acoustic force on a liquid droplet in an acoustic stationary wave

Received 8 February 1971 10.2

Acoustic Force on a Liquid Droplet in an Acoustic Stationary Wave*

[.^wRr. xc•,: A. CRUM

Michelson Physical Laboratory, U.S. Naval Academy, Annapolis, Maryland 21402

This paper considers the time-averaged acoustic force exerted on a spherical liquid droplet in an acoustic stationary wave. Experimental and theoretical values are presented for the minimum acoustic pressure amplitudes required to trap small individual droplets of various liquids near the pressure antinodes of a stationary sound field in a cylinder filled with water. The liquids used were i)araldehyde, hexane, benzene, toluene, chlorobenzene, and carbon tetrachloride. Droplet radii ranged from 400 to 800u, and acoustic- pressure amplitudes required to trap the droplets ranged from 1 to 15 bars. Calculated and observed values for the acoustic force are in substantial agreement, provided the compressibility of the liquid droplets is considered.

INTRODUCTION

Much theoretical and experimental work has been performed on the time-averaged force exerted by a sound field on a spherical inclusion. B jerknes • was one of the first to consider such a force, and he investigated the effects on a spherical air bubble placed in an oscillating liquid. King • published a detailed study of the radiation pressure exerted on a sphere, but neglected the effects of the compressibility of the sphere. Westervelt :• derived a general expression for the radiation pressure on an object of any shape and having arbitrary normal boundary impedance. Yosioka and Kawasima 4 extended the theoretical work of King to compressible spheres and found good agreement between theory and experiment for air bubbles in water. Eller • also reported agreement between theory and experiment for values of the acoustic pressure amplitude needed to trap air bubbles in an acoustic stationary wave. Later, Gor'kov • used an elegant fluid dynamics approach to obtain the result of Yosioka and Kawasima. Embleton 7,s extended the

work of King to spherical waves and produced experimental measurements that showed good agreement between theory and experiment for the radiation pressure exerted on rigid glass spheres in air. Recently, Nyborg, 9 who is interested in the acoustic radiation pressures exerted on biological particles, has calculated the force exerted by a sound field on a small rigid sphere. Also, a recent paper by Dysthe, •ø although containing some errors, gives a comprehensive treatment of the force

exerted on solid, liquid, and gas inclusions in a stationary sound field.

So far, the experimental work has been confined to the extreme cases of rigid spheres, 7,s,•2 or highly compressible spheres such as air bubbles. 4':',•:• Experimental measurements of the acoustic force exerted on glass spheres have been obtained by suspending the st)here from a fine thread and measuring the detlection of the thread-sphere system from the vertical. Since a glass sphere is very rigid, the compressibility of the sphere was not considered in the theory.

Measurements have also been obtained of the acoustic force exerted on an air bubble in water. Eller:' has shown

that an air bubble can be trapped near the pressure antinodes of a stationary sound field, provided it is driven below its resonant frequency. In applying the theory to the case of an air bubble trapped in water, there arise two terms in the force equation. One term results from the radiation pressure exerted on the finite- size inclusion in the sound field and is similar in form to

the term obtained for rigid spheres. A second term arises, however, because of the finite compressibility of the bubble, and is very much larger than the first term. Thus, for an air bubble, the finite-compressibility term, being much larger than the zero-compressibility term, completely dominates the force equation.

A liquid droplet lies midway between the two ex- tremes of rigid, or noncompressible, spheres such as glass, and soft, or highly compressible, spheres such as air bubbles. It is shown in Sec. II that, for a liquid

The Journal of the Acoustical Society of America ]57

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VF OSCILLATOR

[I

AMPLIF! ER FI LTE R

II

I

Iviv• I

L. A. CRUM

MICROLITER SYRINGE

O- RING SEAL

REMOVABLE CAB

O-RING SEAL BRASS RING

THREADS

THIN-WALLED GLASS TUBE

PILL TRANSDUCER

TRAPPED LIQUID DROPLET

FIG. 1. Schematic diagram of experimental apparatus.

BRASS RING

PZT-4 TRANSDUCER

droplet trapped by a stationary sound field in water, the zero-compressibility and the finite-compressibility terms are of nearly equal magnitude and both terms must be considered in the theory. No quantitative experimental measurements of the acoustic force required to trap a liquid droplet in a stationary sound field are available in the literature other than those presented here.

