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An empirical model for rigid frame porous materials with high porosity
Vol. 51,No. 2, pp. 181-198,1997 0 1997ElsevierScience Ltd. All rightsreserved Printedin Great Britain 0003-682X/97/S17.00 + 0.00 SOOO3-682X(96)00052-7 Applied Acoustics,
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An Empirical Model for Rigid Frame Porous Materials with High Porosity N. Voronina Research Institute of Building Physics, Moscow, Russia (Received for publication
16 October 1996)
ABSTRACT The acoustic behaviour of porous materials with a rigid frame has been investigated experimentally. Characteristic impedances and propagation constants have been measured for three porous materials making diflerent pore diameters and porosity. From comparative analysis of experimental results an empirical equation for the structural characteristic has been derived as a function of frequency, porosity and the pore diameter. Empirical formulas have been obtained for acoustic parameters in terms of the structural characteristic. Some interesting behaviour of the phase constant and sound velocity in a porous medium have been remarked upon. This model can be used to predict values of the acoustic impedance and the sound absorption coefficient. Two models for rigid frame porous materials andjbrous materials have been compared when their structural characteristics are equal. 0 1997 Elsevier Science Ltd Keywords: author
to supply.
NOTATION B C CO r” H k 1
coefficient in eqns (6) and (8) sound velocity in a porous material sound velocity in air pore diameter frequency porosity wave number thickness of a material layer 181
182
N. Voronina
m
dimensionless parameter, 103po/pf structural characteristic (eqn (1)) W characteristic impedance WCI real part of W imaginary part of W wi X dimensionless variable (eqn (7)) dimensionless parameter, &% Y 2 dimensionless parameter,,/a attenuation coefficient phase constant B propagation constant Y dynamic viscosity coefficient P density of air PO density of rigid frame Pf bulk density of material Pm
e
INTRODUCTION Porous sound-absorbing materials with a rigid frame are usually used for hanging ceiling and facing walls of buildings. In problems associated with the noise control in enclosures it is of primary importance to be able to predict the acoustic behaviour of these materials which may be determined from the knowledge of two basis parameters: the characteristic impedance, W, and the propagation constant, y (m-l) in their complex forms; W = W, - i Wi and y = a + zj9where W,, Wi are the real and the imaginary parts of W; cz,p are the attenuation coefficient and the phase constant. There are a number of modern theoretical models1-7 that describe acoustic properties of porous media. However, in practical calculations of sound absorption coefficient simple empirical equations for the acoustic parameters Wand y could be used also. Previous researchessT9 have been derived empirical models of sound propagation in a fibrous medium. Now, a similar model is considered for rigid frame porous materials. Three porous materials: a foam-gypsum, a foam-slag-concrete and a foamceramic have been investigated. The foam-gypsum is an artificial porous medium manufactured by mixing a foamer and water with ground gypsum. The foam-slag-concrete and the foamceramic have been manufactured by similar technologies. The macrostructure of each material has been changed according to given technological requirements. As a result many samples having different physical parameters have been obtained for the foam-gypsum and the foam-slag-concrete.
Emperical model for rigid frame porous materials
183
The purpose of this paper is the derivation of the empirical formulas for the acoustic parameters W and y in terms of physical constants characterising the macrostructure of the porous material.
PHYSICAL
PARAMETERS
OF POROUS
MATERIALS
It is known that in porous materials an air viscous flow through microscopic channels is a cause of sound energy damping. In Rayleigh’s theory for a capillary tube model of a porous medium the fractional losses have been described by two relations. The first of them is a relation between the pore diameter, D(m), and the viscous boundary layer thickness. The second relation is that between the value of D and the sound wave length. In this connection two dimensionless physical parameters z and y have been introduced for a quantitative estimation of energy losses in one of the channels. They have been defined as:
where p = 1.85 x lo-’ is the dynamic viscosity coefficient (Pas); p0 is air density (kg /m3); co is sound velocity in air (m s-l); k = 2nf/c, is wave number ( m-l); f is frequency (Hz). The value of D corresponds to the maximum of the pore diameter distribution obtained experimentally by a standard method. For example, Fig. 1 shows the pore diameter distribution determined for a foam-gypsum having density 27 kg/m3. One can see that the maximum is observed at D = 60 x 10e6m. ‘.”
I
I
I
40
60
80
Pore diameter (m x lo-“) Fig. 1. Pore diameter
distribution
for a foam-gypsum
having density 270 kg/m3.
N. Voronina
184
It should be noted that the parameters z and y describe a sound propagation in one of the material channels. The summary effect depends on a pore volume per unit material volume or porosity. Therefore, porosity, H, has also been introduced as the physical parameter. It is easily found from the measured bulk density, pm (kgm3) of the porous material and the density, pf (kgm3) of the rigid frame (matrix) according to the formula: H = 1 - pmlpy In Table 1 the numerical values of the physical parameters D, H, pm, p_ are presented for eight samples of materials.
STRUCTURAL
CHARACTERISTICS
FOR POROUS
MATERIALS
The characteristic impedance and the propagation constant for eight samples of the porous materials (Table 1) have been measured by means of standard methods. The numerical values of W and y are presented in Table 2. The experimental results have been examined to estimate quantitatively the structural characteristic, Q, introduced earlier8,9 as an acoustic parameter Q= IV,--1. In Table 2 th e numerical values of Q obtained from the experimental data for the real part of characteristic impedance are presented. Now, it is interesting to examine how the structural characteristic depends on porosity, the pore diameter and frequency, or the dimensionless parameters z and y (Tables 1 and 2). From Table 2 one can see that the structural characteristic is inversely proportional to the square root of the frequency for each of the material samples under the conditions: H = constant and D = constant. From the comparison of the results for sample 1 and sample 2 (foam-gypsum) having equal porosity and different pore diameters (01 = 200 x 1O-6 m and TABLE 1 Physical Parameters of Porous Sample
dklm3)
Foam-gypsum (pi = 1600 kg/m3) 1 250 2 270 3 300 4 300 350 5 Foam-slag-concrete (pf = 2850 kg/m3) 280 6 350 7 Foam
Materials H
D x 10m6(m)
0.84 0.83 0.81 0.81 0.78
180 60 95 160 60
0.9 0.88
100 85
0.86
85
Emperical model for rigid frame porous materials
185
& = 60 x 10m6m) it is clear that the value of Q varies as D-’ at f = constant. If the pore diameter remains constant (samples 2 and 5) but porosity decreases from Hz = 0.83 to H5 = 0.78 then the structural characteristic increases according to the relation (1 -H)/H at f= constant. As a result taking into account the parameters z, y and after the choice of the numerical coefficient the final empirical formula has been derived as:
14.1(1-H)
Q=
1 -H
The values of Q calculated by eqn (1) are given in Table 2 and Fig. 2. There is a close agreement between the computed and measured data. Equation (1) represents an improvement on that found earlieriO-l2 for rigid frame porous materials. It is important to note that eqn (1) is similar to the expression obtained for fibrous materials8*9 as: Qf =
$j-f(1+s-J
(2)
where d is a fibre diameter (m); q0 = (1 + 2 x 10d4(1 - H)‘)-’ is a factor characterising an influence of a flexible frame for fibrous materials with high porosity.
