New suggestion concerning the origin of sonoluminescence

Physica D 216 (2006) 136–156www.elsevier.com/locate/physd

New suggestion concerning the origin of sonoluminescence

Bishwajyoti Deya, Serge Aubryb,∗

a Department of Physics, University of Pune, Pune 411 007, Indiab Laboratoire Leon Brillouin (CEA-CNRS), CEA Saclay, 91191 Gif-sur-Yvette Cedex, France

Available online 10 March 2006

Abstract

We suggest a new mechanism where sonoluminescence is produced by the tremendously large adiabatic pressure pulse (shock wave) generatedby the close to supersonic (or above) impact of the fluid on the hard core bubble. The light flash is mostly emitted by the fluid surrounding thebubble.

More generally, the emission spectrum of any material submitted to a large adiabatic compression is (roughly) globally dilated by someGruneisen coefficient γ . Temperature simultaneously increases by the same factor, which increases the power of the emitted radiation by a factorγ 4. A rigorous lower bound for the sound velocity in the compressed region at impact is obtained with purely kinematic arguments only assumingthe existence of a non-negative Van der Waals volume for the fluid. For supersonic impacts, the increase of the sound velocity reaches at leastone order of magnitude (and, with reasonable assumptions, much more), which yields an estimation of the Gruneisen coefficient γ and indicatesit may become very large. Then, during the pressure pulse, the thermal infrared (IR) radiation of the compressed fluid can be extended up tovisible–ultraviolet (UV) simultaneously with an intense brightness.

The dynamics of collapsing bubbles have been analyzed taking into account fluid compressibility. Shock waves are generated when the bubble,at a minimum radius, suddenly becomes almost incompressible. For impacts close to supersonic (or above), an intense pressure is briefly generatedin a sphere which extends beyond the central bubble and which thus mostly contains surrounding fluid compacted to near its Van der Waals volume.This compacted fluid generates an intense emission of UV–visible light which suddenly disappears when the fluid expands from its Van der Waalvolume. This situation occurs when the sphere of compacted fluid reaches a critical size of a few minimum bubble radii. Next, this pressurepulse radially propagates through the fluid, initially at highly supersonic velocities, which decay to the normal sound velocity as it simultaneouslyspreads out.c© 2006 Elsevier B.V. All rights reserved.

Keywords: Sonoluminescence; Nonlinear acoustics; Shock waves; Gruneisen coefficient

1. Introduction

Sonoluminescence (SL) is a process by which light isemitted from collapsing ultrasound-driven bubbles. There aretwo main types of SL: multiple bubble sonoluminescence(MBSL) and single bubble sonoluminescence (SBSL). SL ischaracterized by the emission of very short flashes of broad-band UV light that is synchronous with the periodic acousticdriving field. The study of SL gained momentum after thediscovery of SBSL by Gaitan [1]. SBSL is much easier to study,since the bubble can be extremely stable and glow for many

∗ Corresponding author. Tel.: +33 33169086128; fax: +33 3169088261.E-mail addresses: [email protected] (B. Dey),

[email protected] (S. Aubry).

0167-2789/$ - see front matter c© 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.physd.2005.12.012

minutes, making it possible to study both the bubble and thelight that it emits. In contrast, it is very difficult to study MBSL,as individual bubbles last only for a few acoustic cycles and arein constant motion. In the years since SBSL was discovered,a lot of studies have been carried out, both experimental andtheoretical, to understand how and why SL occurs. Variousmechanisms, such as black-body radiation, Bremsstrahlungradiation from a hot plasma core, electrical mechanisms such asfractoluminescence, the Casimir effect, radiation from quantumtunneling, collision-induced emission, etc., have been proposedas the mechanism for light emission. Although these modelsgo a long way to describing various aspects of SL, recentexperiments have exposed serious limitations of the presentunderstanding of SL.

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B. Dey, S. Aubry / Physica D 216 (2006) 136–156 137

We do not consider that the processes of black-body,Bremsstrahlung, collision-induced, cooperative emission froma shock wave induced plasma inside the bubble etc. are themechanisms responsible for SL. This is because, as mentionedbelow, the observations/predictions of most of these theoriesinvolving one or more of these mechanisms do not agree withlater experimental findings. In this paper, we propose a newinterpretation for SL. Our theory is consistent with the findingsof the most recent experiments (see Section 2 below) of SBSLor MBSL, in either aqueous or non-aqueous media.

Unlike other models (see Section 3 below) which place thelocation of light emission only inside the bubble, we believethat, in the case of SL, light is mostly emitted by the fluidsurrounding the collapsing bubble. Actually, we suggest that SLis a particularly favorable experimental situation for studying amore general phenomenon that may occur in conditions otherthan bubble collapse. The emission of brief flashes of lightshould be observed in any material (solid, fluid, gas, or evencold plasma) providing that a tremendously high adiabaticpressure pulse could be produced. Such pressure pulses aretypically obtained at shock waves generated by impacts closeto or above supersonic.

Under a tremendously large pressure that is applied briefly,the whole frequency spectrum and the temperature of a givenmaterial are roughly multiplied by the same large factor γ � 1.This assumption is fulfilled under the common assumption thatthe Gruneisen parameter of a mode ν, which is usually definedas γν = −d ln ων/d ln V (where ων is the mode frequencyand V is the volume), is the same for all modes. Then, theexcitation energies Eν = hων are simply multiplied by thesame factor. Thus, the compressed material becomes hotter andtherefore brighter. It emits a flash of thermal radiation overan extended range of frequencies (multiplied by the factor γ

compared with its normal spectrum) with a larger intensity(roughly proportional to γ 4).