There has recently arisen a need for such a study. The possibility of using trapped liquid droplets as a new means of obtaining more realistic transient cavitation thresholds is rather intriguing. ApfeP 4.1a has performed experiments showing that it is possible to obtain liquid droplets free of solid impurities or motes without ex- tensive and elaborate filtration schemes. The liquid droplet, suspended in a clean host liquid, is perfectly isolated and thus remains quite free of contamination. Also, the position of the droplet in the sound field can be used as a measure of the acoustic pressure amplitude in the near vicinity of the droplet. Finally, the splendid isolation of the droplet, together with its small volume and the catastrophic effects of transient cavitation, allow an accurate visual criterion to be made for such an event.

Accordingly, it is the purpose of this paper to present measurements of the acoustic pressure amplitude required to trap small spherical liquid droplets in an acoustic stationary wave in water and to compare the results with theory. A more detailed description of the work is available in Ref. 16.

I. EXPERIMENTAL MEASUREMENTS

AND OBSERVATIONS

A. Construction and Calibration of the Resonator

Preliminary calculations indicated that it would be necessary to obtain acoustic-pressure amplitudes on the

order of 10 bars in order to trap the liquid droplets. Accordingly, a cylindrical resonator was constructed consisting of a small thin-walled glass cylinder that was mounted via a brass ring to a ceramic transducer. A schematic diagram of the experimental apparatus is shown in Fig. 1. The glass cylinder was approximately 15 cm long with an outside diameter of 3.2 cm and a thickness of 0.2 cm. A brass ring was ]nounted to the top of the glass cylinder and a removable cap machined that could be screwed down on the mounting ring. The cap had a small hole with a collapsable O-ring seal which allowed either a small probe or a hypodermic syringe to be inserted into the water and then sealed off from the outside air. A tubing connection for circulation of the water was also attached to the cap. The water was circulated by a stainless-steel bellows pump through a Millipore Corporation filter holder with a 25-mm- diam filter that had a pore size of 5 u.

The resonator described above was used in the measurements of all liquid droplets except carbon tetrachloride and chlorobenzene. A second resonator, identical to the first except that the a.d. was reduced to 2.54 cm, was used for these liquids. The two resonators had several resonant cylindrical modes that were investigated for possible usefulness. The (r,O,z) = (1,0,3) mode was found most convenient to use. The resonant frequencies for this mode in the two resonators were 41.7 and 50.5 kHz. The wavelengths along the z axis for these two frequencies were 7.8 and 5.8 cm, respectively.

It was found necessary to utilize some external means for obtaining the acoustic-pressure amplitude within the resonator. An internal probe would contaminate the system unless sealed off from the outside, and cavitation would often occur on the surface of the probe. Ac-

158 Volume 50 Number 1 (Part 2) 1971


ACOUSTIC FORCE ON A LIQUID DROPLET

cordingly, a small ceramic transducer was securely mounted to the side of the resonator directly outside the location of the pressm'e antinode and calibrated in terms of an independently calibrated internall}- located probe hydrophone. Subsequent measurements of the acoustic-pressure amplitude were then made with the calibrated external transducer.

B. Measurement of the Liquid-Droplet Radius

The liquid droplet to be trapped was inserted into the suspending fluid, water, by means of a hypodermic syringe. The syringe was graduated in microliters with a least count of 0.1 microliter. The radii of the droplets could be obtained from the known volume, owing to the spherical shape of the droplets. With this small-volume syringe, very small droplets, with radii on the order of a few hundred microns, could be injected into the liquid.

C. Measurement of the Sound Velocity

One of the most critical parameters in the equation for the acoustic force on the liquid droplet, Eq. 10, is the velocity ratio, •r=c*/c. Here, c* is the sound velocity in the droplet liquid and c is the sound velocity in the host liquid. It is important to know this ratio accurately in order to obtain good theoretical values for the acoustic force. Because there was no temperature control of the liquids in the resonator, the measurements were made at slightly different temperatures for each of the liquid droplets. Thus, it was considered necessary to obtain the absolute magnitude and the temperature dependence of the sound velocity for the various liquids used. Hand- book values were incomplete for the temperature range desired, 20ø-30øC. Accordingly, an ultrasonic inter- ferometer was utilized to make measurements of the

sound velocity of each liquid. These measurements, together with those for water taken from Carnevale et al., •7 were used in the calculations for the acoustic force.