EMPIRICAL
FORMULAS
FOR ACOUSTIC
PARAMETERS
Now, it is necessary to derive empirical equations for acoustic parameters W and y in terms of the structural characteristic. For the real part of the characteristic impedance the simple formula can be written: Wa=l+Q (3) In Fig. 3 the experimental values of Wi, a/k and /3/k provided on the basis of measured data (Table 2) are plotted vs corresponded values of Q calculated by eqn (1). First, the results shown in Fig. 3(a),(b) have been examined. One can see that when Q < 1 there are the following approximate relations: Wi = Q/2 and (IL= kQ. In the case where Q > 1 these paameters can be presented by:
Q Wj=-’
2+a’
kQ
a=2+a
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186
Experimental
f(Hz)
Foam-gypsum Sample 1 (H=0.84; 250 500 1000 2000 4000
TABLE 2 Values of Acoustic Parameters W= W,-iWi
D=60~lO-~m)
Sample 3 (H=0.81; 250 500 1000 2000 4000
D=55x10p6m)
Sample4(H=0.81; 250 500 1000 2000 4000
D=l60xlO-“m)
Sample 5 (H=0.78; 250 500 1000 2000 4000
D=60~10-~m)
Sample 7 (H=0.88; 250 500 1000 2000 4000
Materials
y = 01+ i#?(I/m)
-____ Q Exp.
Calc.
D= 180x10-6m)
Sample 2 (H=0.83; 250 500 1000 2000 4000
Foam-slag-concrete Sample 6 (H=0.9; 250 500 1000 2000 4000
for Porous
2.5-i0.62 1.95-i0.47 1.73-i0.38 1.53-i0.18 1.38-iO.15
6+ill 8+il9 12+i34 17+i64 25+illl
1.5 0.95 0.73 0.53 0.38
1.44 1.02 0.72 0.51 0.36
5.6-il.6 4.2-il.4 3.3-d 2.7-i0.7 2.2-i0.5
17+i26 24 + i42 36 + i69 51 +il29 72+il86
4.6 3.2 2.3 1.7 1.2
4.6 3.25 2.3 1.6 1.15
4.46-il.4 3.36-i0.9 2%i0.75 2.2-i0.58 1.8-i0.25
12+i20 19+i33 27 + i56 41+ ilO 59+i168
3.46 2.36 1.8 1.2 0.8
3.4 2.4 1.7 1.2 0.85
3.1-i0.92 2.4-i0.6 2.0-i0.45 1.74-i0.35 1.53-i0.19
8+i14 ll+i23 17+i41 23 + i77 36+il32
2.1 1.4 1.0 0.74 0.53
2.0 1.4 1.0 0.7 0.5
7.4-i2.7 5.7-i2.0 4.4-d .3 3.4-il.0 2.8-i0.7
22 + i36 32 + i54 51 +i85 73 + i145 llO+i276
6.4 4.7 3.4 2.4 1.8
6.8 4.8 3.4 2.4 1.7
2.7-i0.65 2. I -iO.45 I .8-i0.34 1.5-i0.17 1.4-iO.16
6+i12 Il+i20 13+i34 16+i60 26+illl
1.7 1.1 0.8 0.5 0.4
1.6 1.1 0.8 0.55 0.4
3.2-il.1 2.7-i0.73 2.3-i0.52
8+il6 13+i25 19+i42 26 + i72 38+i126
2.2 1.7 1.3 0.9 0.6
2.4 1.7 1.2 0.84 0.6
D= 100x10-6m)
D=85xlO-“m)
1.9-i0.36 I .6-i0.32
continued
Emperical model for rigid frame porous materials
187
Table 2-contd Foam-ceramic Sample 8 (H=0.86; 250 500 1000 2000 4000
D=85~10-~rn) 3.7-il.2
lO+i16 lS+i25 21 +i43 32 + 175 4O+i135
2.8-i0.8 2.3-i0.46
1.9-20.43 1.6-io.26
2.7 1.8 1.3 0.9 0.6
2.6 1.8 1.3 0.9 0.65
where a is a factor that tends to 0 at lower frequencies. The value of a has been found in a form: a = Q/U + @>’ Taking into account this expression the empirical formulas for the parameters Wi and a! can be written as: Wi =
(II=
Q
2+Q/
(4)
+,@I2
2kQ 2+Q/’
.LL
2t
1000
Frequency
(5)
(Hz)
Fig. 2. Structural characteristic as a function of frequency for a foam-gypsum (1) pm=250kg/m3, D= 180~10-~m; (2) p,=300kg/m3, D= 160~10-~m; (3) p,=300kg/m3, D=9x10e6m; (4) p,=270kg/m3, D=60~10-~m.
188
N. Voronina
The functions Wj(Q) and a(Q) computed from eqns (4), (5) and plotted by solid lines in Fig. 3(a), (b) are confirmed experimentally for all tested samples of porous materials. Therefore, eqns (3)--(5) may be included into the empirical model of a sound propagation through rigid frame porous media. The most difficult problem of this study is to obtaln an analytical formula for the phase constant in terms of the structural characteristic. Figure 3(c) shows that the experimental values of (B/k) - 1 (points) lie much above the same function calculated by: /?/k - 1 = Q
In this case the phase constant may be defined as: /3 = k(l + Q(1 + B))
(6)
where B is a coefficient that depends on the structural characteristic. The numerical values of B obtained from the experimental data for the acoustic parameter fi are presented in Table 3 and in Fig. 4. The results (Fig. 4) indicate that the functions BCf) have peaks for samples l-2 in a frequency range 2500-3000 Hz. Two other curves 3 and 4 have the same tendency also. It is possible that their maxima could appear at frequencies higher than 4000 Hz. Now, the united function B(x) is needed to describe acoustic behaviour of materials. It is important to find the variable x by means of which all different curves B(f) can be combined into one curve B(x). From Fig. 4 one can see that the curves 2 (sample 2) and 3 (sample 5) having equal pore diameters but different structural characteristics (Table 2) may be combined if the variable x is inversely proportional to (1 + Q). From the comparison of curves 1 and 2 obtained for samples 1 and 2 having different pore diameters it has been found that x - l/a. To combine curve 3 with curve 4 determined for the foam-gypsum (sample 5) and the foam-slagconcrete (sample 7), having different frame density the variable x must be divided by the value of pf Then introducing the parameters z and WI= 103p,/p~ the resulting equation has been written as: 120m
x=z(l
+Q>
where the numerical factor 120 has been defined to provide the maximum of the function B(x) at x = 1.
