problems with other methods described in Section 1. Investigations continue regarding its accuracy and robustness. The reasons for discrepancies between Field-EA and Field-Avg will be clarified in a subsequent paper.
Appendix A Relationships between ZEAcom(h,x) [the component of ZEA(x) at incident angle h] and ZN(h,x) [the normal surface impedance for oblique incidence at angle h] and between ZEA(x) [the impedance obtained by this method] and ZN(0,x) [the normal impedance at normal incidence (i.e., acoustic impedance)] are given in this appendix. ZN(h,x) can be calculated from Eq. (3). On the other hand, ZEAcom(h,x) can be calculated from Eq. (7). In both equations, the transfer function Hab(h,x) is equal. Then, the relationship between ZEAcom(h,x) and ZN(h,x) can be obtained from Eqs. (3) and (7): Z EAcom ðh;xÞ ¼
Z N ðh;xÞcosh ejkl A1 D1 Z N ðh;xÞcosh ejklcosh C 1 B1 þ qcejklcosh C 1 B2 qcejklcosh A1 D2 ; qcZ N ðh;xÞcosh ejkl A2 D1 qcZ N ðh;xÞcosh ejklcosh C 2 B1 þ q2 c2 ejklcosh C 2 B2 q2 c2 ejklcosh A2 D2
ðA:1Þ where A1 ¼ ð1 þ e2jkd Þ;
ðA:2aÞ
A2 ¼ ð1 e
2jkd
B1 ¼ ð1 þ e
2jkd cos h
Þ;
ðA:2cÞ
B2 ¼ ð1 e
2jkd cos h
Þ;
ðA:2bÞ Þ;
ðA:2dÞ
C 1 ¼ ð1 þ e
2jkðlþdÞ
Þ;
ðA:2eÞ
C 2 ¼ ð1 e
2jkðlþdÞ
Þ;
ðA:2fÞ
D1 ¼ ð1 þ e
2jkðlþdÞ cos h
Þ;
ðA:2gÞ
D2 ¼ ð1 e
2jkðlþdÞ cos h
Þ:
ðA:2hÞ
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Assuming that the incident condition of signals to materials in this method is random incidence, the impedance ZEA(x) obtained by this method can be calculated by averaging the admittance bEAcom(h,x) = 1/ZEAcom(h,x) over all angles of incidence h (h = 0–p/2: random incidence). Hence: Z p=2 bEA ðxÞ ¼ 2 bEAcom ðh; xÞ sin h cos h dh; ðA:3Þ 0
and Z EA ðxÞ ¼
1 ; bEA ðxÞ
ðA:4Þ
where bEA(x) is the admittance averaged bEAcom(h,x) over the angle of incidence h (h = 0–p/2). Similarly, using ZN(h,x), the impedance Zrand(x) averaged over all angles of incidence h (h = 0–p/2) is given by Z p=2 bN ðh; xÞ sin h cos h dh; ðA:5Þ brand ðxÞ ¼ 2 0
and Z rand ðxÞ ¼
1 ; brand ðxÞ
ðA:6Þ
where bN(h,x) = 1/ZN(h,x) is the admittance at oblique incidence of angle h, and brand(x) is the admittance averaged bN(h,x) over the angle of incidence h (h = 0–p/ 2). Here, each ZN(h,x) becomes equal to the normal surface impedance at normal incidence ZN(0,x) when the locally reacting assumption is applied. The admittance at oblique incidence bN(h,x) is given by: bN ðh; xÞ ¼
1 1 ¼ ¼ bN ð0; xÞ: Z N ðh; xÞ Z N ð0; xÞ
ðA:7Þ
Substituting bN(0,x) for bN(h,x) into Eq. (A.5), brand(x) is obtained as Z p=2 Z p=2 bN ð0; xÞ sin h cos h dh ¼ 2bN ð0; xÞ sin h cos h dh brand ðxÞ ¼ 2 0
¼ bN ð0; xÞ;
0
ðA:8Þ
and Z rand ðxÞ ¼
1 1 ¼ ¼ Z N ðxÞ: brand ðxÞ bN ðxÞ
ðA:9Þ
Hence, Zrand(x) is equal to ZN(0,x) for locally reacting materials. In case of locally reacting materials, substituting ZN(0,x) for ZN(h,x) into Eq. (3), Hab(h,x) at each angle is derived. ZEAcom(h,x) can be calculated from Eq. (7) using the Hab(h,x). Here, ZN(0,x) is obtained using the regression model derived by Miki. Figs. 22–24 show comparisons between ZN(h,x) [=ZN(0,x)] and ZEAcom(h,x) and between absorption coefficients aN(h,x) [=aN(0,x)] and aEAcom(h,x), calculated from
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Angle of incidence θ =15˚ Absorption coefficient
2000
1.0 0.8 0.6 0.4
αN
0.2
α EAcom
0.0 0
(a)
1000 500 Frequency [Hz]
Impedance [Pa s/m]
Absorption coefficient
1.2
Impedance ZN
1000
Real part
ZEAcom
0 -1000 Imaginary part
-2000 1500 0 (b)
1500
1000 500 Frequency [Hz]
Fig. 22. Comparisons between ZN(h,x) [=ZN(0,x)] and ZEAcom(h,x) and between aN(h,x) [=aN(0,x)] and aEAcom(h,x) at angle of incidence h = 15 for the locally reacting materials: (a) absorption coefficient; (b) impedance. Angle of incidence θ =45˚ Absorption coefficient
2000
1.0 0.8 0.6 0.4
αN
0.2
α EAcom
0.0 0
(a)
1000 500 Frequency [Hz]
Impedance [Pa s/m]
Absorption coefficient
1.2
Impedance ZN
1000
Real part
ZEAcom
0 -1000 Imaginary part
-2000 1500 0 (b)
1500
1000 500 Frequency [Hz]
Fig. 23. Comparisons between ZN(h,x) [=ZN(0,x)] and ZEAcom(h,x) and between aN(h,x) [=aN(0,x)] and aEAcom(h,x) at angle of incidence h = 45 for the locally reacting materials: (a) absorption coefficient; (b) impedance. Angle of incidence θ =75˚ Absorption coefficient
2000
1.0 0.8 0.6 0.4
αN
0.2
α EAcom
0.0 0
(a)
500 1000 Frequency [Hz]
Impedance [Pa s/m]
Absorption coefficient
1.2
Impedance ZN
1000
Real part
ZEAcom
0 -1000
-2000 1500 0 (b)
Imaginary part
500 1000 Frequency [Hz]
1500
Fig. 24. Comparisons between ZN(h,x) [=ZN(0,x)] and ZEAcom(h,x) and between aN(h,x)] [=aN(0,x)] and aEAcom(h,x) at angle of incidence h = 75 for the locally reacting materials: (a) absorption coefficient; (b) impedance.
