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Sound attenuation in ducts lined with non-isotropic material
Journal of Sound and Vibration (1972) 24 (2), 177-187
SOUND ATTENUATION
IN DUCTS LINED WITH
NON-ISOTROPIC
MATERIAL
U. J. KURZE~ANDI. L. VBR Bolt Beranek and Newman Inc. 50 Moulton Street, Cambridge, Massachusetts 02138, U.S.A. (Received 8 April 1972)
This paper deals with the sound attenuation in rectangular ducts with non-isotropic lining. The non-isotropy may be due to the basic structure of the fibrous lining material or it may he deliberately created by rigid partitioning of the lining perpendicular to the duct axis. The propagation constant in the axial direction is obtained in the form of a transcendental equation which includes as special cases the locally reacting lining investigated by Morse [l] and Cremer [2] for infinite flow resistance in the direction of the duct axis, and the homogeneous lining as treated by Scott [4]. A simple approximate solution of the transcendental equation for low frequencies indicates that optimum attenuation of the fundamental mode is achieved by using a non-isotropic lining with a flow resistance in the direction of the duct axis that increases with increasing frequency. At middle and high frequencies, the propagation constant is evaluated numerically by using a NewtonRaphson scheme for several typical silencer geometries. 1.
INTRODUCTION
The attenuation of sound waves propagating in lined ducts has been treated analytically by two different methods. Morse [l] and Cremer [2] considered a locally reacting lining where sound propagation in the lining parallel to the duct axis is prohibited. The acoustical properties of such a lining can be described by the ratio of sound pressure to particle velocity perpendicular to the surface of the lining material. Tbis ratio is called the wall impedance. The assumption of a locally reacting lining is based on the high flow resistance of dense, porous, sound-absorbing materials restricting the sound propagation in the lining parallel to the duct axis at not-too-low frequencies. From his investigation of grazing sound incidence, Cremer concluded that the locally reacting lining is desirable, because it permits one to achieve maximum peak attenuation in the vicinity of weakly damped resonances in the lining. If sound propagation in the lining parallel to the duct axis is prevented, any lining can be made locally reacting. This is commonly obtained by rigid partitioning oriented perpendicularly to the duct axis. The second method used to describe the sound propagation in lined ducts takes into account the wave propagation in the lining material parallel to the duct axis. Willms [3] investigated the case of very thick lining material, while Scott [4] studied the sound propagation in ducts lined with lighter fibrous materials of finite thickness. Scott’s analytical results for isotropic materials have been experimentally verified by several investigators
P-81. The use of thin fiberglass blankets as linings in air-conditioning ducts (which usually provides both thermal and sound isolation) and as parallel, partition-free baffles in mufflers has motivated the current interest in the acoustical properties and the geometrical configurations of duct lining materials. The purpose of this paper is to shed more light on those t Now at Battelle-Institut e.V., 6WOFrankfurt am Main 90, Am Roemerhof 35, Germany. 177
178
U. J. KURZE AND I. L. VI&
effects of the non-isotropic properties of porous linings on the sound attenuation in lined ducts which have been observed experimentally [6,7] but have not been explained theoretically. Of particular interest is the attenuation in lined ducts or in parallel-baffle silencers at low frequencies, which experimentally has been proven to be superior without, rather than with, partitions in the material [4]. 2. SOUND PROPAGATION IN NON-ISOTROPIC MATERIAL The propagation of a plane sine wave in the x-direction in a fibrous material results in a pressure differential
(1) where p is the pressure, v, is the face particle velocity, and v,/u, the particle velocity inside the material, is higher than v, due to the porosity 0 < 1. The structure factor x, by which the density p of the air is multiplied, accounts for the velocity distribution in narrow “channels” through the material and for constrictions and blind passages which increase the effective mass of the air [9]. B is the specific dynamic flow resistance that at low frequencies is equal to the ratio of pressure difference per unit thickness to the face velocity of steady through-flow of air [IO]. At high frequencies, S is larger because the gradient of velocity inside the material tends to steepen at the fibers and to increase the viscous force on the fibers as the frequency is increased [4, 9 see p. 1371. In addition to these properties of a material with a rigid frame, the structure factor x may account for vibrations of fibers in fibrous materials or of cell membranes of foam materials [ 11,121. In these cases, x decreases with increasing frequency due to the decreasing motion of the frame with increasing inertia. The increase of relative motion between the air and the frame is generally too small to account for any considerable increase of the dynamic flow resistance at high frequencies [4]. It is well known that the flow resistance of fibrous materials is higher in the direction perpendicular to the plane of fibers than it is parallel to this plane. There is also experimental evidence that the structure factor has the same characteristics. Table 1 gives data reduced from measurements by Bokor [6]. TABLE 1
Structure factor and normalized specific Jlow resistance for Fiberglas Superfine BlOO with soundpropagation normal to the plane of fibers (y-direction) and parallel to the plane of fibers (z-direction). Data reduced from measurements by Bokor [6].
04
0.6 0.8 1 The non-isotropy
2.56 1.96 1.74 1.53
0.278 0.272 0.250 0.258
1.64 1.48 1.37 1.34
0.106 0.122 0.122 0.135
can be described by the following generalized form of equation (1): - gradp
= (@p/c)
13ivil+ CWd,
where the subscript i indicates the respective coordinate.
(2)
179
SOUND ATTENUATION IN DUCTS
The second equation describing the sound field in the material relates the particle velocity differential to the compression and by linearization to the pressure: -divv
= (joa/K)p.
(3)
Compression losses are small and the bulk modulus K is assumed to be real. At low frequencies, the compression in fibrous materials is isothermal and K equals the ambient static pressure P,,. Only at high frequencies does K assume the free-field value for adiabatic compression, i.e. K = 1.4.P, = pc2, where c is the free-field sound velocity. Combination of equations (2) and (3) results in the modified wave equation for a nonisotropic material [ 13,141 (4) where jka, is the bulk propagation constant in the i direction with ai = JC(pc2/K)
xi1JCU - i) W~mil,
(9
and k = w/c is the free-field wavenumber. For simplification, the structure factor xr may be combined with the factor pc2/K to form a modified structure factor, again denoted as xr. In addition, the in!luence of deviations from the adiabatic compression on the flow resistance can be neglected. Then equation (5) becomes
ai = JCh - 8 Wlpckl = JCX~- AM) dl.
