acoustic impedance of perforations in contact with ﬁbrous ...engine.osu.edu/asjp/jp/j68.pdf ·...

Acoustic impedance of perforations in contactwith fibrous material

Iljae Lee and Ahmet Selameta�

Department of Mechanical Engineering,and The Center for Automotive Research, The Ohio State University, 930 Kinnear Road,Columbus, Ohio 43212

Norman T. HuffOwens Corning, 4600 Humboldt Drive, Novi, Michigan 48377

�Received 27 September 2005; revised 22 February 2006; accepted 24 February 2006�

Acoustic impedance of perforations in contact with fibrous material, as well as air alone, isexperimentally determined in the absence of mean flow. Different porosities �2.1, 8.4, 13.6, and25.2 %� and hole diameters �0.249 and 0.498 cm� are applied for perforations, along with two fiberfilling densities �100 and 200 kg/m3� to illustrate the effect of variations in such parameters on theperforation impedance. A modified impedance tube setup is developed for the measurement of theacoustic impedance of perforations in contact with fibrous material, and the air alone. The complexwave-number and characteristic impedance of the fibrous material are also experimentally obtainedin this study to utilize such properties for the calculation of acoustic impedance of perforations incontact with the material. The experimental results show that both resistance and reactance of theperforations in contact with air decrease as the porosity increases. It is also shown that the fibrousmaterial significantly increases both the resistance and reactance. For a given filling density, bothresistance and reactance decrease as the porosity increases, except for the specific samples withhighest porosity �25.2%�. © 2006 Acoustical Society of America. �DOI: 10.1121/1.2188354�

PACS number�s�: 43.50.Gf, 43.58.Bh �DKW� Pages: 2785–2797

I. INTRODUCTION

Perforated ducts are often used in reactive silencers tocontrol the acoustic performance, guide the flow and reducethe flow losses, and influence the sound quality through sup-pression of flow noise from abrupt cross-sectional areachanges. In dissipative silencers, perforated ducts are alsoutilized to protect the absorbing material, while modifyingthe acoustic characteristics by adding mainly reactance to theimpedance. The predictions for both types of silencers relyon the acoustic impedance of perforations. This impedance isdependent on the perforation geometry �porosity, shape andsize of holes, wall thickness, and distance between holes�,the flow field �grazing and through flow�, sound pressurelevel, and the medium in contact with the perforation. Thecomplexity of the perforation impedance confines the ana-lytical approaches only to simple perforations, and experi-mental methods are often used to obtain the impedance.While the acoustic characteristics of perforations in contactwith air alone are relatively well established primarily byexperimental work �refer, for example, to Sullivan andCrocker, 1978�, only limited studies on perforations facingabsorbing material are available.

The acoustic impedance of perforations in contact withair alone has been extensively investigated, for example, byIngard �1953�, Sullivan and Crocker �1978�, Melling �1973�,Dickey et al. �2000, 2001�, Dean �1974�, Rao and Munjal�1986�, and Cummings �1986�. An empirical formulation of

a�
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J. Acoust. Soc. Am. 119 �5�, May 2006 0001-4966/2006/119�5

the perforation impedance by Sullivan and Crocker �1978�has been widely used in the linear regime in the absence ofmean flow. Melling �1973� and Dickey et al. �2000� consid-ered the impact of high amplitudes on the perforation imped-ance. In the nonlinear regime, the resistance and reactance ofthe perforation impedance are shown to increase and de-crease, respectively, with the acoustic velocity. The effect ofgrazing mean flow on the impedance was studied by Dean�1974�, Rao and Munjal �1986�, Cummings �1986�, andDickey et al. �2001�. Grazing flow is generally observed todecrease reactance and increase resistance of the perforationimpedance.

Studies have also been conducted on the acoustic imped-ance of perforations facing fibrous material in the absence ofmean flow, which may be categorized as analytical, experi-mental, and semiempirical. The analytical work has consid-ered the effect of fibrous material on the perforation imped-ance. For example, Bolt �1947� considered the massreactance of perforations facing the absorbing material. Heconcluded that the addition of perforations to the absorbingmaterial generally increased the absorption coefficient at lowfrequencies, while decreasing it at high frequencies. Later,Ingard and Bolt �1951� illustrated the effectiveness of perfo-rations at low frequencies through an increase of the resis-tance. They concluded that a combination of absorbing ma-terial and perforations formed a Helmholtz resonator. Ingard�1954� emphasized the increase of resistance by the additionof perforations and the effect of an air gap between the per-forations and absorbing material. His analysis showed that
the air gap between the perforations and facing material nar-
© 2006 Acoustical Society of America 2785�/2785/13/$22.50

