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Sound field of a baffled sound source covered by an anisotropic rigid-porous material
Applied Acoustics 66 (2005) 866–878 www.elsevier.com/locate/apacoust
Sound field of a baffled sound source covered by an anisotropic rigid-porous material Chao-Nan Wang *, Hai-Ming Cho Department of Engineering Science and Ocean Engineering, National Taiwan University, 1, Sec. 4, Roosevelt Rd., Taipei 106, Taiwan, ROC Received 10 September 2001; received in revised form 23 March 2003; accepted 2 March 2004 Available online 21 November 2004
Abstract A theoretical approach for the sound field of a piston sound source covered by a finite thickness layer of anisotropic rigid-porous material is presented. The formulation is an extension of the method worked out by Amedin et al. [Sound field of a baffled piston source covered by a porous medium layer. J Acoust Soc Am 1995;98(3):1757]. First, in the present study the sound field of a point source is described by cylindrical waves. Then, with the proper boundary conditions, the sound pressure radiated from a piston source covered by a layer of anisotropic porous material can be calculated. The effects of frequency and bulk density of material on the sound propagation in an anisotropic porous material are studied. Finally, the effect of anisotropy is discussed. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Sound; Porous material; Anisotropic
1. Introduction Porous materials are used widely in noise control engineering. Thus the understanding of the acoustical characteristics of porous absorbing material is *
Corresponding author. E-mail address: [email protected] (C.-N. Wang).
0003-682X/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2004.03.008
C.-N. Wang, H.-M. Cho / Applied Acoustics 66 (2005) 866–878
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important. Many theoretical studies [1–5] have been made on this topic. However, due to the complexity of the structure of the porous material, several parameters are used in these theoretical models and several experimental measurements are required to identify these parameters. As a result these theoretical approaches are not easy to use directly. Delany and Bazley [6], Wu [7] and Allard and Champoux [8] present empirical formulations, which can be used easily, for fibrous materials and open-cell foams. However, for particular materials or new products, these empirical formulations may not give sufficiently accurate results. To get the acoustical characteristics of these porous materials experimental measurements are still required. Various experimental methods have been proposed for measuring the acoustical characteristics of porous absorbing materials. Impedance tube measurements [9,10] are typical and widespread. There are also free-field measurement methods [11,12]. However, due to the basic assumptions or sample limitations, these methods are suitable only for laboratory measurements. Champoux and Ross [13] have developed an in situ technique to monitor the thermal resistivity of the porous material while it is being manufactured. Recently, Amedin et al. [14,15] have proposed an acoustical model that extends the applicability of the Champoux and Ross technique to measure the characteristic impedance and sound propagation constant of porous absorbing materials. However, since their theoretical framework is based on an isotropic analysis, it will cause some discrepancies when applied to an anisotropic porous material. The object of the present study is to extend the theory proposed by Amedin et al. [14] to develop a theoretical model for the sound transmitted through an anisotropic rigid-porous layer to the semi-infinite half-space above the layer. First, the sound field for a unit source located in an open space formed by anisotropic porous material is developed. Then the sound pressure distribution in air and porous layer is calculated. Further, the difference of the sound field between isotropic and anisotropic porous material is compared.
2. Sound field of a unit source in anisotropic porous materials Consider a point sound source located at R0 radiating sound into an infinite anisotropic porous material. Assuming that the sound propagation speed in z-direction is cN that is different from the sound speed cP in x- and y-directions (or r,h direction). Then the sound at any point R denoted by gx(R,R0) is governed by " # 2 cP o2 o2 o2 2 þ ð1Þ þ 2 þ k N gx ðR; R0 Þ ¼ 4pdðR R0 Þ; cN ox2 oy 2 oz where complex propagation constant kN = x/cN, x is the angular frequency. Following the procedures (Fourier transform, cylindrical coordinate transform and series expansion) in [16] the sound field gx(R,0), for a point source located at origin, can be expressed as
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! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
gx ðR; 0Þ ¼
Z 0
1
l rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J 0 ðlrÞe 2 2 cP 2 l kN cN
jzj
cP cN
l2 k 2N
dl;
ð2Þ
where r is the distance between the projection point of R on x, y plane and the origin and J0 is the zero order of Bessel function of first kind. If the porous material is isotropic, cN equals to cP, and the above equation is the same as that in [14,15]. For simplicity, the equation is rewritten as Z 1 l J 0 ðlrÞ expðgjzjÞ dl; gx ðR; 0Þ ¼ ð3Þ g 0 where s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 cP g¼ l2 k 2N : cN
ð4Þ
3. Governing equations and cylindrical wave solution Suppose a layer of anisotropic porous absorbing material is placed on a horizontal, infinite rigid baffle plane (SI) as shown in Fig. 1. The layer thickness is h and the surface of the porous material is denoted by SII. The sound source S0 lies in the baffle plane, is assumed to be a circular piston of radius a and vibrates with transverse velocity v0exp(jxt). The governing equations for the sound pressure in porous material (p1) and air (p2) in cylindrical coordinate are: " # 2 cP o2 1 o 1 o2 o2 þ þ ð5Þ þ 2 þ k 2N p1 ðr; h; zÞ ¼ 0; 0 6 z 6 h; cN or2 r or r2 oh2 oz
Fig. 1. Configurations of the problem.