(6)
Note that both the structure factor xi and the specific flow resistance r; = B,/pc are slightly frequency dependent. 3. SOUND PROPAGATION IN LAYERED MEDIA In a manner similar to that followed by Scott in his solution [4] of the wave equation for isotropic material, the transcendental equation for the propagation constants of sound in a rectangular duct lined with non-isotropic, porous, sound-absorbing material may now be derived. Consider the duct shown in Figure 1 with an open width h and material of thickness d attached to one wall. The rigid duct walls are planes of symmetry with respect to the sound pressure field and permit the construction of images which may form either a duct with symmetrical lining on two walls or a parallel-baffle silencer. Thus, the solutions to be derived apply to these configurations as well. Non- isotropic /porous lining
‘Open rectangular
Figure 1. Longitudinal
duct
section of a duct lined with non-isotropic
material.
180
U. J. KURZE
AND I. L. Vl?R
The sound pressure field pl in the free space 0 < y < h obeys the wave equation
a% a%+ k’p, = 0,
ayZ + -8.2
and the sound pressure field pII in the non-isotropic material in h < y < h + d obeys the modified wave equation
1 a2h
1 ah
7-_-I.+--
ay ay
a2
*
a22
+
k2Pll= 0,
(8)
where a, and a, are the complex factors which modify the free-field wavenumber k according to equation (6). Symmetrical solutions of the differential equationswhich satisfy theboundary conditions at the duct walls and the condition of a uniform axial propagation constant y can be written as pI -
emyr ~0s {ky4?
e-yzcos (ka,(y
pII -
+ W21>,
(9)
- h - d) J[ 1 + (y/a&)‘]}.
(10)
The ratio of the constants of proportionality in equations (9) and (10) is determined by the condition of equal pressure at the interface y = h: PI
~0s &u 42
E = cos {kh J[l
+ W)21~
~0s @,kdJCl +
+ (y/k)‘]} cos {a$O, - h - d)J[l
Wzk)21) + (r/azk)2]} ’
The final condition determining the propagation constant y follows from the requirement of equal particle velocity perpendicular to the interface y = h:
Combination of equations (11) and (12) yields
-43
+ W)21 tanWJCl + W)23~ = p
JC1 + (r/azk>21 tan{a,kdJCl + WA"lh
(13)
which is a transcendental equation for y. The set of solutions of this equation describes the propagation constants of a set of modes, of which the fundamental mode may be defined as that with the smallest real part of y and, hence, the least attenuation. Equation (13) is the generalized form of the equation derived by Scott [4] for homogeneous material. It also includes the models investigated by Morse Cl] and Cremer L-23,since for a, + cc the right-hand side of equation (13) equals the normalized wall impedance of a locally reacting wall. Tn practice, large values of a, are obtained either by a large structure factor xz in the direction of the duct axis or by a high flow resistance &, as assumed by Morse for dense fibrous materials. Of particular interest is the low-frequency range, where the material thickness is small compared to the wavelength: layIkd 4 1.
(14)
SOUND ATTENUATION
IN DUCTS
181
When the effect of the porous material on the sound propagation (i.e. on attenuation and propagation speed) in the free duct is assumed to be less than that given by the coefficient a, for propagation in the material itself, i.e.
lrl4
(15)
< 19
then the right-hand side of equation (13) can be approximated by
akd[l + (y/a&)‘] (1 + 3 (u,k42Cl+ tr/wV21+ . . .I,
(16)
which is small and thus permits application of a series .expansion procedure to the lefthand side of equation (13). Thus, for low frequencies, where the free duct width h is small compared to the wavelength of sound, the left-hand side becomes -kh[l
+ (y/k)21 (1 + 3 (kh)2[1 + (r/k>2] + . . .}.
(17)
Consideration of the first terms of the series expansion (16) and (17) yields y=jk
1 + (ad/h) 1 + (cJd/hu,2)
+ *
(18)
Consistent with experimental observations by Bokor [6, 71, only the z-parameters of the material enter into the result for low frequencies. The attenuation of sound propagating along the duct, which is proportional to the real part of y, is very small if either waves can propagate freely in the material (i.e. a, = l), or wave propagation in the material is inhibited by partitions (i.e. a, + co). Considerable attenuation can be expected only in the frequency range where the imaginary part of ui, which is B,a/pck, is comparable to the real part of the denominator in equation (18). The optimum flow resistance for lowfrequency attenuation is roughly
sz- A
PC (k/4
JCtt-Mk) + 11.
(19)
This optimum flow resistance is proportional to frequency and assumes very small values at very low frequencies. Note, however, that the maximum attenuation obtainable is proportional to the flow resistance. Thus, the low-frequency attenuation following from equation (18) is generally small, although it is much larger than the attenuation obtainable with a partitioned (i.e. locally reacting) wall. (An order of magnitude in difference between the two obtainable attenuations is presumed for the validity of the approximation leading to equation (18).) Solutions of equation (13) for middle and higher frequencies, where the sound wavelength is comparable to the duct width h and to the lining thickness d, are feasibly obtained by numerical computation only. It was found that, with a Newton-Raphson scheme [ls), a few iterations yield rapid convergence at the fundamental mode for kh < 7c and with some modifications still yield convergence up to kh = 37~ This is the upper frequency limit of practical interest since the beaming effect reduces the attenuation obtainable to very small values for higher frequencies. 4. EXAMPLES
The attenuation of the fundamental mode in a lined duct calculated by numerical evaluation of equation (13) is plotted in Figures 2, 3 and 4 for the purpose of comparing the theories of Morse/Cremer, Scott, and the new approach for non-isotropic material.
182
U. J. KURZE AND I. L. VfiR
The first example deals with a relatively narrow duct, h = 1 in, t and a lining that covers 50 % of the cross-sectional area, i.e. d = 1 in. It is also applicable on 2in thick parallel baffles which are placed 4 in on center in a rectangular duct. The normalized specific flow resistance perpendicular to the lining is ri = 1 in-‘, which is typical for most porous sound-absorbing materials. The porosity of the material is somewhat smaller than unity: D = 0.95. The structure factors, xY= xz = 1.4, were assumed to be independent of direction. Figure 2 shows the attenuation 8.7 Re {yh} dB vs. a logarithmic frequency scale for the above-described narrow duct.