D. Measurement of the Minimum Trapping Pressure and the Acoustic Force

It is shown in Sec. II that a convenient parameter with which to measure the magnitude of the acoustic force is the minimum trapping pressure, that is, the smallest acoustic pressure amplitude at the nearest antinode of the stationary-wave system that is required to trap a given liquid droplet. The procedure whereby the minimum trapping pressure is obtained is now described.

First, the sound field of the resonator was probed with a small ceramic hydrophone in order to ascertain the spatial dependence of the acoustic-pressure amplitude. The suspending fluid, water, was then degassed and filtered so as to be as free of impurities as possible and to increase the cavitation threshold. A high cavitation threshold was desirable to prevent cavitation in the host liquid during the measurements. Next, the droplets

were introduced by the microliter syringe into the suspending fluid. The pressure amplitude was set to be sufficiently large to overcome the buoyant and gravitational forces exerted on the droplet and to allow the droplet to be trapped by the sound field. The input voltage to the driving transducer was then slowly reduced as the droplet was observed with a cathetometer. When the driving voltage was reduced to the point that the droplet could no longer be trapped, the output voltage from the external pill transducer was used as a measure of the acoustic-pressure amplitude at the pressure antinode. More droplets of the same liquid were injected and an average taken over several measurements. The microliter syringe was then withdrawn and loaded with the next liquid, and the procedure repeated; each time an average minimum trapping pressure for each of the liquids was obtained. The acoustic force exerted on the droplet was obtained by using Eq. 10 (Sec. II), which relates the acoustic force to the acoustic-pressure amplitude.

II. THEORETICAL CALCULATION OF THE

ACOUSTIC FORCE EXERTED ON

A LIQUID DROPLET

Nyborg • has obtained the following result for the force exerted on a noncompressible sphere by a stationary sound field'

F at)

where •-• and (' are the second-order approximations to the time-averaged kinetic and potential energy densities, respectively, and V0 is the equilibrium volume of the sphere. The bar over the letter designates time average, and only these forces are considered in the analysis. Also,

B =3 (b-- 1)/(2b+l), (2)

where (3=p*/p and p*, p are the densities of the sphere and the host medium, respectively. To treat the case of a liquid droplet, however, the effect of the compressibility of the sphere must also be considered. A simple method is presented here, whereby this term is easily obtained. Although the method used here to obtain the expression for the acoustic force is not a unified approach, but rather the assimilation of two independent approaches, there appears to be no redundancy and the anal}'sis is quite simple. Moreover, the resulting expression obtained for the acoustic force is in complete agreement with the previous rather complicated ap-

, 6 proaches of Yosioka and Kawasima 4 and Gor kov. Eller:' has shown that the time-averaged acoustic

force exerted on a sphere at a position • in a stationary sound field due to its compressibility alone is given by

( =- V(t)-•-•-(z,t)), (3) The Journal of the Acoustical Society of America 159


L. A. CRUM

D8.0

6.0

4.0

2.0

' I ' I ' I

- o ThEOrY. EQ. 13

, I 5OO

, I 600

DROPLET RADIUS

I 7O0

(/.Z)

I i

I I 8OO

Fro. 2. Variation of the minimum

trapping pressure with droplet radius for droplets of hexane in water. The solid line corresponds to Eq. 13, with 5=0.656, a=0.719, and k•=0.81 cm-L

where V (t) is the instantaneous volume and Pa (Z,/) is the instantaneous acoustic pressure along the z axis.

For a compressible liquid sphere, V(t) can be written as 16

v = v0l- where

a = -œ• sin (tezZ)/p*c '2, (5)

and co= 2•rf, where f is the oscillation frequency of the sound field. For conditions of present interest, pa(z,t) is given by

p,(z,t) = ])a sin (/ezZ) cos(cot), (6)

where we are restricting ourselves to a stationary wave along the z axis. Also, lez= 2•r/Xz, where Xz is the wavelength of the stationary wave in the z direction, p* is the density, and c* is the velocity of sound in the droplet liquid. If we substitute V(t) and p,(z,t) into Eq. 3 and take the time average, we obtain

VolPa2]ez sin(2kzz)(•!_•) = , (7) 4p,6 2

where •2= (c./c)2. If we now add the force term obtained here, due to the

compressibility alone, to the term obtained by Nyborg, which was obtained with the assumption of zero compressibility, the total acoustic force is given by

Fa = F,.-+-F •

[ O• O• .Pa2j•z sin(2kzZ) (•!•) 1 - v0 B ....... t . ($) Oz Oz 4pc 2

The kinetic-energy density is dependent on the particle velocity u(z,t), and the potential-energy density is re- lated to the acoustic pressure p• (z,t). Specifically, we can write