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Y. Takahashi et al. / Applied Acoustics 66 (2005) 845–865 Absorption coefficient
1.0 0.8 0.6 0.4
αr a n d
0.2
α EA
0.0 0
(a)
500 1000 Frequency [Hz]
Impedance
2000
Impedance [Pa s/m]
Absorption coefficient
1.2
Zr a n d
1000
Real part
ZEA
0 -1000 Imaginary part
-2000 1500 (b)
0
500 1000 Frequency [Hz]
1500
Fig. 25. Comparisons between Zrand(x) and ZEA(x) and between arand(x) and aEA(x) for the locally reacting materials: (a) absorption coefficient; (b) impedance.
their impedance. They are shown for incident angles of 15, 45, and 75. Furthermore, comparisons between ZEA(x) and Zrand(x) and between aEA(x) and arand(x) are shown in Fig. 25.
References [1] Garai M. Measurement of the sound-absorption coefficient in situ: the reflection method using periodic pseudo-random sequences of maximum length. Appl Acoust 1993;39:119–39. [2] ISO 354, Acoustics – measurement of sound absorption in a reverberation room. [3] ISO 10534-1 Acoustics – determination of sound absorption coefficient and impedance in impedance tubes. Part 1: method using standing wave ratio. [4] ISO 10534-2 Acoustics – determination of sound absorption coefficient and impedance in impedance tubes. Part 2: transfer function method. [5] Seybert AF, Ross DF. Experimental determination of acoustic properties using a two-microphone random-excitation technique. J Acoust Soc Am 1977;61:1362–70. [6] Chung JY, Blaser DA. Transfer function method of measuring in-duct acoustic properties. I. Theory. J Acoust Soc of Am 1980;68:907–12. [7] Suzuki H, Yano H, Tachibana H. A new method of measuring normal incident sound absorption characteristics of materials using acoustic tube. J Acoust Soc Jpn E 1981;2:161–7. [8] Utsuno H, Tanaka T, Fujikawa T, Seybert AF. Transfer function method for measuring characteristic impedance and propagation constant of porous material. J Acoust Soc Am 1989;86:637–43. [9] Iwase T, Izumi Y. A new sound tube measuring method for propagation constant in porous material – method without any air space at the back of test material. J Acoust Soc Jpn J 1996;52(6):411–9. [in Japanese]. [10] Iwase T, Sato M, Kawabata R. Study of practical improving of new sound tube measuring method for sound propagation constant – reducing technique of resonant vibration of test porous material and application measurements. J Architect Planning Environ Eng (AIJ) 1998;503:17–24. [in Japanese]. [11] Champoux Y, Nicolas J, Allard JF. Measurement of acoustic impedance in a free field at low frequencies. J Sound Vibr 1988;125(1):313–23. [12] Allard JF, Champoux Y, Nicolas J. Pressure variation above a layer of absorbing material and impedance measurement at oblique incidence and low frequencies. J Acoust Soc Am 1989;86(2):766–70.
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[13] Allard JF, Champoux Y. In situ two-microphone technique for the measurement of the acoustic surface impedance of materials. Noise Contr Eng J 1989;32(1):15–23. [14] Allard JF, Depollier C, Guignouard P. Free field surface impedance measurements of soundabsorbing materials with surface coatings. Appl Acoust 1989;26:199–207. [15] Tachibana H, Yano H, Ishii K. Oblique incidence sound absorption characteristics of various absorbing materials measured by cross-correlation method. J Acoust Soc Jpn J 1978;34(1):11–20. [in Japanese]. [16] Tamura M. Spatial Fourier transform method of measuring reflection coefficients at oblique incidence. I: theory and numerical examples. J Acoust Soc Am 1990;88(5):2259–64. [17] Tamura M, Allard JF, Lafarge D. Spatial Fourier-transform method for measuring reflection coefficients at oblique incidence. II. Experimental results. J Acoust Soc Am 1995;97(4):2255–62. [18] Cramond AJ, Don CG. Reflection of impulses as a method of determining acoustic impedance. J Acoust Soc Am 1984;75(2):382–9. [19] Li JF, Hodgson M. Use of pseudo-random sequences and a single microphone to measure surface impedance at oblique incidence. J Acoust Soc Am 1997;102(4):2200–10. [20] Iwase T. A measuring method for sound reflection or absorption coefficient by using two closed microphones with different sensitivities. Proc Inter-noise 1994;94:1951–4. [21] Takahashi Y, Otsuru T, Tomiku R. In situ measurements of absorption characteristics by using two microphones and Environmental Anonymous Noise. Acoust Sci Technol 2003;24(6):382–5. [22] Miki Y. Acoustical properties of porous materials – generalizations of empirical models. J Acoust Soc Jpn E 1990;11:25–30.