C.-N. Wang, H.-M. Cho / Applied Acoustics 66 (2005) 866–878
o2 1 o 1 o2 o2 2 þ þ þ þ k p2 ðr; h; zÞ ¼ 0; or2 r or r2 oh2 oz2
869
z P h;
ð6Þ
where k is the wave number in air and kN is the same as in Section 2. On the rigid baffle plane SI, the boundary condition is m1 ðr; h; 0Þ ¼ 0;
r P a:
ð7Þ
Across S0 the continuity of velocity gives Xm1 ðr; h; 0Þ ¼ m0 ;
0 6 r < a;
ð8Þ
where X is the porosity of the porous material. On the contact surface of air and porous material, i.e., SII, the pressure and volume velocity must be equal: p1 ðr; h; hÞ ¼ p2 ðr; h; hÞ;
ð9Þ
Xm1 ðr; h; hÞ ¼ m2 ðr; h; hÞ:
ð10Þ
The sound pressure radiated into an infinite medium from a baffled piston source of surface S0 with normal velocity m0 is [17] Z jqxm0 gx ðM; N Þ dSðN Þ; ð11Þ pðMÞ ¼ 2p S0 where x is the angular frequency and N denotes the arbitrary point on the source surface. With this equation the sound pressure in the infinite porous material and air can be illustrated. When considering the reflection on the boundary, the sound pressure in the porous material layer and air can be expressed as: Z Z 1 jq xm0 l p1 ðMÞ ¼ 1 J 0 ðlrMN Þ expðgzÞ dl g 2pX S 0 0 Z 1 l þ Q11 ðlÞ J 0 ðlrMN Þ expðgzÞ dl g Z0 1 l þ Q12 ðlÞ J 0 ðlrMN Þ expðþgzÞ dl dSðN Þ; ð12Þ g 0 Z Z 1 jq xv0 l p2 ðMÞ ¼ 0 Q2 ðlÞ J 0 ðlrMN Þ expðg1 zÞ dl dSðN Þ; ð13Þ g1 2pX S 0 0 where rMN is the distance between the projection point of M on x, y plane and the source point N, Q11 is the reflection coefficient from the rigid baffle plane S1, Q12 and Q2 are the reflection and transmission coefficient of the air and porous material contact p surface ffiffiffiffiffiffiffiffiffiffiffiffiffiffiS ffi 2, q1 and q0 are the density of porous material and air, and g1 ¼ l2 k 2 since the sound is propagated in air. Applying (12) and (13) on the boundary conditions (7), (9) and (10), one can get: Q11 ¼ Q12 ¼
ðq1 g1 Xq0 gÞ exp½ðg þ g1 Þh ; ðq1 g1 Xq0 gÞ expððg þ g1 ÞhÞ þ ðq1 g1 þ Xq0 gÞ exp½ðg g1 Þh
ð14Þ
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Q2 ¼
2Xq1 g1 : ðq1 g1 Xq0 gÞ exp½ðg þ g1 Þh þ ðq1 g1 þ Xq0 gÞ exp½ðg g1 Þh
ð15Þ
Fig. 2. The comparison of the magnitude of the on-axis transfer function with [14]: present method, - - - - -; Ref. [14], ––––.
Fig. 3. The comparison of the phase of the on-axis transfer function with [14]: present method, - - - - -; Ref. [14], ––––.
Fig. 4. The comparison of the magnitude of the off-axis transfer function with [14]: present method, - - - - -; Ref. [14], ––––.
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871
Then following the procedures in [14], applying the integration over the piston source surface and the expansion and integration of Bessel function; one can get the final expressions for the sound pressure in air and porous material:
Fig. 5. The comparison of the phase of the off-axis transfer function with [14]: present method, - - - - -; Ref. [14], ––––.
Fig. 6. The effect of the frequency on the transfer function. Material bulk density 24 kg/m3, rP/rN = 0.6.
