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Acoustic methods for determining the effective flow resistivity of fibrous materials
Journal of Sound and Vibration (1992) 153(l), 186-191
ACOUSTIC METHODS FOR DETERMINING THE EFFECTIVE FLOW RESISTIVITY 0~ FIBROUS MATERIALS R. WOODCOCK Dkpartement de gknie ticanique,
Universit& de Sherbrooke, Sherbrooke, Qubbec, Canada Jl K 2Rl AND
M. HoDosoNt Institute for Research in Construction, National Research Council of Canada, Ottawa, Ontario, Canada KlA OR6 (Received 24 July 1990 and injnalform
3 June 1991)
1. INTRODUCTION
The acoustic performance of materials can be predicted using an appropriate prediction model and applicable values of the prediction parameters. For example, in the case of fibrous materials the characteristic impedance and the propagation constant can be predicted using the Delany and Bazley [l] empirical formulae. This requires a knowledge of a single parameter characterizing the material, the effective flow resistivity, cr, assumed not to vary with frequency. In this note, two methods which, in their general application, can be used to determine the parameters characterizing the acoustic properties of arbitrary poro-elastic materials are proposed. The methods, the two-cavity and two-thickness methods, are presented and illustrated by their application to the determination of the effective flow resistivity of fibrous materials. Their more general application, which will be the subject of future reports, is discussed briefly. The methods are based on two corresponding methods, with the same names, for determining the characteristic impedance and propagation constant of a material from measurements of the acoustic impedance at the surface of the material when backed by zero and infinite impedances: the two-cavity method of Yaniv [3] and the two-thickness method of Smith and Parott [4]. The theory forming the basis of these methods, in combination with the Delany and Bazley expressions relating the characteristic impedance and propagation constant to the effective flow resistivity, constitutes the proposed new, methods. Of course, the extent to which the parameter values determined by these methods represent true physical properties of the material under study depend on the accuracy of the prediction model. For example, the flow resistivity of a material is conventionally determined by non-acoustic methods, by measuring the pressure drop across a sample of the material [2]. However, this flow resistivity is not necessarily equal to the effective flow resistivity used in the Delany and Bazley model. The two are related by the porosity of the material. It should be mentioned that various researchers have developed more refined or improved methods for measuring the characteristic impedance and propagation constant, or the flow resistance related to the real part of the characteristic impedance, of materials [5-71. Since the objective of the present work was to demonstrate the validity of a procedure and not to apply the procedure, it was beyond the scope of the work to work with more refined techniques. t Presentaddress: Departmentof MechanicalEngineering,University of BritishColumbia, Vancouver,BC, Canada V6T 124. 186 OO22-46OX/92/04O186 +06 $03.00/O
Q 1992 Academic Press Limited
LE-ITERS
2.
TO THE
THEORETICAL
187
EDITOR
DEVELOPMENT
2.1. Delany and Bazley model According to the Delany and Bazley model, the characteristic impedance (Zc= Z,+jZ,) and propagation constant ( y = yl + j yi)of fibrous materials are predicted from the effective flow resistivity, cr, and the frequency,f, according to the following expressions: Z,=p&J[l
+o~051(cr/f)0’75],
y,=0~175k&r/f)0’59,
zj = -0*077p0co( o/f)“‘73, yi=kO[ 1 + 0’086(0/f)0’7],
(la, b) (2a, b)
in which o is expressed in m.k.s. rayl/m, poco is the specific impedance of air, and k. is the wavenumber in air. Note that the temporal dependence e+jcu’has been adopted. Using equations (1) and (2) the effective flow resistivity can be related to the characteristic impedance and propagation constant by four different expressions: o= [(z,-poco)/(o~051poc~f-o’75)]“o~75, o = [y&O* 175kOf-0.59)]“0.59,
o= [-Zi/(0~077~~c~f-0’73)]“0~73, o=[(yi-ko)/(O~086k~f-“‘7)]“0’7.
