Bjerknes force and bubble levitation under single-bubble ...

Bjerknes force and bubble levitation under single-bubble sonoluminescence conditions Thomas J. Matula, Sean M. Cordry,a) Ronald A. Roy,b) and Lawrence A. Crum Applied Physics Laboratory, University of Washington, 1013 NE 40th Street, Seattle, Washington 98105

~Received 21 January 1997; accepted for publication 28 May 1997! Bubble levitation in an acoustic standing wave is re-examined here for the case of single-bubble sonoluminescence. The equilibrium position of the bubble is calculated by equating the average Bjerknes force with the average buoyancy force. The predicted values, as a function of pressure amplitude, are compared with experimental measurements. Our measurements indicate that the equilibrium position of the bubble shifts away from the pressure antinode as the drive pressure increases, in qualitative agreement with calculations, but unexpected when only linear theory is considered @A. Eller, J. Acoust. Soc. Am. 43, 170 ~1968!#. The Bjerknes force also provides an upper limit to the drive pressure in which a bubble can be levitated near ~and above! a pressure antinode, even in the absence of shape instabilities. © 1997 Acoustical Society of America. @S0001-4966~97!05209-0# PACS numbers: 43.35.Ei @HEB#

INTRODUCTION

In 1989, Gaitan discovered that a single air bubble trapped at a pressure antinode in a standing wave could be made to emit light.1 This discovery of single-bubble sonoluminescence, or SBSL, has led to many remarkable experimental measurements of the light ~and sound! emitted from single bubbles.2–8 Of fundamental importance for generating cavitation bubbles that can remain stable for hours at a time is the process of bubble levitation. Previous studies of bubble levitation in a stationary wave have been carried out for bubbles larger than resonance size ~where the bubble is levitated near a pressure node!9,10 and for bubbles smaller than resonance size,11,12 in both cases driven into small amplitude oscillations. In this paper we will consider in more detail the levitation of bubbles used for SBSL experiments ~bubbles below resonance size, driven into highly nonlinear oscillations!. In particular we will look at the role of the Bjerknes force on a bubble ~described later!, since it is this force that causes a bubble to be levitated against the gravitational ~buoyancy! force. Although little is mentioned about the light emission from a sonoluminescing bubble in this paper, our calculations and measurements were carried out in the parameter space where the stable sonoluminescence of single, air-filled bubbles occur. Our calculations show that for low driving pressures, a small increase in pressure leads to a larger ~in magnitude! Bjerknes force, as expected. This causes the bubble to be pulled closer to the antinode. However, at larger driving pressures, the magnitude of the Bjerknes force actually becomes smaller, and the bubble’s equilibrium levitation position shifts away from the antinode. We will show how this occurs, with experimental measurements of the levitation position of a sonoluminescence bubble that shows qualitative agreement with our calculations. Quantitative agreement can a!

Current address: York College, York, NE 68467-2699. Current address: Dept. of Aerospace and Mechanical Engineering, Boston University 110 Cummington Street, Boston, MA 02215.

b!

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J. Acoust. Soc. Am. 102 (3), September 1997

be obtained if one assumes that the presence of the bubble perturbs the pressure profile in the near vicinity of the bubble. If true, this may have implications for building stable SBSL levitation cells. In the following sections we make extensive use of the radius-time profile calculated for SBSL bubbles. The bubble’s volume mode pulsations are calculated using the Keller equation, given by13

S D 12

S D S

R˙ 3 R˙ 1 R˙ R d RR¨ 1 R˙ 2 12 5 11 1 c 2 3c r c c dt

D

3 @ p b 2 p a ~ t !# , where

S

p b5 p 01

2s 2pv R0

DS

R 30 k ~ R 3 2a 3 ! k

D

2

R˙ 2s 24 m . R R

~1!

~2!

