Measurement and Calculation of Acoustic Propagation Constants in Arrays of Small Air-Filled Rectangular Tubes Heui-Seol Roh, W

Measurement and calculation of acoustic propagation constants in arrays of small air-filled rectangular tubes Heui-Seol Roh, W. Patrick Arnott, James M. Sabatier, and Richard Raspet

Citation: The Journal of the Acoustical Society of America 89, 2617 (1991); View online: https://doi.org/10.1121/1.400700 View Table of Contents: http://asa.scitation.org/toc/jas/89/6 Published by the Acoustical Society of America

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A simplified model for linear and nonlinear processes in thermoacoustic prime movers. Part II. Nonlinear oscillations The Journal of the Acoustical Society of America 102, 3497 (1998); 10.1121/1.420396 Measurement and calculation of acoustic propagation constants in arrays of small air-filled rectangular tubes Heui-Seol Roh, W. Patrick Arnott, and James M. Sabatier NationalCenter for PhysicalAcoustics, University of MississippLUniversity, Mississippi 38677 Richard Raspet Departmentof Physicsand Astronomy, University of MississippLUniversity, Mississippi 38677 (Received18 July 1990;accepted for publication20 January1991 ) An experimentaland theoreticalinvestigation of soundpropagation in a poroussample composedof capillarytubes with rectangularcross sections is describedin this paper.An experimentaltechnique valid for low flow resistivityand high porosityporous samples was developedto measurethe attenuationand phasevelocity in the porousmaterial. This technique usestransmission of a shortpulse in a large tube throughthe poroussample and subsequent frequencydomain analysis in the range200-1300 Hz. Good agreementwas obtainedif an anomaloustortuosity factor of 1.1 is usedin the theory. A scalingfactor for relatingcylindrical and squaretube capillarytheories, known as the dynamicshape factor, was investigated. Propagationconstants computed from useof a near unity dynamicshape factor in the cylindricalpore theory agreefavorably with calculationsbased on the squarepore theory for

the frequenciesand pore radii usedin the experiment. _ PACS numbers:43.20.Mv, 43.55.Ev, 43.28.Fp, 43.50.Vt

INTRODUCTION constantsand characteristicimpedances for porousmedia The interactionof soundwith porousmedia has many consistingof rectangularpores. The effectsof a slightcapil- practicalapplications and a long history.Porous media are lary tube curvature or tortuosity are accountedfor in this ideal sound absorbers for use in architectural acoustics and model. For the singlepore, a seriessolution is usedfor the anechoicchambers. The porousnature of the earth'ssurface particlevelocity, pressure, density, and temperature. The ex- and oceanbottoms can greatlyinfluence the propagationof tensionof the single-poretheory to bulk mediaconsisting of soundin the air and oceans.An exampleof the diverseuses an arrayof poresis developedusing standard techniques. 2's of soundpropagation in porousmedia comes from our labo- This is discussedin Sec.I. The single-poresolution for rectangular poreshas beendeveloped independently by Stin- ratory, wherewe recentlyused measurements and theoryfor 7 soundpropagation in the porousground to determinephys- son. ical parametersthat arerelated to the agriculturalsuitability The rectangularpore model is comparedto attenuation of the soilsinvestigated. • and phasevelocity measurements. The ceramicporous sam- Severalmodels 2-5 for porousmedia are basedon the plesused in the experimentconsisted of nominallystraight adaptationof the solutionfor soundpropagation in cylindri- capillarytubes having square cross sections. These low flow cal capillarytubes to poresof irregulargeometries. The basic resistivity,high porositysamples may be usefulfor low-fre- fluid field equationsused in thesemodels are a simplified quencysound absorption. A more thoroughdescription of versionof thelinearized Navier-Stokes equations 3 for a flu- the poroussamples and experimentaltechnique is givenin id. The approximationemployed in this calculationis that Sec.II. Comparisonof theoryand experimentis discussedin the transverse fluid velocities are much smaller than the lon- Sec. III. gitudinalfluid velocity.Zwikker and Kosten5 werefirst to A generaltheory for arbitrarypore shape 8 wasdevel- obtain the solutionbased on the simplifiedversion of the oped by introducinga scalefactor known as the dynamic fluid model equations and showed that these solutions shapefactor to scalebetween different pore geometries. The agreedwith Kirchhoff's exact solution 4 in thelimit of high limiting casesfor this scalingfactor are circularpores and andlow frequencies.Tijdeman 6 andStinson 7 investigated parallel slits. The limiting casesof rectangularpores are the rangeof validity of the Zwikker and Kosten solutionin squarepores and parallelslits, which giverectangular pore comparisonto the morerigorous Kirchhoff solution and de- theorya widerange of applicability.The dynamicshape fac- terminedthat the conditionfor the approximationto hold tor for squarepores is frequencydependent, just as it is for wasthe conditionon velocitieslisted above. Tijdeman refers parallelslits. 8 A discussionof thedynamic shape factor for to the Zwikker and Kosten theory as the "low reducedfre- squarepores is given in Sec. IV. quency approximation." We havedeveloped a porousmedia model for rectangular cross-sectioncapillary tubes. The rectangularpore calcu- I. PROPAGATION IN RECTANGULAR PORE MEDIA lationis based on the lowreduced frequency approximation. Soundpropagation in a singlerectangular capillary tube Specifically,the modelallows one to computepropagation isdeveloped first. The acousticfield in the poreis specified to

