TECHNICAL NOTES AND RESEARCH BRIEFS

TECHNICAL NOTES AND RESEARCH BRIEFS

Paul B. Ostergaard

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Editor's Note: Original contributions to the Technical Notes and Research Briefs section are always welcome. Manuscripts should be double-spaced, and ordinarily not longer than about 1500 words. There are no publication charges, and consequently, no free reprints; however, reprints may be purchased at the usual prices.

Correlation between tortuosity and

transmission coefficient of porous media at high frequency [43.20.Jr, 43.35.Mr, 43.35.Zc]

Manuel Melon and Bernard Castagn•de

Laberatoire d'Acoustique de I'Universitd du Maine, Ave. O. Messiaen, BP 535, 72017 Le Marts, France

The tortaosity and the transmission coeJ:ficient of porous media are mea- sured with an automatic 2-D .¾catt technique usittg ttarrow-battd piezoelec- tric tranxducers with a 39-kHz central frequeno: The results .show a strong correlation betoken the t•,o measured parameters. This can be explained front a high-frequenr3' analysis which predicts dependence of transmission

on tortaosity and characteristic lengths.

INTRODUCTION

At high frequency, the skeleton of porous media. such as reticulated foams, glass wools, or felts, does not vibrate. The Blot theory can be re- placed by the theory developed by Johnson • and AIlard 2 for a rigid frame case. This last theory has already been used to study the reflection versus ansieft Reasonable agreement has been obtained between predictions and measurements with. however. values of the characteristic lengths slightly different from the ones used at audible frequencies. Another noteworthy point is the intrinsic heterogeneity, due to the fabrication process, of these types of materials whose importance increases with frequency. Previous measurements performed at 39 kHz on a 20-cmx30-cm sample show varia- tions in the order of 4% for the tortuosity and 30% for the transmission

I. HIGH-FREQUENCY APPROXIMATIONS

The equations presented below are valid at normal incidence with an e -/ø'• time dependence. The characteristic impedance of porous media at high frequency can be calculated to a first-order approximation by using Eqs. (6), (9), and 04) from Reft 4:

Z c= paCo • I 8( I -j)

where P0, Co.)'. B •, a•, $. A. and A', respectively, are density, sound wave speed in air, the ratio of the specific heats, the Prandlt number of air. tortu- osity. viscous skin depth, characteristic viscous. and thermal lengths of the porous media.

The viscous skin depth is given by

•= q•7•tpo•O. (2)

where r/is the viscosity of air.

At high frequency, the first-order expression for the complex wave veclor of the material is provided in Ref. 4:

. (3)

The transmission coefficient can be calculated from the transfer matrix

of a porous media slab immersed in air:

( cos(kL) j•.Zc. lCP)sin(kL) I

A = j(cD/Zc)sin(kL ) cos(kL) ]' (4)

where L and cD are the thickness and porosity of the material.

When ta tends to infinity, the coefficient of transmission can be rewrit-

ten:

2

T - (5)

2cos(kL)+j sinekL)X(Zc/Zc,C•+Zcoq•/Zc) ' where Zco is the characteristic impedance of air.

For many porous media, porosity and tortuosity are close to one. The next approximation:

xf•/• + • l x/'•= 2. (6)

is valid in the range of 0.9-1.0 for porosity and 1.0-1.4 for tortuosity.

Accordingly, the wave speed C in the material is close to Co. Using this approximation, the absolute value of the transmission coefficient becomes:

r =exp{ o• •--[8[ I + ¾-1\] ]

which depends, at high frequency, mostly on three parameters of the mate- rial, i.e., the tortuosity and the two characteristic lengths, This simple for- mula is valid for plane waves. It does not take into account diffraction or scattering effects which can be induced by the heterogeneous nature of the material. Figure I highlights the validity of the approximation for frequen- cies larger than 20 kHz.

II. EXPERIMENTAL SETUP AND MEASUREMENTS

Some 2-D scan measurements have been performed using two fixed narrow-band piezoelectric transducers (39 kHz), while the sample tested is mounted on a movable frame (as shown in Fig. 2). A Panametrics 5058 PR pulser receiver is used to provide the large dynamic range needed for the measurements due to the strong damping of the acoustical waves crossing the porous sample. A 30 to 40 dB per cm (or Neper) attenuation is quite

1

0.4 0.2

0.6

10 t 10 2

Frequency (kHz)

FIG. !. Transmission coefficient of a 4-cm-lhick porous material at normal incidence. --:no approximation; ---: approximation of the wave vector;

.... : approximation on the wave vector and on the transmission coefficient.