Emperical model for rigid frame porous materials
189
(a)
(b)
Structural
characteristic
(Q)
I” I
a A
A, .
1.25
Structural
characteristic
l 0
2
(Q)
Fig. 3. Acoustic parameters as a function of structural characteristic: (a) the imaginary part of characteristic impedance; (b) the attenuation coefficient; (c) the phase constant for porous materials listed in Table 1. (1) sample 1; (2) sample 2; (3) sample 3; (4) sample 4; (5) sample 5;
190
N. Voronina
Experimental
Q
f (Hz)
Foam-gypsum z = 70.0 250 500 1000 2000 4000
TABLE 3 values of coefficient
B
X
B Exp.
Calc.
(m = 0.765) 1.44 1.02 0.72 0.51 0.36
0.54 0.65 0.76 0.87 0.96
0.07 0.1 0.21 0.46 0.58
0.07 0.12 0.21 0.4 0.61
z=36.5 250 500 1000 2000 4000
4.6 3.25 2.3 1.62 1.15
0.45 0.57 0.76 0.96 1.17
0.03 0.06 0.21 0.55 0.35
0.05 0.08 0.21 0.61 0.3
z=46.0 250 500 1000 2000 4000
3.4 2.4 1.7 1.2 0.85
0.46 0.59 0.74 0.91 1.08
0.02 0.09 0.22 0.46 0.53
0.05 0.08 0.19 0.5 0.55
2=59.5 250 500 1000 2000 4000
2.0 1.4 1.0 0.7 0.5
0.52 0.65 0.78 0.92 1.05
0.03 0.07 0.25 0.57 0.6
0.06 0.12 0.23 0.53 0.59
z= 36.5 250 500 1000 2000 4000
6.8 4.8 3.4 2.4 1.7
0.33 0.45 0.58 0.76 0.96
0 0.02 0.08 0.24 0.63
0.04 0.05 0.08 0.21 0.61
1.56 1.1 0.78 0.55 0.39
0.44 0.54 0.63 0.73 0.81
0.03 0.07 0.12 0.18 0.32
0.05 0.07 0.1 0.18 0.26
2.4 1.7 1.2 0.85 0.6
0.35 0.45 0.55 0.65 0.75
0.04 0.04 0.08 0.14 0.2
0.04 0.05 0.07 0.12 0.2 continued
Foam-slag-concrete z = 47.0 250 500 1000 2000 4000 z=43.0 250 500 1000 2000 4000
(m = 0.43)
191
Emperical model for rigid frame porous materials
Table S-contd Foam-ceramic z=43.0 250 500 1000 2000 4000
(m = 0.47) 2.56 1.8 1.28 0.9 0.64
0.37 0.46 0.57 0.69 0.8
0 0 0.06 0.16 0.31
0.04 0.05 0.08 0.15 0.25
Taking account of the variable x calculated by eqn (7) the formula for the coefficient B(x) has been derived as: B= (60x2-
120x+61.5)-’
(8)
Figure 5 shows a good correspondence between the values of B(x) calculated by eqn (8) and determined experimentally (Table 3). Therefore, eqn (8) may be used to compute the coefficient B(x) presented in eqn (6) for the phase constant. From Fig. 5 one can see that when x < 0.5 or x > 1.5 the coefficient B is equal approximately to 0. Then the phase constant can be written as: B=Wl
+e>
(9)
Over the range 0.5 < x < 1.5 the acoustic parameter B depends on the function B(x) having a maximum at x = 1. It should be noted that the same
Frequency
(Hz)
Fig. 4. Experimental coefficient B as a function of frequency for a foam-gypsum (1) p,,, =250 kg/m3, D= 180x 10e6m; (2) p,,, = 270kg/me6, D=6Ox 10e6m; (3) p,,, = 350 kg/m3, D = 60 x lop6 m; a foam-slag-concrete (4) p,,, = 350 kg/m3, D = 85 x 10W6m.
192
N. Voronina
behaviour of the phase constant has been observed earlier9 for super-thin fibrous materials. The increase of this acoustic parameter in some frequency ranges means that a sound velocity, c (m SC’), in porous media can be considerably smaller than the same defined from eqn (9) as:
co “=l+Q Figure 6 illustrates this fact for fibreglass and foam-gypsum having equal structural characteristics (Qimo = 1.7 atf = 1000 Hz). One can see that there is a region where the sound velocity is not dependent on frequency. This region is 500-800Hz for fibreglass and 2000-3000Hz for the foam-gypsum (sample 3). At present it is difficult to explain the tendency of the sound velocity in fibrous and porous media. It is interesting to examine how the sound absorption coefficient, 1~0,at normal incidence of a plane sound wave on a material layer with a thickness, I,(m) depends on the effect of the function B(x). The sound absorption coefficient of sample 3 (a foam-gypsum) at I= 0.02 m has been computed by means of eqns (3)-(6) for two cases: when B # 0 using three variables z, Q, kl and when B = 0 using two varables Q, kl. From Fig. 7 it is clear that there is a difference between the curves 1 (B # 0) and 2 (B = 0) over the frequency
Parameter
(x)
Fig. 5. Coefficient B as a function of a parameter x calculated by eqn (8) and determined experimentally for porous materials (Table 1). (1) sample 1; (2) sample 2; (3) sample 3; (4) sample 4; (5) sample 5; (6) sample 6; (7) sample 7.
193
Emperical model for rigid frame porous materials
range higher than 1OOOHz. The difference is confirmed experimentally. At frequencies lower than 1OOOHzboth variants of the coefficient B lead to equal results. Thus, the empirical model derived here for rigid frame porous materials with high porosity (H > 0.75) predicts the acoustic parameters IV, y and also the sound-absorption coefficient rather exactly. When x < 0.5 this model is more simple (B = 0). It is valid for porous materials having large frame density pj> 2000 kg/m3 (m < 0.5). In this case a nomogram has been plotted for the determinaton of the sound-absorption coefficient. Figure 8 presents some curves of equal sound absorption coefficients in a co-ordinate system of /
/
IO
Frequency (Hz) Fig. 6. Sound velocity as a function of frequency for a foam-gypsum (1) p,,, = 300 kg/m3, D = 95 x 10m6m and a super fibreglass (2) P,,, = 20kg/m3, D = 1O-6 m.
^.