rowed the effective frequency range of absorption coeffi-cients. The experimental work on the acoustic characteristicsof perforations combined with an absorbent is confined todetermining the surface acoustic impedance or the absorptioncoefficient, rather than the perforation impedance. Havingconsidered the perforations facing absorbing material as aHelmholtz resonator, one of the earlier works by Callawayand Ramer �1952� showed that the air gap between the per-forations and absorbing material increased the absorption co-efficient for a very low duct porosity �1.23%�. Later, Davern�1977� experimentally illustrated the effect of porosity andwall thickness of the perforations, absorbing material den-sity, air space, and the impervious layer between perforationsand absorbing material on the absorption coefficient. Thesemiempirical studies first measured the perforation imped-ance in the absence of fibrous material, then empirically in-corporated the acoustic properties of the material to accountfor their effect on the perforation impedance. Kirby andCummings �1998� presented a semiempirical impedance for-mulation for the perforation backed by absorbing material inthe presence of grazing mean flow. They concluded that theabsorbing material increased the perforation impedance andthe results were strongly dependent on the density of absorb-ing material immediately adjacent to perforations. They alsoargued that the interaction among holes could be neglectedand thus the formulation for single hole could be used for amultihole perforation. Selamet et al. �2001� adapted the con-cept of Kirby and Cummings by modifying the formulationof Sullivan and Crocker �1978� for the perforations facingair-fibrous material. They illustrated the significant impact ofperforation impedance on the transmission loss of dissipativesilencers particularly with low porosity. The foregoing stud-ies with the absorbent, however, are limited to developingexpressions for the acoustic impedance of perforations byintuitive modifications of the available no-fiber relationships.Hence, the predictive studies on the acoustic performance ofdissipative silencers have thus far used the perforation im-pedance with air, or its alternatives with such heuristic modi-fications.

The present study experimentally determines the acous-tic impedance of perforations facing air-air �air on bothsides� and air-fibrous material �air on one side and fibrousmaterial on the other side� directly in the absence of meanflow. First, the measurement methodology developed in thisstudy is applied to the perforations facing air-air, and theresults are compared with the available literature. Then, theimpedance of perforation in contact with air-fibrous materialof � f =100 and 200 kg/m3 is also experimentally acquired.Since accurate acoustic properties of the fibrous material arenecessary for the measurement of acoustic impedance of per-forations in contact with the material, the characteristic im-pedance and wave number of the material are also experi-mentally obtained in this work.

Following this Introduction, Sec. II presents the defini-tions of perforation impedance and existing formulations andSec. III describes the experimental setups and data reduction.Section IV gives the results of acoustic properties of fibrousmaterial and acoustic impedance of perforations followed by
the concluding remarks in Sec. V.
2786 J. Acoust. Soc. Am., Vol. 119, No. 5, May 2006

II. DEFINITIONS OF PERFORATION IMPEDANCE ANDEXISTING FORMULATIONS

A. Definitions of perforation impedance

The acoustic impedance is the ratio of the pressure dif-ference across perforations to the particle velocity �Fig. 1�.Two available definitions of the impedance differ by the waythe particle velocity is specified. One uses the particle veloc-ity uh at the hole in Fig. 1, and the other the velocity u1 or u2

in the duct. These definitions and the relationship in betweenare introduced next.

The specific acoustic impedance of a hole is defined as

Zh =p1 − p2

uh= Rs + i�Ls = Rs + i��0�eff

= Rs + ik0�0c0�eff, �1�

where p1 and p2 are the acoustic pressures before and afterthe perforated plate, respectively, as shown in Fig. 1; �0 theair density; c0 the speed of sound in air; k0 the wave numberin air; Rs the specific resistance; Ls the specific inertance;and

�eff = tw + �dh �2�

the effective length of the hole, where tw is the wall thick-ness, dh the hole diameter, and � the end correction coeffi-cient. The coefficient � is dependent on the hole geometryand the characteristics of medium in contact with the hole.The nondimensional acoustic impedance of a hole is thengiven by

�h =p1 − p2

Z0uh= R + ik0�tw + �dh� , �3�

where Z0=�0c0 is the characteristic impedance of air and R isthe dimensionless resistance.

The nondimensional acoustic impedance of a perforatedplate can be defined as

�p =p1 − p2

Z0u1, �4�

where the particle velocity u1 in the duct is related to uh

FIG. 1. A perforated plate in a duct.

through conservation of mass by

Lee et al.: Perforation impedance with fibrous material

˜

u1 =nhAh

A1uh = �uh, �5�

nh being the number of holes, Ah the area of a hole, and � theporosity. Equations �3�–�5� yield the relationships betweenthe impedance of a hole and a plate as

�p =p1 − p2

Z0uh�=

�h

�=

R + ik0�tw + �dh��

. �6�

While the impedance of a hole is representative of the acous-tic characteristics of an isolated single perforation, it is theimpedance of a plate typically used in the analyses, for ex-ample, in transmission loss predictions. The impedance of aplate �p is usually measured first, the impedance of a hole �h

is then calculated using Eq. �6�.