The parameters used for the calculation are: a•= 1.2, cD=0.98, A= 1.2X l0 4, A-5.0x l0 4 which are common parameters for acoustical materials. 4

1228 d. Acoust. Soc. Am. 98 (2), Pt. 1, August 1995 0001-4966/95/98(2)/1228/4/$6.00 ¸ 1995 Acoustical Society of America 1228

FIG. 2. Experimental setup.

common at 39 kHz ibr many porous materials. The signals are captured by a DeCroy 9310 digiti/ing oscilloscope. A Macintosh Quadra 950 computer running LabVIEW 3.0 is used to drive the robot via the serial prat and to acquire dm signals flora the oscilloscope via the parallel pore The ,neasure- ments am completely automated.

The sample tested is a 2-era-thick glass wool slab in the form of a 20-cmX30-cm rectangular plate. The spatial step between two measure- meats is I cm, the diameter of the transducer is 2 in. (5.08 cm). The trans- mission coefficient is obtained from Ihe ratio of amplitudes with and without the sample. The propugation index is derived lYmn the wave speed in the fluid of the porous media, s which •s given by

C,o= I/( Co + 'r/L), (8)

where r is time delay evaluated flora au intemorrclation algorithm between the ultrasonic pulses with and xxithout the sample. The squared propagation index n, • is then given by

n• = (Co IC,,,)L (9)

Because of the low dispersion obser,•ed on high porosity acoustical media, 6 u, • at 39 kHz is close to ot•. This last techuique replaces the electrical

measurement of the tmluosity wilh a gain of time and precision. Moreover, the sample is not damaged by the measmement. With the use of broadband

16

14

12

• 8

6

4

2

Transmission ratio'

isound amphtudes with and without sample material)

5 10 15 2O 25

Length (cm)

0.3 0.35 0.4 0.45

FIG. 3. Inlcnsity map ot the transmis4on coefficient of a 2-cm-thick glass wool. Frequency: 39 kHz. Sample size: 28 cm b) 17 cm.

16

14

12

'• 8

6

4

2

Tortuosity (squared root of wavespeed with and without sample)

5 10 15 20 25

Length (cm)

1.09 1.1 1.11 1.12 1.13 1.14 1.15

FIG. 4. Intensity map of the torluos]ty of a 2-era-thick glass wool. Fre- quency: 39 kill Sample size 28 cm hy 17 cm.

transducers, torluosity can be obtained wilh highm precision b) cm•4dering lhe dispersion ctuvefi

The ]extdts me presented wilh a Jillcar interlxfiatim• belwecn each n]ea- suremerit point. Figme 3 pros tilex a churacteriqic example flu the spalial

miniran) are in the range lmm one •o two. Figure 4 presents the spatial variation ol Ihe tortuosi•y at 39 kill These two inlensily maps are qrongly to,elated. as p•edicted. The &adation• on the transmission coefficient which are mote pronounced (3(1•) than on dm IoHuosily (3.4rk) are related to the exponential dependence as shown m Eq. (7). Sexeml other materials such as fi•ams and 1•11% have been lesled confiHning this relalionxhip.

A 4topic error analyq% on Eq. (8) qmws the fluUualions ol the sam- ple's thickness do not account fl,' all the observed heterogencity of the tortuosity. The observed variations in tommsity •ould require thickness xariutions of the ordm of AC•l•xLl(I C0/•)=6.5 mm with the wave wed variations AC•l•= 1.7•. Co=342 m/s and lilt mean wave speed in the porous media •=324 m/s. This is significantly more than Ihe cqimated Ihickness variations of • I mm on the gin%% wool sample used for the men-

Hox&ever if the uansmission coelbcicm is calculated &• ith Ihe help of Eq. (7) ovm Ihe fi]II grid of the map, then variations oF Ihe torluosily are not impoaant enough to obtain Ihe observed variatious ol the translmssiou co- elficient. In lact, the attenuation predicted is smaller than thai meaqured.