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p1 ðMÞ ¼
jq1 axv0 X
jq axv0 p2 ðMÞ ¼ 0 X
Z
1
1 þ ð1 þ expð2gzÞÞQ11 ðlÞ J 0 ðlrM ÞJ 1 ðlaÞ expðgzÞ dl; g ð16Þ
1
Q2 ðlÞ J 0 ðlrM ÞJ 1 ðlaÞ expðg1 zÞ dl: g1
0
Z 0
ð17Þ
These involve the integration of singular function between 0 and 1. The integration routine QAGP provided in [18] is used for the numerical evaluation.
4. Numerical results and discussion Obviously the propagation constant (kN) perpendicular to the planar direction, the complex density (q1), and (cP/cN) need to be specified first in order to calculate the sound pressure distribution in porous material and air. For simplicity, the empirical formulations of Delany and Bazley [6] for fibrous porous material have been used, i.e.:
Fig. 7. The effect of the frequency on the transfer function. Material bulk density 24 kg/m3, rP/rN = 0.6.
C.-N. Wang, H.-M. Cho / Applied Acoustics 66 (2005) 866–878
k c ¼ ðx=cÞ½1 þ 0:0978ðq0 f =rÞ Z c ¼ q0 c½1 þ 0:0571ðq0 f =rÞ
0:700
0:754
j0:189ðq0 f =rÞ
j0:087ðq0 f =rÞ
0:595
0:732
;
;
873
ð18Þ ð19Þ
where kc is the complex propagation constant,Zc is the characteristic impedance, c is the sound speed in air, and r is the flow resistivity of the material in normal direction. It should be noted that these equations are suitable for MKS units. Thus the complex density is given by q1 ¼ Z c k c X=x:
ð20Þ
The value of (cP/cN) is estimated by Eq. (18) if the ratio between the flow resistivities of planar and normal direction is known. According to the study of Allard et al. [19], the ratio of the planar (rP) to normal (rN) flow resistivity for fibrous material is about 0.6. Therefore only the values of (rP/rN) between 0.6 and 1 are considered in the present study. The parameter (called ‘‘transfer function’’) to describe the result is defined as ZðMÞ ¼
pðMÞ ; m0
ð21Þ
Fig. 8. The effect of the bulk density on the transfer function. Frequency 2000 Hz, rP/rN = 0.6.
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where p(M) is the sound pressure at point M, m0 is the normal velocity of the sound source and Z(M) is similar to the acoustic impedance. 4.1. The accuracy of numerical results To verify the accuracy of the program and the infinite range of integration, the numerical results compare to AmedinÕs are illustrated in Figs. 2–5. The flow resistivity of the isotropic porous materials to be considered is 5, 20, 35, and 50 cgs (g s1 cm3) and the thickness of the porous layer is 5 cm. Figs. 2 and 3 are the magnitude and phase of the transfer function for the on-axis point that is located on the vertical axis and 1 cm above the isotropic porous material. Figs. 4 and 5 are the results of point located at 1 cm above the material and 5 cm apart from the center axis of the cylinder. It can be seen that in either case the agreement are very good for all flow resistivities. This means that the program is reliable. 4.2. The frequency effect A hypothetical anisotropic porous material layer (bulk density 24 kg/m3 , thickness 5 cm, normal flow resistivity about 7 cgs and (rP/rN) = 0.6) is used to illustrate
Fig. 9. The effect of the bulk density on the transfer function. Frequency 2000 Hz, rP/rN = 0.6.
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the effect of frequency on the transfer function. Four frequencies, i.e., 500, 1000, 1500 and 2000 Hz are analyzed. For the off-axis case, the calculation points are located at 5 cm above the porous material and move away z-axis horizontally from 0 to 30 cm. The numerical results are shown in Figs. 6 and 7 for the case of on-axis and off-axis, respectively. It is found that the magnitude of the transfer function increases when the frequency increases. This phenomenon indicates that the energy concentrates gradually in front of the sound source, i.e., the sound source becomes more directional when the frequency increases. Further, for 2000 Hz, the same phase angle change at different distances between on-axis (Fig. 6) and off-axis (Fig. 7) is also an expression for the anisotropic property of the material. 4.3. The influence of material density Three fibrous materials with bulk density 24, 32, and 48 kg/m3 are analyzed to illustrate the effect of material density on the sound propagation. According to Bies and Hansens [20], the corresponding flow resistivities for the porous material, required in calculating the complex propagation constant and the characteristic imped-
Fig. 10. The anisotropic effect on the transfer function. Material bulk density 48 kg/m3, frequency 2000 Hz, rP/rN = S.