(4
*
250
(b)
500
Frequency (Hz)
1000
1000
2000
3000
4000
Frequency (Hz)
Fig. 7. Plots of a sound absorption coefficient for a foam-gypsum having p,,, = 300 kg/m3 and D = 95x 10e6 m calculated using the condition: (1) x > 0.5 (B#O); (2) x < 0.5 (B= 0) at layer thickness 0.02 m. (a)f< 1000 Hz; (b) f> 1000 Hz.
N. Voronina
194
two dimensionless variables Q, kl. A numerical value of (~0can be determined from a point with coordinates Q, kl calculated at the frequency of interest. Figure 8 illustrates some points obtained for a granular mineral wool plate ‘Akmigran’ (pm = 400 kg/ms, pf = 2600 kg/m3, H = 0.85, D = 75 x lop6 m) with layer thickness 0.02 m at frequencies 500, 1000, 2000,400O Hz. In Fig. 7 the results of determining values a!~in this way are presented by the curve 2. There is a close agreement between computed and measured data. It should be noted that this nomogram (Fig. 8) can be useful to technologies for manufacturing different porous materials. On the one hand the sound absorption coefficient may be determined without measurements by means of an impedance tube method. On the other hand by optimization of the structural parameters H and D it is possible to provide the desired value of cx,-,at known parameter kl. More detailed consideration of this important question is not the subject of this paper.
COMPARISON
OF EMPIRICAL MODES FOR POROUS AND FIBROUS MATERIALS
In this study the empirical model for porous materials has been compared with the same derived earlier9 for fibrous materials in the form: W=
1 +Qr-iQfF
Parameter
Fig. 8. Nomogram
(!x/)
for a determination of a sound absorption layer thickness, I (x < 0.5).
coefficent
for porous
material
Emperical model for rigid frame porous materials
Frequency
195
(Hz)
Fig. 9. Acoustic parameters as a function of a frequency: (a) the real part of characteristic impedance; (b) the imaginary part of characteristic impedance; (c) the attenuation coefficient; (d) the phase constant measured and calculated (1) by eqns (3-6) for foam-slag-concrete (sample 7, P,,, = 350 kg/m3 , D = 85 x low6 m); (2) by eqn (1 l), (12) for fibreglass (pm = 80 kg/m3, d=4x 10Wm).
N. Voronina
196
Y=kQy
Qr>Cl + B.) +ik(l +Qdl 1+Qr
1 + FU +
+B’))
(12)
where Qf is the structural characteristic calculated by eqn (2); the coefficients F and Bf are defined as: for for for
x1 < 1 1 < x1 < 4 x1 > 4
F = -0.2x: + 0.7~~ + 0.25&(1 F= ,/X/(1 +4&) F=4 fi/(2+4 fi)
Bf = ,/&(lOz&
- 12.6zlx, + 421
+ 2x:)-2
4(1 + 40) 1 +ylQ2
where x1
=
x0
ym/(l =
Yi(l
+
qo(l
Qjj2)
+
yI
+g$%F)
z,=
=
mx
lo2
Ji&f
Two acoustic materials having equal structural characteristics (Q = Qf) have been compared. They are: the foam-slag-concrete (sample 7) with
0.6
0
’
I 1000
I 2000
Frequency (Hz) Fig. 10. Plots
concrete
of a sound absorption coefficient measured and calculated for (1) foam-slagD = 85x 10e6 m; (2) fibreglass with pm = 80 kg/m3, having pm = 350 kg/m3, d=4x 10-6m at layer thickness 0.04m.
Emperical model for rigid frame porous materials
197
p,,, = 350 kg/m 3; D =85x 10P6m and the fibreglass with pm = 80 kg/m3; d =4x 10m6m. Figure 6 shows the acoustic parameters IV, y calculated by eqns (3)-(6) using (1) for this porous material and by eqns (10)--Q 1) using (2) for the fibrous material. There is no difference between the real parts of characteristic impedance [Fig. 9(a)] for both media. It is valid also for the phase constants [Fig. 9(d)]. However, for fibreglass the imaginary part of characteristic impedance [Fig. 9(b)] and the attenuation coefficient [Fig. 9(c)] are more than the same for the porous material. Therefore, it can be assumed that an acoustic similarity is not possible between fibrous and porous materials. This discrepancy between two models has an effect on the soundabsorption coefficient. From Fig. 10 one can see that in a frequency range lower then 2000 Hz (kl < n/2) the value of aygfor the foam-slag+oncrete is more than the value for the fibreglass. At frequencies higher then 2OOOHZ (kl > n/2) the sound-absorption coefficient depends on the characteristic impedance only and, therefore, the curves 1 and 2 coincide.
CONCLUSION An experimental study of acoustic parameters has been carried out for rigid frame porous materials having different physical constants which characterise the medium macrostructure. The empirical model has been derived by comparative analyses of these results. This model can be used to predict values of the acoustic impedance and the sound-absorpton coefficient for the porous material layer with known thickness. Two models for porous and fibrous materals, having equal structural characteristics, have been compared. One can see that acoustic similarity is not possible between porous and fibrous materials.
REFERENCES 1. Attenborough, K., Acoustical characteristics of rigid fibrous absorbents and 2.
granular materials. Journal Acoust. Sot. Am., 1983, 73(3), 785-799. Attenborough, K., On the acoustic slow wave in air-filled granular media.
Journal Acoust. Sot. Am., 1987, 81(l), 93-102. 3. Attenborough, K., Model of the acoustical properties of air-saturated granular media. Acta Acustica, 1993, 1, 213-226. 4. Yamamoto, T. and Turgut, A., Acoustic wave propagation through porous media with arbitrary pore size distributions. Journal Acoust. Sot. Am., 1988, 83(5), 17441751. 5. Stinson, M. R., The propagation of plane sound waves in narrow and wide circular tubes, and generalization to uniform tubes of arbitrary cross-sectonal shape. Journal Acousf. Sot. Am., 1991, 89(2), 55C558.
198
N.
Voronina
6. Stinson, M. R. and Champoux, Y., Propagation of sound and the assignment of shape factors of model porous materials having simple pore geometries. Journal Acoust. Sot. Am., 1992, 91(2), 685695. 7. Wilson, D. K., Relaxion-matched modelling of propagation through porous media, including fractal pore structure. Journal Acoust. Sot. Am., 1993, 94(2), 1136-l 145. 8. Voronina, N., Acoustic proporties of fibrous materials. Appl. Acoust., 1994, 42, 165-174. 9. Voronina, N., Improved empirical model of sound propagation through a fibrous material. Appl. Acoust., 1996, 48(2), 121-132. 10. Voronina, N., Determination of structural characteristic for foam-gyps. Tr./ NIISF, Building Acoustics, 1979, 25-3 1. 11. Voronina, N., Physic-mathematical model of a porous absorbing material. Tr./ NIISF, Building and acoustic means of noise-protection, 1988, 44-53 12. Voronina, N., Physic-mathematical model of sound waves propagation through a porous media. Rep.lXI All-Union Acoustic Conference, Moscow, 1991.
