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Characteristic dimensions and prediction at high frequencies of the surface impedance of porous layers
Denis Lafarge, Jean F. Allard, and Bruno Brouard
Laboratoired91coustique, C.N.R.S. URA No. 1101, Facult• desSciences Mans, B.P. 535, Ave. Olivier du
Messiaen, F-72017 Le Mans C•dex, France

ChristineVerhaegen and Walter Lauriks
LaboratoriumvoorAkoestiek, KatholiekeUniversiteitLeuven,Celestijnenlaan D, B-3001, Heverlee, 200 Belgium

(Received 15 May 1992; revised19 January 1993; accepted27 January 1993) The surfaceacoustic impedance a glasswool and a reticulatedfoam is measured a free field of in up to 20 000 Hz. The characteristic dimensions and A' can be calculatedfor the glasswool, A and the surfaceimpedancecan be predictedwith no adjustableparameters.The motion of the frameis not taken into account.The agreement betweenmeasurement and predictionis good. For the foam, the characteristicdimensions cannot be calculated,because the geometryof the frame is not simple. A correct choiceof A and A' allows a preciseprediction of the surface impedancefor a large range of frequencies and different thicknesses. PACS numbers: 43.20.Gp, 43.20.Jr, 43.20.Rz, 43.55.Ev

INTRODUCTIONIn animportant paper, Johnson aL1gave simple et a
expression the effective for densityp of a fluid saturating a porousmedium, which simultaneously takes into account the inertial and the viscousforces,and presentsa correct asymptoticbehavior at high frequencies. indicated by As
Johnson,the exact expression the effectivedensity canfor not be calculatedfor the case of porous materials with aframe having a complicatedgeometry,but the asymptotic high-frequency behaviorcan be predictedif the tortuosity

Both Eqs. (1) and (2) are identical when the square velocitiesare removed. For the caseof identical cylindrical pores, v(M) is a constant that can be removed from Eq. ( 1) and the two characteristic dimensions equal. If the are poresdo not have a constantcrosssection,the dimension

Ais smallerthan A', because weightingby the square the velocity givesa larger contribution of the constrictionsof the poresin Eq. (1).

a oo, the characteristic and viscous dimension are known A, (the tortuositywas previouslydenotedby ks by Zwikker

andKosten, q2byAttenborough, canbeevalu2 and 3and becausegenerally most of the fibers lie inplanar planes atedfromelectrical conduction measurements4'5). parallel to the surfaceof the samples.A layer is repreThecharacteristic dimensionis given 1 A by

Fibrous materials such as glasswool are anisotropic

2 fs[v2(M)IdS A-- f vIo2(M) [dV'


where v(M) is the microscopic velocityfield related to the steady flow of an inviscid fluid in the material, and the integralsin the numeratorand the denominatorare performed over the contact surface between the fluid and the

sentedin Fig. 1. The direction normal to the surfaceis x 3 and planar planesare parallelto the XlX2 plane.The characteristicdimensions are different for waves propagating parallel and perpendicularto the planar planes and these different characteristicdimensions must be known to predict the surfaceimpedanceofthe layer at oblique inci-

dence. wasshown AllardandChampoux the It by 7 that
characteristic dimensions could be calculated for the case

frame, and the fluid volume, respectively. For the case of materials having parallel cylindrical pores,A is two times as large as the ratio of the area and the perimeterof the cross section,and is equalto the radius of the poresif they are circularcross-sectional shaped.In a

of glasswools, the fibersbeing modeled as circular crosssectional-shaped cylindersof total length L per unit volume, and of radius R, lying in planar planes.The dimen-

sionsA/v and A•v relatedto propagation the normal in

direction given 7 are by
A/v= 1/2rrLR, A•v= 2A/v. (3)
If the fibers are parallel in the planar direction, two orthogonaldirectionsparallel to the...
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