(3a, b) (3~ d)
2.2. Two-cavity method Yaniv [3] proposed a method based on transmission-line theory for determining the characteristic impedance and propagation constant of any material which can be considered as a modified fluid, of which a fibrous material is an example. This relies on measuring the surface impedances, Z”p and Z?, of a sample of thickness d of the material when backed by infinite and zero impedances, respectively, and on the following expressions (using the notation later proposed by Smith and Parott [4]): zc= (Z;” * z$“*,
r=()/2d)
ln [(l +a)/(1 -a)l,
(4a, b)
in which a= (Z~/Zq3)“*, Backing impedances of infinity and zero can be obtained by locating an infinite-impedance surface at distances of, respectively, zero and a quarterwavelength behind the sample, as illustrated in Figure 1.
z?
(a) Figure
(b)
I. Measurement configurations
for the two-cavity
method
This method, in combination with the Delany and Bazley model, constitutes an acoustic method for determining the effective flow resistivity of the material: measurements are made of Z”p and Zy, for example in an impedance tube; equations (4) are used to determine Z,, Zi, y, and y/i four estimates of o are obtained using equations (3).
(a) Figure 2. Measurement
(b)
configurations
for the two-thickness
method.
LETTERS
188
TO THE 6
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6
4
4 $ oz
EDITOR
2
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1 \_
0
2 0
-2 iIii!iz
-2 2
4
0
-2
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(b)
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-2
m
lIz?il
2
0
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500 1000 1500 2000 2500
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500 1000 1500 QOOO 2500 Cd)
(c)
%equency(Hz) Figure 3. Normalized surface impedances for a material backed by a quarter-wavelength cavity: (a) 600 Hz; (b) 1000 Hz; (c) 1500 Hz; (d) 1800 Hz. -, Average measured curves; - - -, Delany and Bazley prediction (CT=I5 000 m.k.s. rayl/m).
2.3. Two-thickness method Smith and Parott [4] proposed another method for determining the characteristic impedance and propagation constant of a material. Measurements are made of the acoustic impedances, Z1 and Zz, at the surface of, respectively, single and double thicknesses of the material with infinite-impedance backings, illustrated in Figure 2. They derived the following expressions : zc= [Z,(2Z*-Z,)]“2,
r=(1/2d)
In I(1 +a)/(1 -a)],
with a = [(2Z2- Z,)/Z,] I’*. Again, these equations can be combined with equations (3), thus constituting a second acoustic method for determining the effective flow resistivity of a material from impedance-tube measurements.
LEITERS
TO THE
189
EDITOR
‘r-l---l
-2 0
500
1000 1500 2000 2500 (a)
1 0
500
1000 I500
2000 2500
(b)
Ffequency(Hz)
Figure 4. Normalized surface impedances for single and double thicknesses of a material: (a) d= 2.5 cm; (b) d= 5.0 cm. -, Average measured curves; - - -, Delany and Bazley prediction (u= 15 000 m.k.s. rayl/m).
3.
EXPERIMENTAL
VALIDATION
In order to validate the two methods, measurements of the acoustic surface impedance were made in a 10 cm diameter impedance tube using the transfer-function method of Chung and Blaser [S]. Such measurements are valid over the frequency range 500-2000 Hz. The material tested was a mineral wool with flow resistivity, as measured by conventional means, of about 15 000 m.k.s. rayl/m. Measurements were made at 600, 1000, 1500 and 1800 Hz. To test the two-cavity method, measurements were made on a 2.5 cm thick sample with appropriate (zero or quarter wavelength at the test frequency) air-cavity depths. To test the two-thickness method, measurements were made on 2.5 cm and 5.0 cm thick samples. In each case, three independent measurements of the relevant surface impedances were made; the average measured curves are shown in Figures 3 and 4. The averages of the various measured impedances at the four test frequencies are shown in Table 1. TABLE
1
Averages of the measured impedances used to evaluate the eflective j?ow resistivity of a material by the two-cavity and the two-thickness methods Two-cavity method
I
z”p Frequency (Hz) 600 1000 1500 1800
Two-thickness method
Z?