In situ measurements of surface impedance and absorption coefficients of porous materials using two microphones and ambient noise Y. Takahashi a
a,*
, T. Otsuru b, R. Tomiku
c
Graduate School of Engineering, Oita University, 700 Dannoharu, Oita 870-1192, Japan b Faculty of Engineering, Oita University, 700 Dannoharu, Oita 870-1192, Japan c Venture Business Laboratory, Oita University, 700 Dannoharu, Oita 870-1192, Japan
Received 3 April 2003; received in revised form 10 November 2004; accepted 10 November 2004 Available online 20 January 2005
Abstract An in situ measurement method is proposed for obtaining the normal surface impedance and absorption coefficient of porous materials using two microphones located close to the material without a specific sound source such as a loudspeaker. Ambient environmental noise that does not excite distinct modes in the sound field is employed as the sound source. Measurements of the normal surface impedance of glass wool and rockwool have been made using this method in various sound fields. The repeatability and wide applicability of the method are demonstrated by comparing results of measurements in one room with different noise conditions and in three other environments (corridor, cafeteria and terrace). The assumed diffuse nature of the sound field on the material is validated by using absorption characteristics obtained experimentally at oblique incidence. This method allows simple and efficient in situ measurements of absorption characteristics of materials in a diffuse field. 2004 Elsevier Ltd. All rights reserved. Keywords: Impedance; In situ; Diffuse field; Two microphones; Ambient noise
*
Corresponding author. Tel.: +81 97 554 6406; fax: +81 97 554 7918. E-mail address: [email protected] (Y. Takahashi).
0003-682X/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2004.11.004
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1. Introduction Numerous methods have been proposed to measure material absorption characteristics. Garai [1] has classified the basic techniques as (i) reverberation room methods, (ii) tube methods, and (iii) reflection methods. The reverberation room method [2] allows measurement of the diffuse incidence absorption coefficient, which is used widely as a practical absorption index. However, creation of a diffuse sound field requires a large and costly reverberation room. A completely diffuse sound field can be achieved only rarely. Moreover, an accurate value of complex impedance cannot be derived from the absorption coefficient alone. The tube method has many variations [3–10]. Usually either the standing wave method or the transfer function method [4] is used. These methods can be used to measure accurate normal acoustic impedance at normal incidence with rather small equipment. However, Champoux et al. [11] and Garai [1] have pointed out some limitations of these tube methods. As well as these problems, the absorption characteristics obtained by the tube methods do not correspond to those in practical conditions. To predict and control acoustics in real environments, it is crucial to measure material absorption characteristics subject to various placement conditions in situ. Garai [1] has indicated that the reverberation room and tube methods are essentially laboratory methods so they are unsuitable for measuring absorption characteristics in situ. Various reflection methods [11–20] have been devised and developed to carry out in situ measurement in the expectation that they can solve the problems of other methods. In particular, the methods proposed by Allard et al. [11–14] or Garai [1] constitute efficient and useful reflection methods. Allard et al. describe reflection methods using two microphones. Their methods allow measurement of a small sample, area about 1 m2, in a free field. Moreover, they allow in situ measurement of a sample area of about 6 m2 at frequencies of 500 Hz or higher but the required sample size increases as the frequency decreases. GaraiÕs method for measuring the absorption coefficient in situ uses MLS as a test signal to improve background noise immunity. However, some problems remain in these methods, including (a) low-frequency limitation, (b) inaccuracy with weakly absorbing materials, (c) the need for complicated equipment when used in situ and (d) the practicality of the relative positioning of a sound source, microphones and sample. It is concluded that present methods do not allow in situ measurement of material absorption characteristics both simply and efficiently. The existing modes inside a room may cause measurement difficulties with the conventional reflection methods when they are excited by a loudspeaker source. The method presented here avoids the influence of room modes and enables simple and efficient in situ measurements of material absorption characteristics at low frequencies below 500 Hz, whereas such measurements are very difficult using conventional reflection methods. Background environmental noise is employed as the sound source in this measurement. We examine the basis and effectiveness of this measurement method and clarify details of measured absorption characteristics.
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Firstly, we describe a procedure for measuring absorption characteristics of a material. Secondly, glass wool and rockwool measurements are made in a room to verify the methodÕs reliability. Then, we compare results measured in different environmental noise conditions to verify the repeatability of measurements. We also demonstrate the wide applicability of measurements through comparison of results measured in various sound fields. Finally, we discuss details of the measured absorption characteristics and clarify the incidence condition of the signals used in the measurement.
2. Description of the method 2.1. The measurement setup The authors recently proposed a new simple and efficient method for in situ measurement of material absorption characteristics using two microphones and ambient noise [21]. The method is based on measurement of the transfer function between two microphones using a spectrum analyzer. Fig. 1 shows a schematic diagram of the measurement setup. The two microphones are located at Ma and Mb. The distance d from the material surface to Mb is 10 mm and the space l between the two microphones is 13 mm. Two identical 1/2-in. (1.27 cm) microphones are employed to measure the sound pressure. When the microphones have different sensitivities, they are calibrated relative to each other using the standard switching technique. The FFT resolution is set to 1.0 Hz and a Hamming window is employed. Measured data are averaged 30 times. The material to be measured is backed by a rigid wall. Ambient noise around the material, so-called background noise, is employed as the sound source instead of a specific sound source such as a loudspeaker. Previously [21], the authors have called such noise ‘‘environmental anonymous noise’’ (EA-Noise) to indicate that the noise does not excite distinct modes. It is assumed that EA-Noise represents ambient noise with no strong
Fig. 1. A schematic diagram of the measurement setup.