This difibrence has been ah'eady obtained on cemenled gla•s beuds specimens ? and sandstones s at ultrasonic f]equencies. To explain Illis dis- crepancy. one must also consider thai Ihe charuct•riMic dimensions are themselves scalar fields over the I-cm step grid. Unformuately. these tgo parsmeier% are not easy to measure I(•ally. nm fo• Ibc [hei• aven•ge xulues.

Consequeully, co,cottons in the predictions ol Fq. (7) based on the local variatious of A and •' cannot be computed.

III. CONCLUSION

AI high frequency, heterogeneity ol acoustical pnrous media is signifi- cant. Measurements presented here show Ilml variulions of the transmission coefficient with location are smugly correlaled •ith Ihe xmialions of the torluosity. 'I his result is wflidated b) the lush Ikequency approximations of the pmpagalion laws. Nevertheless, Imtuo•ity is not entirely reHxmsible for all the vatiulions of wave transmission. In fi•cl Ihe sligbl changes in the geometry ot the porous network mudit 5 bolh 1o•luoqily and the churacte•is- lic lengths.

1229 d. Acoust. Soc. Am., Vol. 98, No. 2, Pt. 1, August 1995 Technical Notes and Research Briefs 1229

D. L. Johnson, J. Koplik, and R. Dashen, "Theory of dynamic permeabil- ity and the tortuosity in fluid saturated porous media," J. Fluid Mech. 176, 379-402 (1987).

j. E Allart, Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials (Chapman and Hall, London, 1993).

B. Castagn•de, N. Brown, and M. Melon, "Mesare du coefficient de rt- flexion de mat•riaux acoustiques dans I'air • I'aide d'ultrasons hesse frt- quence," C. R. Aced. Sci. Paris, 318 1453-1457 (1994).

4D. Lafarge, I. F. Allard, and B. Binnard, "Characteristic dimensions and predictions at high frequencies of the surface impedance of porous lay- ers," J. Acoust. Soc. Am. 93 2474-2478 (1993).

5J. F. Allard, B. CasLagn[de, M. Henry, and W. Lauriks, "Evaluation of the

tortuosity in acoustic porous materials saturated by air," Rev. Sci. [nstrum 65, 754-755 (1994).

6N. Brown, B. Castagn•de, W. Lauriks, and M. Melon, "Experimental study of the dispersion of ultrasonic waves in •x)rous media," C. R. Acad.

Sci. Paris 319, 393-399 (1994).

?P. B. Nagy, "Slow wave propagation in air-filled permeable solids," J.

Acoust. Soc. Am. 93, 3224-3234 (1993).

SG. A. Gist, "Fluid effects of velocity and attenuation in sandstones," J.

Aeoust. Soc. Am. 96, 1158-1173 (1994).

Advanced-degree diserrtations in acoustics

Editorb note: Abstracts of Doctoral and Master's theses will be wel-

comed at all times. Please note that they must be double spaced, limited to 200 words, must include the appropriate PACS classification numbers, and formatted as shown below (don't make the editor retype them, please!). The address for obtaining a copy of the thesis is helpful. Please submit two copies.

Design of piezoactive motors using modeling [43.3&Ar, 43.38.Fx, 43.35.Yb]--Ronan Le Letty, Cedrat Recherche Comp. 10, Ch. du Pr•

Carrd, ZIRST 4301, F38943 Meylan Cedex, France, September 1994 (Ph.D.). Piezoactive motors based on piezoelectric, electrostrictive, or mag-

forces. They offer miniaturization possibilities and characteristics leading to integrated devices (motor and brake). The work objective is related to their design using modeling. Three different phases have been followed: the first is the use of the finite element software ATILA, which takes the 3-D elec- tromechanical coupling effects into account, for •he stator analysis and de- sign. The results, which agree with experiments in most cases, show the software ability in design work. In the second phase, the traditional models

mechanisms between stator and rotor. A normal mode expansion method has been proposed and tested on two dift•rent ultrasonic motors (a traveling wave type and a mode conversion one), showing the method versatility. The third phase is related to experimental studies, where a motor prototype has been built and completely evaluated. Hence the analysis of both theoretical and experimental results shows the way to obtain better characteristics for prototypes.

Thesis advisor: Paul Gonnard.