B. Existing formulations

The empirical relationship of Sullivan and Crocker�1978� given by

�p =0.006 + ik0�tw + 0.75dh�

��7�

has been widely used for the perforation impedance withair-air contact. This expression was obtained using a perfo-ration with �=4.2%. In Eq. �7�, the coefficient 0.75 isassociated with the end correction of the perforation.While the theoretical maximum value is 8 /3��0.85� fora single hole with an infinite flange, the coefficient maybe smaller for multiholes due to the interaction amongholes. Melling �1973� graphically illustrated such interac-tion between two holes.

Kirby and Cummings �1998� suggested a semiempirical

acoustic impedance �̃p for perforated plates facing air-fibrousmaterial by modifying �p that faces air-air as

�̃p = �p +

ik00.425�− 1 +Z̃

Z0

k̃

k0�dh

�. �8�

In Eq. �8�, half of the end correction for a perforation facingair-air �the first term inside the parentheses� is combinedwith a second term to account for the effect of the fibrousmaterial. However, the use of Eq. �8� for dissipative silencersleads to erroneous transmission loss predictions possiblydue, according to Kirby �2001�, to an overestimated perfora-tion impedance. Such overestimation may be attributed to theignored interaction among holes. Selamet et al. �2001�adapted the concept by applying Eq. �7� to the perforationsfacing air-fibrous material as follows:

�̃p =

0.006 + ik0�tw +0.75

2�1 +

Z̃

Z0

k̃

k0�dh

�. �9�

In Eq. �9�, the effect of absorbent is considered with the

complex characteristic impedance Z̃ and wave-number k̃ ofthe material, thus both real and imaginary parts of the im-
pedance can be modified in Eq. �9�. Since it is obtained by
J. Acoust. Soc. Am., Vol. 119, No. 5, May 2006

modifying Eq. �7� based on �=4.2%, Eq. �9� is expected tobe valid near this particular porosity.

III. EXPERIMENTAL SETUP AND DATA REDUCTION

An impedance tube setup shown in Fig. 2 �Refer, forexample, to Dickey et al., 2000 or Selamet et al., 1994, fordetails� is modified for the measurements of characteristicimpedance and wave number of the absorbing material, andthe acoustic impedance of perforations in the absence ofmean flow, particularly in contact with the material. The re-sulting setups and the data reduction for such measurementsare described next.

A. Measurement of characteristic impedance andwave number of fibrous material

A transfer function method with two different termina-tion conditions �Fig. 3� in conjunction with the impedancetube of Fig. 2 is developed to acquire experimentally thecomplex characteristic impedance and wave number of thefibrous material in the absence of mean flow. A sample ofabsorbing material is mounted inside the main duct boundedby wire screens with a porosity higher than 75% defining thelocations a and b in Fig. 3. Transfer functions among themicrophones and two measurements with distinctly differentterminations are required in the present method to determinethe complex characteristic impedance and wave number ofthe absorbing material in the frequency domain. Anechoicand fully reflective boundaries are applied as two differenttermination conditions.

The transfer matrix, which expresses the relationshipsbetween acoustic pressure and velocities at the surfaces aand b of the sample, is given for the first termination in terms

of the complex characteristic impedance Z̃, the wave-number

k, and the length of the sample � f as

FIG. 2. Schematic of the impedance tube setup in the absence of mean flow.

FIG. 3. Schematic of the experimental setup for obtaining the acoustic prop-
erties of absorbing material in the absence of mean flow.
Lee et al.: Perforation impedance with fibrous material 2787

�pa

ua = cos�k̃� f� iZ̃ sin�k̃� f�

isin�k̃� f�

Z̃cos�k̃� f� ��pb

ub , �10�

and for the second termination �designated by prime�

�pa�

ua� = cos�k̃� f� iZ̃ sin�k̃� f�

isin�k̃� f�

Z̃cos�k̃� f� ��pb�

ub� . �11�

Combining Eqs. �10� and �11� and rearranging yields

cos�k̃� f�

iZ̃ sin�k̃� f�

isin�k̃� f�

Z̃

cos�k̃� f�� =

pb ub 0 0

0 0 pb ub

pb� ub� 0 0

0 0 pb� ub��

−1

pa

ua

pa�

ua��

= T1

T2

T3

T4

� , �12�

where T1=T4 is noted. In Eq. �12�, acoustic pressures�pa , pb , pa� , pb�� and velocities �ua ,ub ,ua� ,ub�� at the samplesurfaces a and b can be calculated from the pressuremeasurements at four microphones �pm1 , pm2 , pm3 , pm4 ,pm1� , pm2� , pm3� , pm4� � in Fig. 3. The transfer matrix between thesample surface a and microphone 2 �for a straight pipe ofconstant cross-sectional area �Munjal, 1987�� is given by, forthe first termination

�pa

ua = cos�k0�2� iZ0sin�k0�2�

i1

Z0sin�k0�2� cos�k0�2� �

−1

�pm2

um2 . �13�

In Eq. �13�, the acoustic velocity um2 at microphone 2 can beexpressed in terms of pressure measurements, pm1 and pm2 atmicrophones 1 and 2. In light of