Z
&
\
* Re
Im
Re
Im
Re
Im
Re
Im
388.3 482.0 648.9 721.5
207.8 222.4 313.6 371.8
326.7 272.6 258.2 250.4
-1237.3 -699.6 -425.9 -299.4
299.6 265.5 259.7 253.5
-1107.3 -606.7 -349.6 -235.7
346.9 375.5 457.7 531.3
-477.6 -204.9 -46.8 2.4
In Table 2 are shown the four effective flow resistivities of the material estimated using equations (3a-d), the data in Table 1 and the two methods. Averaging over all equations and frequencies, the effective flow resistivities are, from the two-cavity method, 14 400 m.k.s. rayl/m, and from the two-thickness method, 14 200 m.k.s. rayl/m.
190
LETTERS TO THE EDITOR TABLE 2
EfSectiveJow
resistivities (in m.k.s. rayl/m) of a material estimatedfrom the measured data in Table 1 using the two-cavity and two-thickness methods Two-cavity method
Frequency (Hz) 600 1000 1500 1800
Equation
(3a)
17600 16 100 22 200 21 700
Equation (3b)
Equation (3~)
13 000 15 500 14 900 10000
13500 15 700 14400 12 900
Two-thickness Frequency (Hz) 600 1000 1500 1800
Equation
(3a)
12600 11 300 14700 16500
Equation (3d) 12300 10500 10000 9200
method
Equation (3b)
Equation (3~)
13 700 14 200 12800 11400
16 300 18000 17 300 16300
Equation (3d) 12800 13 100 13700 15900
The two methods give similar average values for the effective flow resistivity, and values which are in good agreement with that of about 15 000 m.k.s. rayl/m obtained by conventional methods. Individual estimates vary by up to 60% from the average value. No consistent variation with either frequency or equation is apparent, except that equation (3) gives somewhat higher values in the case of the two-cavity method. Averaging over all four equations and several test frequencies appears necessary to obtain high accuracy. 4. CONCLUSIONS The proposed two-cavity and two-thickness methods, acoustic methods for determining the effective flow resistivities of fibrous materials, have been shown to give values of effective flow-resistivity that are consistent with those measured by conventional nonacoustic techniques. Given the lower uncertainties and individual variations associated with the two-thickness method, and the fact that the associated measurements are easier to perform, this method should be preferred in practice. Only experience will tell whether these methods are to be preferred over conventional methods. The main interest of the proposed methods is in their application to other materials. In the case of other materials, similar methods could, in principle, be developed using other prediction models. In the general case of arbitrary “poro-elastic” materials, it should be possible to develop similar methods based on prediction models which describe the material by more than one parameter. The methods could then be used to determine the values of the various parameters of different poro-elastic materials, values which can only be estimated at present. Given that there are four estimation equations available, it should be possible to work with up to four parameters. Such methods have been developed and tested by the authors, and will be the subject of future reports. REFERENCES
105-116. Acoustic properties of fibrous absorbent materials. 2. ASTM STANDARD C522-87. Standard test method for airflow resistance of acoustical materials. I. M. E. DELANY and E. N. BAZLEY 1971 Applied Acousrics 3,
3. S. L. YANIV 1973 Journal of the Acoustical Society of America 54, 1138-l 142. Impedance tube measurement of propagation constant and characteristic impedance of porous materials.
LETTERS TO THE EDITOR
191
4. C. D. SMITH and T. L. PAROTT 1983 Journal of the Acoustical Society of America 74, 1577-l 582. Comparison of three methods for measuring acoustic properties of bulk materials, 5. C. BORDONE-SACERDOTE and G. G. SACERD~TE 1975 Aclcstica 34, 77-80. A method for measuring the acoustical impedance of porous materials. 6. K. U. INGARD and T. A. DEAR 1985 Journal of Sound and Vibration 103,567-572. Measurement of acoustic flow resistance. 7. H. UTSLJNO, T. TANAKA, T. FUJIKAWA and A. F. SEYBERT 1989 Journal of the Acoustical Society of America 86,637-643. Transfer function method for measuring characteristic impedance and propagation constant of porous materials. 8. J. Y. CHIJNG and D. A. BLASER 1980 Journal of the Acoustical Society of America (X$907-921. Transfer-function method of measuring air-duct acoustic properties. I. Theory. II. Experiment.