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directivity. The required characteristics of the noise will be clarified in following sections. 2.2. Impedance measurement Let pi(x) and pr(x) represent the incident and reflected sound pressure in the frequency domain at the material surface. Assuming plane waves (implied time dependence is ejxt), with oblique incidence at angle h, the sound pressures pa(h,x) and pb(h,x) measured at Ma and Mb (see Fig. 2) are given by pa ðh; xÞ ¼ pi ðxÞejkðlþdÞ cos h þ pr ðxÞejkðlþdÞ cos h
ð1aÞ
pb ðh; xÞ ¼ pi ðxÞejkd cos h þ pr ðxÞejkd cos h :
ð1bÞ
and
Then, the reflection coefficient r(h) and the normal surface impedance of material ZN(h,x) are calculated from rðhÞ ¼
pr ðxÞ p ðh; xÞ pb ðh; xÞejkl cos h 2jkd cos h ¼ a e pi ðxÞ pb ðh; xÞejkl cos h pa ðh; xÞ
ð2Þ
and Z N ðh; xÞ ¼ ¼
qc 1 þ rðhÞ cos h 1 rðhÞ qc H ab ðh; xÞð1 e2jkðlþdÞ cos h Þ ejkl cos h ð1 e2jkd cos h Þ ; cos h H ab ðh; xÞð1 þ e2jkðlþdÞ cos h Þ ejkl cos h ð1 þ e2jkd cos h Þ
ð3Þ
where k is the wavelength constant, q is the density of air, c is the velocity of sound, and Hab(h,x) is the transfer function pb(h,x)/pa(h,x) in the direction defined by microphones at the angle of incidence h. The absorption coefficient aN(h,x) is calculated from the normal surface impedance ZN(h,x). When EA-Noise is utilized as the sound source, the sound pressure paEA(x) and pbEA(x) measured at Ma and Mb are values averaged over time and incident angle. Consequently, using averaged sound pressures, the transfer function between two
Fig. 2. A schematic of the impedance measurement at oblique incidence of angle h.
Y. Takahashi et al. / Applied Acoustics 66 (2005) 845–865
849
microphones becomes HabEA(x) = pbEA(x)/paEA(x). So for measurements using EANoise, substituting HabEA(x) for Hab(h,x) and an average value of h in Eq. (3), the impedance can be calculated. However, it is difficult to ascertain an average value of incident angles of sound in measurements because of the difficulty in obtaining the instantaneous incident angle of sound for each FFT used in the measurements. Moreover, the first object of this study is to carry out simple and stable measurements for absorption characteristics of materials in situ. Therefore, the first stage of this study considers normal incidence, i.e., cosh = 1 in Eq. (3). In this case, the impedance can be expressed as Z EA ðxÞ ¼ qc
H abEA ðxÞð1 e2jkðlþdÞ Þ ejkl ð1 e2jkd Þ : H abEA ðxÞð1 þ e2jkðlþdÞ Þ ejkl ð1 þ e2jkd Þ
ð4Þ
The absorption coefficient aEA(x) is calculated from the impedance ZEA(x). The relationship between ZN(h,x) and ZEA(x) is shown in Appendix A.
3. Method reliability This section discusses the repeatability and applicability of the method. Using the proposed method, the absorption characteristics of glass wool (density = 32 kg/m3) and rockwool (density = 150 kg/m3) have been measured in four rooms during different ambient noise conditions, with various types of incident sound fields, and the results have been compared. The sound pressure levels (Lp) of noise during measurement were monitored at one-second intervals. The area and thickness of each material was 0.3 · 0.6 m2 and 0.05 m, respectively. Table 1 shows details of the environments, i.e., volumes and reverberation times (500 Hz, one octave band). Measurements were ceased when any of the one-second Lp exceeded 10 dB above the mean during the measurements. In following sections, measured values are averaged by frequency range and plotted in 100 Hz steps to show any differences distinctly. 3.1. Measurements in a room Measurements of two materials were carried out in a room with three different ambient noise conditions (Symbol: R-L) to investigate repeatability and robustness. Fig. 3 shows a plan view of the room with furniture layout and sample location. In
Table 1 Outlines and reverberation time (one octave band, center frequency = 500 Hz) of sound fields used in measurement Symbol
Sound field
Volume (m3)
Reverberation time (s)
R-L Cr Caf Tr
Laboratory Corridor Cafeteria Terrace
70 177 2062 (outdoor)
0.17 1.57 1.63
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Fig. 3. Plan view of a room (R-L) with furniture layout (dimensions in mm).
condition (i), nine personal computers were turned on, including the one used in measurement, all windows and a door were open, and two people operating measurement instruments were in the room. In condition (ii), the computers and people were the same as in condition (i) but all windows and the door were closed. In condition (iii), the computers and people were the same as in condition (i), but six additional people were present either performing light tasks using computers or chatting quietly around the material. Fig. 4 shows the Lp monitored during the measurement p Þ and variances period for each room condition. Table 2 lists the averaged values ðL (r2(Lp)). The mounting and locations of the material were identical for each measurement. In each case the ambient noise can be regarded as EA-Noise. Figs. 5 and 6 show the measured absorption coefficients aEA (x) and complex impedance values ZEA(x). For glass wool, mean deviations (MDs) of absorption coefficients are less than 0.015 at frequencies above 200 Hz. MDs for real parts of impedances are about 100 Pa s/m at 200 Hz and less than 42 Pa s/m at frequencies above 300 Hz, whereas those for imaginary parts are about 125 Pa s/m at 200 Hz and less than 35 Pa s/m at frequencies above 300 Hz. As for rockwool, MDs of absorption coefficients are about 0.02 at 200 Hz and less than 0.02 at frequencies
Fig. 4. Monitored Lp at three room conditions in R-L: (a) glass wool measurement; (b) rockwool measurement.
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Table 2 Averaged values and variances of the monitored Lps at three room conditions in R-L r2(Lp) (dB)
(a) Glass wool measurement Condition (i) Condition (ii) Condition (iii)
58.2 54.2 60.3
6.8 4.1 10.9
(b) Rockwool measurement Condition (i) Condition (ii) Condition (iii)
57.9 55.5 59.6
3.4 3.0 6.3
Glass wool
1.2 1.0 0.8 0.6
Condition (i)
0.4 Condition (ii)
0.2
Condition (iii)
0.0 0
500 1000 Frequency [Hz]
(a)
Absorption coefficient αEA
Absorption coefficient α EA
p (dB) L
Rock wool
1.2 1.0 0.8 0.6
Condition (i)
0.4 Condition (ii)
0.2
Condition (iii)
0.0 0
1500
500 1000 Frequency [Hz]
(b)
1500
Fig. 5. Comparisons of the absorption coefficients measured at three room conditions in R-L: (a) glass wool; (b) rockwool.