We choose parameters that match our laboratory conditions; a viscosity of m 51 cP, a surface tension coefficient of s 572 dyn/cm, the sound speed in water c51490 m/s, a vapor pressure at room temperature of p y 52660 Pa, and an ambient pressure of p 0 51.0133105 Pa. We also assume that the gas ~with k 51.4! has a van der Waals hard core, with R 0 /a58.54, where a is the excluded radius and R 0 55 m m is the equilibrium radius of the bubble. We begin by examining the Bjerknes force on a small bubble, which is levitated just above the antinode of a standing wave. Then the equilibrium position of the bubble, assuming the average buoyancy force just balances the Bjerknes force, is considered. Again, we choose parameters that match laboratory conditions for our experimental system, in this case a rectangular levitation cell with dimensions 8.538.5-cm cross section, and a water column ( r 51000 kg/cm3) of 14.6 cm. The drive frequency for this system is 19.5 kHz, being operated in the ~1,1,3! mode. That is, there are three pressure antinodes along the vertical direction. We choose one of the antinodes as the origin of our coordinate system. For the bubble to be in stable equilibrium,

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FIG. 1. For small drive pressures, the pressure force attracting the bubble towards the pressure antinode during the first half of the acoustic cycle is greater than the force pushing the bubble away during the second half of the acoustic cycle, since the volume of the bubble is greater in the former case ~illustrated by the size of the arrows!. Thus over an acoustic cycle, the average force is directed toward the pressure antinode. The solid lines refer to the drive pressure, the dashed lines refer to the pressure gradient, and the vertical line marks the location of the antinode. The z axis corresponds to the vertical axis.

the horizontal radiation forces must be zero, thus we consider only forces in the vertical direction in which gravity acts. Note that with three antinodes, the vertical wavelength is l52/3314.6'9.73 cm. I. THE BJERKNES FORCE

In an acoustic standing wave, bubbles can be levitated against the gravitational force of buoyancy by the wellknown Bjerknes force, F Bjerk5 ^ F p & 5 ^ 2V(t)“ P & , where F p is the instantaneous pressure force on the bubble, “ P is the gradient of the applied acoustic pressure, and ^•••& denotes the time average.14–16 In our levitation cell the standing wave pressure profile ~along the vertical axis! is given by P(z,t)52 P a cos(kzz)sin(vt). The magnitude of this pressure gradient is then “ P5k z P a sin(kzz)sin(vt), so that F p 5 ~ 24 p /3! R 3 k z P a sin~ k z z ! sin~ v t ! .

~3!

Here k z 52 p /l z is the vertical wave number, l z 59.73 cm is the vertical wavelength, z is the position above the antinode v 52 p f , where f is the drive frequency and P a is the drive pressure amplitude. Note that the sign of the pressure force changes as the drive pressure changes. Physically, the Bjerknes force arises from a pressure difference ~gradient! across the bubble. Figure 1 illustrates this for the case of small driving pressures ~and for drive frequencies below the bubble’s natural resonance frequency!. During the negative portion of the sound field, the bubble grows. There is a pressure force on the bubble due to a slight difference in pressure exerted on either side of the bubble’s surface. This force directs the bubble towards the pressure antinode. During the compressive phase of the sound field, the bubble is small, and the force is directed away from the 1523

J. Acoust. Soc. Am., Vol. 102, No. 3, September 1997

FIG. 2. ~a! Calculated steady-state radius-time curves for a 5-mm bubble driven with a pressure amplitude of 1.3, 1.5, and 1.7 atm using Eq. ~1!. Note that as the pressure amplitude is increased, the bubble collapses later in the acoustic cycle, such that it remains relatively large even after the pressure changes phase. The arrows illustrate the direction of the Bjerknes force, towards the pressure antinode during the first half-cycle, and away from the pressure antinode during the second half cycle. ~b! This instantaneous pressure force for bubbles driven as shown in ~a!.