T=pu2(z,t)/2, U=pa2(Z,t)/2pC 2. (9)

If these expressions for T and U are substituted into Eq. 8, the expression for the average acoustic force

exerted on a liquid droplet in a stationary sound field is given by

VoPa2tez sin(2tezz)I1 4pc • •r • x,•-7-•-•/D' (10) The first term in the brackets is due to the finite

compressibility of the droplet, and lhe second term is the result that would be obtained if the droplet were as- sumed to be incompressible, or a rigid sphere. Let us call the first term the compressibility term and the second term the rigid-sphere term. The compressibility term as given here does not apply rigorously to air bubbles, however. The case of an air bubble cannot be treated properly with the simple approach given here, but must be analyzed in more detail. For various approaches, see Refs. 4, 5, and 6. For the case of liquid droplets, b, • 1, and the two terms in brackets in Eq. 10 are the same order of magnitude. It is shown in Sec. III in the comparison of theory and experiment that both terms must be considered in order to obtain

good agreement with the experimental results. A liquid droplet can be trapped at a position z in the

sound field if the acoustic force exerted on the droplet balances the vector sum of the buoyant force Fb of the liquid on the droplet and the gravitational force Fa on the droplet. Thus, a condition for equilibrium of the droplet is that

Although positions of both stable and unstable equilibrium exist (see Sec. III-B), it was found that, owing to the nature of the stationary-wave system used, individual liquid droplets of visible size could be trapped at positions of stable equilibrium only if the acoustic force was directed toward the pressure antinodes. The absolute-value signs are introduced to account for the fact that the droplet may be trapped either above or below the pressure antinode, depending upon whether the




Fro. 3. Variation of the acoustic-

pressure amplitude at the antinode with droplet position for droplets of paraldehyde in water. The solid line corresponds to Eq. 12 with 5=0.994, •-0.776, and kz=0.81 cm -1. The position of the droplet is measured in terms of the dis-

placement from the pressure antinode, at the left on the graph.

2.4

2.0

1.2

o.e

0.4

I I I I I I I

/ --

/

© •••O • THEORY, EQ. 12 / - MTP

!

I -- I I

I I

/ / --

/ /

//

I I

Xz/32 Xz/•6 I i i i i

3XZ/32 XZ/8 5XZ/32 3XZ/16 7XZ/32 DROPLET POSITION

droplet liquid is respectively less or more dense than the suspending fluid.

Solving for the pressure amplitude required to trap the droplet, we obtain

.Pa 2-- . (12) kz sin(2kzz)[-(1/6o '2) -(56-2)/(26+ 1)•

Equation 12 suggests that there exist an infinite number of trapping pressures depending upon the position z of the droplet in the standing-wave system. However, it is possible to choose a unique value of the trapping pressure that is readily measurable by trapping the droplet at a position z such that sin(2kzz)= 1. This maximum value of sin(2k_.z) occurs when z is equal to Xz/8, where X.. is the wavelength of the stationary wave in the z direction. The pressure amplitude required to trap a droplet at this point is then a minimum. Let us call this value of the pressure amplitude the minimum trapping pressure P,,', which is seen from Eq. 12 to be given by

4 8 1--6[gp2c 2 } • _Pa t= . (13)

There exist liquids that have values of 6 and a such that the expression in brackets in the denominator of Eq. 13 is negative. These droplets are forced toward the pressure nodes of a stationary-wave system. The droplets that are drawn toward, and can be trapped near, a pressure antinode are those whose values of 6 and a obey the equation

1 56--2

--->---- (14) 6• 2 26-t-1

or for all droplets whose value of 6< 2/5.

Thus, for the great majorit 5' of liquids, the above equation asserts that, in order for a liquid droplet to be trapped near a pressure antinode by a stationary sound field, the compressibility term in the force equation must be larger than the rigid-sphere term.