Glass wool Real part
1000 0
Condition (i) -1000 Imaginary part
0
(a)
Condition (ii) Condition (iii)
-2000
Rockwool
2000
500 1000 Frequency [ Hz]
Impedance ZEA [Pa s/m]
Impedance ZEA [Pa s/m]
2000
1000
Real part
0 Condition (i) -1000 Imaginary part
Condition (iii)
-2000
1500
0
(b)
Condition (ii)
500 1000 Frequency [ Hz]
1500
Fig. 6. Comparisons of the impedance measured at three room conditions in R-L: (a) glass wool; (b) rockwool.
above 300 Hz. At frequencies above 200 Hz, MDs for the real part of impedance are less than 85 Pa s/m, whereas those for the imaginary part are less than 90 Pa s/m. Thus, although the averaged values and variance of the monitored Lp differ in each measurement, as shown in Table 2, the overall tendencies of three results agree well at frequencies above 200 Hz.
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Furthermore, Fig. 7 compares the absorption coefficients aEA(x) measured four times in condition (i) and shows the averaged values and variance of the monitored Lp in each measurement. MDs of absorption coefficients for glass wool are less than 0.02 within measurement frequency range. For rockwool, they are less than 0.025 within the frequency range. Though discrepancies among measured results are observed because of slight differences of noise conditions in each measurement, the four results agree well within the frequency range. We have confirmed also that MDs of 15 results over the three conditions have the same values as the four for room condition (i). Thus, the proposed measurement method has been shown to give repeatable results for different ambient noise conditions in a given room. 3.2. Measurements in various sound fields
Glass wool
1.2 1.0 0.8
L p [dB] σ (L p ) [dB] 58.2 6.8 57.1 2.5 57.8 3.5 58.2 2.7 2
0.6 0.4 0.2 0.0 0
(a)
500 1000 Frequency [Hz]
Absorption coefficient α EA
Absorption coefficient αEA
To investigate the general applicability of the method, measurements of the two materials have been carried out in three other environments [a corridor (Symbol: Cr), a cafeteria (Symbol: Caf) and a terrace (Symbol: Tr)], volumes and reverberation times of which are shown in Table 1. Plan views of furniture layouts and material locations in the corridor and cafeteria are shown in Figs. 8 and 9. Figs. 10 and 11 show the measurement conditions in the cafeteria and the Terrace. In the corridor, windows 1 and 2, shown in Fig. 8, were opened and there were two people inside. In the cafeteria, there were many people chatting, as shown in Fig. 10. On the terrace, material samples were located at 2 m distance from a building. Fig. 12 shows the impedance ZEA(x) measured in the three sound fields. Results for R-L given in the previous section are shown also. The monitored Lp values in each sound field are shown in Fig. 13 and Table 3 shows their averaged values and variances. For glass wool, MDs of impedance are less than 60 Pa s/m at frequencies above 200 Hz except for a MD of about 145 Pa s/m at 200 Hz for the imaginary part. At frequencies above 200 Hz, MDs of real parts are less than 70 Pa s/m whereas those of imaginary parts are less than 100 Pa s/m for rockwool. Moreover, we have found
1500
Rock wool
1.2 1.0 0.8
L p [dB] σ (L p ) [dB] 57.9 3.4 57.8 2.3 58.7 4.9 57.1 2.9 2
0.6 0.4 0.2 0.0 0
(b)
500 1000 Frequency [Hz]
1500
Fig. 7. Comparisons of the absorption coefficients among four times measurement in R-L with condition (i) and the averaged values and variances of monitored Lp: (a) glass wool; (b) rockwool.
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Fig. 8. Plan view of the corridor (Cr) with sample location (dimensions in mm).
Fig. 9. Plan view of the cafeteria (Caf) with furniture layout and sample location (dimensions in mm).
Fig. 10. Condition of the cafeteria (Caf) used for measurements.
that MDs of absorption coefficients at frequencies above 200 Hz are less than 0.025 for glass wool and less than 0.02 for rockwool, but these absorption coefficient values are not shown here. The overall tendency is that the impedances measured in the four sound fields agree well with each other at frequencies above 200 Hz.
854
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Fig. 11. Condition of the terrace (Tr) used for measurements.
Glass wool Real part
1000 0 -1000 Imaginaryp art
-2000 0
(a)
2000
R-L
Caf
Cr
Tr
500 1000 Frequency [Hz]
Impedance ZEA [Pa s/m]
Impedance ZEA [Pa s/m]
2000
Rock wool
1000
Real part
0 Imaginary part
-1000 -2000
1500
0
(b)
R-L
Caf
Cr
Tr
500 1000 Frequency [Hz]
1500
Fig. 12. Comparisons of the impedances measured in four sound fields: (a) glass wool; (b) rockwool.
Fig. 13. Monitored Lp in four sound fields: (a) glass wool measurement; (b) rockwool measurement.
These results indicate that certain types of material absorption characteristics can be measured universally using the method even if the incident sound fields are different. Furthermore, it is noteworthy that results measured in the cafeteria are
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Table 3 Averaged values and variances of the monitored Lps in four sound fields p (dB) L
r2(Lp) (dB)
(a) Glass wool measurement R-L Cr Caf Tr
58.2 55.9 71.6 59.3
6.8 1.8 4.7 6.4
(b) Rockwool measurement R-L Cr Caf Tr
57.9 55.0 71.2 59.1
3.4 2.3 0.6 1.1
indistinguishable from the others despite great differences in the averaged values of one-second Lp. Consequently, the method may be considered to have sufficient reliability as an in situ method for measuring absorption characteristics of porous materials. In following sections, values obtained in the cafeteria are chosen as representative values of absorption characteristics measured by the method; they are referred to as ‘‘EA-Caf’’.