ACOUSTIC METHODS FOR DETERMINING THE EFFECTIVE FLOW RESISTIVITY 0~ FIBROUS MATERIALS R. WOODCOCK Dkpartement de gknie ticanique,
Universit& de Sherbrooke, Sherbrooke, Qubbec, Canada Jl K 2Rl AND
M. HoDosoNt Institute for Research in Construction, National Research Council of Canada, Ottawa, Ontario, Canada KlA OR6 (Received 24 July 1990 and injnalform
3 June 1991)
1. INTRODUCTION
The acoustic performance of materials can be predicted using an appropriate prediction model and applicable values of the prediction parameters. For example, in the case of fibrous materials the characteristic impedance and the propagation constant can be predicted using the Delany and Bazley [l] empirical formulae. This requires a knowledge of a single parameter characterizing the material, the effective flow resistivity, cr, assumed not to vary with frequency. In this note, two methods which, in their general application, can be used to determine the parameters characterizing the acoustic properties of arbitrary poro-elastic materials are proposed. The methods, the two-cavity and two-thickness methods, are presented and illustrated by their application to the determination of the effective flow resistivity of fibrous materials. Their more general application, which will be the subject of future reports, is discussed briefly. The methods are based on two corresponding methods, with the same names, for determining the characteristic impedance and propagation constant of a material from measurements of the acoustic impedance at the surface of the material when backed by zero and infinite impedances: the two-cavity method of Yaniv [3] and the two-thickness method of Smith and Parott [4]. The theory forming the basis of these methods, in combination with the Delany and Bazley expressions relating the characteristic impedance and propagation constant to the effective flow resistivity, constitutes the proposed new, methods. Of course, the extent to which the parameter values determined by these methods represent true physical properties of the material under study depend on the accuracy of the prediction model. For example, the flow resistivity of a material is conventionally determined by non-acoustic methods, by measuring the pressure drop across a sample of the material [2]. However, this flow resistivity is not necessarily equal to the effective flow resistivity used in the Delany and Bazley model. The two are related by the porosity of the material. It should be mentioned that various researchers have developed more refined or improved methods for measuring the characteristic impedance and propagation constant, or the flow resistance related to the real part of the characteristic impedance, of materials [5-71. Since the objective of the present work was to demonstrate the validity of a procedure and not to apply the procedure, it was beyond the scope of the work to work with more refined techniques. t Presentaddress: Departmentof MechanicalEngineering,University of BritishColumbia, Vancouver,BC, Canada V6T 124. 186 OO22-46OX/92/04O186 +06 $03.00/O
Q 1992 Academic Press Limited
LE-ITERS
2.
TO THE
THEORETICAL
187
EDITOR
DEVELOPMENT
2.1. Delany and Bazley model According to the Delany and Bazley model, the characteristic impedance (Zc= Z,+jZ,) and propagation constant ( y = yl + j yi)of fibrous materials are predicted from the effective flow resistivity, cr, and the frequency,f, according to the following expressions: Z,=p&J[l
+o~051(cr/f)0’75],
y,=0~175k&r/f)0’59,
zj = -0*077p0co( o/f)“‘73, yi=kO[ 1 + 0’086(0/f)0’7],
(la, b) (2a, b)
in which o is expressed in m.k.s. rayl/m, poco is the specific impedance of air, and k. is the wavenumber in air. Note that the temporal dependence e+jcu’has been adopted. Using equations (1) and (2) the effective flow resistivity can be related to the characteristic impedance and propagation constant by four different expressions: o= [(z,-poco)/(o~051poc~f-o’75)]“o~75, o = [y&O* 175kOf-0.59)]“0.59,
o= [-Zi/(0~077~~c~f-0’73)]“0~73, o=[(yi-ko)/(O~086k~f-“‘7)]“0’7.