pressure antinode. However, since the corresponding volume is smaller, this force is smaller, and hence, over an acoustic cycle the average ~or Bjerknes! force directs the bubble towards the antinode. This argument on the direction of the force applies only to a bubble that is driven below its natural resonance frequency. For bubbles driven above their natural resonance frequency, a different phase response requires them to be forced away from the pressure antinode and toward a node. Though valid at lower drive pressure amplitudes, the results of this description of the Bjerknes force must be modified at higher driving pressures. Consider Fig. 2~a! which shows steady-state radius-time curves for a bubble (R 0 55 m m) with three different driving pressures. The normalized driving pressure is also shown. Recall that the pressure force changes sign with the drive pressure. This demarcation line is shown near t'25 m s for the particular parameters used in the calculation. The pressure force is directed towards the pressure antinode during the tensile portion of the sound field, and away from the antinode during the positive portion. For small drive pressures the volume of the bubble at its maximum size ~where it contributes most to the pressure force! occurs during the negative portion of the sound field. However, as the drive pressure increases, the bubble colMatula et al.: Bjerknes force and bubble levitation

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FIG. 3. The Bjerknes force, buoyancy force, and equilibrium bubble position for bubbles with an equilibrium radius of 3 mm ~heavy lines! and 5 mm ~light lines!. For small drive pressures, the Bjerknes force is small and negative. At about 1.78 atm the force becomes positive, and the system can no longer support a stable bubble near the pressure antinode. Note that the equilibrium position of SBSL bubbles ~bubbles driven above the lower threshold of 1.2 bar! shifts away from the pressure antinode as the drive pressure is increased. Also note that since the bubble is not far from the antinode, the drive pressure at the bubble is nearly identical to the pressure at the antinode.

lapses later in the cycle and can still be large after the pressure turns positive. With the bubble’s volume near its maximum after the pressure turns positive, the pressure force contributes a relatively large component directed away from the pressure antinode. This is illustrated in Fig. 2~b! which shows the instantaneous pressure force calculated for the three R-t curves shown in Fig. 2~a!. The Bjerknes force is the average pressure force, averaged over an acoustic cycle, 24 p k P sin~ k z z ! F Bjerk5 ^ F p & 5 3T z a

E

T

0

R ~ t ! sin~ v t ! dt, 3

~4!

where T51/f is the acoustic period. This force is plotted in Fig. 3 as a function of drive pressure amplitude ~assuming the equilibrium position z51 mm!. For small pressure amplitudes, the Bjerknes force is small and negative ~but not zero!, directing the bubble towards the pressure antinode. As the pressure amplitude increases beyond 1.2 bar ~near the cavitation threshold! the magnitude of the Bjerknes force increases rapidly, resulting from the rapidly increasing volume-mode pulsations of the bubble; however, note that at even higher pressure amplitudes, the slope of the force changes sign, and the magnitude of the Bjerknes force begins to decrease. The change in slope near 1.6 bar results from the bubble collapsing later in the acoustic cycle, as shown in Fig. 2. The integral in Eq. ~4! begins to contribute a large positive component to the average force, which competes with, and eventually dominates the negative components in front of the integral. Eventually ~in this case, for P a '1.78 bar! the Bjerknes force becomes positive, and consequently the bubble is pushed away from the antinodal region. This im1524


plies that there exists an extinction threshold for the drive pressure amplitude, such that above this threshold a ~sonoluminescing! bubble cannot be levitated above the pressure antinode. We note, however, that there does exist, at least theoretically, stable levitation positions below the pressure antinode above this extinction threshold. In this region, the Bjerknes force pushes the bubble down and away from the antinode, while the buoyancy force pulls the bubble up and toward the antinode, such that there can exist an equality between the Bjerknes and buoyancy forces. Also note that the location of the zero-crossing of the Bjerknes force does not depend on the position of the bubble. This fact can be seen from Eq. ~4!. The zero-crossing shown in Fig. 3 depends only on the frequency and the equilibrium bubble size R 0 @and to a lesser extent, the surface tension and viscosity, as they relate to the radius R(t)#. Calculations over a large range of these values show only minor influences on the location of the zero-crossing. Experimentally, this extinction threshold does not appear to be reached. Instead, apparently shape instabilities are generated near a pressure amplitude of 1.5 bar, causing the bubble to selfdestruct.17 II. EQUILIBRIUM BUBBLE POSITION