4. Clarification of measured absorption characteristics What kind of absorption characteristics are measured by this method? In this section, incident condition of signals to materials in measurement is investigated. A particular incident condition is assumed and the assumption is verified experimentally. 4.1. Investigation on incident condition of signals 4.1.1. Comparison with values obtained by the Miki model Fig. 14 shows comparisons between EA-Caf and absorption characteristics obtained using the regression model by Miki [22] (MikiÕs-eq) for glass wool. For absorption coefficients, distinct differences can be found between EA-Caf and MikiÕs-eq within measurement frequency range. The values of the imaginary part of impedance are close to MikiÕs-eq, while real part values are not. Clearly, the incident condition in the present method differs from normal incidence because EA-Noise is utilized as the sound source. Therefore, the differences between EA-Caf and MikiÕs-eq are likely to result from differences in the incident conditions. 4.1.2. Impedance averaged on angle of incidence Hereafter, the authors assume that the incident condition in the measurements is diffuse. Then, we consider that the impedance at an incident condition corresponding to a diffuse field can be obtained according to the calculation procedure for the random incidence absorption coefficient.
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Y. Takahashi et al. / Applied Acoustics 66 (2005) 845–865 Absorption coefficient
1.0 0.8 0.6 0.4
EA-Caf
0.2 Miki's-eq
0.0 0
(a)
Impedance
2000
Impedance [Pa s/m]
Absorption coefficient
1.2
500 1000 Frequency [Hz]
Real part
1000 0 -1000
Imaginary part
EA-Caf Miki's-eq
-2000 1500
0
(b)
500 1000 Frequency [Hz]
1500
Fig. 14. Comparisons between EA-Caf and MikiÕs-eq for glass wool: (a) absorption coefficient; (b) impedance.
When normal surface impedance at oblique incidence ZN(h,x) is given, the impedance in a diffuse field Zfield(x) can be expressed as following equations by averaging admittance bN(h,x) = 1/ZN(h,x) over the angle of incidence h (h = 0–78: field incidence): R 78 bN ðh; xÞ sin h cos h dh bfield ðxÞ ¼ 0 R 78 ; ð5Þ sin h cos h dh 0 and Z field ðxÞ ¼
1 ; bfield ðxÞ
ð6Þ
where bfield(x) is the admittance in a diffuse field. 4.2. Experimental verification 4.2.1. Experimental procedure of obtaining absorption characteristics at an incident condition of field incidence This section explains an experimental procedure to derive the impedance ZEA(x) measured by the method. The previous section showed that the impedance at field incidence, Zfield(x), is obtainable from the impedance ZN(h,x) calculated by Eq. (3). However, it is considered that the impedance Zfield(x) differs from impedance ZEA(x) because the impedance ZN(h,x) is different from the component of the impedance ZEA(x) at incident angle h. If this component is denoted by ZEAcom(h,x), it is considered that Z EAcom ðh; xÞ ¼
H ab ðh; xÞð1 e2jkðlþdÞ Þ ejkl ð1 e2jkd Þ qc: H ab ðh; xÞð1 þ e2jkðlþdÞ Þ ejkl ð1 þ e2jkd Þ
ð7Þ
The transfer function Hab(h,x) in Eq. (7) is equal to that in Eq. (3). Using the impedance ZEAcom(h,x) and Eq. (5), the impedance in a diffuse field ZEAfield(x) is obtained by averaging admittance bEAcom(h,x) = 1/ZEAcom(h,x) over the angle of incidence h (h = 0–78). Hence
Y. Takahashi et al. / Applied Acoustics 66 (2005) 845–865
R 78 bEAfield ðxÞ ¼
0
bEAcom ðh; xÞ sin h cos h dh ; R 78 sin h cos h dh 0
857
ð8Þ
and Z EAfield ðxÞ ¼
1 ; bEAfield ðxÞ
ð9Þ
bEAfield(x) is the admittance in a diffuse field obtained using ZEAcom(h,x). We consider that the impedance ZEA(x) is equal to impedance ZEAfield(x) when the incident condition in the method is diffuse. For that reason, we infer that comparisons between ZEA(x) and ZEAfield(x), the latter of which is obtained experimentally, should verify the type of the measured absorption characteristics. Measurements of absorption characteristics at oblique incidence have been carried out in an anechoic room using the technique described by Allard et al. [12] (AllardÕs method). A sound source was placed 2.5 m distant from the material surface as shown in Fig. 15. The incident angle hi was varied in 15 steps from 0 to 75 with respect to the normal to the material surface. The receiving system settings were the same as those described in Section 2. The impedance at each angle of incidence was calculated using Eq. (7). Thereby, six discrete impedance spectra were obtained. A discrete approximation of equation of Eq. (8) is P5 b ðhi ; xÞm sinðhi Þ cosðhi ÞDh bEAfield ðxÞ ¼ i¼0 EAcom ; ð10Þ P5 i¼0 m sinðhi Þ cosðhi ÞDh where hi ¼ i Dh;
Dh ¼ 15;
m ¼ 1=2 ði ¼ 0; 5Þ; m ¼ 1 ði ¼ 1; 2; 3; 4Þ;
and
bEAcom ðhi ; xÞ ¼ 1=Z EAcom ðhi ; xÞ: Using the six measured discrete impedances and Eqs. (10) and (9), the impedance at field incidence ZEAfield(x) and hence the absorption coefficient at field incidence
Fig. 15. Experimental setup of the normal surface impedance measurement at oblique incidence of angle hi.