Figure 2. Calculated attenuation D,,of the fundamental mode over a length I along the axis of the duct that is equal to the free duct width h. Duct width h = 1 in, lining thickness d = 1 in, gas temperature 70°F (21.1T). Material data: porosity Q = 0.95, structure factors xY = xz = 1.4, normalized specific flow resistance normal to the lining ri = SJpc = 1 in-‘. Parameter: normalized specific flow resistance parallel to the duct axis, r:.
If it is assumed that there are rigid partitions inside the material perpendicular to the surface, then the lining is locally reacting and characterized by P, + co, so that results according to the theory of Morse [l] and Cremer [2] are obtained. The attenuation is very small below f = 1400 Hz, where the lining thickness d is smaller than one-eighth of the sound wavelength 1= c/,/j$in the material. Figure 2 shows an attenuation peak at the quarter-wavelength resonance in the lining, where d = n/4. Additional peaks occur at odd multiples of the first resonance frequency, and valleys at even multiples. Another curve in Figure 2 represents isotropic material with ri = 1 in-’ and shows results according to Scott’s theory. The first attenuation peak is shifted to higher frequencies. This shift can be attributed to an increased stiffness of the lining at oblique sound incidence. Although the attenuation at low frequencies (below 14OOHz) is small compared to the maximum attenuation, the increase of attenuation relative to that of the locally reacting lining can be of considerable practical interest. For instance, 40 in long parallel baffles of t 1 in - 234cm.
SOUND ATTENUATION
183
IN DUCTS
isotropic material yield at least 20 dB attenuation at 700 Hz rather than 4 dB predicted by the theory for locally reacting linings! (In both cases, additional attenuation due to end effects, i.e. absorption of higher order modes, is neglected.) In order to obtain the same attenuation of 20 dB with a locally reacting lining, one has to increase the flow resistance IS,by a factor of 5. This would result in a considerable reduction of the attenuation peak at the quarter-wavelength resonance. A third curve in Figure 2 for r: = 2 in- ’ gives results for non-isotropic material. The flow resistance in the z-direction is larger than optimum for high attenuation below f = 1000 Hz and smaller than optimum above f = 1600 Hz. The first attenuation peak is higher and occurs at a lower frequency than for the isotropic material. Evidently, this non-isotropic material yields an attenuation intermediate between those mentioned. Note the still significant improvement achieved at low frequencies compared with the locally reacting lining. The envelope of the three curves in Figure 2 suggests that optimum attenuation performance in the frequency region below 3 kHz could be obtained by using a non-isotropic lining with a flow resistance r: which increases with increasing frequency.
I
100
2
3
4
I,,,,
S67S9,m
Frequency
2
3
4
5
6789,4000
1Hz 1
Figure 3. Calculated attenuation functions. Same data as in Figure 2, but h = 3 in, d = 3 in, r; = l-5 in- *.
Less difference in attenuation performance between locally reacting material and isotropic material can be observed in Figure 3 for a typical geometry of parallel baffles for medium-frequency mufflers. The duct width h is 3 in and the lining thickness d is 3 in. Structure factor and porosity values are as before, xY= xZ = 1.4 and D = 0.95, and the specific flow resistance perpendicular to the lining is somewhat higher: ri = l-5 in- ‘. Note, however, that the total flow resistance is considerably higher (namely, ry = rid = 4.5 compared with ry = 1 in the previous example) which causes an increase in attenuation at low frequencies according to the theory for locally reacting linings. A further increase in attenuation at low frequencies by finite values of ri is marginal. For isotropic material, the attenuation following from Scott’s theory is only marginally different from results according to Morse/Cremer as illustrated in Figure 3, except for frequencies in the vicinity of the resonance in the material which coincides with kh x T(. Non-isotropic material with a small flow resistance ri = 0.33 in yields somewhat higher attenuation forf < 300 Hz andff> 3400 Hz, and less attenuation for intermediate frequen-
184
U. J. KURZE AND I. L. VfiR
ties. The attenuation peaks shown in Figure 3 are due to maximum coupling between waves in the free duct and retarded waves in the material. From consideration of the results of Figure 3, there appears no need to solve the more complex equation of Scott for duct linings of large thickness d or of high flow resistance, since for such cases approximate results can be taken from the theory for locally reacting linings. However, materials with less flow resistance require the refined analysis.
2
3
4
56766,000
Fmquency
Figure 4. Calculated attenuation
2
5
4
5 6 76!
(Hz)
functions. Same data as in Figure 3, but rl = 0.75 in- I.
Figure 4 gives calculated data for ri = 0.75 in-l, which may be representative for the lower end of the flow resistance range of glass fiber and mineral wool materials typically used in parallel-baffle mufflers. When compared with those of Figure 3, the attenuation functions plotted in Figure 4 show more pronounced extremes for both the locally reacting lining (rz --t co) and the isotropic material (ri = 0.75 in-‘). However, the low-frequency attenuation performance of the non-isotropic material (r: = 0.33 in) is almost unchanged, which is consistent with the approximation given in equation (18), which yields y as a function of ri but independent of I;. Of the material properties involved in these calculations, least is usually known about the magnitude of the structure factor, since it can be determined only by dynamic tests, whereas the flow resistance and porosity are at least defined by static test procedures. (Note, however, that the dynamic flow resistance can be smaller than the static flow resistance for materials with elastic frame or larger for materials with considerable nonlinear behavior.) A good estimate for the structure factor of fibrous materials at intermediate frequencies is the value, as used above, of x = 1.4. However, x can be much higher at low frequencies for materials with an elastic frame, and it approaches the lower vahre x = 1 at high frequencies [12]. For the numerical evaluation of the influence of the structure factor on the attenuation performance, the same duct lining as before may be considered, with d = h = 3 in, r; = 0.75 in-’ and r: = 0.33 in- ‘, but with the structure factor xr = xz varied between 1 and 2. Calculated attenuation curves shown in Figure 5 indicate a relatively small influence of the structure factor xz at low frequencies and thus eliminate the necessity of considering a possibly strong frequency dependence of the structure factor. Attenuation peaks at higher
185
SOUND ATTFMJATION IN DUCTS
frequencies become more pronounced with increasing structure factor. However, the bandwidth of relatively high attenuation (for example, 8.7 Re {yh} dB Z 1.5 dB) is maximum for the smallest structure factor x = 1. This result is consistent with the general characteristics of resonant duct lining absorbers, which yield maximum attenuation bandwidth for maximum volume participating in dissipation, i.e. for 1/4 resonators rather than Helmholtz resonators.