�pm2

um2 = cos�k0�1� − iZ0sin�k0�1�

− i1

Z0sin�k0�1� cos�k0�1� ��pm1

um1 ,

�14�

um2 is given by

um2 =pm1 − pm2cos�k0�1�

. �15�
iZ0sin�k0�1�

Substitution of Eq. �15� into Eq. �13� yields

�pa

ua = cos�k0�2� iZ0sin�k0�2�

i1

Z0sin�k0�2� cos�k0�2� �

−1

� pm2

pm1 − pm2cos�k0�1�iZ0sin�k0�1�

� , �16�

which expresses the acoustic pressure and velocity at thesample surface a in terms of pressures alone measured atmicrophones 1 and 2. Similarly, the acoustic pressure andvelocity at the sample surface a for the second termination isgiven by

�pa�

ua� = cos�k0�2� iZ0sin�k0�2�

i1

Z0sin�k�2� cos�k0�2� �

−1

� pm2�

pm1� − pm2� cos�k0�1�iZ0sin�k0�1�

� . �17�

The acoustic pressure and velocity at sample surface bcan be similarly expressed in terms of pressures measured atmicrophones 3 and 4 for the first termination as

�pb

ub = cos�k0�3� iZ0sin�k0�3�

i1

Z0sin�k0�3� cos�k0�3� �

� pm3

pm3cos�k0�4� − pm4

iZ0sin�k0�4�� , �18�

and for the second termination

�pb�

ub� = cos�k0�3� iZ0sin�k0�3�

i1

Z0sin�k0�3� cos�k0�3� �

� pm3�

pm3� cos�k0�4� − pm4�

iZ0sin�k0�4�� . �19�

The complex characteristic impedance and wave numberof the sample can then be obtained from Eq. �12� and Eqs.

�16�–�19�. The complex characteristic impedance Z̃ is given,in terms of T2 and T3 of Eq. �12�, by

Z̃ =�T2

T3�20�

and the complex wave number, in terms of T1 or T4 of Eq.�12�, by

k̃ =1

� fcos−1�T1� =

1

� fcos−1�T4� . �21�

Measured T1 and T4 of Eq. �12� are expected to be identical.
However, that may not be exactly true in experiments due,

for example, to inhomogeneity in the filling and imperfectsample surfaces. Thus, the complex wave number is obtainedin the present study using an average of T1 and T4 as

k̃ =1

� fcos−1�T1 + T4

2� . �22�

B. Measurement of acoustic impedance ofperforations

Once the acoustic properties of the absorbing materialare obtained, the acoustic impedance of perforations in con-tact with the absorbent on one side can be also determinedexperimentally. Unlike the acoustic properties of the absorb-ing material, only a single experiment with one terminationis adequate for the measurement of perforation impedance.An anechoic termination is applied to reduce the interfer-ences at microphones by the reflected wave from the termi-nation. In view of Fig. 4, replacing u1 of Eq. �4� by uc givesthe acoustic impedance of a perforated plate as

�̃p =pc − pd

Z0uc, �23�

where pc and pd are the acoustic pressures at the perforationsurfaces contacting air and absorbing material, respectively,and uc is the acoustic velocity in the duct at the perforationsurface. The quantities pc and uc are determined in terms ofpressures measured at microphones 1 and 2 from Eq. �16� as

�pc

uc = cos�k0�2� iZ0sin�k0�2�

i1

Z0sin�k0�2� cos�k0�2� �

−1

� pm2

pm1 − pm2cos�k0�1�iZ0sin�k0�1�

� . �24�

The acoustic pressure and velocity at the perforation surfacein contact with the fibrous material are expressed in terms ofa series of transfer matrices and the pressures measured atmicrophones 3 and 4 as

�pd

ud = �Tab��Tb3� pm3

pm3cos�k0�4� − pm4 � , �25�

FIG. 4. Schematic of the experimental setup for obtaining the acoustic im-pedance of perforations in the absence of mean flow.

iZ0sin�k0�4�


where

�Tab� = cos�k̃� f� iZ̃ sin�k̃� f�

isin�k̃� f�

Z̃cos�k̃� f� � �26�

is the transfer matrix between surfaces a and b, and

�Tb3� = cos�k0�3� iZ0sin�k0�3�

i1

Z0sin�k0�3� cos�k0�3� � �27�

is the transfer matrix between surface b and microphone 3.Finally, the perforation impedance can be determined fromEqs. �23�–�27� and sound pressure measurements at four mi-crophones. In Eq. �23�, experimentally determined acousticvelocities uc and ud at the perforation surface may not beexactly identical in experiments. Thus, an average of the twovelocities is used for the calculation of perforation imped-ance as

�̃p =pc − pd

Z0�uc + ud�

2

. �28�

The same setup and the data reduction technique are alsoused for the measurements of acoustic impedance of perfo-rations without the fibrous material.