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ance, are 7, 11 and 19 cgs rayls. Figs. 8 and 9 are the on-axis and off-axis (the calculation points are the same as in Figs. 6 and 7) results for 2000 Hz sound propagated in an anisotropic porous material in thickness 5 cm with rP/rN is 0.6. It is seen that the magnitude of the transfer function decreases when the bulk density increases since the resistance is increased. Further, in Fig. 8 the phase results reveal that the wavelength (also the speed) of the sound wave in the normal direction is different in material. In Fig. 9 it can also find the apparent difference in the wavelength for the case of 32 and 48 kg/m3 in the horizontal direction. 4.4. The anisotropic effect Usually porous fibrous materials are anisotropic in the normal and planar directions. This fact obviously will affect the sound propagation in the material. In the present study three values (i.e., 0.6, 0.8 and 1.0) of the ratio of the planar (rP) to normal (rN) flow resistivity are considered to investigate the anisotropy effect. The numerical results for 2000 Hz sound propagated in a fibrous material of bulk density 48 kg/m3 and 5 cm in thickness are shown in Figs. 10 and 11. For the on axis case
Fig. 11. The anisotropic effect on the transfer function. Material bulk density 48 kg/m3, frequency 2000 Hz, rP/rN = S.
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877
Fig. 12. The comparison of the phase derivative with respect to radial distance of anisotropic materials and wave number in air.
(Fig. 10) there is a little change in the magnitude of the transfer function and the phase is almost the same. However, the phase changes obviously in off-axis case (Fig. 11) since the calculated points are on the horizontal plane and thus the effect of planar flow resistivity appears. Fig. 12 shows the phase derivatives with respect to radial distance for the three anisotropic materials. It is found that close to the source axis the phase derivatives are smaller than the wave number in air and the anisotropy effect is revealed. However, as the radial distance increases the phase derivatives converge to the wave number in air.
5. Conclusions A theoretical formulation for the analysis of sound propagation in an anisotropic rigid-porous material is developed in the present study. The approach is based on the study of Amedin et al. [14]. As a result of the present study the sound field of a piston source covered by a porous material, either isotropic or anisotropic, can be evaluated. From the results of numerical analysis, it is found that the frequency and material density have the stronger effects. Nevertheless the effect of anisotropy should not be ignored.
References [1] Attenborough K. Acoustical characteristics of porous materials. Phys Rep 1982;82:179–227. [2] Biot MA. The theory of propagation of elastic waves in a fluid-saturated porous solid, I. Low frequency range, II. Higher frequency range. J Acoust Soc Am 1956;28:168–91. [3] Johnson DL, Koplik J, Dashen R. Theory of dynamic permeability and tortuosity in fluid-saturated porous media. J Fluid Mech 1987;176:379–402. [4] Champoux Y, Allard JF. Dynamic tortuosity and bulk modulus in air-saturated porous media. J Appl Phys 1991;70:1975–9.
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[5] Allard JF, Herzog P, Lafarge D, Tamura M. Recent topics concerning the acoustics of fibrous and porous materials. Appl Acoust 1993;39:3–21. [6] Delany ME, Bazley EN. Acoustical properties of fibrous absorbent materials. Appl Acoust 1970;3:105–16. [7] Wu Q. Empirical relations between acoustical properties and flow resistivity of porous plastic opencell foams. Appl Acoust 1986;25:141–8. [8] Allard JF, Champoux Y. New empirical equations for sound propagation in rigid frame fibrous materials. J Acoust Soc Am 1992;91(6):3346–53. [9] Smith CD, Parrott TL. Comparison of three methods for measuring acoustic properties of bulk materials. J Acoust Soc Am 1983;74:1577–82. [10] ASTM C 384-90, Standard test method for impedance and absorption of acoustical materials by the impedance tube method; revised in 1990. [11] Allard JF, Sieben B. Measurement of acoustic impedance in a free field with two microphones and a spectrum analyzer. J Acoust Soc Am 1985;77:1617–8. [12] Cramon AJ, Don CG. Reflection of impulses as a method of determining acoustic impedance. J Acoust Soc Am 1984;75:382–9. [13] Champoux Y, Ross M-J. Measurement of thermal resistivity of insulating material using an acoustical technique. Noise Control Eng J 1993;40:151–8. [14] Amedin CK, Berry A, Champoux Y, Allard JF. Sound field of a baffled piston source covered by a porous medium layer. J Acoust Soc Am 1995;98(3):1757–66. [15] Amedin CK, Champoux Y, Berry A. Acoustical characterization of absorbing porous materials through transmission measurements in a free field. J Acoust Soc Am 1997;102(4):1982–94. [16] Morse PM, Ingard KU. Theoretical acoustics. New York: McGraw-Hill; 1968. [17] Fahy F. Sound and structural vibration. Orlando: Academic; 1987. [18] Piessens R, De Doncker-Kapenga E, Uberhuber CW, Kahaner DK. QUADPACK, a subroutine package for automatic integration. Berlin: Springer-Verlag; 1983. [19] Allard JF, Bourdier R, LÕEsperance A. Anistropy effect in glass wool on normal impedance in oblique incidence. J Sound Vib 1987;114:233–8. [20] Bies DA, Hansens CH. Flow resistance information for acoustical design. Appl Acoust 1980;13:357–91.