In the analysis above, we assumed that the position of the bubble was 1 mm above the pressure antinode. This is due to the additional force of buoyancy exerted on a levitated bubble, directed vertically upward, against the gravitational field. This force is given by F buoy5 r gV(t), where r is the density of the fluid, and g is the gravitational acceleration. Here, we calculate the average levitation position of the bubble, and find how the equilibrium bubble position18 is dependent on the drive pressure amplitude ~assuming the equilibrium radius does not change!. A bubble can be stably levitated if the Bjerknes and average buoyancy forces are equal. Thus

rg T

E

T

0

V ~ t ! dt5

vkzPa sin~ k z z ! T

E

T

0

V ~ t ! sin~ v t ! dt. ~5!

A simple expression for the equilibrium levitation position of the bubble can be obtained provided one assumes the bubble is near the pressure antinode. Then sin(kzz)'kzz, and z'

rg L1 , v k 2z P a L 2

~6!

where L 1 5 * T0 V(t)dt and L 2 5 * T0 V(t)sin(vt)dt. Figure 3 shows the equilibrium bubble position @using Eq. ~6!# above the antinode as a function of the applied pressure amplitude. For small drive pressures, the equilibrium position of the bubble is nearly inversely related to the drive pressure, as seen in Eq. ~6!. However, as the drive pressure is increased further, the bubble position begins to shift away from the antinode, the slope of the curve changing signs. Beyond about 1.78 bar the bubble can no longer be levitated above the pressure antinode, since the Bjerknes force now pushes the bubble away from the antinode. In the next section we describe our experimental apparatus and method for measuring the equilibrium bubble position, and attempt to confirm elements of the above analysis, specifically the region that shows the equilibrium position of Matula et al.: Bjerknes force and bubble levitation

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FIG. 4. The measured pressure profile in the levitation cell with ~3! and without ~s! a bubble present. The arrow marks the approximate location of the bubble during this measurement. Also shown is a fit to A1B cos(kzz 1f) for both cases. In the absence of the bubble, l59.73 cm and corresponds to the third eigenmode of our system. With the bubble present, we set l 8 53.539.73 cm to take into account the possible flattening of the local pressure gradient, as discussed in the text. The particular value for l 8 comes from determining the scaling factor that was needed to match the data with the calculations in Fig. 5. Note, however, that the variation between the two cases may simply involve acoustic radiation damping or scattering ~the high-frequency acoustic signal emitted from the bubble was removed by passing the signal through a low-pass filter!.

FIG. 5. Measurements of the levitation position of the bubble are shown together with the calculated results. The solid line comes directly from Eq. ~6!, using l59.73 cm (k z '64.6 m21). The measurements, shown as circles, show qualitative agreement, but appear to be offset from the calculated values by a scaling factor. As discussed in the text, quantitative agreement can be obtained if one assumes that the bubble modifies the local pressure gradient, such that the bubble ‘‘sees’’ a wavelength of l 8 53.5l. The dashed curve represents a calculation that was modified to take this possible flattening of the pressure gradient into account. The inset shows a blow up of the measured region. The error bars are a result of bubble motion, as seen through the Ga¨ertner microscope.

the bubble shifting further from the antinode as the drive pressure is increased. Parenthetically, we mention that in an experiment one cannot follow a bubble’s position entirely along a single curve shown in Fig. 3. Each curve represents the equilibrium position for a particular bubble with a particular equilibrium size ~3 and 5 mm!. It is known that the equilibrium radius of a bubble changes as one changes the pressure amplitude.17,19

The dashed line in Fig. 4 is a fit to A1B cos(kzz1f) with l59.73 cm. This corresponds to the vertical wavelength of our cell ~recall that our cell is 14.6 cm long, and is operated in the third eigenmode!. Note that with the bubble present, the pressure profile in the vicinity of the bubble varies slightly from that with no bubble present. The physical mechanism for this variation may be important, and is discussed in the next section.