PII: ELSEVIER
An Empirical Model for Rigid Frame Porous Materials with High Porosity N. Voronina Research Institute of Building Physics, Moscow, Russia (Received for publication
16 October 1996)
ABSTRACT The acoustic behaviour of porous materials with a rigid frame has been investigated experimentally. Characteristic impedances and propagation constants have been measured for three porous materials making diflerent pore diameters and porosity. From comparative analysis of experimental results an empirical equation for the structural characteristic has been derived as a function of frequency, porosity and the pore diameter. Empirical formulas have been obtained for acoustic parameters in terms of the structural characteristic. Some interesting behaviour of the phase constant and sound velocity in a porous medium have been remarked upon. This model can be used to predict values of the acoustic impedance and the sound absorption coefficient. Two models for rigid frame porous materials andjbrous materials have been compared when their structural characteristics are equal. 0 1997 Elsevier Science Ltd Keywords: author
to supply.
NOTATION B C CO r” H k 1
coefficient in eqns (6) and (8) sound velocity in a porous material sound velocity in air pore diameter frequency porosity wave number thickness of a material layer 181
182
N. Voronina
m
dimensionless parameter, 103po/pf structural characteristic (eqn (1)) W characteristic impedance WCI real part of W imaginary part of W wi X dimensionless variable (eqn (7)) dimensionless parameter, &% Y 2 dimensionless parameter,,/a attenuation coefficient phase constant B propagation constant Y dynamic viscosity coefficient P density of air PO density of rigid frame Pf bulk density of material Pm
e
INTRODUCTION Porous sound-absorbing materials with a rigid frame are usually used for hanging ceiling and facing walls of buildings. In problems associated with the noise control in enclosures it is of primary importance to be able to predict the acoustic behaviour of these materials which may be determined from the knowledge of two basis parameters: the characteristic impedance, W, and the propagation constant, y (m-l) in their complex forms; W = W, - i Wi and y = a + zj9where W,, Wi are the real and the imaginary parts of W; cz,p are the attenuation coefficient and the phase constant. There are a number of modern theoretical models1-7 that describe acoustic properties of porous media. However, in practical calculations of sound absorption coefficient simple empirical equations for the acoustic parameters Wand y could be used also. Previous researchessT9 have been derived empirical models of sound propagation in a fibrous medium. Now, a similar model is considered for rigid frame porous materials. Three porous materials: a foam-gypsum, a foam-slag-concrete and a foamceramic have been investigated. The foam-gypsum is an artificial porous medium manufactured by mixing a foamer and water with ground gypsum. The foam-slag-concrete and the foamceramic have been manufactured by similar technologies. The macrostructure of each material has been changed according to given technological requirements. As a result many samples having different physical parameters have been obtained for the foam-gypsum and the foam-slag-concrete.
Emperical model for rigid frame porous materials
183
The purpose of this paper is the derivation of the empirical formulas for the acoustic parameters W and y in terms of physical constants characterising the macrostructure of the porous material.
PHYSICAL
PARAMETERS
OF POROUS
MATERIALS
It is known that in porous materials an air viscous flow through microscopic channels is a cause of sound energy damping. In Rayleigh’s theory for a capillary tube model of a porous medium the fractional losses have been described by two relations. The first of them is a relation between the pore diameter, D(m), and the viscous boundary layer thickness. The second relation is that between the value of D and the sound wave length. In this connection two dimensionless physical parameters z and y have been introduced for a quantitative estimation of energy losses in one of the channels. They have been defined as:
where p = 1.85 x lo-’ is the dynamic viscosity coefficient (Pas); p0 is air density (kg /m3); co is sound velocity in air (m s-l); k = 2nf/c, is wave number ( m-l); f is frequency (Hz). The value of D corresponds to the maximum of the pore diameter distribution obtained experimentally by a standard method. For example, Fig. 1 shows the pore diameter distribution determined for a foam-gypsum having density 27 kg/m3. One can see that the maximum is observed at D = 60 x 10e6m. ‘.”
I
I
I
40
60
80
Pore diameter (m x lo-“) Fig. 1. Pore diameter
distribution
for a foam-gypsum
having density 270 kg/m3.
N. Voronina
184
It should be noted that the parameters z and y describe a sound propagation in one of the material channels. The summary effect depends on a pore volume per unit material volume or porosity. Therefore, porosity, H, has also been introduced as the physical parameter. It is easily found from the measured bulk density, pm (kgm3) of the porous material and the density, pf (kgm3) of the rigid frame (matrix) according to the formula: H = 1 - pmlpy In Table 1 the numerical values of the physical parameters D, H, pm, p_ are presented for eight samples of materials.
STRUCTURAL
CHARACTERISTICS
FOR POROUS
MATERIALS
The characteristic impedance and the propagation constant for eight samples of the porous materials (Table 1) have been measured by means of standard methods. The numerical values of W and y are presented in Table 2. The experimental results have been examined to estimate quantitatively the structural characteristic, Q, introduced earlier8,9 as an acoustic parameter Q= IV,--1. In Table 2 th e numerical values of Q obtained from the experimental data for the real part of characteristic impedance are presented. Now, it is interesting to examine how the structural characteristic depends on porosity, the pore diameter and frequency, or the dimensionless parameters z and y (Tables 1 and 2). From Table 2 one can see that the structural characteristic is inversely proportional to the square root of the frequency for each of the material samples under the conditions: H = constant and D = constant. From the comparison of the results for sample 1 and sample 2 (foam-gypsum) having equal porosity and different pore diameters (01 = 200 x 1O-6 m and TABLE 1 Physical Parameters of Porous Sample
dklm3)
Foam-gypsum (pi = 1600 kg/m3) 1 250 2 270 3 300 4 300 350 5 Foam-slag-concrete (pf = 2850 kg/m3) 280 6 350 7 Foam
Materials H
D x 10m6(m)
0.84 0.83 0.81 0.81 0.78
180 60 95 160 60
0.9 0.88
100 85
0.86
85
Emperical model for rigid frame porous materials
185
& = 60 x 10m6m) it is clear that the value of Q varies as D-’ at f = constant. If the pore diameter remains constant (samples 2 and 5) but porosity decreases from Hz = 0.83 to H5 = 0.78 then the structural characteristic increases according to the relation (1 -H)/H at f= constant. As a result taking into account the parameters z, y and after the choice of the numerical coefficient the final empirical formula has been derived as:
14.1(1-H)
Q=
1 -H
The values of Q calculated by eqn (1) are given in Table 2 and Fig. 2. There is a close agreement between the computed and measured data. Equation (1) represents an improvement on that found earlieriO-l2 for rigid frame porous materials. It is important to note that eqn (1) is similar to the expression obtained for fibrous materials8*9 as: Qf =
$j-f(1+s-J
(2)
where d is a fibre diameter (m); q0 = (1 + 2 x 10d4(1 - H)‘)-’ is a factor characterising an influence of a flexible frame for fibrous materials with high porosity.