Sound field of a baffled sound source covered by an anisotropic rigid-porous material Chao-Nan Wang *, Hai-Ming Cho Department of Engineering Science and Ocean Engineering, National Taiwan University, 1, Sec. 4, Roosevelt Rd., Taipei 106, Taiwan, ROC Received 10 September 2001; received in revised form 23 March 2003; accepted 2 March 2004 Available online 21 November 2004
Abstract A theoretical approach for the sound field of a piston sound source covered by a finite thickness layer of anisotropic rigid-porous material is presented. The formulation is an extension of the method worked out by Amedin et al. [Sound field of a baffled piston source covered by a porous medium layer. J Acoust Soc Am 1995;98(3):1757]. First, in the present study the sound field of a point source is described by cylindrical waves. Then, with the proper boundary conditions, the sound pressure radiated from a piston source covered by a layer of anisotropic porous material can be calculated. The effects of frequency and bulk density of material on the sound propagation in an anisotropic porous material are studied. Finally, the effect of anisotropy is discussed. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Sound; Porous material; Anisotropic
1. Introduction Porous materials are used widely in noise control engineering. Thus the understanding of the acoustical characteristics of porous absorbing material is *
Corresponding author. E-mail address: [email protected] (C.-N. Wang).
0003-682X/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2004.03.008
C.-N. Wang, H.-M. Cho / Applied Acoustics 66 (2005) 866–878
867
important. Many theoretical studies [1–5] have been made on this topic. However, due to the complexity of the structure of the porous material, several parameters are used in these theoretical models and several experimental measurements are required to identify these parameters. As a result these theoretical approaches are not easy to use directly. Delany and Bazley [6], Wu [7] and Allard and Champoux [8] present empirical formulations, which can be used easily, for fibrous materials and open-cell foams. However, for particular materials or new products, these empirical formulations may not give sufficiently accurate results. To get the acoustical characteristics of these porous materials experimental measurements are still required. Various experimental methods have been proposed for measuring the acoustical characteristics of porous absorbing materials. Impedance tube measurements [9,10] are typical and widespread. There are also free-field measurement methods [11,12]. However, due to the basic assumptions or sample limitations, these methods are suitable only for laboratory measurements. Champoux and Ross [13] have developed an in situ technique to monitor the thermal resistivity of the porous material while it is being manufactured. Recently, Amedin et al. [14,15] have proposed an acoustical model that extends the applicability of the Champoux and Ross technique to measure the characteristic impedance and sound propagation constant of porous absorbing materials. However, since their theoretical framework is based on an isotropic analysis, it will cause some discrepancies when applied to an anisotropic porous material. The object of the present study is to extend the theory proposed by Amedin et al. [14] to develop a theoretical model for the sound transmitted through an anisotropic rigid-porous layer to the semi-infinite half-space above the layer. First, the sound field for a unit source located in an open space formed by anisotropic porous material is developed. Then the sound pressure distribution in air and porous layer is calculated. Further, the difference of the sound field between isotropic and anisotropic porous material is compared.
2. Sound field of a unit source in anisotropic porous materials Consider a point sound source located at R0 radiating sound into an infinite anisotropic porous material. Assuming that the sound propagation speed in z-direction is cN that is different from the sound speed cP in x- and y-directions (or r,h direction). Then the sound at any point R denoted by gx(R,R0) is governed by " # 2 cP o2 o2 o2 2 þ ð1Þ þ 2 þ k N gx ðR; R0 Þ ¼ 4pdðR R0 Þ; cN ox2 oy 2 oz where complex propagation constant kN = x/cN, x is the angular frequency. Following the procedures (Fourier transform, cylindrical coordinate transform and series expansion) in [16] the sound field gx(R,0), for a point source located at origin, can be expressed as
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! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
gx ðR; 0Þ ¼
Z 0
1
l rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J 0 ðlrÞe 2 2 cP 2 l kN cN
jzj
cP cN
l2 k 2N
dl;
ð2Þ
where r is the distance between the projection point of R on x, y plane and the origin and J0 is the zero order of Bessel function of first kind. If the porous material is isotropic, cN equals to cP, and the above equation is the same as that in [14,15]. For simplicity, the equation is rewritten as Z 1 l J 0 ðlrÞ expðgjzjÞ dl; gx ðR; 0Þ ¼ ð3Þ g 0 where s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 cP g¼ l2 k 2N : cN
ð4Þ
3. Governing equations and cylindrical wave solution Suppose a layer of anisotropic porous absorbing material is placed on a horizontal, infinite rigid baffle plane (SI) as shown in Fig. 1. The layer thickness is h and the surface of the porous material is denoted by SII. The sound source S0 lies in the baffle plane, is assumed to be a circular piston of radius a and vibrates with transverse velocity v0exp(jxt). The governing equations for the sound pressure in porous material (p1) and air (p2) in cylindrical coordinate are: " # 2 cP o2 1 o 1 o2 o2 þ þ ð5Þ þ 2 þ k 2N p1 ðr; h; zÞ ¼ 0; 0 6 z 6 h; cN or2 r or r2 oh2 oz
Fig. 1. Configurations of the problem.