2617 J. Acoust.Soc. Am. 89 (6),June 1991 0001-4966/91/062617-08500.80 @ 1991Acoustical Society of America 2617 first order in the acousticvariables. Boundary conditionsat p• (x,y,z)= --po13T•(x,y,z) + (•//c2)p•(2), (7) the pore wall are that the walls are rigid and thus the total and particlevelocity is taken aszero. Due to the high heatcapac- ity and thermal conductivityof the pore wall, the tempera- -- icopoC•T• (x,y,z) = -- ico]3Top•(z) ture of the fluid in the pore at the boundaryis takento be the sameas the pore wall. We do not assumeany internal mean + tc•x 2 + T•(x,y,z). (8) flow.9 The single-poretheory is then usedto developthe theoryfor a porousmaterial consisting of an array of rectan- These relationsare given in the frequencydomain where gular capillary tubespossibly having a slight longitudinal •/3t is replacedby -- ico,where t is time and cois angular curvature or tortuosity. frequency.Response functions and transportcoefficients are cp,the constant pressure heat capacity per unit mass; y isthe A. Acoustical disturbances in a single rectangular ratio of specificheats; c is the adiabatic sound speed; capillary tube /3= -- (•p/•T) p/Po,is the thermal expansion coefficient; r/ is the viscosity;and tcis the thermal conductivity.In order, The coordinatesystem shown in Fig. 1 has the z axis theseequations express the z componentof the equationof parallel to the tube axis. The transverse dimension is motion, continuityor massconservation, equation of state spannedby an x-y coordinatesystem with the origin at the for density,and heat transfer. Equations(5)-(8) are the lower left cornerof the rectangle.First-order acoustic vari- sameset of equationsused by Zwikker and Kosten. 5 In using ablesare the real parts of these approximateequations the assumptionis that the p(z,t) = Po q- P• (z) exp( -- icot), ( 1 ) transversevelocity components vx and vy are much less than v(x,y,z,t)= [ v• (x,y,z),v•(x,y,z),v z(x,y,z) ] the longitudinalvelocity Vz. Further discussionsof theseap- proximationscan be foundin AppendixB of Ref. 6, Appen- X exp ( -- icot), (2) dix A of Ref. 10, and Ref. 7. T(x,y,z,t) = To + T• (x,y,z) exp(--kot), (3) The following notation will be used to facilitate com- and parisonwith Attenborough'sresults. 2'8 A dimensionless "shear-wavenumber" that is proportionalto the ratio of the p(x,y,z,t) =Po + P• (x,y,z) exp(--kot). (4) pore radius and the viscousboundary layer thicknessis Subscriptzero refersto ambient values;subscript 1 implies A = R (pocO/rI) •/2. Here R is a characteristictransverse di- first order; and in Eq. (2) the x, y, and z componentsof mensionof the pore. For definitenesswe take R to be twice particlevelocity are vx, vy,and Vz,respectively. Equations the transversepore area dividedby the transversepore pe- (1), (3), and (4) are the acousticpressure, temperature, rimeter.Thus R, whichis twice the hydraulic radius, 1• isthe and density.Acoustical disturbances within the poreare tak- tube radius for a cylindrical pore and the semiwidthfor a en to satisfythe following relations: squarepore. "Wide tubes"with the sameR are acoustically equivalent,i.e., they have the samepropagation constants. 3 --icopoVz (X,y,z)=- •dp• (z) + '1 + Vz(x,y,z), A dimensionlessnumber proportional to the ratio of the pore (•) radius to the thermal boundary layer thickness is Ar = R(po o9Cp/tc ) •/2Or A r -- •t•, • Pr•/2 , whereNpr '--' •]cp/tcis - i•p• (x,y,z) the Prandtl number. To make rapid progress,denote the z componentof the +po (.•v• (x,y,z) •vy(x,y,z) •Vz(X,y,z) ) =O, particle velocity vz by (6) Vz(x,y,z) = F(x,y;X ) dp• (z) . (9) icOpo dz Similarly,denote the acousticpore temperature by T, (x,y,z)= [(y-- 1)/C2pol3]p, (z)F(x,y;Ar). (10) Thethermodynamic relation To/3 2/cp = (y- 1)/c 2 can be usedin Eq. (8) for T•. The particlevelocity and temperature, Eqs. ( 5 ) and (8), reduceto the simpleforms F(x,y;A) + •iA •x2 + F(x,y;A) = 1 (11 )