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aEAfield(x) have been obtained. The resulting absorption characteristics are denoted hereafter as ‘‘Field-EA’’. Likewise, a discrete approximate of Eq. (5) can be defined. bfield and Zfield(x) can be obtained from that equation. Those absorption characteristics are denoted by ‘‘Field-Avg’’. 4.2.2. Comparisons with experimental averaged absorption characteristics Field-EA on glass wool and rockwool are obtained experimentally with the procedure described in the previous section. Figs. 16 and 17 show comparisons between EA-Caf and Field-EA. In the experiments at oblique incidence, the accuracy of values obtained at low frequencies below 500 Hz is insufficient because of the small sample areas and the influence of the positioning of the sound source, microphones and sample. However, despite the slight differences between EA-Caf and Field-EA, the values of EA-Caf correspond well with those of Field-EA at frequencies above 400 Hz for both glass wool and rockwool. Thus, the incident condition assumption of the method described in Section 4.1 is confirmed to be sufficiently appropriate. More specifically, the method enables us to measure impedance of materials in a diffuse field, ZEAfield(x). Absorption coefficient
1.0 0.8 0.6 0.4 EA-Caf
0.2
Field-EA
0.0 0
(a)
500 1000 Frequency [Hz]
Impedance
2000
Impedance [Pa s/m]
Absorption coefficient
1.2
Real part
1000 0 EA-Caf
-1000
Imaginary part
Field-EA
-2000 0
1500
(b)
500 1000 Frequency [Hz]
1500
Fig. 16. Comparisons between EA-Caf and Field-EA for glass wool: (a) absorption coefficient; (b) impedance.
Fig. 17. Comparisons between EA-Caf and Field-EA for rockwool: (a) absorption coefficient; (b) impedance.
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Fig. 18. Comparisons between the absorption characteristics obtained by Eq. (3) [ZN(h,x)] and those obtained by Eq. (7) [ZEAcom(h,x)] at angle of incidence h = 15 for glass wool: (a) absorption coefficient; (b) impedance.
Fig. 19. Comparisons between the absorption characteristics obtained by Eq. (3) [ZN(h,x)] and those obtained by Eq. (7) [ZEAcom(h,x)] at angle of incidence h = 45 for glass wool: (a) absorption coefficient; (b) impedance.
4.2.3. Comparisons between two absorption characteristics at oblique incidence The impedance ZEAcom(h,x) differs from the normal surface impedance at oblique incidence ZN(h,x). To discern differences between ZEAcom(h,x) and ZN(h,x) at every incident angle h, the absorption characteristics for glass wool obtained as described at incident angles of 15, 45, and 75 are compared with those at same angles calculated by Eq. (3) using the same measured transfer functions. These results are averaged by frequency range and are shown in Figs. 18–20 at 100 Hz steps. Averaged residuals of results are shown with the figures. Although some differences are observed for impedance at angle of incidence h = 75, averaged residuals of absorption coefficients are less than 0.03 at all incident angles and differences between aEAcom(h,x) and aN(h,x) are relatively small. Comparisons between Field-EA and Field-Avg are shown with their averaged residuals in Fig. 21. For both real and imaginary parts of impedance, averaged residuals are less than 75 Pa s/m. They are less than 0.02 for absorption coefficients. It seems
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Fig. 20. Comparisons between the absorption characteristics obtained by Eq. (3) [ZN(h,x)] and those obtained by Eq. (7) [ZEAcom(h,x)] at angle of incidence h = 75 for glass wool: (a) absorption coefficient; (b) impedance.
Fig. 21. Comparisons between Field-EA and Field-Avg for glass wool: (a) absorption coefficient; (b) impedance.
that the differences between absorption characteristics obtained by Eq. (3) and those obtained by Eq. (7) become larger as the angle of incidence h increases. Nevertheless, differences between Field-EA and Field-Avg are relatively small compared to those of Fig. 20. Hence, it is concluded that the impedance obtained by this method ZEA(x) is close to Zfield(x) which represents the average of ZN(h,x) over the angle of incidence h (h = 0–78) where ZN(h,x) is the normal surface impedance at oblique incidence.
5. Conclusions A new in situ measurement method for obtaining absorption characteristics of porous materials at field incidence using EA-Noise has been presented. The effectiveness and validity of the method have been examined. The measurement method offers good repeatability and robustness: MDs of absorption coefficients are less than
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0.025 and those of impedance are less than 100 Pa s/m at frequencies above 200 Hz. Also, it is applicable in different sound fields: MDs of absorption coefficients are less than 0.025 and those of impedance are less than 145 Pa s/m at frequencies above 200 Hz. It is interesting to note that the method can measure absorption characteristics of porous materials at frequencies below 500 Hz even when measurements are carried out on small samples in non-free field conditions. The nature of the incident condition of signals to materials in this measurement has been examined experimentally. It has been shown that the incident condition in this measurement can be regarded as field or random incidence. We conclude that this method offers a simple and efficient in situ measurement method for obtaining the impedance of porous materials in a diffuse field, Zfield(x), and eliminates the problems with other methods described in Section 1. Investigations continue regarding its accuracy and robustness. The reasons for discrepancies between Field-EA and Field-Avg will be clarified in a subsequent paper.