I
2
I
3
4
I
I
I
I,!,
2
s6799,0M)
Frequency
3
4
5
9 7 9 91opoo
(Hz 1
Figure 5. Calculated attenuation functions for d = h = 3 in, ri = 0.75 in-‘, Parameter is the structure factor, x.
r: = 0.33 in-‘,
a = 0.95.
The theoretical data calculated from equation (13) are related to the fundamental mode of sound propagating in narrow ducts or in parallel-baffle silencers. The attenuation measured across finite-length silencers is higher partly due to the absorption of sound energy in higher order modes. This absorption takes place predominantly at both ends of the silencer. Experimental results obtained with various parallel-baffle configurations in a square duct indicated that these effects can be explained by the change of the free duct area at both ends of the silencer. The attenuation was determined from sound intensity measurements [14] at both sides of the test section with the same cross sectional area according to Atten. = 10 log !iliI $ 2
dB,
(20)
2
where lpll and ]p21 are the amplitudes of the sound pressure at locations in the entrance and at the exit duct respectively, and A+, and A& are phase differences evaluated by shifting the microphone over a short distance AZ along the duct axis at these respective locations. Generally, the measured attenuation data exhibited a frequency dependence similar to that predicted theoretically from the structure factor, porosity and the non-isotropic flow resistance of the porous material employed. However, the measured attenuation data were consistently higher than those predicted, even after accounting for the end effects and the energy loss through the duct wall (the latter have been determined by measurements in an empty duct). This remaining excess attenuation can be explained by compressional
186
U. J. KURZE AND I. L. Vl%R
losses in the material which were not included in the theory. It should be noted that this excess attenuation, AL, is typically a few decibels in long narrow ducts. If expressed in terms of the attenuation per duct width, AD,, it would be less than O-1dB.
t
Frequency
(Hz)
Figure 6. Attenuation in a duct lined with Fiberglas SupertIne BlOOof 1 in thickness. Parameter h is the half airspace between two linings. @-@, Measured by Bokor [6]; O---O, calculated from data in Table 1.
Higher attenuation values than predicted by equation (13) have been measured by Bokor [6] in ducts that are wide compared to the lining thickness. Figure 6 shows Bokor’s data compared with results calculated from the material data in Table 1. While good agreement is found for the narrow duct with h = 1 cm, the experimental results are considerably higher in wider ducts with h = 5 and 8 cm. Bokor explains this discrepancy by the curvature of the wave-front near the sound source which results in higher order modes. 5. CONCLUSIONS The attenuation performance of ducts with porous linings or parallel-baffle silencers cannot generally be predicted by the theory for locally reacting duct linings. For fibrous linings with low specific flow resistance, in particular, one has to consider the non-isotropic properties of the material. As the experimental data obtained by Bokor have already indicated, the low-frequency attenuation is primarily determined by the material properties for sound propagation parallel to the axis of the duct. There is an optimum flow resistance of the material in that direction which yields maximum attenuation at low frequencies (note, however, that this attenuation is low compared with the peak attenuation obtained at higher frequencies). This optimum flow resistance increases in proportion with frequency. At higher frequencies, where the lining thickness becomes comparable to the sound wavelength, attenuation peaks occur, which are pronounced if, due to a large structure factor, the propagation speed of sound in the material is considerably lower than in free air. However, maximum bandwidth of high attenuation is obtained by materials with a structure factor near unity.
SOUND ATTENUATIONIN DUCTS
187
The high-frequency attenuation performance of porous materials is slightly better than that of locally reacting linings, but it is also limited by the beaming effect of sound in the center of wide ducts. From comparisons with experimental data, the theoretical results appear to be conservative. Higher attenuation occurs whenever inhomogeneities of the lined duct give rise to higher order modes. ACKNOWLEDGEMENT The aut.hors wish to thank Dr Uno Ingard, Massachusetts Institute of Technology, for helpful suggestions during the preparation of the manuscript. REFERENCES 1. P. M. MORSE1939 Journal of the Acoustical Society of America 11,205-210. The transmission of sound inside pipes. 2. L. CREMER 1940 Akustische Zeitschrift 5, 57-76. Nachhallzeit und Dlmpfungsmass bei streifendem Einfall. 3. W. WILLMS1941 Akustische Zeitschrif 6, M-165. Die Schalld%mpfung in Absorptionsrohren. 4. R. A. SCOTT 1946 Proceedings of the Physical Society 58,358-368. The propagation of sound between walls of porous material. 5. E. A. LESKOV,G. L. OSIPOVand E. J. YUDIN 1970 Applied Acoustics 3,47-56. Experimental investigations of splitter duct silencers. 6. A. BOKOR 1969 Journal of Sound and Vibration 10, 390403. Attenuation of sound in lined ducts. 7. A. BOKOR 1971 Journal of Sound and Vibration 14, 367-373. A comparison of some acoustic duct lining material, according to Scott’s theory. 8. J. WALSDORFF1971 Seventh International Congress on Acoustics, Budapest, paper 25 A 6. Absorptionsschalld%npfer ohne Kassettierung. 9. L. CREMER1950 Wellentheoretische Raumakustik. Leipzig: S. Hirzel Verlag. See p. 145. 10. V. N. BARANOVAand K. A. VELIZHNINA1957 Soviet Physics Acoustics 3, 107-111. Acoustic parameters of certain sound absorbent materials. (In this paper an impedance tube method for the experimental determination of the bulk acoustical properties of porous materials is described.) 11. L. L. BERAN~K 1947 Journal of the Acoustical Society of America 19, 556-568. Acoustical properties of homogeneous isotropic rigid tiles and flexible blankets. 12. R. A. SCOTT 1946 Proceedings of the Physical Society 58, 165-183. The absorption of sound in a homogeneous porous medium. 13. P. M. MORSEand U. K. INGARD 1961 In Handbuch der Physik XI/l (Fltigge, ed.) p. 17. Berlin: Springer-Verlag. Linear Acoustic Theory. 14. U. KURZE 1969 Acustica 21, 74-85. Schallausbreitung im Kanal mit periodischer Wandstruktur. 15. F. B. HILDEBRAND1956 Introduction to Numerical Analysis. New York: McGraw-Hill Book Company. See p. 451.