The boundary element method 'for predicting acoustic performance of mufflers [43.50.Gf, 3,3.20.Bi}--Ji Zhenlin, Department

of Marine Power Engineering, Harbin Shipbuilding Engineering Institute, Harbin 150001, P. R. China, October 1993 (Ph.D.). In this thesis, the

boundary element method (BEM) has been developed systematically for

predicting and analyzing acoustic performance of mufflers without and with flow and temperature gradients. The formulations of three-dirheo•ional and

axisymmetric BEM are derived, the problems •n singular integra[io• and

comer point are handled, and the applicatiop of symmetry. is studied. A substrticmre boundary elemeot-transfer impedance matrix method has been presented to predict acoustic performance of c0..mplicated structure silencing system. This method has the advantage of a considerable saving of computer memory and computing time, an increase of calculation accuracy over the conventional BEM, and is suitable to a case with singular boundaries. A boundary element approach to sound transmission-radiation systems has also been developed. A number of calculations for predicting acoustic per-

formance of mufflers including various effects have been conducted, and the calculated results by the present methods agree with those by o, ther pub- lished methods and experimental ones very well. For example, prediction of the insertion loss of an exhaust muffler and sound pressure level radiated

1230 J. Acoust. Soc. Am., Vol. 98, No. 2, Pt. 1, August 1995

from the open end was fulfilled on a diesel engine by combining the present BEaM methodology with the characteristics method, and the comparison of predicted .results with measured ones showed good agreement.

Thesis advisor: Zhang Zhihua.

Thesis may be obtained from the author at Institute of Acoustics, Nanjing

University, Nanjing 210093, P. 1•. China.

Reflection and transmissiott of a focused finite amplitude sound beam incident on a curved interface [43.25.Jh, 43.80.Qf, 43.80.Sh]--lnder Raj S. Makin, Biomedical Engineering Program, Th4•

University of Texas at Austin, Austin, TX Y8712, July 1994 (Ph:D.

Biomedical Engineering). Reflection and transmission of a finite amplitude

focused sound beam at a weakly curved interface separating two fluid-like media are investigated. The KZK parabolic wave equation, which accounts for thermoviscous absorption, diffraction, and nonlinearity, is used to 'de- scribe the high intensity focused beam: T),e first part of the work deal.s with the quasilinear analysis of a weakly nonlinear beam after its reflection and transmission from a curved interface. A Green's function approach is used to define the field integrals describing the primary and the nonlinearly gener-

primary and second harmonic beams when a Gaussian amplitude distribu- tion at the source is assumed. The second part of th6 research uses a nu-

merical frequency domain solution of the KZK equatio,n for a fully nonlin-

ear analysis of the reflected and transmitted fields. Both piston and Gauss!an sources are considered. Harmonic components generated in the medium due

to propagation of the focused beam are evaluated, and formation of sho•ks in the reflected and transmitted beams is investigated. A finite amIi..litede

focused beam is observed to be modified due to reflection and transmission from a curved interface in a manner distinct from that in the case of a small

signal beam. Propagation curves, beam patterns, phase plots and time wave- forms for various parameters defining .the source and media pairs are pre- sented, highlighting the effect of the interface curvature on the reflected and transmitted beams. Relevance of the current work to biomedical applications

Thesis advisor: Mark F. Hamilton.

Copies of this thesis may be obtained from lnder Raj S. Makin, Department of Engineering, Swarthmore College, Swarthmore, PA 19081-1397.

Chara•terizatio• .of acoustic recoupling materials using a

progressive wave duct [43.20.Mv, 43.20..Ye, 43.35.MrJ--Gill.es Bessard, Laborattire d'Acoustique, I.E.M.N. (U.M.R. C.N.R.S. 9929), Institute Sup•rieur d'Electronique du Nord, 41 Bd Vanban, 59046 Lille Cedex, France, January 1995 (Doctorate). In the field of underwater acous- tics and at low .frequencies, the increase of detection efficiency techniques requires the improvement of decoupling materials. To assess the decoupling efficiency of new materials, the acoustic characteristics must be well known.

Therefore, a progressive wave duct is developed to characterize these ma- terials as a function of frequency and pressure and to determine their reflec- tion, transmission, masking, and anechoic efficiencies. The device uses two Tonpilz transducers located at both ends of the waveguide. At one end, a

Technical Notes and Research Bdefs 1230