and 2b F(x,y;A r) + iA• + F(x,y;A r) = 1, (12) respectively,subject to the boundarycondition F = 0 at the poreboundary. This is the particle velocityand excesstem- peratureboundary condition. The solutionfor F(x,y;A)for FIG. 1. Coordinatesystem and geometryfor the singlepore calculation. rectangularpore boundaries is •2

2618 J. Acoust. Soc. Am., Vol. 89, No. 6, June 1991 Roh ot aL' Acousticpropagation in small rectangulartubes 2618 16 sin(mrrx/2a) sin(nrry/2b) B. Extension to bulk media: Propagation constants and F( x,y;A) = • rr2m.•odd mn Ym.. (A ) characteristic impedance (13) Consider a fluid half-spaceoverlying a porous half- where spacesaturated by the same fluid. The pores are taken to havea rectangularcross section and we let the capillarytube Ym,.(A) = 1 + (iw•/A2) [ (b2m2 q-a2rt2)/(a q- b)2] axisof eachpore be at an angle0 with respectto the surface (14) normal. The tortuosityq- 1/cos 0 for sucha poroussam- and twice the ratio of transversepore area to perimeter is ple.2'5'8'• The openvolume divided by the total volumeis R = 2ab/(a q- b). [Recallthat A = R(poco/rl)'/2.]In the the porosity1• of the sample.The boundaryconditions are sums,m and n are odd numbersranging from 1-o•. For later continuityof volumevelocity (from massconservation ) and use,the averageF(A ) = [ 1/ (4ab) ] œF(x,y;A) dx dy over the the continuity of pressure(from Newton's third law) at the pore crosssection is porousinterface. :'3'5'8 Since fluid only flows into the pores, the bulk particle velocity V•b in the porous media is 64 1 (15) V•t, = •v•/q. TM In orderto accountfor propagationin a m(/l,)--7rn,n•oddm2H2yrn,n(/•) ß slantedpore (or other tortuouspath), the differentialdz in •u•.(I) ,• ( ,• 1 tll tortuosity.z•'8 Thusthe bulk acoustical equations are The z componentof the particlevelocity is givenby Eq. (9) Vzo(z) dp, (z) with F(x,y;A) in Eq. ( 13) and the excesstemperature is giv- ico•q.... 0 (23) en by Eq. (10) with the replacementof A in Eq. ( 13) by Ar. 11 qdz To derive a wave equation for the pressure,the fluid and equationsin Eqs. (5)-(8) are averagedover the pore cross dVzo(Z) section.Denote by Pl (z) = [ 1/(4ab) ]œPl(x,y,z)dx dy the • -- io.)Cpl(Z) = O. (24) averageof the acousticdensity in the pore and usesimilar notationfor vz(z) and T1 (z) for the transversearea average DifferentiatingEq. (23) by z and eliminatingVzt, with Eq. ofvz (x,y,z) and T1 (x,y,z). UseofEqs. (9) and (10) for thez (24) give an expression componentof the particlevelocity and temperatureand the fluid equations(5)-(8) resultsin a set of averagedequa- + (z) = 0 (25) tions: for the pressure in the porous media. Assuming kOpo dp• ( z ) p• o:exp (ikz) givesthe dispersionrelation •Vz(Z) =•, (16) F(A ) dz • = ro•q• = (roVc•)q•([( • - r)F(•) + r]/F(•)) dvz(Z) (26) -- iwp, (z) + Po •=0, (17) dz for the complexwave number k. From Eq. (23), the characteristic impedanceis and Z --•cOq2 /90 q c ,o,(z) =([(1 -- y)F(Xr) + y]/c2)p,(z). (18) •k F(A)'/2 • [ (1 -- •/)F(Ar) + •/] 1/2 The boundaryconditions, Vx (x,y,z) = 0 and vy(x,y,z) = 0 (27) at the boundary,were usedin obtainingEq. (17) from the The plane-wavepressure reflection coefficient rp for a wave continuityequation (6). Also, Eq. (10) for T1 (z) was used normally incident from the fluid half-spaceon the porous in the equationof state (7) to obtain Eq. (18). Following sampleis Attenborough2 we definea complexdensity from Eq. (16) and complexcompressibility from Eq. (18). Z - poC (F(•) [ ( • -- r)F(•) + r] }- '/• -- a/q z +poC (F(•) [ (• - r)F(•) + r]} -'/• + a/q' t5 =po/F(A) (19) (28) and This relation will be used in the next section.Principal results of this sectionare the propagationconstant (26) and C= ( 1Oo)(Pl •1 ) -- [ ( 1 -- •/)F(Ar) + •/]Ooc2. characteristicimpedance (27), which for rectangular pore (20) porousmedia are to be evaluatedwith the function F(A) given in Eq. (15). Eliminatingpl from the continuityand state equations (17) and (18) and usingEqs. (19) and (20) we obtain II. MEASUREMENT OF THE PHASE VELOCITY AND dp• (z) icotSVz(Z)--•=0 (21) ATTENUATION dz A. Description of the porous media and A schematicdrawing of the poroussample is shownin dvz(Z) • -- icoCp•(z) = 0. (22) Fig. 2. Each subsectionof the compositesample was of the nominallength 7.68 cm. Figure 2 indicatesa compositesam-