EMPIRICAL
FORMULAS
FOR ACOUSTIC
PARAMETERS
Now, it is necessary to derive empirical equations for acoustic parameters W and y in terms of the structural characteristic. For the real part of the characteristic impedance the simple formula can be written: Wa=l+Q (3) In Fig. 3 the experimental values of Wi, a/k and /3/k provided on the basis of measured data (Table 2) are plotted vs corresponded values of Q calculated by eqn (1). First, the results shown in Fig. 3(a),(b) have been examined. One can see that when Q < 1 there are the following approximate relations: Wi = Q/2 and (IL= kQ. In the case where Q > 1 these paameters can be presented by:
Q Wj=-’
2+a’
kQ
a=2+a
N. Voronina
186
Experimental
f(Hz)
Foam-gypsum Sample 1 (H=0.84; 250 500 1000 2000 4000
TABLE 2 Values of Acoustic Parameters W= W,-iWi
D=60~lO-~m)
Sample 3 (H=0.81; 250 500 1000 2000 4000
D=55x10p6m)
Sample4(H=0.81; 250 500 1000 2000 4000
D=l60xlO-“m)
Sample 5 (H=0.78; 250 500 1000 2000 4000
D=60~10-~m)
Sample 7 (H=0.88; 250 500 1000 2000 4000
Materials
y = 01+ i#?(I/m)
-____ Q Exp.
Calc.
D= 180x10-6m)
Sample 2 (H=0.83; 250 500 1000 2000 4000
Foam-slag-concrete Sample 6 (H=0.9; 250 500 1000 2000 4000
for Porous
2.5-i0.62 1.95-i0.47 1.73-i0.38 1.53-i0.18 1.38-iO.15
6+ill 8+il9 12+i34 17+i64 25+illl
1.5 0.95 0.73 0.53 0.38
1.44 1.02 0.72 0.51 0.36
5.6-il.6 4.2-il.4 3.3-d 2.7-i0.7 2.2-i0.5
17+i26 24 + i42 36 + i69 51 +il29 72+il86
4.6 3.2 2.3 1.7 1.2
4.6 3.25 2.3 1.6 1.15
4.46-il.4 3.36-i0.9 2%i0.75 2.2-i0.58 1.8-i0.25
12+i20 19+i33 27 + i56 41+ ilO 59+i168
3.46 2.36 1.8 1.2 0.8
3.4 2.4 1.7 1.2 0.85
3.1-i0.92 2.4-i0.6 2.0-i0.45 1.74-i0.35 1.53-i0.19
8+i14 ll+i23 17+i41 23 + i77 36+il32
2.1 1.4 1.0 0.74 0.53
2.0 1.4 1.0 0.7 0.5
7.4-i2.7 5.7-i2.0 4.4-d .3 3.4-il.0 2.8-i0.7
22 + i36 32 + i54 51 +i85 73 + i145 llO+i276
6.4 4.7 3.4 2.4 1.8
6.8 4.8 3.4 2.4 1.7
2.7-i0.65 2. I -iO.45 I .8-i0.34 1.5-i0.17 1.4-iO.16
6+i12 Il+i20 13+i34 16+i60 26+illl
1.7 1.1 0.8 0.5 0.4
1.6 1.1 0.8 0.55 0.4
3.2-il.1 2.7-i0.73 2.3-i0.52
8+il6 13+i25 19+i42 26 + i72 38+i126
2.2 1.7 1.3 0.9 0.6
2.4 1.7 1.2 0.84 0.6
D= 100x10-6m)
D=85xlO-“m)
1.9-i0.36 I .6-i0.32
continued
Emperical model for rigid frame porous materials
187
Table 2-contd Foam-ceramic Sample 8 (H=0.86; 250 500 1000 2000 4000
D=85~10-~rn) 3.7-il.2
lO+i16 lS+i25 21 +i43 32 + 175 4O+i135
2.8-i0.8 2.3-i0.46
1.9-20.43 1.6-io.26
2.7 1.8 1.3 0.9 0.6
2.6 1.8 1.3 0.9 0.65
where a is a factor that tends to 0 at lower frequencies. The value of a has been found in a form: a = Q/U + @>’ Taking into account this expression the empirical formulas for the parameters Wi and a! can be written as: Wi =
(II=
Q
2+Q/
(4)
+,@I2
2kQ 2+Q/
.LL
2t
1000
Frequency
(5)
(Hz)
Fig. 2. Structural characteristic as a function of frequency for a foam-gypsum (1) pm=250kg/m3, D= 180~10-~m; (2) p,=300kg/m3, D= 160~10-~m; (3) p,=300kg/m3, D=9x10e6m; (4) p,=270kg/m3, D=60~10-~m.
188
N. Voronina
The functions Wj(Q) and a(Q) computed from eqns (4), (5) and plotted by solid lines in Fig. 3(a), (b) are confirmed experimentally for all tested samples of porous materials. Therefore, eqns (3)--(5) may be included into the empirical model of a sound propagation through rigid frame porous media. The most difficult problem of this study is to obtaln an analytical formula for the phase constant in terms of the structural characteristic. Figure 3(c) shows that the experimental values of (B/k) - 1 (points) lie much above the same function calculated by: /?/k - 1 = Q
In this case the phase constant may be defined as: /3 = k(l + Q(1 + B))
(6)
where B is a coefficient that depends on the structural characteristic. The numerical values of B obtained from the experimental data for the acoustic parameter fi are presented in Table 3 and in Fig. 4. The results (Fig. 4) indicate that the functions BCf) have peaks for samples l-2 in a frequency range 2500-3000 Hz. Two other curves 3 and 4 have the same tendency also. It is possible that their maxima could appear at frequencies higher than 4000 Hz. Now, the united function B(x) is needed to describe acoustic behaviour of materials. It is important to find the variable x by means of which all different curves B(f) can be combined into one curve B(x). From Fig. 4 one can see that the curves 2 (sample 2) and 3 (sample 5) having equal pore diameters but different structural characteristics (Table 2) may be combined if the variable x is inversely proportional to (1 + Q). From the comparison of curves 1 and 2 obtained for samples 1 and 2 having different pore diameters it has been found that x - l/a. To combine curve 3 with curve 4 determined for the foam-gypsum (sample 5) and the foam-slagconcrete (sample 7), having different frame density the variable x must be divided by the value of pf Then introducing the parameters z and WI= 103p,/p~ the resulting equation has been written as: 120m
x=z(l
+Q>
where the numerical factor 120 has been defined to provide the maximum of the function B(x) at x = 1.