III. COMPARISON OF THEORY AND

EXPERIMENT

A. Variation of the Minimum Trapping Pressure with Droplet Radius

It was observed in Sec. II that the expression for the minimum trapping pressure, Eq. 13, did not involve the droplet radius. To test this prediction of the theory, individual droplets of various liquids were inserted into the resonator by the microliter syringe. Figure 2 shows the results of a measurement of the effect of droplet size on the minimum trapping pressure for droplets of hexane (6=0.656, a=0.719) in water. Droplet size was on the order of a few hundred microns and ranged from approximately 400 to 800 u. The solid line corresponds to Eq. 13. Within the limits of experimental error, it is seen that the minimum trapping pressure is independent of droplet size. The fact that the pressure amplitude required to trap a droplet is independent of radius is due to the fact that the acoustic, the buoyant, and the gravitational forces are all proportional to the volume. This result is useful in that the droplet radius is the most difficult parameter to ascertain or control, and thus the level of difficulty of the measurements is con- siderably reduced.

B. Variation of the Trapping-Pressure Amplitude with Droplet Position

It was predicted in Sec. II that a liquid droplet could be trapped at an infinite number of positions in the

The Journal of the Acoustical Society of America 161


L. A. CRUM

TABLE I. Experimental and theoretical values of the minimum trapping pressure for the liquids studied.

MTP MTP MTP

Velocity Density exptl. theor. theor. Liquid ratio ratio (bars) (comp.) (total)

Paraldehyde 0.776 0.994 1.1 0.62 0.98 Hexane 0.719 0.656 4.2 3.55 3.94 Benzene 0.861 0.874 4.5 2.97 4.47 Toluene 0.862 0.861 4.9 3.09 4.57

Chlorobenzene 0.848 1.101 7.6 2.53 6.93 Carbon 0.619 1.585 15 5.33 14.4

tetrachloride

sound field. The actual dependence of the trapping- pressure amplitude upon the droplet position was shown in Eq. 12 to vary inversely as the square root of sin(2k•z).

To test this part of the theory, measurements were made of the variation of the pressure amplitude required to trap a droplet with the position of the droplet in the sound field. These measurements were made for a

variety of liquids and a typical result is shown in Fig. 3. Here, a plot is shown of pressure amplitude versus droplet position for paraldehyde (5=0.994, g=0.776). The pressure antinode is to the left on the graph. It is implied from the theoretical curve that it would require an infinite pressure amplitude to trap a droplet exactly at the antinode, for the force is zero there.

The experimental results begin to deviate from theory close to the antinode, probably owing to the finite size of the droplet, but they still verify the prediction that a large pressure amplitude is necessary in order to trap a droplet very close to a pressure antinode. It is seen also that, as the pressure amplitude is reduced, the droplet moves away from the antinode, following the theoretical curve. As the droplet approaches a distance of X•/8 from the antinode, the pressure amplitude is nearly independent of position. In this region, as the pressure is further reduced, the droplet tends to drift away from the antinode and escapes from the sound field. In the region X•/8 to X•/4, as shown by the dashed curve in Fig. 3, the droplet can be trapped, but only at a point of unstable equilibrium, and thus this does not represent a

T^BLF, II. Experimental and theoretical values of the acoustic force required to trap droplets of the liquids studied.

Liquid

MTP Force a Force a

Velocity Density exptl. exptl. theor. ratio ratio (bars) (dynes) (dynes)

Paraldehyde Hexane

Benzene

Toluene

Chlorobenzene

Carbon tetrachloride

0.776 0.994 1.1 0.004 0.003

0.719 0.656 4.2 0.201 0.176

0.861 0.874 4.5 0.065 0.065

0.862 0.861 4.9 0.082 0.071

0.848 1.101 7.6 0.062 0.052

0.619 1.585 15 0.325 0.300

For a radius of 500 •.

region that can be easily tested experimentally. The pressure amplitude required to trap the droplet near Xz/8 and yet just prevent the droplet from escaping is called the minimum trapping pressure and serves as a convenient point to compare theory with experiment.

C. Minimum Trapping Pressures for the Liquids Studied

The results of the measurements of the minimum

trapping pressure for the various liquids together with some liquid properties and theoretical predictions are shown in Table I. The minimum trapping pressure is denoted by MTP for convenience. The first column shows the liquids studied. These liquids were chosen for their accessibility and also because they are essentially insoluble in water.

It was predicted in Sec. II that for liquid droplets, both the rigid-sphere term and the compressibility term would need to be considered in order to obtain good agreement between theory and experiment. The last three columns of Table I show that this prediction is verified by the experimental results. The second-to-last column shows the theoretically predicted values of the minimum trapping pressure based upon the compressibility term alone. Recall that for a droplet to be trapped near a pressure antinode the compressibility term must be larger than the rigid-sphere term in the force equation. The last column shows the theoretical values of the

minimum trapping pressure when both terms are considered. It is evident from the consideration of these last

three columns that good agreement is obtained between theory and experiment, provided both terms are considered in the force equation.