III. EXPERIMENT

Our experimental system was described above. It consisted of a rectangular levitation cell with dimensions 8.5 38.5-cm cross section, and a water column ( r 51000 kg/m3) of 14.6 cm. The drive frequency for this system was 19.5 kHz, being operated in the ~1,1,3! mode. We measured the position of the bubble with a Sony Handycam video camera, focused at infinity, looking into a Ga¨ertner microscope. Images were captured and stored on a computer for later processing. With filars of 10- and 100-mm separations ~calibrated using a calibration microscope slide!, resolutions as fine as 2 mm were obtainable. Backlighting from a fiber-optic illuminator was used as a ~white! light source. Our results for the equilibrium position of the bubble will be shown in the next section. The position of the sonoluminescence bubble was measured for different drive pressure amplitudes ('1.3521.50 60.05 bar). A calibrated needle hydrophone20 was used to measure the approximate pressure amplitude near the antinode, with and without a bubble in place. Figure 4 shows the vertical pressure profile for the two cases. In both cases the hydrophone was offset approximately 4 mm from the center line of the cell, so that with the bubble present, we could still map out the vertical pressure profile in the cell, without disturbing the bubble. 1525


IV. RESULTS AND DISCUSSION

Figure 5 shows the experimentally obtained equilibrium position of the bubble for various pressure amplitudes, together with the calculated equilibrium position. Note that there is good qualitative agreement between the calculated and measured equilibrium positions, showing that the equilibrium position of sonoluminescence bubbles actually shifts away from the pressure antinode as the drive pressure is increased. However, the quantitative agreement is poor. There appears to be a multiplicative scaling discrepancy of approximately 12.25. We now discuss possible sources for the observed discrepancy between the measured and calculated bubble position. ~Parenthetically, we mention that in many experiments, one can generate sonoluminescing bubbles that move in almost any conceivable direction, and over relatively large distances, as the drive pressure is changed. Only after extreme care is taken to fine tune the driving frequency, does the bubble behave as shown here.! Experimental errors include the fact that the absolute z coordinate of the bubble is estimated, since the exact location of the pressure antinode is only known as an average over the finite size of the hydrophone ~'1-mm diameter!. Thus there may be a systematic offset to the bubble positions Matula et al.: Bjerknes force and bubble levitation

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measured with the video system. This error represents an error in the accuracy of the measurement; however, the relative error ~error in precision! is small, since resolutions as fine as 2 mm were obtainable. In our case, the discrepancy between the measured and calculated positions appears to be that of scaling, and not from an offset error ~note the log scale in Fig. 5!. Also, consider that in going from Eq. ~5! to Eq. ~6! we assume that the bubble is near the pressure antinode, and we take sin(kzz)'kzz, where z is the distance above the antinode. Experimentally, the position of the bubble is z 8 5z1c, where c is an offset resulting from our inability to exactly determine the antinode position ~due to the finite size of the hydrophone!. Even an offset error as large as a few millimeters affects this approximation only minimally. A sensitivity analysis of Eq. ~6! in other parameters such as the equilibrium radius of the bubble and drive pressure amplitude shows only small variations and do not account for the observed scaling discrepancy. A potential experimental source for error that could result in a scaling error would be a miscalibration of the fiducial markings on the microscope eyepiece; however, these markings were calibrated against a calibration microscope slide. A video camera looking into the microscope eyepiece will not affect the calibration. Furthermore, a second set of experiments done at the fundamental frequency of 13.3 kHz ~the 1,1,1 mode! also shows discrepancies, but with a different scaling ('7.3). 21 One would expect that miscalibrated fiducial markings would result in similar scaling discrepancies. It may be that the bubble itself, as an object in a sound field, provides a boundary which distorts the pressure profile in the cell, most profoundly near the bubble itself. Consider Fig. 4 which shows that the pressure profile with the bubble present appears flattened near the bubble ~near the pressure antinode!. This flattening of the pressure gradient may appear to the bubble as a longer localized wavelength. If so, then the bubble does not ‘‘see’’ a vertical wavelength of 9.73 cm, as calculated by the size of the water column in the levitation cell, but instead the flattening of the pressure gradient appears as a longer wavelength ~at least in the near vicinity of the bubble!. If one assumes that the presence of the bubble affects the local gradient, resulting in a modified wavelength l 8 53.5l @which becomes a scaling factor of 12.25 in Eq. ~6!, since k z 52 p /l#, then quantitative agreement is obtained between the calculated and measured equilibrium position ~as shown in Fig. 5!. The introduction of a scaling factor via a modified wavelength is also self consistent with the measured pressure profile in the levitation cell near the bubble. Figure 4 shows that near the pressure antinode, the pressure profile can be fit with a wavelength of l 8 53.5l. Though not shown here, experiments done at 13.3 kHz also show quantitative agreement with the introduction of a modified wavelength of l 8 52.7l. The analysis above did not take into account that a bubble in a stationary sound field can remove energy from the incident sound field through various mechanisms such as thermal, viscous, and radiative damping. For small bubbles 1526