IV. ACOUSTIC PROPERTIES OF FIBROUS MATERIAL

A modified impedance tube setup described in the pre-ceding section is used to experimentally obtain the complexcharacteristic impedance and wave number of fibrous mate-rial. The present study focuses on two filling densities �� f

=100 and 200 kg/m3 representing approximate bounds inpractical applications�, and experimentally determines theiracoustic properties. These properties are then directly appliedto the measurements of acoustic impedance of perforation incontact with the fibrous material. The precise measurementsof the distances between the sample surfaces and micro-phones or among microphones are critical in the experi-ments. The accuracy within 0.1 mm for the materials withwell-defined surfaces is recommended by ASTM 1050-98�1998�. However, such a precise physical measurement maynot be possible due, for example, to the difficulty in findingthe centers of the microphones with their diameter of6.35 mm �1/4 in. �. Thus, the distances �1 through �4 and thesample length � f in Figs. 3 and 4 are acoustically estimatedusing quarter wave resonances �Katz, 2000�. The estimateddistances by this method are as follows: �1=�4

=3.55 cm,�2=12.81 cm,�3=12.71 cm, and � f =10.16 cm.When � f is too short, the impact of inhomogeneous fillingtends to magnify. On the other hand, too long of a sample isdetrimental to transferring sufficient sound energy to thedownstream microphones due to wave dissipation throughthe material, particularly at high frequencies. Therefore, � f

was chosen to balance these competing factors. An accurate


estimation of speed of sound is also important for the mea-surements, which is calculated here by

c0 = 20.047�273.15 + TC �m/s� , �29�

as suggested by ASTM 1050-98 �1998�. All experimentswere performed at the room temperature, TC=19±1 °C.

A. Fiber samples

Advantex® continuous-strand glass fiber developed byOwens Corning �Novi, MI� is used in the experiments. Thetexturization process separates filament roving strands of fi-ber glass into individual filaments by turbulent air flow�SilentexTM process�. The average diameter of the individualfilaments in the roving strand is 24 �m. In addition to theacoustic properties, high thermal and corrosion resistance ofthe Advantex® glass fiber may also be desirable particularlyfor the exhaust systems. The physical and chemical proper-ties of various absorptive materials and their relative durabil-ity in automotive silencer operations have been described byBeranek �1988� and Huff �2001�.

Absorbent samples, 10 cm long and 4.9 cm in diameterand with two different filling densities � f =100 and200 kg/m3, are placed in the test tube �Fig. 2�. Densitiessignificantly lower than � f =100 kg/m3 may cause the move-ment of fibers inside the silencers and higher than � f

=200 kg/m3 may be undesirable in terms of cost, weight,and in some cases acoustic attenuation.

Random filling is applied to achieve overall isotropicand homogeneous conditions. However, partially or locallynonisotropic or inhomogeneous regions may exist due, forexample, to the hand filling of the absorbent into the testtube. Thus, experiments with five different samples for eachfilling density are performed to examine the effect of varia-tions in filling conditions. An average of five individual ex-periments is then used as the final result. The present studyuses a well-texturized condition, which has mostly individualfilaments. The front surfaces of the absorbent sample alsoneed to be well defined in the experiments. The surfaces ofthe sample perpendicular to the direction of wave propaga-tion are identified by two screens. The screens have a highporosity not to influence the surface properties of the absor-bent and sufficiently high tension to hold the absorbent flat atthe surface.

B. Characteristic impedance and wave number

The normalized empirical complex characteristic imped-

ance Z̃ and the wave-number k̃ of Delany and Bazley �1970�for fibrous materials in mks units

Z̃

Z0= �1 + 0.0511�f/Rf�−0.75� + i�− 0.0768�f/Rf�−0.73�

�30�


k̃

k0= �1 + 0.0858�f/Rf�−0.70� + i�− 0.1749�f/Rf�−0.59� �31�

have been widely used due to its simplicity and accuracy.Here f is the frequency in hertz and Rf the flow resistivitywhich is primarily dependent on the fiber diameter, layerorientation, and filling density. Although the Delany and Ba-zley formulation is well representative of the acoustic behav-ior of fibrous material in general, the present study experi-mentally determines the complex characteristic impedanceand wave number of specific fibrous material since the pro-cess of data reduction for perforation impedance requires ac-curate �rather than simply somewhat representative� acousticproperties of the material in contact with the perforations.The present study has retained the forms of Eqs. �30� and�31�, since they provide suitable fits. The curve-fitted com-plex characteristic impedance and wave number based on theaveraged values of five experiments are expressed by

Z̃

Z0= �1 + a1f−b1� − ia2f−b2, �32�

k̃

k0= �1 + a3f−b3� − ia4f−b4, �33�

with the coefficients presented in Table I for both fillingdensities of � f =100 and 200 kg/m3. The function “fit” ofMATLAB® is applied for all curve fits in the frequency rangeof 100–3200 Hz. At lower frequencies, two distinct termi-nation conditions may not be achieved, which are neces-sary for accurate results, due to the imperfect anechoicboundary. The accuracy of the measurement at low fre-quency is also limited by the microphone spacing �ASTM1050-98, 1998�. Figure 5 shows, for example, the charac-teristic impedance and wave number for � f =100 kg/m3.The curve fits represent most of the data points well, par-ticularly for the wave number.