(3a, b) (3~ d)
2.2. Two-cavity method Yaniv [3] proposed a method based on transmission-line theory for determining the characteristic impedance and propagation constant of any material which can be considered as a modified fluid, of which a fibrous material is an example. This relies on measuring the surface impedances, Z”p and Z?, of a sample of thickness d of the material when backed by infinite and zero impedances, respectively, and on the following expressions (using the notation later proposed by Smith and Parott [4]): zc= (Z;” * z$“*,
r=()/2d)
ln [(l +a)/(1 -a)l,
(4a, b)
in which a= (Z~/Zq3)“*, Backing impedances of infinity and zero can be obtained by locating an infinite-impedance surface at distances of, respectively, zero and a quarterwavelength behind the sample, as illustrated in Figure 1.
z?
(a) Figure
(b)
I. Measurement configurations
for the two-cavity
method
This method, in combination with the Delany and Bazley model, constitutes an acoustic method for determining the effective flow resistivity of the material: measurements are made of Z”p and Zy, for example in an impedance tube; equations (4) are used to determine Z,, Zi, y, and y/i four estimates of o are obtained using equations (3).
(a) Figure 2. Measurement
(b)
configurations
for the two-thickness
method.
LETTERS
188
TO THE 6
8 :I ::
6
4
4 $ oz
EDITOR
2
I /’
1 \_
0
2 0
-2 iIii!iz
-2 2
4
0
-2
-4
-6
(b)
(a) 4
4 \ I
2
,’ 8’
2
5 a
0
0
-2
-2
m
lIz?il
2
0
-2
0
500 1000 1500 2000 2500
0
500 1000 1500 QOOO 2500 Cd)
(c)
%equency(Hz) Figure 3. Normalized surface impedances for a material backed by a quarter-wavelength cavity: (a) 600 Hz; (b) 1000 Hz; (c) 1500 Hz; (d) 1800 Hz. -, Average measured curves; - - -, Delany and Bazley prediction (CT=I5 000 m.k.s. rayl/m).
2.3. Two-thickness method Smith and Parott [4] proposed another method for determining the characteristic impedance and propagation constant of a material. Measurements are made of the acoustic impedances, Z1 and Zz, at the surface of, respectively, single and double thicknesses of the material with infinite-impedance backings, illustrated in Figure 2. They derived the following expressions : zc= [Z,(2Z*-Z,)]“2,
r=(1/2d)
In I(1 +a)/(1 -a)],
with a = [(2Z2- Z,)/Z,] I’*. Again, these equations can be combined with equations (3), thus constituting a second acoustic method for determining the effective flow resistivity of a material from impedance-tube measurements.
LEITERS
TO THE
189
EDITOR
‘r-l---l
-2 0
500
1000 1500 2000 2500 (a)
1 0
500
1000 I500
2000 2500
(b)
Ffequency(Hz)
Figure 4. Normalized surface impedances for single and double thicknesses of a material: (a) d= 2.5 cm; (b) d= 5.0 cm. -, Average measured curves; - - -, Delany and Bazley prediction (u= 15 000 m.k.s. rayl/m).
3.
EXPERIMENTAL
VALIDATION
In order to validate the two methods, measurements of the acoustic surface impedance were made in a 10 cm diameter impedance tube using the transfer-function method of Chung and Blaser [S]. Such measurements are valid over the frequency range 500-2000 Hz. The material tested was a mineral wool with flow resistivity, as measured by conventional means, of about 15 000 m.k.s. rayl/m. Measurements were made at 600, 1000, 1500 and 1800 Hz. To test the two-cavity method, measurements were made on a 2.5 cm thick sample with appropriate (zero or quarter wavelength at the test frequency) air-cavity depths. To test the two-thickness method, measurements were made on 2.5 cm and 5.0 cm thick samples. In each case, three independent measurements of the relevant surface impedances were made; the average measured curves are shown in Figures 3 and 4. The averages of the various measured impedances at the four test frequencies are shown in Table 1. TABLE
1
Averages of the measured impedances used to evaluate the eflective j?ow resistivity of a material by the two-cavity and the two-thickness methods Two-cavity method
I
z”p Frequency (Hz) 600 1000 1500 1800
Two-thickness method
Z?