Emperical model for rigid frame porous materials
189
(a)
(b)
Structural
characteristic
(Q)
I” I
a A
A, .
1.25
Structural
characteristic
l 0
2
(Q)
Fig. 3. Acoustic parameters as a function of structural characteristic: (a) the imaginary part of characteristic impedance; (b) the attenuation coefficient; (c) the phase constant for porous materials listed in Table 1. (1) sample 1; (2) sample 2; (3) sample 3; (4) sample 4; (5) sample 5;
190
N. Voronina
Experimental
Q
f (Hz)
Foam-gypsum z = 70.0 250 500 1000 2000 4000
TABLE 3 values of coefficient
B
X
B Exp.
Calc.
(m = 0.765) 1.44 1.02 0.72 0.51 0.36
0.54 0.65 0.76 0.87 0.96
0.07 0.1 0.21 0.46 0.58
0.07 0.12 0.21 0.4 0.61
z=36.5 250 500 1000 2000 4000
4.6 3.25 2.3 1.62 1.15
0.45 0.57 0.76 0.96 1.17
0.03 0.06 0.21 0.55 0.35
0.05 0.08 0.21 0.61 0.3
z=46.0 250 500 1000 2000 4000
3.4 2.4 1.7 1.2 0.85
0.46 0.59 0.74 0.91 1.08
0.02 0.09 0.22 0.46 0.53
0.05 0.08 0.19 0.5 0.55
2=59.5 250 500 1000 2000 4000
2.0 1.4 1.0 0.7 0.5
0.52 0.65 0.78 0.92 1.05
0.03 0.07 0.25 0.57 0.6
0.06 0.12 0.23 0.53 0.59
z= 36.5 250 500 1000 2000 4000
6.8 4.8 3.4 2.4 1.7
0.33 0.45 0.58 0.76 0.96
0 0.02 0.08 0.24 0.63
0.04 0.05 0.08 0.21 0.61
1.56 1.1 0.78 0.55 0.39
0.44 0.54 0.63 0.73 0.81
0.03 0.07 0.12 0.18 0.32
0.05 0.07 0.1 0.18 0.26
2.4 1.7 1.2 0.85 0.6
0.35 0.45 0.55 0.65 0.75
0.04 0.04 0.08 0.14 0.2
0.04 0.05 0.07 0.12 0.2 continued
Foam-slag-concrete z = 47.0 250 500 1000 2000 4000 z=43.0 250 500 1000 2000 4000
(m = 0.43)
191
Emperical model for rigid frame porous materials
Table S-contd Foam-ceramic z=43.0 250 500 1000 2000 4000
(m = 0.47) 2.56 1.8 1.28 0.9 0.64
0.37 0.46 0.57 0.69 0.8
0 0 0.06 0.16 0.31
0.04 0.05 0.08 0.15 0.25
Taking account of the variable x calculated by eqn (7) the formula for the coefficient B(x) has been derived as: B= (60x2-
120x+61.5)-’
(8)
Figure 5 shows a good correspondence between the values of B(x) calculated by eqn (8) and determined experimentally (Table 3). Therefore, eqn (8) may be used to compute the coefficient B(x) presented in eqn (6) for the phase constant. From Fig. 5 one can see that when x < 0.5 or x > 1.5 the coefficient B is equal approximately to 0. Then the phase constant can be written as: B=Wl
+e>
(9)
Over the range 0.5 < x < 1.5 the acoustic parameter B depends on the function B(x) having a maximum at x = 1. It should be noted that the same
Frequency
(Hz)
Fig. 4. Experimental coefficient B as a function of frequency for a foam-gypsum (1) p,,, =250 kg/m3, D= 180x 10e6m; (2) p,,, = 270kg/me6, D=6Ox 10e6m; (3) p,,, = 350 kg/m3, D = 60 x lop6 m; a foam-slag-concrete (4) p,,, = 350 kg/m3, D = 85 x 10W6m.
192
N. Voronina
behaviour of the phase constant has been observed earlier9 for super-thin fibrous materials. The increase of this acoustic parameter in some frequency ranges means that a sound velocity, c (m SC’), in porous media can be considerably smaller than the same defined from eqn (9) as:
co “=l+Q Figure 6 illustrates this fact for fibreglass and foam-gypsum having equal structural characteristics (Qimo = 1.7 atf = 1000 Hz). One can see that there is a region where the sound velocity is not dependent on frequency. This region is 500-800Hz for fibreglass and 2000-3000Hz for the foam-gypsum (sample 3). At present it is difficult to explain the tendency of the sound velocity in fibrous and porous media. It is interesting to examine how the sound absorption coefficient, 1~0,at normal incidence of a plane sound wave on a material layer with a thickness, I,(m) depends on the effect of the function B(x). The sound absorption coefficient of sample 3 (a foam-gypsum) at I= 0.02 m has been computed by means of eqns (3)-(6) for two cases: when B # 0 using three variables z, Q, kl and when B = 0 using two varables Q, kl. From Fig. 7 it is clear that there is a difference between the curves 1 (B # 0) and 2 (B = 0) over the frequency
Parameter
(x)
Fig. 5. Coefficient B as a function of a parameter x calculated by eqn (8) and determined experimentally for porous materials (Table 1). (1) sample 1; (2) sample 2; (3) sample 3; (4) sample 4; (5) sample 5; (6) sample 6; (7) sample 7.
193
Emperical model for rigid frame porous materials
range higher than 1OOOHz. The difference is confirmed experimentally. At frequencies lower than 1OOOHzboth variants of the coefficient B lead to equal results. Thus, the empirical model derived here for rigid frame porous materials with high porosity (H > 0.75) predicts the acoustic parameters IV, y and also the sound-absorption coefficient rather exactly. When x < 0.5 this model is more simple (B = 0). It is valid for porous materials having large frame density pj> 2000 kg/m3 (m < 0.5). In this case a nomogram has been plotted for the determinaton of the sound-absorption coefficient. Figure 8 presents some curves of equal sound absorption coefficients in a co-ordinate system of /
/
IO
Frequency (Hz) Fig. 6. Sound velocity as a function of frequency for a foam-gypsum (1) p,,, = 300 kg/m3, D = 95 x 10m6m and a super fibreglass (2) P,,, = 20kg/m3, D = 1O-6 m.
^.
(4
*
250
(b)
500
Frequency (Hz)
1000
1000
2000
3000
4000
Frequency (Hz)
Fig. 7. Plots of a sound absorption coefficient for a foam-gypsum having p,,, = 300 kg/m3 and D = 95x 10e6 m calculated using the condition: (1) x > 0.5 (B#O); (2) x < 0.5 (B= 0) at layer thickness 0.02 m. (a)f< 1000 Hz; (b) f> 1000 Hz.