driven into small amplitude oscillations, thermal damping dominates.22 However, for large drive pressure amplitudes, radiative damping may dominate. These damping mechanisms may also show up as a change in the measured pressure profile. We were not able to conclusively determine whether the change in the pressure profile was due to damping or whether the bubble itself distorts the pressure gradient. Nor did we consider the issue of bubble translation in this study. The translation may induce microstreaming to such an extent that it affects the bubble’s equilibrium position. The issue of microstreaming from single-bubble sonoluminescence has only recently been considered.23 V. CONCLUSION

Measurements of the equilibrium position of a sonoluminescence bubble in a stationary wave shows qualitative agreement with the calculated equilibrium position; as the drive pressure amplitude increases, the bubble shifts away from the pressure antinode. We explained this shift as an equilibrium between the competing buoyancy and acoustic pressure forces. The delayed collapse of a sonoluminescence bubble at high-pressure amplitudes greatly affects the Bjerknes force. Furthermore, we found that at sufficiently high drive pressure amplitudes, there exists an upper pressure threshold for bubble levitation, even in the absence of surface instabilities. This threshold appears when the Bjerknes force no longer holds the bubble against the gravitational force ~near 1.78 bar!. At higher drive pressures, stable bubble levitation can only exist below the pressure antinode, where the Bjerknes and buoyancy forces again act in opposite directions ~assuming the bubble remains spherical and does not develop other instabilities!. These results may have implications for experiments performed in a microgravity environment, where the buoyancy force is negligible. Quantitative agreement between the measured and calculated equilibrium positions of a sonoluminescence bubble can be obtained if one assumes that the bubble itself acts to distort the pressure profile in the immediate vicinity of the bubble. If true, then explicitly, how does the bubble affect the local pressure profile? Also, this may provide some insight into building stable levitation cells for single-bubble sonoluminescence experiments. ACKNOWLEDGMENTS

We wish to thank M. Averkiou and J. Ketterling for their assistance with some measurements, and Phil Marston and Detlef Lohse for helpful discussions. This research is supported by NSF. 1

D. F. Gaitan, L. A. Crum, C. C. Church, and R. A. Roy, ‘‘Sonoluminescence and bubble dynamics for a single, stable, cavitation bubble,’’ J. Acoust. Soc. Am. 91, 3166–3183 ~1992!. 2 B. P. Barber, R. Hiller, K. Arisaka, H. Fetterman, and S. Putterman, ‘‘Resolving the picosecond characteristics of synchronous sonoluminescence,’’ J. Acoust. Soc. Am. 91, 3061–3063 ~1992!. 3 M. J. Moran, R. E. Haigh, M. E. Lowry, D. R. Sweider, G. R. Adel, J. T. Carlson, S. D. Lewia, A. A. Atchley, D. F. Gaitan, and X. K. Marugama, ‘‘Direct observations of single sonoluminescence pulses,’’ Nucl. Instrum. Methods Phys. Res. B 96, 651–656 ~1995!. Matula et al.: Bjerknes force and bubble levitation