2619 J. Acoust.Soc. Am., Vol. 89, No. 6, June 1991 Roh eta/.' Acousticpropagation in small rectangulartubes 2619 38.4cm (a) (b) Digital Scope& FFT Analyzer

14.2 cm Differential Singlecycle tone Amplifier burstgenerator

SIDE VIEW: END VIEW Power Amplifier & 5 SUBSECTION- MicrophonePower COMPOSITE SAMPLE Supply 125 Volts D. C. PowerSupply

FIG. 2. (a) Sideand (b) end view of a squarepore porous sample. To form the total sampleseveral pieces of the nominallength 7.68 cm wereput to- MIC1I • MIC2I gether.It was not possibleto align the squaresfrom subsectionto subsec- 14.6 cm tion. I fcutoff=1376 Hz• SQUARE PORE CAPACITIVE COMPOSITE SAMPLE DRIVER ple madefrom five individualpieces. Individual pieceswere < 6.09meters taped togetherat the joint. A sheetof Teflon was wrapped F-60m? 1--60m--I around the compositeto facilitate insertionof the sample into the plane-wavemeasurement tube and sealthe sample- FIG. 3. Blockdiagram of theapparatus used to determinethe phasevelocity tube interface. Individual pieceswere ceramicsmade by and attenuationof soundin low flow resistivity,high porositysamples. Corning.13 The pores of each piece were nominally square in crosssection and nominally straight in the longitudinaldi- rection. of duration 1.3 ms (for a centerfrequency of '750Hz) was Three different square pore media were investigated. generatedusing a functiongenerator. The signalwas ampli- Table I listscharacteristics of each.In Table I, porositywas fied and was addedto a dc polarizingvoltage of 125 V. The estimatedusing 1• = (numberof pores/unitarea) X (2a) 2, capacitivedriver consistedof an aluminized mylar mem- where a is the squaresemiwidth. Flow resistivitywas com- branestretched over a groovedbackplate. TM A Teflonring putedusing (see Ref. 8) tr = 8•q2s/a2•,where s is a steady aroundthe perimeterof the driver wasused to hold the my- flowshape factor s = 0.89for squarepores TM and a tortuo- lar in placeand sealthe driver insidethe tube. The tubewas sity q = 1.1was estimated from fittingthe theoryand experi- made of aluminum and had a length of 6.09 m, an inside diameter of 14.6 cm, and a wall thicknessof 1.11 cm. Holes ment for the propagationconstants as discussedbelow. In were made and threaded in the tube 60 cm from each end and comparisonwith other porous media, 8 thesquare pore sam- pleshave low flow resistivityand high porosity.The average themicrophones were inserted to beflush with the innertube semiwidths are listed in Table I for the 200- and 400- wall. A minicomputerwith a 12-bit analog-to-digitalboard pores/in.2 material,for whichthe pore cross-sectional shape was usedto record and analyze the amplifiedmicrophone is well approximatedas a square.However, due apparently signals.The digitizingrate was 300 kHz. The functiongener- to differencesin the manufacturingprocess, the poresof the ator was used to trigger the minicomputerand 30 pulses 300-pores/in.2 samples were not well approximatedby a were averagedin the time domain for each measurement. The purposeof microphone2 was to give a referencetime squareshape. Two oppositecorners of the otherwisesquare shapewere rounded. The semiwidthofa = 0.50 mm listedin andspace location for a pulsetraveling in the emptytube, so that the ambientsound speed could be determinedfrom the TableI for the 300-pores/in.2samples was determined from pulsearrival time at microphone1. the shortestdiagonal length divided by (23/2).No explicit A singlemicrophone measurement method was used to usewas made of the calculatedflow resistivityin the theory determinethe propagationconstants in the squarepore me- for the propagationconstants. dia. Figure4 (a) showsmicrophone 1 measurementof pulses with and withouta samplepresent in the tube.As expected, B. Experimental apparatus the pulsemeasured with the samplepresent is delayedin A block diagramfor the experimentalapparatus used to time and attenuatedon accountof passagethrough the po- determinethe attenuationand phasevelocity of sound in rousmedia. The Fouriertransform of a typicalpulse indicat- porousmedia is shownin Fig. 3. A singlecycle of a sinewave ed that the pressurelevel was about 20 dB abovethe back- groundfor frequenciesin the range200-1300 Hz. The cutoff frequency3 above which nonplanar modes can propagate in the tube is 1376 Hz. TABLE I. Geometricalproperties of the three poroussamples used. C. Determination of attenuation and phase velocity by Pores/unit area Semiwidtha Porosity• Flow resistivity (in. -2) (mm) (%) (N m -4 s) the transfer function method

200 0.77 73 368 Denoteby Po(t) the incidentpressure pulse at the right 300 0.50 47 1356 end of the sampleand denoteby Po(f) the Fourier trans- 400 0.57 81 606 form ofpo(t). The spectrump m( f ) afterpassage through a sampleof length D m is