V. ACOUSTIC IMPEDANCE OF PERFORATIONS

The acoustic impedance of perforations in contact withair-fibrous material is different than the one facing air-air dueto the influence of absorbing material. The fiber filling den-sity and texturization condition in contact with the perfora-tion are particularly important. Ingard �1954� claims that thefiber properties only within a distance of one hole diameterare significant. Thus, the air gap between the perforation andthe fibrous material may also alter the perforation imped-ance. The present study considers only the perforations di-

TABLE I. Fit coefficients of the measured characteristic impedance andwave number.

Fillingdensity�kg/m3� a1 b1 a2 b2 a3 b3 a4 b4

100 33.20 0.7523 28.32 0.6512 39.20 0.6841 38.39 0.6285200 25.69 0.5523 71.97 0.7072 56.03 0.6304 62.05 0.5980

rectly in contact with the fibrous material without the air gap.


Acoustic impedance of perforations facing air only andair-fibrous material is measured in the present study usingthe impedance tube presented in Fig. 4 and methodologydescribed in Sec. II. The averaged results from five experi-ments with different fibrous material samples are utilized inthe data reduction.

A. Perforation plate samples

Table II presents 11 samples of circular plates consid-ered in this study with different porosity �, wall thickness tw,and hole diameter dh to investigate the effect of such param-eters on the perforation impedance. Figures 6 and 7 provide,respectively, the detailed dimensions and the photos of the

FIG. 5. Characteristic impedance andwave number of the fibrous materialwith � f =100 kg/m3; �a� real part and�b� imaginary part.

circular perforated plates used in the present study. Perfora-


TABLE II. Perforation samples.

SampleNo.

Hole diameter,dh �cm�

Wall thickness,tw �cm�

Porosity,� �%�

A1 0.249 0.08 2.1A2 8.4A3 13.6A4 25.2B1 0.16 2.1B2 8.4B3 13.6B4 25.2C2 0.498 0.08 8.4C3 13.6C4 25.2


tions facing air-air with higher than 10% porosity have notbeen, in general, considered in the literature since the effectof varying perforations at high porosities on the transmissionloss of perforated silencers is insignificant. However, with

FIG. 6. Dimensions of perforated

the fibrous material, the transmission loss may be affected by


the perforation impedance even at higher porosities. Thus,four porosities ��=2.1,8.4,13.6, and 25.2 %� are used in thepresent study, defined by the area ratio of all holes to theplate. The samples are categorized into three groups �Table

samples �unit: centimeters �inch��.
plate
II�; �1� group A is the base with dh=0.249 cm and tw


=0.08 cm; �2� group B has the same hole diameter as groupA and doubles the wall thickness to tw=0.16 cm; and �3�group C has the same wall thickness as group A, while dou-bling the hole diameter to dh=0.498 cm. The numbers �1–4�after the group identification by �A–C� in Table II designatedifferent duct porosities. The holes on the plates are distrib-uted to achieve similar distances between the holes in thevertical and horizontal directions. For impedance measure-ments, the circular sample plates are installed in the imped-ance tube of 4.9 cm inner diameter, as shown in Fig. 4. Thedistance from the tube wall to the nearest hole is chosen tobe longer than the distance between holes in order to mini-mize the effect of the interaction between holes and the wall.

B. Acoustic impedance of perforations in contactwith air alone

The dimensionless resistance R and the end correctioncoefficient � in Eq. �6� of the perforation samples with air-aircontact are experimentally determined using the impedancetube setup depicted in Fig. 4. While this setup is originally

FIG. 7. Pictures of perforated plate samples.

designed to measure the impedance of perforation facing


fibrous material, it can also be utilized for the perforation incontact with air only. For each perforation plate, five experi-ments are performed, and their averaged values are then usedto obtain R and �. The end correction coefficients are calcu-lated from the curve-fitted reactance in the frequency rangeof 100–3200 Hz for most samples. The only exception is A1and B1 with �=2.1% which are curve fitted in the range of100–2600 Hz due to the unstable signals at frequencieshigher than 2600 Hz. The mean values of resistance in thesame frequency range are also obtained.

Figure 8 presents the measured R and � �also given inTable III� for the perforation samples of Table II, facing air-air. Figure 8�a� shows that the resistance R decreases as theporosity increases, particularly from �=2.1 to 8.4 %. GroupB �tw=0.16 cm� has higher resistance than group A �tw

=0.08 cm�, due to the doubled wall thickness. The resistance0.006 obtained by Sullivan and Crocker �1978� for �=4.2% falls in between the range of �=2.1 and 8.4 % in Fig.8�a�. Figure 8�b� illustrates the effect of porosity, wall thick-ness, and hole diameter on �, thus reactance of the perfora-tion impedance. As the porosity increases, � decreases be-cause of the shorter distance between the holes, hence astronger interaction among the holes. Groups A and B havesimilar values of � indicating that the end correction coeffi-cient is essentially independent of the wall thickness. Group

FIG. 8. Comparison of measurements of the acoustic impedance of a hole incontact with air-air; �a� resistance and �b� end correction coefficient.