C.-N. Wang, H.-M. Cho / Applied Acoustics 66 (2005) 866–878
o2 1 o 1 o2 o2 2 þ þ þ þ k p2 ðr; h; zÞ ¼ 0; or2 r or r2 oh2 oz2
869
z P h;
ð6Þ
where k is the wave number in air and kN is the same as in Section 2. On the rigid baffle plane SI, the boundary condition is m1 ðr; h; 0Þ ¼ 0;
r P a:
ð7Þ
Across S0 the continuity of velocity gives Xm1 ðr; h; 0Þ ¼ m0 ;
0 6 r < a;
ð8Þ
where X is the porosity of the porous material. On the contact surface of air and porous material, i.e., SII, the pressure and volume velocity must be equal: p1 ðr; h; hÞ ¼ p2 ðr; h; hÞ;
ð9Þ
Xm1 ðr; h; hÞ ¼ m2 ðr; h; hÞ:
ð10Þ
The sound pressure radiated into an infinite medium from a baffled piston source of surface S0 with normal velocity m0 is [17] Z jqxm0 gx ðM; N Þ dSðN Þ; ð11Þ pðMÞ ¼ 2p S0 where x is the angular frequency and N denotes the arbitrary point on the source surface. With this equation the sound pressure in the infinite porous material and air can be illustrated. When considering the reflection on the boundary, the sound pressure in the porous material layer and air can be expressed as: Z Z 1 jq xm0 l p1 ðMÞ ¼ 1 J 0 ðlrMN Þ expðgzÞ dl g 2pX S 0 0 Z 1 l þ Q11 ðlÞ J 0 ðlrMN Þ expðgzÞ dl g Z0 1 l þ Q12 ðlÞ J 0 ðlrMN Þ expðþgzÞ dl dSðN Þ; ð12Þ g 0 Z Z 1 jq xv0 l p2 ðMÞ ¼ 0 Q2 ðlÞ J 0 ðlrMN Þ expðg1 zÞ dl dSðN Þ; ð13Þ g1 2pX S 0 0 where rMN is the distance between the projection point of M on x, y plane and the source point N, Q11 is the reflection coefficient from the rigid baffle plane S1, Q12 and Q2 are the reflection and transmission coefficient of the air and porous material contact p surface ffiffiffiffiffiffiffiffiffiffiffiffiffiffiS ffi 2, q1 and q0 are the density of porous material and air, and g1 ¼ l2 k 2 since the sound is propagated in air. Applying (12) and (13) on the boundary conditions (7), (9) and (10), one can get: Q11 ¼ Q12 ¼
ðq1 g1 Xq0 gÞ exp½ðg þ g1 Þh ; ðq1 g1 Xq0 gÞ expððg þ g1 ÞhÞ þ ðq1 g1 þ Xq0 gÞ exp½ðg g1 Þh
ð14Þ
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Q2 ¼
2Xq1 g1 : ðq1 g1 Xq0 gÞ exp½ðg þ g1 Þh þ ðq1 g1 þ Xq0 gÞ exp½ðg g1 Þh
ð15Þ
Fig. 2. The comparison of the magnitude of the on-axis transfer function with [14]: present method, - - - - -; Ref. [14], ––––.
Fig. 3. The comparison of the phase of the on-axis transfer function with [14]: present method, - - - - -; Ref. [14], ––––.
Fig. 4. The comparison of the magnitude of the off-axis transfer function with [14]: present method, - - - - -; Ref. [14], ––––.
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871
Then following the procedures in [14], applying the integration over the piston source surface and the expansion and integration of Bessel function; one can get the final expressions for the sound pressure in air and porous material:
Fig. 5. The comparison of the phase of the off-axis transfer function with [14]: present method, - - - - -; Ref. [14], ––––.