N. Voronina
194
two dimensionless variables Q, kl. A numerical value of (~0can be determined from a point with coordinates Q, kl calculated at the frequency of interest. Figure 8 illustrates some points obtained for a granular mineral wool plate ‘Akmigran’ (pm = 400 kg/ms, pf = 2600 kg/m3, H = 0.85, D = 75 x lop6 m) with layer thickness 0.02 m at frequencies 500, 1000, 2000,400O Hz. In Fig. 7 the results of determining values a!~in this way are presented by the curve 2. There is a close agreement between computed and measured data. It should be noted that this nomogram (Fig. 8) can be useful to technologies for manufacturing different porous materials. On the one hand the sound absorption coefficient may be determined without measurements by means of an impedance tube method. On the other hand by optimization of the structural parameters H and D it is possible to provide the desired value of cx,-,at known parameter kl. More detailed consideration of this important question is not the subject of this paper.
COMPARISON
OF EMPIRICAL MODES FOR POROUS AND FIBROUS MATERIALS
In this study the empirical model for porous materials has been compared with the same derived earlier9 for fibrous materials in the form: W=
1 +Qr-iQfF
Parameter
Fig. 8. Nomogram
(!x/)
for a determination of a sound absorption layer thickness, I (x < 0.5).
coefficent
for porous
material
Emperical model for rigid frame porous materials
Frequency
195
(Hz)
Fig. 9. Acoustic parameters as a function of a frequency: (a) the real part of characteristic impedance; (b) the imaginary part of characteristic impedance; (c) the attenuation coefficient; (d) the phase constant measured and calculated (1) by eqns (3-6) for foam-slag-concrete (sample 7, P,,, = 350 kg/m3 , D = 85 x low6 m); (2) by eqn (1 l), (12) for fibreglass (pm = 80 kg/m3, d=4x 10Wm).
N. Voronina
196
Y=kQy
Qr>Cl + B.) +ik(l +Qdl 1+Qr
1 + FU +
+B’))
(12)
where Qf is the structural characteristic calculated by eqn (2); the coefficients F and Bf are defined as: for for for
x1 < 1 1 < x1 < 4 x1 > 4
F = -0.2x: + 0.7~~ + 0.25&(1 F= ,/X/(1 +4&) F=4 fi/(2+4 fi)
Bf = ,/&(lOz&
- 12.6zlx, + 421
+ 2x:)-2
4(1 + 40) 1 +ylQ2
where x1
=
x0
ym/(l =
Yi(l
+
qo(l
Qjj2)
+
yI
+g$%F)
z,=
=
mx
lo2
Ji&f
Two acoustic materials having equal structural characteristics (Q = Qf) have been compared. They are: the foam-slag-concrete (sample 7) with
0.6
0
’
I 1000
I 2000
Frequency (Hz) Fig. 10. Plots
concrete
of a sound absorption coefficient measured and calculated for (1) foam-slagD = 85x 10e6 m; (2) fibreglass with pm = 80 kg/m3, having pm = 350 kg/m3, d=4x 10-6m at layer thickness 0.04m.
Emperical model for rigid frame porous materials
197
p,,, = 350 kg/m 3; D =85x 10P6m and the fibreglass with pm = 80 kg/m3; d =4x 10m6m. Figure 6 shows the acoustic parameters IV, y calculated by eqns (3)-(6) using (1) for this porous material and by eqns (10)--Q 1) using (2) for the fibrous material. There is no difference between the real parts of characteristic impedance [Fig. 9(a)] for both media. It is valid also for the phase constants [Fig. 9(d)]. However, for fibreglass the imaginary part of characteristic impedance [Fig. 9(b)] and the attenuation coefficient [Fig. 9(c)] are more than the same for the porous material. Therefore, it can be assumed that an acoustic similarity is not possible between fibrous and porous materials. This discrepancy between two models has an effect on the soundabsorption coefficient. From Fig. 10 one can see that in a frequency range lower then 2000 Hz (kl < n/2) the value of aygfor the foam-slag+oncrete is more than the value for the fibreglass. At frequencies higher then 2OOOHZ (kl > n/2) the sound-absorption coefficient depends on the characteristic impedance only and, therefore, the curves 1 and 2 coincide.
CONCLUSION An experimental study of acoustic parameters has been carried out for rigid frame porous materials having different physical constants which characterise the medium macrostructure. The empirical model has been derived by comparative analyses of these results. This model can be used to predict values of the acoustic impedance and the sound-absorpton coefficient for the porous material layer with known thickness. Two models for porous and fibrous materals, having equal structural characteristics, have been compared. One can see that acoustic similarity is not possible between porous and fibrous materials.
REFERENCES 1. Attenborough, K., Acoustical characteristics of rigid fibrous absorbents and 2.
granular materials. Journal Acoust. Sot. Am., 1983, 73(3), 785-799. Attenborough, K., On the acoustic slow wave in air-filled granular media.
Journal Acoust. Sot. Am., 1987, 81(l), 93-102. 3. Attenborough, K., Model of the acoustical properties of air-saturated granular media. Acta Acustica, 1993, 1, 213-226. 4. Yamamoto, T. and Turgut, A., Acoustic wave propagation through porous media with arbitrary pore size distributions. Journal Acoust. Sot. Am., 1988, 83(5), 17441751. 5. Stinson, M. R., The propagation of plane sound waves in narrow and wide circular tubes, and generalization to uniform tubes of arbitrary cross-sectonal shape. Journal Acousf. Sot. Am., 1991, 89(2), 55C558.
198
N.
Voronina
6. Stinson, M. R. and Champoux, Y., Propagation of sound and the assignment of shape factors of model porous materials having simple pore geometries. Journal Acoust. Sot. Am., 1992, 91(2), 685695. 7. Wilson, D. K., Relaxion-matched modelling of propagation through porous media, including fractal pore structure. Journal Acoust. Sot. Am., 1993, 94(2), 1136-l 145. 8. Voronina, N., Acoustic proporties of fibrous materials. Appl. Acoust., 1994, 42, 165-174. 9. Voronina, N., Improved empirical model of sound propagation through a fibrous material. Appl. Acoust., 1996, 48(2), 121-132. 10. Voronina, N., Determination of structural characteristic for foam-gyps. Tr./ NIISF, Building Acoustics, 1979, 25-3 1. 11. Voronina, N., Physic-mathematical model of a porous absorbing material. Tr./ NIISF, Building and acoustic means of noise-protection, 1988, 44-53 12. Voronina, N., Physic-mathematical model of sound waves propagation through a porous media. Rep.lXI All-Union Acoustic Conference, Moscow, 1991.