C has higher values of � than those of group A presumably


due to the larger hole diameter resulting in a longer distanceamong the holes, thereby reducing the interaction. The endcorrections obtained in the present study are lower than 0.75of Sullivan and Crocker �1978� for all porosities, possiblydue to different distance between holes.

C. Acoustic impedance of perforations in contactwith fibrous material

The experimental results for R and � of the perforationsfacing an absorbent with � f =100 and 200 kg/m3 as well asair alone are shown in Figs. 9–11, for groups A, B, and C,respectively, and Table III. Figures 9–11 show that fibrousmaterial substantially increases both resistance and end cor-rection coefficients for all three groups. The resistance R of� f =200 kg/m3 are higher than those of lower filling density� f =100 kg/m3 for all samples. The effect of different fillingdensity on � diminishes for the samples A4 and B4. Theaverage increases of R over all 11 samples �recall Table II�due to the fibrous material are reflected by factors of nearly 5and 10 relative to the baseline with no fiber for � f =100 and200 kg/m3, respectively, whereas the average � is approxi-mately only doubled for both fiber densities. Thus, the im-pact of fibrous material on the resistance is more significantthan on the end correction coefficient. The imaginary part ofthe perforation impedance depends also on the frequency, thehole diameter, and the wall thickness �recall Eq. �3��.

For groups A �Fig. 9� and B �Fig. 10�, as the porosityincreases, � decreases for both filling densities except forone sample �A4 with �=25.2% and � f =100 kg/m3�. For thesame groups, R decreases with porosity up to �=13.6%, andthen increases to that of �=25.2%. Group C �Fig. 11� withdh=0.498 cm exhibits some differences from groups A and Bwith dh=0.249 cm; R of �=25.2% is lower than that of �=13.6%, and the change of � from �=13.6 to 25.2 % ismore dramatic than from �=8.4 to 13.6 %.

Table IV presents relative standard deviations �R /R and� /�� for R and � calculated from five repeat experimentswith air alone and with fibrous material. R /R is usually

TABLE III. Measured R and � of the perforations.

SampleNo.

R

Air alone� f =100�kg/m3�

��

A1 0.007624 0.03054 0A2 0.005101 0.02996 0A3 0.004437 0.02298 0A4 0.004500 0.02716 0B1 0.008429 0.03313 0B2 0.006074 0.02793 0B3 0.005744 0.02444 0B4 0.005539 0.02657 0C2 0.005318 0.04728 0C3 0.005013 0.04598 0C4 0.004395 0.03973 0

higher than � /�, and both deviations, in general, increase


with the porosity. R /R and � /� for � f =200 kg/m3 are, inmost cases, greater than those for � f =100 kg/m3, possiblydue to relatively nonhomogeneous filling conditions, includ-ing texturization of the fiber, for the higher density. The tex-turization of the absorbent may substantially affect theacoustic impedance of perforations in contact with this ma-terial �Lee, 2005�. The present study has provided the imped-ance of perforation in contact with well texturized fibrousmaterial.

�

03�

Airalone

� f =100�kg/m3�

� f =200�kg/m3�

5 0.5350 0.6989 0.77695 0.4409 0.6471 0.74120 0.2506 0.4576 0.49997 0.1286 0.4717 0.45900 0.5179 0.6514 0.76630 0.4224 0.6026 0.67582 0.2666 0.4661 0.50486 0.1066 0.3951 0.39805 0.4707 0.6206 0.71678 0.4473 0.6269 0.71424 0.2471 0.4504 0.4926

FIG. 9. Acoustic impedance of a hole, group A; �a� resistance and �b� end

f =20kg/m

.0493

.0457

.0361

.0444

.0577

.0490

.0414

.0507

.0901

.0930

.0760

correction coefficient.


VI. CONCLUSIONS

The understanding of the acoustic behavior of the fi-brous material and perforations is critical for the design ofdissipative silencers. Therefore, the present study has devel-oped experimental setups to measure: �a� the complex char-acteristic impedance and the complex wave number of thefibrous material, and �b� the acoustic impedance of perfora-tions, particularly in the presence of absorbing material.

The characteristic impedance and wave number of thefibrous material with two filling densities �� f =100 and200 kg/m3� are obtained using a modified impedance tubesetup with four microphones and two different terminationconditions. The averaged values from five samples are curvefitted as a function of frequency. These curve fits representmost of the data points well, particularly for the wave num-ber at both filling densities. The characteristic impedance andwave number inferred from the experiments are then used forthe measurement of acoustic impedance of perforations incontact with the fibrous material.