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V. H. Arakeri, ‘‘Effect of dissolved gas content on single-bubble sonoluminescence,’’ Pramana, J. Phys. 40, L145 ~1993!. 5 R. Hiller, K. Weninger, S. J. Putterman, and B. P. Barber, ‘‘Effect of noble gas doping in single bubble sonoluminescence,’’ Science 445, 248– 250 ~1994!. 6 T. J. Matula, R. A. Roy, P. D. Mourad, W. B. McNamara III, and K. S. Suslick, ‘‘Comparison of multibubble and single-bubble sonoluminescence spectra,’’ Phys. Rev. Lett. 76, 2602–2606 ~1995!. 7 J. B. Young, T. Schmiedel, and W. Kang, ‘‘Sonoluminescence in high magnetic fields,’’ Phys. Rev. Lett. 77, 4816 ~1996!. 8 I. M. Hallaj, T. J. Matula, R. A. Roy, and L. A. Crum, ‘‘Measurements of the acoustic emission from glowing bubbles,’’ J. Acoust. Soc. Am. 100, 2717~A! ~1996!. 9 T. J. Asaki and P. L. Marston, ‘‘Acoustic radiation force on a bubble driven above resonance,’’ J. Acoust. Soc. Am. 96, 3096 ~1995!. 10 T. J. Asaki and P. L. Marston, ‘‘Equilibrium shape of an acoustically levitated bubble driven above resonance,’’ J. Acoust. Soc. Am. 97, 2138 ~1995!. 11 L. A. Crum and A. I. Eller, ‘‘Motion of bubbles in a stationary wave,’’ J. Acoust. Soc. Am. 48, 181 ~1969!. 12 L. A. Crum and A. Prosperetti, ‘‘Nonlinear oscillations of gas bubbles in liquids: An interpretation of some experimental results,’’ J. Acoust. Soc. Am. 73, 121 ~1983!. 13 V. Kamath, A. Prosperetti, and F. N. Egolfopoulos, ‘‘A theoretical study of sonoluminescence,’’ J. Acoust. Soc. Am. 94, 248–260 ~1993!.

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A. Eller, ‘‘Force on a bubble in a standing acoustic wave,’’ J. Acoust. Soc. Am. 43, 170 ~1968!. 15 A. Prosperetti, ‘‘Bubble phenomena in sound fields: part two,’’ Ultrasonics 22, 115 ~1984!. 16 L. A. Crum, ‘‘Bjerkness forces on bubbles in a stationary sound field,’’ J. Acoust. Soc. Am. 57, 1363 ~1975!. 17 R. G. Holt and D. F. Gaitan, ‘‘Observation of stability boundaries in the parameter space of single bubble sonoluminescence,’’ Phys. Rev. Lett. 77, 3791 ~1996!. 18 The bubble actually translates vertically during an acoustic cycle. This is due to a change in the buoyancy and Bjerknes force with a change in the bubble’s volume. In our work we simply average over an acoustic cycle, and do not consider the instantaneous position of the bubble. 19 B. P. Barber, C. C. Wu, R. Lo¨fstedt, P. H. Roberts, and S. J. Putterman, ‘‘Sensitivity of sonoluminescence to experimental parameters,’’ Phys. Rev. Lett. 72, 1380 ~1994!. 20 The hydrophone calibration is discussed in S. M. Cordry, ‘‘Bjerknes forces and temperature effects in single-bubble sonoluminescence,’’ Ph.D. thesis, University of Mississippi, 1995. 21 S. M. Cordry, ‘‘Bjerknes forces and temperature effects in single-bubble sonoluminescence,’’ Ph.D. thesis, University of Mississippi, 1995. 22 A. Prosperetti, ‘‘Thermal effects and damping mechanisms in the forced radial oscillations of gas bubbles in liquids,’’ J. Acoust. Soc. Am. 61, 17 ~1977!. 23 M. Longuet-Higins ~private communication!.

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