D. The Acoustic Force Exerted on the Liquid Droplets

Although the parameter measured experimentally is the acoustic-pressure amplitude at the antinode required to trap a droplet in the sound field, it is possible to use these measurements to obtain both experimental and theoretical values for the acoustic force exerted on

the liquid droplet. Since the acoustic force exerted on a trapped droplet

must always equal the vector sum of the buoyant and gravitational forces, theoretical values of the acoustic force can be most easily obtained directly from Eq. 11, using the measured values for the droplet volume and the droplet-liquid density ratio. Experimental values of the acoustic force can be obtained in an indirect way by substituting the measured pressure amplitude required to trap a particular droplet at a given position in the sound field directly into Eq. 10.

Table II shows a comparison of theoretical and experimental results for the acoustic force exerted on a trapped liquid droplet in a stationary sound field. It is seen from this table that good agreement is obtained between theory and experiment for the acoustic force.




The acoustic force varies two orders of magnitude between paraldehyde and carbon tetrachloride. The large variation in the force is due to the large variation in the density ratio, which determines the difference between the buoyant and gravitational forces.

It is also seen that there is not a direct correlation

between the magnitude of the acoustic force and the magnitude of the minimum trapping pressure. This is explained by noting that the acoustic force given in Eq. 10 depends upon the square of the acoustic-pressure amplitude--and also upon two terms involving the density and velocity ratios.

The values tabulated in Tables I and II are the basic

results of this study.

IV. CONCLUSIONS

The principal conclusions of this paper are as follows'

(1) It is possible to trap small individual droplets of various liquids in an acoustic stationary wave. The liquids that can be trapped must possess restricted values of the density ratio • and the velocity ratio rr as determined by Eq. 14.

(2) The theoretical calculations of the force exerted on a compressible sphere by a stationary sound field correctly predict the acoustic-pressure amplitudes required 1o trap a liquid droplet in the sound field.

ACKNOWLEDGMENTS

The author would like to gratefully acknowledge the many useful and enlightening discussions with R. Apfel and A. Eller, and the financial support granted by the Office of Naval Research and the Naval Academy Re- search Council. Special thanks are due S. Elder for his assistance in obtaining research support.

* Presented in part at the 78th meeting of the Acoustical Society of America, San Diego, California, 4-7 November 1969 [-L. Crum, J. Acoust. Soc. Amer. 47, 82(A) (1970)-].

• V. F. K. Bjerknes, Fields of Force (Columbia U. P., New York, 1906).

•' L. V. King, Proc. Roy. Soc. (London) A147, 212-240 (1934). a p. j. Westervelt, J. Acoust. Soc. Amer. 23, 312-315 (1951). 4 K. Yosioka and ¾. Kawasima, Acustica 5, 167-178 (1955). • A. I. Eller, J. Acoust. Soc. Amer. 43, 170-171 (L) (1968). • L. P. Gor'kov, Soy. Phys. Dokl. 6, 773-775 (1962). * T. F. W. Embleton, J. Acoust. Soc. Amer. 26, 40-46 (1954). 8 T. F. W. Embleton, J. Acoust. Soc. Amer. 26, 46-50 (1954). • W. L. Nyborg, J. Acoust. Soc Amer. 42, 947-952 (1967). •0 K. B. Dysthe, J. Sound Vibration 10, 331-339 (1969). n L. A. Crum, J. Sound Vibration 12, 253-254 (L) (1970). • T. Hasegawa and K. Yosioda, J. Acoust. Soc. Amer. 46, 1139-

1144 (1969). •a R. K. Gould, J. Acoust. Soc. Amer. 43, 1185-1186 (L) (1968). • R. E. Apfel, J. Acoust. Soc. Amer. 49, 145-155 (1971). • R. Apfel (private communication). • L. A. Crum, "The Acoustic Radiation Pressure on a Liquid

Droplet in a Stationary Sound Field," Michelson Physical Lab., U.S. Naval Academy, Tech. Rep. No. C-1 (1970).

•* A. Carnvale, P. Bowen, M. Basileo, and J. Sprenke, J. Acoust. Soc. Amer. 44, 1098-1102 (1968).

The Journal of the Acoustical Society of America 163


acoustic force on a liquid droplet in an acoustic stationary wave

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