Fig. 6. The effect of the frequency on the transfer function. Material bulk density 24 kg/m3, rP/rN = 0.6.
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p1 ðMÞ ¼
jq1 axv0 X
jq axv0 p2 ðMÞ ¼ 0 X
Z
1
1 þ ð1 þ expð2gzÞÞQ11 ðlÞ J 0 ðlrM ÞJ 1 ðlaÞ expðgzÞ dl; g ð16Þ
1
Q2 ðlÞ J 0 ðlrM ÞJ 1 ðlaÞ expðg1 zÞ dl: g1
0
Z 0
ð17Þ
These involve the integration of singular function between 0 and 1. The integration routine QAGP provided in [18] is used for the numerical evaluation.
4. Numerical results and discussion Obviously the propagation constant (kN) perpendicular to the planar direction, the complex density (q1), and (cP/cN) need to be specified first in order to calculate the sound pressure distribution in porous material and air. For simplicity, the empirical formulations of Delany and Bazley [6] for fibrous porous material have been used, i.e.:
Fig. 7. The effect of the frequency on the transfer function. Material bulk density 24 kg/m3, rP/rN = 0.6.
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k c ¼ ðx=cÞ½1 þ 0:0978ðq0 f =rÞ Z c ¼ q0 c½1 þ 0:0571ðq0 f =rÞ
0:700
0:754
j0:189ðq0 f =rÞ
j0:087ðq0 f =rÞ
0:595
0:732
;
;
873
ð18Þ ð19Þ
where kc is the complex propagation constant,Zc is the characteristic impedance, c is the sound speed in air, and r is the flow resistivity of the material in normal direction. It should be noted that these equations are suitable for MKS units. Thus the complex density is given by q1 ¼ Z c k c X=x:
ð20Þ
The value of (cP/cN) is estimated by Eq. (18) if the ratio between the flow resistivities of planar and normal direction is known. According to the study of Allard et al. [19], the ratio of the planar (rP) to normal (rN) flow resistivity for fibrous material is about 0.6. Therefore only the values of (rP/rN) between 0.6 and 1 are considered in the present study. The parameter (called ‘‘transfer function’’) to describe the result is defined as ZðMÞ ¼
pðMÞ ; m0
ð21Þ
Fig. 8. The effect of the bulk density on the transfer function. Frequency 2000 Hz, rP/rN = 0.6.
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where p(M) is the sound pressure at point M, m0 is the normal velocity of the sound source and Z(M) is similar to the acoustic impedance. 4.1. The accuracy of numerical results To verify the accuracy of the program and the infinite range of integration, the numerical results compare to AmedinÕs are illustrated in Figs. 2–5. The flow resistivity of the isotropic porous materials to be considered is 5, 20, 35, and 50 cgs (g s1 cm3) and the thickness of the porous layer is 5 cm. Figs. 2 and 3 are the magnitude and phase of the transfer function for the on-axis point that is located on the vertical axis and 1 cm above the isotropic porous material. Figs. 4 and 5 are the results of point located at 1 cm above the material and 5 cm apart from the center axis of the cylinder. It can be seen that in either case the agreement are very good for all flow resistivities. This means that the program is reliable. 4.2. The frequency effect A hypothetical anisotropic porous material layer (bulk density 24 kg/m3 , thickness 5 cm, normal flow resistivity about 7 cgs and (rP/rN) = 0.6) is used to illustrate
Fig. 9. The effect of the bulk density on the transfer function. Frequency 2000 Hz, rP/rN = 0.6.
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the effect of frequency on the transfer function. Four frequencies, i.e., 500, 1000, 1500 and 2000 Hz are analyzed. For the off-axis case, the calculation points are located at 5 cm above the porous material and move away z-axis horizontally from 0 to 30 cm. The numerical results are shown in Figs. 6 and 7 for the case of on-axis and off-axis, respectively. It is found that the magnitude of the transfer function increases when the frequency increases. This phenomenon indicates that the energy concentrates gradually in front of the sound source, i.e., the sound source becomes more directional when the frequency increases. Further, for 2000 Hz, the same phase angle change at different distances between on-axis (Fig. 6) and off-axis (Fig. 7) is also an expression for the anisotropic property of the material. 4.3. The influence of material density Three fibrous materials with bulk density 24, 32, and 48 kg/m3 are analyzed to illustrate the effect of material density on the sound propagation. According to Bies and Hansens [20], the corresponding flow resistivities for the porous material, required in calculating the complex propagation constant and the characteristic imped-
Fig. 10. The anisotropic effect on the transfer function. Material bulk density 48 kg/m3, frequency 2000 Hz, rP/rN = S.