2620 J. Acoust. Soc. Am., Vol. 89, No. 6, June 1991 Roh ot a/.' Acousticpropagation in small rectangulartubes •620 2 (a) Eq. (28) weexpect r v •0. Measurementsof rv withmicro- phone2 in Fig. 3 for a nominalfrequency of 750 Hz gave 21• (0.03, 0.05, and0.03) for the (200, 300,and 400)- pores/in.2 samples,respectively. The phasevelocity and attenuationconstant are com- putedfrom k = cO/Cphd- ia, Cph(f) = 2•rf/[ (Dm-- D1 ) --1Im ln(h,.) + ko], (31) 0.77 mm SEMI-WIDTH SAMPLE and 0 2 x10-3 TIME (seconds) a(f) = -- [20(D,• --D 1)- 1/ln(10)] Reln(h,•), (32) (b) wherecph is the phasevelocity; a is the attenuationin MIC1 [ Pm(f) Po(f) dB/cm; and Re In (h,•) and Im In (h ,• ) refer to the real and • Po(t) imaginaryparts of In (h,•), respectively. Equations( 30)-( 32 ) were usedto analyzethe time domain pulsesto obtainexperimentally the phasevelocity and FIG. 4. (a) Samplewaveforms for microphone1 with and without the po- attenuationfor the squarepore media. In all cases,we aver- rous material in the tube. The porousmedia was constructedas shown in agedover 30 time domain pulsesbefore taking transforms. Fig. 2. (b) Expandedschematic view of the tubeand poroussample in Fig. We alsoused five subsections,so that m = 5 in Eqs. (30)- 3. The Fourier transformafter the incidentpulse passes through the sample ispm(f). (32) and (D,• -- D• ) = 30.7 cm nominally. Sincea transfer functiontechnique was used,it was not necessaryto determinethe pressureabsolutely or the frequencyresponse of the microphone.A centralassumption that wasverified experi- 1 It2 mentally was the repeatabilityof any pulse measurements P.• (f) -- Po( f ) exp(ikDm) •' , sincep• (t) andPl (t) were measuredat differenttimes. The 1 --rp2 exp(2ikDm) experimentalresults for the threedifferent pore sizes given in , (29) Table I are displayedin Fig. 5. The experimentaltechnique whererv is the frequency dependent pressure reflection coef- describedhere is similar to a methodused by Ding• to mea- ficient(28) and k is the complexwave number for the po- sure the reflection coefficient of absorbents. rous media (26). Sincethe tube has a large diameterwe An error analysiscan be made by usingan unapproxi- approximatethe characteristic impedance of thetube by Po c, matedtransfer function for Eq. (30) in Eqs. ( 31 ) and (32). 