A new experimental setup has been developed in thepresent study for the measurement of acoustic impedance ofperforations in contact with the fibrous material as well aswith air alone in the absence of mean flow. The resistance Rand the end correction coefficient � of 11 perforationsamples with different porosities ��=2.1, 8.4, 13.6, and

FIG. 10. Acoustic impedance of a hole, group B; �a� resistance and �b� endcorrection coefficient.

25.2 %�, hole diameters �dh=0.249 and 0.498 cm�, and wall


thickness �tw=0.08 and 0.16 cm� are measured along withthe fibrous material of � f =100 and 200 kg/m3. For the per-forations without fibrous material, the resistance R and theend correction coefficient � decrease as the porosity in-creases. The fibrous material significantly increases both Rand � compared to the perforation in contact with air only.For both filling densities, both R and � decrease as the po-rosity increases, except for some of the samples with highestporosity ��=25.2% �.

FIG. 11. Acoustic impedance of a hole, group C; �a� resistance and �b� endcorrection coefficient.

TABLE IV. Standard deviations relative to R and � �%�.

SampleNo.

R

R

�

�

Air alone� f =100�kg/m3�

� f =200�kg/m3� Air alone

� f =100�kg/m3�

� f =200�kg/m3�

A1 24.5 16.1 25.4 7.4 3.4 7.7A2 10.3 15.7 16.0 4.9 3.7 6.9A3 11.3 25.4 31.8 8.9 7.2 13.8A4 17.5 44.2 38.2 30.8 10.9 25.5B1 7.3 15.3 18.1 3.4 4.2 4.6B2 6.9 17.1 23.0 4.2 4.1 7.5B3 9.4 21.8 31.8 8.4 9.2 16.1B4 15.3 25.8 36.1 35.2 15.9 36.9C2 18.7 19.4 22.1 3.0 4.2 5.9C3 16.6 18.3 26.9 3.5 5.1 6.3C4 20.7 29.0 35.9 10.0 7.8 13.9


The results obtained in the present study can be directlyused for the predictions of acoustic behavior of silencers thatuse similar fibrous material and perforation geometry. Whenthe material and geometry deviate substantially than those ofthe current work, caution should be exercised in the use ofperforation impedance.

Nomenclature

A1 cross-sectional area of the duct before theperforated plate

Ah cross-sectional area of a holea1,2,3,4 b1,2,3,4 coefficients of wave number and charac-

teristic impedance of fibrous materialc̃ complex speed of sound in the absorbing

materialc0 speed of sound in airdh perforate hole diameter

f frequencyi imaginary unit �=�−1�k̃ complex wave number in the absorbing

materialk0 wave number in airLs specific inertance�1 length between microphones 1 and 2�2 length between microphone 2 and sample

upstream surface�3 length between microphone 3 and sample

downstream surface�4 length between microphones 3 and 4

�eff effective length of a hole� f length of a fiber samplenh number of holesp1 acoustic pressure before the perforate

platep2 acoustic pressure after the perforate platepa acoustic pressure at the upstream of the

fiber sample with the first terminationcondition

pa� acoustic pressure at the upstream of thefiber sample with the second terminationcondition

pb acoustic pressure at the downstream ofthe fiber sample with the first terminationcondition

pb� acoustic pressure at the downstream ofthe fiber sample with the second termina-tion condition

pc acoustic pressure at the perforations con-tacting air

pd acoustic pressure at the perforations con-tacting fiber

pm1 acoustic pressure at microphone 1 withthe first termination condition





pm1� acoustic pressure at microphone 1 withthe second termination condition




R dimensionless resistance of perforationimpedance

Rf flow resistivity of the absorbing materialRs specific resistancetw wall thickness of perforated duct

�Tab� transfer matrix of a fiber sample�Tb3� transfer matrix between the downstream

of a fiber sample and microphone 3TC room temperature

T1,2,3,4 matrix elementsu1 particle velocity before the perforated

plateu2 particle velocity after the perforated plateuh particle velocity at a hole

um1 particle velocity at microphone 1 with thefirst termination condition

um2 particle velocity at microphone 2 with thefirst termination condition

ua particle velocity at upstream of the fibersample with the first terminationcondition

ua� particle velocity at upstream of the fibersample with the second terminationcondition

ub particle velocity at downstream of the fi-ber sample with the first terminationcondition

ub� particle velocity at downstream of the fi-ber sample with the second terminationcondition

uc particle velocity at perforations contact-ing air

ud particle velocity at perforations contact-ing fiber

Z0 �0c0, characteristic impedance of air

Z̃ �c̃, complex characteristic impedance ofthe absorbing material

Zh acoustic impedance of a hole� end correction coefficient of perforation

impedance� one standard deviation of �R one standard deviation of R� porosity�0 density of air� f filling or bulk density of fibrous material�̃ effective density of the absorbing

material


� angular velocity�p nondimensionalized acoustic impedance

of perforated plate

�̃p nondimensionalized acoustic impedanceof perforated plate in contact with fibrousmaterial

�h nondimensionalized acoustic impedanceof a hole

�̃h nondimensionalized acoustic impedanceof a hole in contact with fibrous material

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