Appendix A Relationships between ZEAcom(h,x) [the component of ZEA(x) at incident angle h] and ZN(h,x) [the normal surface impedance for oblique incidence at angle h] and between ZEA(x) [the impedance obtained by this method] and ZN(0,x) [the normal impedance at normal incidence (i.e., acoustic impedance)] are given in this appendix. ZN(h,x) can be calculated from Eq. (3). On the other hand, ZEAcom(h,x) can be calculated from Eq. (7). In both equations, the transfer function Hab(h,x) is equal. Then, the relationship between ZEAcom(h,x) and ZN(h,x) can be obtained from Eqs. (3) and (7): Z EAcom ðh;xÞ ¼
Z N ðh;xÞcosh ejkl A1 D1 Z N ðh;xÞcosh ejklcosh C 1 B1 þ qcejklcosh C 1 B2 qcejklcosh A1 D2 ; qcZ N ðh;xÞcosh ejkl A2 D1 qcZ N ðh;xÞcosh ejklcosh C 2 B1 þ q2 c2 ejklcosh C 2 B2 q2 c2 ejklcosh A2 D2
ðA:1Þ where A1 ¼ ð1 þ e2jkd Þ;
ðA:2aÞ
A2 ¼ ð1 e
2jkd
B1 ¼ ð1 þ e
2jkd cos h
Þ;
ðA:2cÞ
B2 ¼ ð1 e
2jkd cos h
Þ;
ðA:2bÞ Þ;
ðA:2dÞ
C 1 ¼ ð1 þ e
2jkðlþdÞ
Þ;
ðA:2eÞ
C 2 ¼ ð1 e
2jkðlþdÞ
Þ;
ðA:2fÞ
D1 ¼ ð1 þ e
2jkðlþdÞ cos h
Þ;
ðA:2gÞ
D2 ¼ ð1 e
2jkðlþdÞ cos h
Þ:
ðA:2hÞ
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Assuming that the incident condition of signals to materials in this method is random incidence, the impedance ZEA(x) obtained by this method can be calculated by averaging the admittance bEAcom(h,x) = 1/ZEAcom(h,x) over all angles of incidence h (h = 0–p/2: random incidence). Hence: Z p=2 bEA ðxÞ ¼ 2 bEAcom ðh; xÞ sin h cos h dh; ðA:3Þ 0
and Z EA ðxÞ ¼
1 ; bEA ðxÞ
ðA:4Þ
where bEA(x) is the admittance averaged bEAcom(h,x) over the angle of incidence h (h = 0–p/2). Similarly, using ZN(h,x), the impedance Zrand(x) averaged over all angles of incidence h (h = 0–p/2) is given by Z p=2 bN ðh; xÞ sin h cos h dh; ðA:5Þ brand ðxÞ ¼ 2 0
and Z rand ðxÞ ¼
1 ; brand ðxÞ
ðA:6Þ
where bN(h,x) = 1/ZN(h,x) is the admittance at oblique incidence of angle h, and brand(x) is the admittance averaged bN(h,x) over the angle of incidence h (h = 0–p/ 2). Here, each ZN(h,x) becomes equal to the normal surface impedance at normal incidence ZN(0,x) when the locally reacting assumption is applied. The admittance at oblique incidence bN(h,x) is given by: bN ðh; xÞ ¼
1 1 ¼ ¼ bN ð0; xÞ: Z N ðh; xÞ Z N ð0; xÞ
ðA:7Þ
Substituting bN(0,x) for bN(h,x) into Eq. (A.5), brand(x) is obtained as Z p=2 Z p=2 bN ð0; xÞ sin h cos h dh ¼ 2bN ð0; xÞ sin h cos h dh brand ðxÞ ¼ 2 0
¼ bN ð0; xÞ;
0
ðA:8Þ
and Z rand ðxÞ ¼
1 1 ¼ ¼ Z N ðxÞ: brand ðxÞ bN ðxÞ
ðA:9Þ
Hence, Zrand(x) is equal to ZN(0,x) for locally reacting materials. In case of locally reacting materials, substituting ZN(0,x) for ZN(h,x) into Eq. (3), Hab(h,x) at each angle is derived. ZEAcom(h,x) can be calculated from Eq. (7) using the Hab(h,x). Here, ZN(0,x) is obtained using the regression model derived by Miki. Figs. 22–24 show comparisons between ZN(h,x) [=ZN(0,x)] and ZEAcom(h,x) and between absorption coefficients aN(h,x) [=aN(0,x)] and aEAcom(h,x), calculated from
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Angle of incidence θ =15˚ Absorption coefficient
2000
1.0 0.8 0.6 0.4
αN
0.2
α EAcom
0.0 0
(a)
1000 500 Frequency [Hz]
Impedance [Pa s/m]
Absorption coefficient
1.2
Impedance ZN
1000
Real part
ZEAcom
0 -1000 Imaginary part
-2000 1500 0 (b)
1500
1000 500 Frequency [Hz]
Fig. 22. Comparisons between ZN(h,x) [=ZN(0,x)] and ZEAcom(h,x) and between aN(h,x) [=aN(0,x)] and aEAcom(h,x) at angle of incidence h = 15 for the locally reacting materials: (a) absorption coefficient; (b) impedance. Angle of incidence θ =45˚ Absorption coefficient
2000
1.0 0.8 0.6 0.4
αN
0.2
α EAcom
0.0 0
(a)
1000 500 Frequency [Hz]
Impedance [Pa s/m]
Absorption coefficient
1.2
Impedance ZN
1000
Real part
ZEAcom
0 -1000 Imaginary part
-2000 1500 0 (b)
1500
1000 500 Frequency [Hz]
Fig. 23. Comparisons between ZN(h,x) [=ZN(0,x)] and ZEAcom(h,x) and between aN(h,x) [=aN(0,x)] and aEAcom(h,x) at angle of incidence h = 45 for the locally reacting materials: (a) absorption coefficient; (b) impedance. Angle of incidence θ =75˚ Absorption coefficient
2000
1.0 0.8 0.6 0.4
αN
0.2
α EAcom
0.0 0
(a)
500 1000 Frequency [Hz]
Impedance [Pa s/m]
Absorption coefficient
1.2
Impedance ZN
1000
Real part
ZEAcom
0 -1000
-2000 1500 0 (b)
Imaginary part
500 1000 Frequency [Hz]
1500
Fig. 24. Comparisons between ZN(h,x) [=ZN(0,x)] and ZEAcom(h,x) and between aN(h,x)] [=aN(0,x)] and aEAcom(h,x) at angle of incidence h = 75 for the locally reacting materials: (a) absorption coefficient; (b) impedance.
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Y. Takahashi et al. / Applied Acoustics 66 (2005) 845–865 Absorption coefficient
1.0 0.8 0.6 0.4
αr a n d
0.2
α EA
0.0 0
(a)
500 1000 Frequency [Hz]
Impedance
2000
Impedance [Pa s/m]
Absorption coefficient
1.2
Zr a n d
1000
Real part
ZEA
0 -1000 Imaginary part
-2000 1500 (b)
0
500 1000 Frequency [Hz]
1500
Fig. 25. Comparisons between Zrand(x) and ZEA(x) and between arand(x) and aEA(x) for the locally reacting materials: (a) absorption coefficient; (b) impedance.
their impedance. They are shown for incident angles of 15, 45, and 75. Furthermore, comparisons between ZEA(x) and Zrand(x) and between aEA(x) and arand(x) are shown in Fig. 25.
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