2 wherec is the adiabaticsound speed in air. The exp(ikDm) Let A -- r • [exp(2ikDm) -- exp(2ikD•) ]. The fractional factor accountsfor propagationthrough the sample,the error in phasevelocity that occursdue to the approximation (1 --rp2 ) factor is for transmissioninto and out of the sam- inEq. (30) is/5½•Im ACph(f)/[2rrf(D• -- D• ) ].Similar- ple,and the denominator accounts for the multiplereflection ly,for attenuation,/5, •0.2 Re A/[ a( f)ln 10(Din-- D•) ], of waveswithin the sample. where (D,•- D• ) is in meters. Use of representative In Eq. (29) the subscriptrn refers to the number of numbersfor the 200-in.- 2 material gives/5½ • 1% and subsectionsused for a measurement.For example,rn = 5 in /5, • 7%. To improve on the error, Dm and D• shouldboth Figs.2 and 3. We may form a transferfunction h,• (f) from be uniformly increased.The errors introducedby making the resultsof two .experimentson differentporous sample the approximationin Eq. (30) are the greatestsource of lengths.The Fourier transformof the time domainpulses error. recordedby microphone1 in Fig. 4 (b) for two differentsam- ple lengthsgives, from useof Eq. (29), a transferfunction III. DISCUSSION OF EXPERIMENTAL AND h,, ( f ) =p,, ( f )/Pl ( f)•exp[i(k- ko)(D,, -- D• ) ], THEORETICAL ATTENUATION AND PHASE VELOCITY (30) The experimentaland theoretical attenuation and phase where ko = co/cis the wavenumber for soundin air and k is velocity were determinedfrom use of Eqs. (30)-(32) and givenin Eq. (26). In Eq. (30) the singlesubsection spec- (26), respectively,and k = co/cp•d- ia. Figure5(a) and trum p• (f) was used as a referenceto divide out the fre- (b) showsthe experimentaland theoretical attenuation con- quencyresponse of the capacitivedriver and microphone, stant and phasevelocity. For the theory, the physicalcon- andthe transmission coefficient ( 1 - r v2) in Eq. (29). The stantsused were y= 1.4,Ne =0.707, Po= 1.2 kg/m3, secondpart ofEq. (30) is an approximationbecause we have •7= 1.85X 10- 5kg/(m s), andthe adiabatic sound speed c assumed[ ( 1 -- r •2 exp2ikD• ) / ( 1 -- r •2 exp2ikDm ) ] • 1. from the propagationtime of a pulsebetween microphones 1 The reflectioncoefficient r• in Eq. (28) is significantlyless and 2 in Fig. 3 with no samplein the tube. than 1 for two reasons.First, IF(A)I• 1 for the frequency To obtainthe acceptableagreement among theory and rangeof the presentexperiment as a consequenceof the low experimentindicated in Fig. 5(a) and (b), a tortuosityof flow resistivityof the squarepore samples used. Second, for q = 1.1 wasused. Referring to Eq. (26) for the propagation the squarepore materialthe porosity11 is large. Henceby constantk, note that the effectof tortuosity (which is q> 1)