SOUND ATTENUATION
IN DUCTS LINED WITH
NON-ISOTROPIC
MATERIAL
U. J. KURZE~ANDI. L. VBR Bolt Beranek and Newman Inc. 50 Moulton Street, Cambridge, Massachusetts 02138, U.S.A. (Received 8 April 1972)
This paper deals with the sound attenuation in rectangular ducts with non-isotropic lining. The non-isotropy may be due to the basic structure of the fibrous lining material or it may he deliberately created by rigid partitioning of the lining perpendicular to the duct axis. The propagation constant in the axial direction is obtained in the form of a transcendental equation which includes as special cases the locally reacting lining investigated by Morse [l] and Cremer [2] for infinite flow resistance in the direction of the duct axis, and the homogeneous lining as treated by Scott [4]. A simple approximate solution of the transcendental equation for low frequencies indicates that optimum attenuation of the fundamental mode is achieved by using a non-isotropic lining with a flow resistance in the direction of the duct axis that increases with increasing frequency. At middle and high frequencies, the propagation constant is evaluated numerically by using a NewtonRaphson scheme for several typical silencer geometries. 1.
INTRODUCTION
The attenuation of sound waves propagating in lined ducts has been treated analytically by two different methods. Morse [l] and Cremer [2] considered a locally reacting lining where sound propagation in the lining parallel to the duct axis is prohibited. The acoustical properties of such a lining can be described by the ratio of sound pressure to particle velocity perpendicular to the surface of the lining material. Tbis ratio is called the wall impedance. The assumption of a locally reacting lining is based on the high flow resistance of dense, porous, sound-absorbing materials restricting the sound propagation in the lining parallel to the duct axis at not-too-low frequencies. From his investigation of grazing sound incidence, Cremer concluded that the locally reacting lining is desirable, because it permits one to achieve maximum peak attenuation in the vicinity of weakly damped resonances in the lining. If sound propagation in the lining parallel to the duct axis is prevented, any lining can be made locally reacting. This is commonly obtained by rigid partitioning oriented perpendicularly to the duct axis. The second method used to describe the sound propagation in lined ducts takes into account the wave propagation in the lining material parallel to the duct axis. Willms [3] investigated the case of very thick lining material, while Scott [4] studied the sound propagation in ducts lined with lighter fibrous materials of finite thickness. Scott’s analytical results for isotropic materials have been experimentally verified by several investigators
P-81. The use of thin fiberglass blankets as linings in air-conditioning ducts (which usually provides both thermal and sound isolation) and as parallel, partition-free baffles in mufflers has motivated the current interest in the acoustical properties and the geometrical configurations of duct lining materials. The purpose of this paper is to shed more light on those t Now at Battelle-Institut e.V., 6WOFrankfurt am Main 90, Am Roemerhof 35, Germany. 177
178
U. J. KURZE AND I. L. VI&
effects of the non-isotropic properties of porous linings on the sound attenuation in lined ducts which have been observed experimentally [6,7] but have not been explained theoretically. Of particular interest is the attenuation in lined ducts or in parallel-baffle silencers at low frequencies, which experimentally has been proven to be superior without, rather than with, partitions in the material [4]. 2. SOUND PROPAGATION IN NON-ISOTROPIC MATERIAL The propagation of a plane sine wave in the x-direction in a fibrous material results in a pressure differential
(1) where p is the pressure, v, is the face particle velocity, and v,/u, the particle velocity inside the material, is higher than v, due to the porosity 0 < 1. The structure factor x, by which the density p of the air is multiplied, accounts for the velocity distribution in narrow “channels” through the material and for constrictions and blind passages which increase the effective mass of the air [9]. B is the specific dynamic flow resistance that at low frequencies is equal to the ratio of pressure difference per unit thickness to the face velocity of steady through-flow of air [IO]. At high frequencies, S is larger because the gradient of velocity inside the material tends to steepen at the fibers and to increase the viscous force on the fibers as the frequency is increased [4, 9 see p. 1371. In addition to these properties of a material with a rigid frame, the structure factor x may account for vibrations of fibers in fibrous materials or of cell membranes of foam materials [ 11,121. In these cases, x decreases with increasing frequency due to the decreasing motion of the frame with increasing inertia. The increase of relative motion between the air and the frame is generally too small to account for any considerable increase of the dynamic flow resistance at high frequencies [4]. It is well known that the flow resistance of fibrous materials is higher in the direction perpendicular to the plane of fibers than it is parallel to this plane. There is also experimental evidence that the structure factor has the same characteristics. Table 1 gives data reduced from measurements by Bokor [6]. TABLE 1
Structure factor and normalized specific Jlow resistance for Fiberglas Superfine BlOO with soundpropagation normal to the plane of fibers (y-direction) and parallel to the plane of fibers (z-direction). Data reduced from measurements by Bokor [6].
04
0.6 0.8 1 The non-isotropy
2.56 1.96 1.74 1.53
0.278 0.272 0.250 0.258
1.64 1.48 1.37 1.34
0.106 0.122 0.122 0.135
can be described by the following generalized form of equation (1): - gradp
= (@p/c)
13ivil+ CWd,
where the subscript i indicates the respective coordinate.
(2)
179
SOUND ATTENUATION IN DUCTS
The second equation describing the sound field in the material relates the particle velocity differential to the compression and by linearization to the pressure: -divv
= (joa/K)p.
(3)
Compression losses are small and the bulk modulus K is assumed to be real. At low frequencies, the compression in fibrous materials is isothermal and K equals the ambient static pressure P,,. Only at high frequencies does K assume the free-field value for adiabatic compression, i.e. K = 1.4.P, = pc2, where c is the free-field sound velocity. Combination of equations (2) and (3) results in the modified wave equation for a nonisotropic material [ 13,141 (4) where jka, is the bulk propagation constant in the i direction with ai = JC(pc2/K)
xi1JCU - i) W~mil,
(9
and k = w/c is the free-field wavenumber. For simplification, the structure factor xr may be combined with the factor pc2/K to form a modified structure factor, again denoted as xr. In addition, the in!luence of deviations from the adiabatic compression on the flow resistance can be neglected. Then equation (5) becomes
ai = JCh - 8 Wlpckl = JCX~- AM) dl.