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ance, are 7, 11 and 19 cgs rayls. Figs. 8 and 9 are the on-axis and off-axis (the calculation points are the same as in Figs. 6 and 7) results for 2000 Hz sound propagated in an anisotropic porous material in thickness 5 cm with rP/rN is 0.6. It is seen that the magnitude of the transfer function decreases when the bulk density increases since the resistance is increased. Further, in Fig. 8 the phase results reveal that the wavelength (also the speed) of the sound wave in the normal direction is different in material. In Fig. 9 it can also find the apparent difference in the wavelength for the case of 32 and 48 kg/m3 in the horizontal direction. 4.4. The anisotropic effect Usually porous fibrous materials are anisotropic in the normal and planar directions. This fact obviously will affect the sound propagation in the material. In the present study three values (i.e., 0.6, 0.8 and 1.0) of the ratio of the planar (rP) to normal (rN) flow resistivity are considered to investigate the anisotropy effect. The numerical results for 2000 Hz sound propagated in a fibrous material of bulk density 48 kg/m3 and 5 cm in thickness are shown in Figs. 10 and 11. For the on axis case
Fig. 11. The anisotropic effect on the transfer function. Material bulk density 48 kg/m3, frequency 2000 Hz, rP/rN = S.
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Fig. 12. The comparison of the phase derivative with respect to radial distance of anisotropic materials and wave number in air.
(Fig. 10) there is a little change in the magnitude of the transfer function and the phase is almost the same. However, the phase changes obviously in off-axis case (Fig. 11) since the calculated points are on the horizontal plane and thus the effect of planar flow resistivity appears. Fig. 12 shows the phase derivatives with respect to radial distance for the three anisotropic materials. It is found that close to the source axis the phase derivatives are smaller than the wave number in air and the anisotropy effect is revealed. However, as the radial distance increases the phase derivatives converge to the wave number in air.
5. Conclusions A theoretical formulation for the analysis of sound propagation in an anisotropic rigid-porous material is developed in the present study. The approach is based on the study of Amedin et al. [14]. As a result of the present study the sound field of a piston source covered by a porous material, either isotropic or anisotropic, can be evaluated. From the results of numerical analysis, it is found that the frequency and material density have the stronger effects. Nevertheless the effect of anisotropy should not be ignored.
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[5] Allard JF, Herzog P, Lafarge D, Tamura M. Recent topics concerning the acoustics of fibrous and porous materials. Appl Acoust 1993;39:3–21. [6] Delany ME, Bazley EN. Acoustical properties of fibrous absorbent materials. Appl Acoust 1970;3:105–16. [7] Wu Q. Empirical relations between acoustical properties and flow resistivity of porous plastic opencell foams. Appl Acoust 1986;25:141–8. [8] Allard JF, Champoux Y. New empirical equations for sound propagation in rigid frame fibrous materials. J Acoust Soc Am 1992;91(6):3346–53. [9] Smith CD, Parrott TL. Comparison of three methods for measuring acoustic properties of bulk materials. J Acoust Soc Am 1983;74:1577–82. [10] ASTM C 384-90, Standard test method for impedance and absorption of acoustical materials by the impedance tube method; revised in 1990. [11] Allard JF, Sieben B. Measurement of acoustic impedance in a free field with two microphones and a spectrum analyzer. J Acoust Soc Am 1985;77:1617–8. [12] Cramon AJ, Don CG. Reflection of impulses as a method of determining acoustic impedance. J Acoust Soc Am 1984;75:382–9. [13] Champoux Y, Ross M-J. Measurement of thermal resistivity of insulating material using an acoustical technique. Noise Control Eng J 1993;40:151–8. [14] Amedin CK, Berry A, Champoux Y, Allard JF. Sound field of a baffled piston source covered by a porous medium layer. J Acoust Soc Am 1995;98(3):1757–66. [15] Amedin CK, Champoux Y, Berry A. Acoustical characterization of absorbing porous materials through transmission measurements in a free field. J Acoust Soc Am 1997;102(4):1982–94. [16] Morse PM, Ingard KU. Theoretical acoustics. New York: McGraw-Hill; 1968. [17] Fahy F. Sound and structural vibration. Orlando: Academic; 1987. [18] Piessens R, De Doncker-Kapenga E, Uberhuber CW, Kahaner DK. QUADPACK, a subroutine package for automatic integration. Berlin: Springer-Verlag; 1983. [19] Allard JF, Bourdier R, LÕEsperance A. Anistropy effect in glass wool on normal impedance in oblique incidence. J Sound Vib 1987;114:233–8. [20] Bies DA, Hansens CH. Flow resistance information for acoustical design. Appl Acoust 1980;13:357–91.