2621 J. Acoust.Soc. Am., Vol. 89, No. 6, June 1991 Roh et al.' Acousticpropagation in small rectangulartubes 2621 0.3 poreswere spacedby about 150/•m. Apparently,the wall poresdid not connectadjacent square pores. Intuitively, it (a)..... ] seemsthat the effectsof porouswalls would be to increase m 0.2 the compressibilityof the gasand hencedecrease the phase z velocity.The attenuationwould also increase, which is what o wasobserved experimentally. Champoux ]6 reporteda tor- z tuosityof 1.2for the200- and 400-pores/in. 2samples using a m 0.1 nonacousticaltechnique. ]7 It was not possibleto align the poresin neighboring subsectionsof the compositesample, as discussedin Sec. 0.0 I I I II A. To investigatethe effectsof misalignment,attenuation 400 600 800 1000 1200 FREQUENCY (Hz) and phasevelocity were measuredfor compositesamples consistingof oneto fivesubsections. In the singlesubsection i i ! i i measurementthe referencefor the transfer function in Eq. (30) wastaken to be the emptytube signal. The discrepancy between these measurements was less than 3 % and showed no systematictrends. Sample misalignment was probably v •.3ooj(b.."""""""" .."""""""""""" not the causeof an apparenttortuosity greater than 1.

ß 280ß IV. DYNAMIC SHAPE FACTOR FOR SQUARE PORES 260 The hypothesisof Attenborough's8 cylindrical capillary-tube-basedporous media theory was that a circular pore of radiusa/n, where n is known as the dynamicshape I , I ! I I • , i 4oo ,oo o'oo doo factor, could be made acousticallyequivalent to another FREQUENCY (Hz) pore of characteristicradius a by proper choiceof n. The conditionfor acousticalequivalence is taken to be FIG. 5. (a) Experimentaland theoretical attenuation and (b) phasevelocity forthe three square pore semiwidths given in TableI. Thesolid symbols Im•Sc(A/n) = Im•Ss(A), (33) are experimentalpoints. The squares,circles, and trianglesare the 200-, 400-, and 300-in.- 2 results.The error in (a) wasestimated to be twicethe wherewe recall that A = a (Poco/z/) 2/2 for a circularpore of sizeof the squaresymbols. A representativeerror bar is shownin (b). radiusa or a squarepore of semiwidtha and subscriptsc and s referto circularand squarepores, respectively. This condition occurssince the imaginarypart of the complexdensity•5 is very large for small/l and thus determinesthe behaviorof thepropagation constants and impedance for small/l.8 is to increasethe attenuationa and decreasethe phaseveloc- The complexdensity is given generallyfor rectangular ity, asone would intuitively expect.The calculatedpropaga- poresin Eq. (19). For squarepores of semiwidtha the function constantfor a tortuosityq = 1 resultsin an attenuation tion Fs(A) is 10% lower than the measuredvalue and a phasevelocity 10% above the measuredvalue. The discrepancybetween 64 1 experimentaland theoreticalphase velocity in Fig. 5(b) for Fs(A) = -•- m,n•od dm2rt2 [1 + (iw'2/4,• 2)(m 2q- rt2) ] the 300-pores/in.2 samples having pores of semiwidth0.50 (34) mm may be due to the irregular shapeof the porecross sec- from Eqs. (14) and ( 15) for the specialcase a = b. For cy- tion, as described in Sec. II A. lindricalpores, 2 Poresthat have a tortuosity other than 1 have a radius whichis not constantalong the pore,a slightcurvature or tilt = - [ 2/,/7 ] [./1 ], with respectto the axis normal to the surface,or are in a (35) materialIn theceramic for whichsquare the porerigid material frame assumption used in theisexperiment, notvalid. s wherethe dynamicshape factor n hasbeen inserted and the J's are Besselfunctions. The range8 of n is thoughtto be the poreswere straightand the pore walls had a densityand 0.5•

2622 J. Acoust.Soc. Am., Vol. 89, No. 6, June 1991 Roh eta/.: Acousticpropagation in small rectangulartubes 2622 i .... i .... i .... i

_ the excesstemperature. A measurementtechnique using frequencydomain analysis of shortpulses propagated through high porosity,low flow resistivitysamples was developed for 0.98 determiningpropagation constants. Propagation constants were measuredfor a ceramicporous media havingstraight capillarytube openings with squarecross sections. Use of an 0.96 anomaloustortuosity factor q = 1.1 resulted in favorable agreementamong experimental and theoretical values of the

0.94 propagationconstants. It wasargued that the nonunitytor- tuosity value was due to the finite porosityof the ceramic porewalls. A dynamicshape factor n = 0.97 wassuggested asthe radiusscaling factor for squareand circularpore theories for the frequencyrange (200-1300 Hz) and pore sizes FIG. 6. Dynamicshape factor n to scalebetween the squareand circular (0.50-0.77-mm semiwidthsquare pores) of this investiga- poresof semiwidtha andradius a/n, respectively. tion. Future work will involvean investigationof the anoma- lous tortuosityand propagation½onstaiit IiicasuIeCIiiCII[S Oil longer samples. ence between the propagation constantscalculated with these theories.