(6)
Note that both the structure factor xi and the specific flow resistance r; = B,/pc are slightly frequency dependent. 3. SOUND PROPAGATION IN LAYERED MEDIA In a manner similar to that followed by Scott in his solution [4] of the wave equation for isotropic material, the transcendental equation for the propagation constants of sound in a rectangular duct lined with non-isotropic, porous, sound-absorbing material may now be derived. Consider the duct shown in Figure 1 with an open width h and material of thickness d attached to one wall. The rigid duct walls are planes of symmetry with respect to the sound pressure field and permit the construction of images which may form either a duct with symmetrical lining on two walls or a parallel-baffle silencer. Thus, the solutions to be derived apply to these configurations as well. Non- isotropic /porous lining
‘Open rectangular
Figure 1. Longitudinal
duct
section of a duct lined with non-isotropic
material.
180
U. J. KURZE
AND I. L. Vl?R
The sound pressure field pl in the free space 0 < y < h obeys the wave equation
a% a%+ k’p, = 0,
ayZ + -8.2
and the sound pressure field pII in the non-isotropic material in h < y < h + d obeys the modified wave equation
1 a2h
1 ah
7-_-I.+--
ay ay
a2
*
a22
+
k2Pll= 0,
(8)
where a, and a, are the complex factors which modify the free-field wavenumber k according to equation (6). Symmetrical solutions of the differential equationswhich satisfy theboundary conditions at the duct walls and the condition of a uniform axial propagation constant y can be written as pI -
emyr ~0s {ky4?
e-yzcos (ka,(y
pII -
+ W21>,
(9)
- h - d) J[ 1 + (y/a&)‘]}.
(10)
The ratio of the constants of proportionality in equations (9) and (10) is determined by the condition of equal pressure at the interface y = h: PI
~0s &u 42
E = cos {kh J[l
+ W)21~
~0s @,kdJCl +
+ (y/k)‘]} cos {a$O, - h - d)J[l
Wzk)21) + (r/azk)2]} ’
The final condition determining the propagation constant y follows from the requirement of equal particle velocity perpendicular to the interface y = h:
Combination of equations (11) and (12) yields
-43
+ W)21 tanWJCl + W)23~ = p
JC1 + (r/azk>21 tan{a,kdJCl + WA"lh
(13)
which is a transcendental equation for y. The set of solutions of this equation describes the propagation constants of a set of modes, of which the fundamental mode may be defined as that with the smallest real part of y and, hence, the least attenuation. Equation (13) is the generalized form of the equation derived by Scott [4] for homogeneous material. It also includes the models investigated by Morse Cl] and Cremer L-23,since for a, + cc the right-hand side of equation (13) equals the normalized wall impedance of a locally reacting wall. Tn practice, large values of a, are obtained either by a large structure factor xz in the direction of the duct axis or by a high flow resistance &, as assumed by Morse for dense fibrous materials. Of particular interest is the low-frequency range, where the material thickness is small compared to the wavelength: layIkd 4 1.
(14)
SOUND ATTENUATION
IN DUCTS
181
When the effect of the porous material on the sound propagation (i.e. on attenuation and propagation speed) in the free duct is assumed to be less than that given by the coefficient a, for propagation in the material itself, i.e.
lrl4
(15)
< 19
then the right-hand side of equation (13) can be approximated by
akd[l + (y/a&)‘] (1 + 3 (u,k42Cl+ tr/wV21+ . . .I,
(16)
which is small and thus permits application of a series .expansion procedure to the lefthand side of equation (13). Thus, for low frequencies, where the free duct width h is small compared to the wavelength of sound, the left-hand side becomes -kh[l
+ (y/k)21 (1 + 3 (kh)2[1 + (r/k>2] + . . .}.
(17)
Consideration of the first terms of the series expansion (16) and (17) yields y=jk
1 + (ad/h) 1 + (cJd/hu,2)
+ *
(18)
Consistent with experimental observations by Bokor [6, 71, only the z-parameters of the material enter into the result for low frequencies. The attenuation of sound propagating along the duct, which is proportional to the real part of y, is very small if either waves can propagate freely in the material (i.e. a, = l), or wave propagation in the material is inhibited by partitions (i.e. a, + co). Considerable attenuation can be expected only in the frequency range where the imaginary part of ui, which is B,a/pck, is comparable to the real part of the denominator in equation (18). The optimum flow resistance for lowfrequency attenuation is roughly
sz- A
PC (k/4
JCtt-Mk) + 11.
(19)
This optimum flow resistance is proportional to frequency and assumes very small values at very low frequencies. Note, however, that the maximum attenuation obtainable is proportional to the flow resistance. Thus, the low-frequency attenuation following from equation (18) is generally small, although it is much larger than the attenuation obtainable with a partitioned (i.e. locally reacting) wall. (An order of magnitude in difference between the two obtainable attenuations is presumed for the validity of the approximation leading to equation (18).) Solutions of equation (13) for middle and higher frequencies, where the sound wavelength is comparable to the duct width h and to the lining thickness d, are feasibly obtained by numerical computation only. It was found that, with a Newton-Raphson scheme [ls), a few iterations yield rapid convergence at the fundamental mode for kh < 7c and with some modifications still yield convergence up to kh = 37~ This is the upper frequency limit of practical interest since the beaming effect reduces the attenuation obtainable to very small values for higher frequencies. 4. EXAMPLES
The attenuation of the fundamental mode in a lined duct calculated by numerical evaluation of equation (13) is plotted in Figures 2, 3 and 4 for the purpose of comparing the theories of Morse/Cremer, Scott, and the new approach for non-isotropic material.
182
U. J. KURZE AND I. L. VfiR
The first example deals with a relatively narrow duct, h = 1 in, t and a lining that covers 50 % of the cross-sectional area, i.e. d = 1 in. It is also applicable on 2in thick parallel baffles which are placed 4 in on center in a rectangular duct. The normalized specific flow resistance perpendicular to the lining is ri = 1 in-‘, which is typical for most porous sound-absorbing materials. The porosity of the material is somewhat smaller than unity: D = 0.95. The structure factors, xY= xz = 1.4, were assumed to be independent of direction. Figure 2 shows the attenuation 8.7 Re {yh} dB vs. a logarithmic frequency scale for the above-described narrow duct.