Z
&
\
* Re
Im
Re
Im
Re
Im
Re
Im
388.3 482.0 648.9 721.5
207.8 222.4 313.6 371.8
326.7 272.6 258.2 250.4
-1237.3 -699.6 -425.9 -299.4
299.6 265.5 259.7 253.5
-1107.3 -606.7 -349.6 -235.7
346.9 375.5 457.7 531.3
-477.6 -204.9 -46.8 2.4
In Table 2 are shown the four effective flow resistivities of the material estimated using equations (3a-d), the data in Table 1 and the two methods. Averaging over all equations and frequencies, the effective flow resistivities are, from the two-cavity method, 14 400 m.k.s. rayl/m, and from the two-thickness method, 14 200 m.k.s. rayl/m.
190
LETTERS TO THE EDITOR TABLE 2
EfSectiveJow
resistivities (in m.k.s. rayl/m) of a material estimatedfrom the measured data in Table 1 using the two-cavity and two-thickness methods Two-cavity method
Frequency (Hz) 600 1000 1500 1800
Equation
(3a)
17600 16 100 22 200 21 700
Equation (3b)
Equation (3~)
13 000 15 500 14 900 10000
13500 15 700 14400 12 900
Two-thickness Frequency (Hz) 600 1000 1500 1800
Equation
(3a)
12600 11 300 14700 16500
Equation (3d) 12300 10500 10000 9200
method
Equation (3b)
Equation (3~)
13 700 14 200 12800 11400
16 300 18000 17 300 16300
Equation (3d) 12800 13 100 13700 15900
The two methods give similar average values for the effective flow resistivity, and values which are in good agreement with that of about 15 000 m.k.s. rayl/m obtained by conventional methods. Individual estimates vary by up to 60% from the average value. No consistent variation with either frequency or equation is apparent, except that equation (3) gives somewhat higher values in the case of the two-cavity method. Averaging over all four equations and several test frequencies appears necessary to obtain high accuracy. 4. CONCLUSIONS The proposed two-cavity and two-thickness methods, acoustic methods for determining the effective flow resistivities of fibrous materials, have been shown to give values of effective flow-resistivity that are consistent with those measured by conventional nonacoustic techniques. Given the lower uncertainties and individual variations associated with the two-thickness method, and the fact that the associated measurements are easier to perform, this method should be preferred in practice. Only experience will tell whether these methods are to be preferred over conventional methods. The main interest of the proposed methods is in their application to other materials. In the case of other materials, similar methods could, in principle, be developed using other prediction models. In the general case of arbitrary “poro-elastic” materials, it should be possible to develop similar methods based on prediction models which describe the material by more than one parameter. The methods could then be used to determine the values of the various parameters of different poro-elastic materials, values which can only be estimated at present. Given that there are four estimation equations available, it should be possible to work with up to four parameters. Such methods have been developed and tested by the authors, and will be the subject of future reports. REFERENCES
105-116. Acoustic properties of fibrous absorbent materials. 2. ASTM STANDARD C522-87. Standard test method for airflow resistance of acoustical materials. I. M. E. DELANY and E. N. BAZLEY 1971 Applied Acousrics 3,
3. S. L. YANIV 1973 Journal of the Acoustical Society of America 54, 1138-l 142. Impedance tube measurement of propagation constant and characteristic impedance of porous materials.
LETTERS TO THE EDITOR
191
4. C. D. SMITH and T. L. PAROTT 1983 Journal of the Acoustical Society of America 74, 1577-l 582. Comparison of three methods for measuring acoustic properties of bulk materials, 5. C. BORDONE-SACERDOTE and G. G. SACERD~TE 1975 Aclcstica 34, 77-80. A method for measuring the acoustical impedance of porous materials. 6. K. U. INGARD and T. A. DEAR 1985 Journal of Sound and Vibration 103,567-572. Measurement of acoustic flow resistance. 7. H. UTSLJNO, T. TANAKA, T. FUJIKAWA and A. F. SEYBERT 1989 Journal of the Acoustical Society of America 86,637-643. Transfer function method for measuring characteristic impedance and propagation constant of porous materials. 8. J. Y. CHIJNG and D. A. BLASER 1980 Journal of the Acoustical Society of America (X$907-921. Transfer-function method of measuring air-duct acoustic properties. I. Theory. II. Experiment.