ACKNOWLEDGMENTS v. CONCLUSION We havedeveloped a modelfor porousmedia consisting We are gratefulto Michael R. Stinsonand Yvan Cham- of rectangularpore capillary tubes.We accountedfor vis- poux for discussionsand tortuositymeasurements on some cousand thermaldissipation. A seriessolution was obtained of the poroussamples used in this investigation.Conversa- for the transversevariation of the longitudinalvelocity and tionswith Keith Attenborough,Henry E. Bass,and Kevin L. Williams are alsoappreciated.

i ß 0.3 (a)

m 0.2 o J. M. Sabatier,H. Hess,W. P. Arnott, K. Attenborough,and M. Rom- kens,"In situmeasurements of soilphysical properties by acousticaltech- niques,"Soil Sci. Soc.Am. J. 84, 658-672 (1990). ua 0.1 K. Attenborough,"Acoustical characteristics ofporous materials," Phys. Rep. 82, 179-227 (1982). A.D. Pierce,Acoustics: An Introductionto Its PhysicalPrinciples and Ap- plications(Acoustical Society of America, New York, 1989), Chap. 10. 0.0 4j. W. S. Rayleigh,The Theory of Sound(Dover, New York, 1945),Vol. , 400i , 600i , 800t, , 1 d00 ' 1200i II. FREQUENCY (Hz) C. Zwikkerand C. W. Kosten,SoundAbsorbing Materials (Elsevier, Am- sterdam, 1949), Chap. 2. H. Tijdeman,"On the propagation of sound waves in cylindricaltubes," J. i (b) Sound Vib. 39, 1-33 (1975). 300 M. Stinson,"The propagationof planesound waves in narrowand wide circulartubes, and generalizationto uniformtubes of arbitrarycross-sec- tionalshape," J. Acoust.Soc. Am. 89, 550-558 ( 1991). K. Attenborough,"Acoustical characteristics of rigid fibrous absorbents G28o and granularmaterials," J. Acoust.Soc. Am. 73, 785-799 (1983). o A. Cummingsand I. J. Chang,"Acoustic propagation in porousmedia with internal mean flow," J. Sound Vib. 114, 565-581 (1987). ]øG. W. Swift,"Thermoacoustic engines," J. Acoust.Soc. Am. 84, 1145- ß-r 260 1180 (1988). P. C. Carman,Flow of GasesThrough Porous Media (Academic,New York, 1956). ]2L. S. Han, "Hydrodynamicentrance lengths for incompressiblelaminar 400 600 800 1 00 1 00 flow in rectangularducts," J. Appl. Mech. 27, 403-409 (1960). The au- FREQUENCY (Hz) thor givesthe solutionfor our Eq. (11 ) for a coordinatesystem at the centerof a rectangle.He usesthe differentialequation for a differentpur- posethan the presentpaper. FIG. 7. (a) Attenuationand (b) phasevelocity computed using a square ]3Theceramics were manufactured by CorningIncorporated, Industrial pore(solid line) of semiwidtha = 0.50mm anda circularpore (dashed ProductsDivision, Corning,New York 14831. line) of radiusa -- 0.50mm/n, where n -- 0.97is a dynamicshape factor. ]4F. D. Shields,H. E. Bass,and L. N. Bolen,"Tube methodof soundab-

2623 J. Acoust. Soc. Am., Vol. 89, No. 6, June 1991 Roh eta/.' Acousticpropagation in small rectangular tubes 2623 sorption measurementextended to frequenciesfar above cut off," J. 16y. Champoux(private communication, 1990). Acoust. Soc. Am. 62, 346-353 (1977). 17y. Champouxand M. R. Stinson,"Measurement of tortuosityof porous y. Ding, "A wave-tubeimpulse method for measuringsound-reflection materialand implicationsfor acousticalmodeling," J. Acoust.Soc. Am. coefficient of absorbents," Acustica 57, 188-190 (1985). Suppl. 1 87, S139 (1990).

2624 J. Acoust.Soc. Am., Vol. 89, No. 6, June1991 Roheta/.: Acousticpropagation in smallrectangular tubes 2624