Figure 2. Calculated attenuation D,,of the fundamental mode over a length I along the axis of the duct that is equal to the free duct width h. Duct width h = 1 in, lining thickness d = 1 in, gas temperature 70°F (21.1T). Material data: porosity Q = 0.95, structure factors xY = xz = 1.4, normalized specific flow resistance normal to the lining ri = SJpc = 1 in-‘. Parameter: normalized specific flow resistance parallel to the duct axis, r:.
If it is assumed that there are rigid partitions inside the material perpendicular to the surface, then the lining is locally reacting and characterized by P, + co, so that results according to the theory of Morse [l] and Cremer [2] are obtained. The attenuation is very small below f = 1400 Hz, where the lining thickness d is smaller than one-eighth of the sound wavelength 1= c/,/j$in the material. Figure 2 shows an attenuation peak at the quarter-wavelength resonance in the lining, where d = n/4. Additional peaks occur at odd multiples of the first resonance frequency, and valleys at even multiples. Another curve in Figure 2 represents isotropic material with ri = 1 in-’ and shows results according to Scott’s theory. The first attenuation peak is shifted to higher frequencies. This shift can be attributed to an increased stiffness of the lining at oblique sound incidence. Although the attenuation at low frequencies (below 14OOHz) is small compared to the maximum attenuation, the increase of attenuation relative to that of the locally reacting lining can be of considerable practical interest. For instance, 40 in long parallel baffles of t 1 in - 234cm.
SOUND ATTENUATION
183
IN DUCTS
isotropic material yield at least 20 dB attenuation at 700 Hz rather than 4 dB predicted by the theory for locally reacting linings! (In both cases, additional attenuation due to end effects, i.e. absorption of higher order modes, is neglected.) In order to obtain the same attenuation of 20 dB with a locally reacting lining, one has to increase the flow resistance IS,by a factor of 5. This would result in a considerable reduction of the attenuation peak at the quarter-wavelength resonance. A third curve in Figure 2 for r: = 2 in- ’ gives results for non-isotropic material. The flow resistance in the z-direction is larger than optimum for high attenuation below f = 1000 Hz and smaller than optimum above f = 1600 Hz. The first attenuation peak is higher and occurs at a lower frequency than for the isotropic material. Evidently, this non-isotropic material yields an attenuation intermediate between those mentioned. Note the still significant improvement achieved at low frequencies compared with the locally reacting lining. The envelope of the three curves in Figure 2 suggests that optimum attenuation performance in the frequency region below 3 kHz could be obtained by using a non-isotropic lining with a flow resistance r: which increases with increasing frequency.
I
100
2
3
4
I,,,,
S67S9,m
Frequency
2
3
4
5
6789,4000
1Hz 1
Figure 3. Calculated attenuation functions. Same data as in Figure 2, but h = 3 in, d = 3 in, r; = l-5 in- *.
Less difference in attenuation performance between locally reacting material and isotropic material can be observed in Figure 3 for a typical geometry of parallel baffles for medium-frequency mufflers. The duct width h is 3 in and the lining thickness d is 3 in. Structure factor and porosity values are as before, xY= xZ = 1.4 and D = 0.95, and the specific flow resistance perpendicular to the lining is somewhat higher: ri = l-5 in- ‘. Note, however, that the total flow resistance is considerably higher (namely, ry = rid = 4.5 compared with ry = 1 in the previous example) which causes an increase in attenuation at low frequencies according to the theory for locally reacting linings. A further increase in attenuation at low frequencies by finite values of ri is marginal. For isotropic material, the attenuation following from Scott’s theory is only marginally different from results according to Morse/Cremer as illustrated in Figure 3, except for frequencies in the vicinity of the resonance in the material which coincides with kh x T(. Non-isotropic material with a small flow resistance ri = 0.33 in yields somewhat higher attenuation forf < 300 Hz andff> 3400 Hz, and less attenuation for intermediate frequen-
184
U. J. KURZE AND I. L. VfiR
ties. The attenuation peaks shown in Figure 3 are due to maximum coupling between waves in the free duct and retarded waves in the material. From consideration of the results of Figure 3, there appears no need to solve the more complex equation of Scott for duct linings of large thickness d or of high flow resistance, since for such cases approximate results can be taken from the theory for locally reacting linings. However, materials with less flow resistance require the refined analysis.
2
3
4
56766,000
Fmquency
Figure 4. Calculated attenuation
2
5
4
5 6 76!
(Hz)
functions. Same data as in Figure 3, but rl = 0.75 in- I.
Figure 4 gives calculated data for ri = 0.75 in-l, which may be representative for the lower end of the flow resistance range of glass fiber and mineral wool materials typically used in parallel-baffle mufflers. When compared with those of Figure 3, the attenuation functions plotted in Figure 4 show more pronounced extremes for both the locally reacting lining (rz --t co) and the isotropic material (ri = 0.75 in-‘). However, the low-frequency attenuation performance of the non-isotropic material (r: = 0.33 in) is almost unchanged, which is consistent with the approximation given in equation (18), which yields y as a function of ri but independent of I;. Of the material properties involved in these calculations, least is usually known about the magnitude of the structure factor, since it can be determined only by dynamic tests, whereas the flow resistance and porosity are at least defined by static test procedures. (Note, however, that the dynamic flow resistance can be smaller than the static flow resistance for materials with elastic frame or larger for materials with considerable nonlinear behavior.) A good estimate for the structure factor of fibrous materials at intermediate frequencies is the value, as used above, of x = 1.4. However, x can be much higher at low frequencies for materials with an elastic frame, and it approaches the lower vahre x = 1 at high frequencies [12]. For the numerical evaluation of the influence of the structure factor on the attenuation performance, the same duct lining as before may be considered, with d = h = 3 in, r; = 0.75 in-’ and r: = 0.33 in- ‘, but with the structure factor xr = xz varied between 1 and 2. Calculated attenuation curves shown in Figure 5 indicate a relatively small influence of the structure factor xz at low frequencies and thus eliminate the necessity of considering a possibly strong frequency dependence of the structure factor. Attenuation peaks at higher
185
SOUND ATTFMJATION IN DUCTS
frequencies become more pronounced with increasing structure factor. However, the bandwidth of relatively high attenuation (for example, 8.7 Re {yh} dB Z 1.5 dB) is maximum for the smallest structure factor x = 1. This result is consistent with the general characteristics of resonant duct lining absorbers, which yield maximum attenuation bandwidth for maximum volume participating in dissipation, i.e. for 1/4 resonators rather than Helmholtz resonators.
