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 Articles you may be interested in Thermoacoustic engines The Journal of the Acoustical Society of America 84, 1145 (1998); 10.1121/1.396617 The propagation of plane sound waves in narrow and wide circular tubes, and generalization to uniform tubes of arbitrary cross-sectional shape The Journal of the Acoustical Society of America 89, 550 (1998); 10.1121/1.400379 Design and experimental verification of a cascade traveling-wave thermoacoustic amplifier Journal of Applied Physics 119, 204906 (2016); 10.1063/1.4952983 Use of complex frequency plane to design broadband and sub-wavelength absorbers The Journal of the Acoustical Society of America 139, 3395 (2016); 10.1121/1.4950708 Stability analysis of a helium-filled thermoacoustic engine The Journal of the Acoustical Society of America 96, 370 (1998); 10.1121/1.410486 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 short pulsein 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 rec- tangular 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 rectangu- lar cross-sectioncapillary tubes. The rectangularpore calcu- I. PROPAGATION IN RECTANGULAR PORE MEDIA lationis based on the low reducedfrequency 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),