I
2
I
3
4
I
I
I
I,!,
2
s6799,0M)
Frequency
3
4
5
9 7 9 91opoo
(Hz 1
Figure 5. Calculated attenuation functions for d = h = 3 in, ri = 0.75 in-‘, Parameter is the structure factor, x.
r: = 0.33 in-‘,
a = 0.95.
The theoretical data calculated from equation (13) are related to the fundamental mode of sound propagating in narrow ducts or in parallel-baffle silencers. The attenuation measured across finite-length silencers is higher partly due to the absorption of sound energy in higher order modes. This absorption takes place predominantly at both ends of the silencer. Experimental results obtained with various parallel-baffle configurations in a square duct indicated that these effects can be explained by the change of the free duct area at both ends of the silencer. The attenuation was determined from sound intensity measurements [14] at both sides of the test section with the same cross sectional area according to Atten. = 10 log !iliI $ 2
dB,
(20)
2
where lpll and ]p21 are the amplitudes of the sound pressure at locations in the entrance and at the exit duct respectively, and A+, and A& are phase differences evaluated by shifting the microphone over a short distance AZ along the duct axis at these respective locations. Generally, the measured attenuation data exhibited a frequency dependence similar to that predicted theoretically from the structure factor, porosity and the non-isotropic flow resistance of the porous material employed. However, the measured attenuation data were consistently higher than those predicted, even after accounting for the end effects and the energy loss through the duct wall (the latter have been determined by measurements in an empty duct). This remaining excess attenuation can be explained by compressional
186
U. J. KURZE AND I. L. Vl%R
losses in the material which were not included in the theory. It should be noted that this excess attenuation, AL, is typically a few decibels in long narrow ducts. If expressed in terms of the attenuation per duct width, AD,, it would be less than O-1dB.
t
Frequency
(Hz)
Figure 6. Attenuation in a duct lined with Fiberglas SupertIne BlOOof 1 in thickness. Parameter h is the half airspace between two linings. @-@, Measured by Bokor [6]; O---O, calculated from data in Table 1.
Higher attenuation values than predicted by equation (13) have been measured by Bokor [6] in ducts that are wide compared to the lining thickness. Figure 6 shows Bokor’s data compared with results calculated from the material data in Table 1. While good agreement is found for the narrow duct with h = 1 cm, the experimental results are considerably higher in wider ducts with h = 5 and 8 cm. Bokor explains this discrepancy by the curvature of the wave-front near the sound source which results in higher order modes. 5. CONCLUSIONS The attenuation performance of ducts with porous linings or parallel-baffle silencers cannot generally be predicted by the theory for locally reacting duct linings. For fibrous linings with low specific flow resistance, in particular, one has to consider the non-isotropic properties of the material. As the experimental data obtained by Bokor have already indicated, the low-frequency attenuation is primarily determined by the material properties for sound propagation parallel to the axis of the duct. There is an optimum flow resistance of the material in that direction which yields maximum attenuation at low frequencies (note, however, that this attenuation is low compared with the peak attenuation obtained at higher frequencies). This optimum flow resistance increases in proportion with frequency. At higher frequencies, where the lining thickness becomes comparable to the sound wavelength, attenuation peaks occur, which are pronounced if, due to a large structure factor, the propagation speed of sound in the material is considerably lower than in free air. However, maximum bandwidth of high attenuation is obtained by materials with a structure factor near unity.
SOUND ATTENUATIONIN DUCTS
187
The high-frequency attenuation performance of porous materials is slightly better than that of locally reacting linings, but it is also limited by the beaming effect of sound in the center of wide ducts. From comparisons with experimental data, the theoretical results appear to be conservative. Higher attenuation occurs whenever inhomogeneities of the lined duct give rise to higher order modes. ACKNOWLEDGEMENT The aut.hors wish to thank Dr Uno Ingard, Massachusetts Institute of Technology, for helpful suggestions during the preparation of the manuscript. REFERENCES 1. P. M. MORSE1939 Journal of the Acoustical Society of America 11,205-210. The transmission of sound inside pipes. 2. L. CREMER 1940 Akustische Zeitschrift 5, 57-76. Nachhallzeit und Dlmpfungsmass bei streifendem Einfall. 3. W. WILLMS1941 Akustische Zeitschrif 6, M-165. Die Schalld%mpfung in Absorptionsrohren. 4. R. A. SCOTT 1946 Proceedings of the Physical Society 58,358-368. The propagation of sound between walls of porous material. 5. E. A. LESKOV,G. L. OSIPOVand E. J. YUDIN 1970 Applied Acoustics 3,47-56. Experimental investigations of splitter duct silencers. 6. A. BOKOR 1969 Journal of Sound and Vibration 10, 390403. Attenuation of sound in lined ducts. 7. A. BOKOR 1971 Journal of Sound and Vibration 14, 367-373. A comparison of some acoustic duct lining material, according to Scott’s theory. 8. J. WALSDORFF1971 Seventh International Congress on Acoustics, Budapest, paper 25 A 6. Absorptionsschalld%npfer ohne Kassettierung. 9. L. CREMER1950 Wellentheoretische Raumakustik. Leipzig: S. Hirzel Verlag. See p. 145. 10. V. N. BARANOVAand K. A. VELIZHNINA1957 Soviet Physics Acoustics 3, 107-111. Acoustic parameters of certain sound absorbent materials. (In this paper an impedance tube method for the experimental determination of the bulk acoustical properties of porous materials is described.) 11. L. L. BERAN~K 1947 Journal of the Acoustical Society of America 19, 556-568. Acoustical properties of homogeneous isotropic rigid tiles and flexible blankets. 12. R. A. SCOTT 1946 Proceedings of the Physical Society 58, 165-183. The absorption of sound in a homogeneous porous medium. 13. P. M. MORSEand U. K. INGARD 1961 In Handbuch der Physik XI/l (Fltigge, ed.) p. 17. Berlin: Springer-Verlag. Linear Acoustic Theory. 14. U. KURZE 1969 Acustica 21, 74-85. Schallausbreitung im Kanal mit periodischer Wandstruktur. 15. F. B. HILDEBRAND1956 Introduction to Numerical Analysis. New York: McGraw-Hill Book Company. See p. 451.