At the present stage, the shape and parameters ofthe pressure pulse cannot be calculated precisely, becausethe equation of state of materials are unknown at thosetremendously large pressure. Nevertheless, simple kinematicarguments allows one to scale the order of magnitude of theGruneisen shift and the pressure that could be expected fromonly the knowledge of the impact velocity and the Van derWaals contraction rate.

This paper is organized as follows: in Section 2, we mentionthe recent experiments that provide some evidence that a largepart of SL light may be emitted from the liquid surrounding thebubble; in Section 3, we mention the main early theories; inSection 4, we justify and discuss the assumptions of our model;in Section 5, we discuss the modelling of bubble dynamics,which refers to the appendices; in Section 6, we explain whyan adiabatic pressure pulse produces light; and finally Section 7is devoted to concluding remarks. The Appendix describes thetechnical calculations supporting our arguments: Appendix Arecalls the Rayleigh–Plesset theory for bubble dynamics anddiscusses its limit of validity; Appendix B describes thepropagation of the pressure front and the supersonic impact inone dimension; Appendix C discusses the 3D spherical case.

2. Experimental facts

The main experimental results that have been obtained sofar from SBSL experiments can be summarized as follows (fordetails, see the recent reviews [2–5]):

(1) The sonoluminiscent bubble oscillates with the oscillationfrequency of the sinusoidal driving force (typically20–40 kHz). The forcing pressure Pa

P0∼ 1.2–1.4, where

Pa is the pressure amplitude of the sound wave and P0 isthe ambient pressure (1 bar).

(2) Once during each oscillation period the bubble collapsesvery rapidly from its maximum radius Rmax (typically10R0, where R0 ∼ 5 µm is the undriven radius of thebubble) to a minimum radius of Rmin (typically R0/10)and the flash of light is emitted. After light emission,the bubble pulsates freely until it comes to rest, and thenthe whole process is repeated in following cycles. Boththe light intensity and amplitude of oscillation of thebubble depend sensitively not only on the forcing pressureamplitude but also on the concentration and type of the gasdissolved in the liquid.

(3) The SL spectrum displays a strong UV spectrum, as wellas sensitivity to temperature. SL in colder water makesfor much larger light emission. Typically, as the water iscooled from 30 to 0 ◦C, the intensity of SL increases by afactor of ∼100.

(4) The effect of changing the acoustic frequency (typically20–40 kHz) appears to be comparatively small.

(5) Flash width is typically of the order of a few hundredpicoseconds and varies with external parameters such asforcing pressure and dissolved concentration of the gas.Flash width is independent of the color of light emitted [6].

(6) The presence of some noble gas increases SL. There is apronounced decrease in SBSL intensity from the heavierto the lighter noble gas species in the order Xe → Kr →

Ar. However, Ne and He spectra are not much different,even though He spectra are sometimes more intense thanNe spectra.

(7) Besides light, there is also sound emission in SL. Thevelocity of the outgoing shock wave in the immediatevicinity of the collapsing bubble (∼4000 m/s) is muchfaster than the speed of sound (c = 1430 m/s) in waterunder normal condition. The high shock speed originatesfrom the strong compression of the fluid around thebubble at collapse. The pressure at the surface of thecollapsing bubble just before it reaches the minimum sizehas been inferred from needle hydrophone measurementsand computation to be over 6000 bar.

(8) Symmetry of the collapsing bubble plays a role. Anincrease in spherical symmetry of the collapsing bubbleunder controlled conditions leads to more light.

(9) Observation of the isotopic effect SL. Most SL spectraexhibit a peak wavelength below 290 nm. However, insystems such as H2 or D2 dissolved in D2O, the peakwavelength is observed to be shifted far to the red, witha peak spectral radiance at 400 nm [7]. The authors [7]could not explain the origin of the remarkably large shift,

138 B. Dey, S. Aubry / Physica D 216 (2006) 136–156

especially in view of the small difference in chemical andelastic properties between light and heavy water.

(10) Recent Mie scattering experiments [8] show that, in thelast nanosecond around the minimum velocities, most ofthe Mie scattering is by the highly compressed wateraround the bubble and not at the bubble wall.

(11) Yet another experiment by Matula et al. [9] on MBSLshowed the spectral lines from metal ions dissolved inwater. Since metal ions are non-volatile, the origin of themetal ion lines is a puzzle. One possibility is that themetal ions get into the bubble on its asymmetric collapse;another is that the SL is produced from the water outsidethe bubble.

(12) The most recent experiment on the measurement of OHlines in the spectrum of SBSL by Young et al. [10] andBaghdassarian et al. [11] supports the latter view. Theauthors could not explain the origin of the OH lines, butsuggested that it may be due to instabilities in the collapseof the large bubble. Alternately, the appearance of the OHin the SL spectra may be a signature of SL produced fromthe water outside the bubble.

(13) The needle hydrophone measurements [5] and computa-tion [12] show that the pressure at the surface of the bub-ble in the final nanosecond before reaching the minimumsize is over 6000 bar.

(14) Didenko et al. [13] generated SBSL in various liquids(aqueous and non-aqueous) such as adiponitrile. More-over, similar to the observed OH lines for water, they [13]observed the excited CN lines, one of the groups that makeup the adiponitrile. Similarly, other non-aqueous liquids(having an OH group) like ethanol and a mixture of waterwith hydrogen peroxide or glycerine also show strong SLemission [14]. Of course, for better SL effect, one has tochoose a liquid with very low vapor pressure and a chem-ical structure that results in dissociation products that areeasily soluble in the liquid. This is because the liquid vaporcan be responsible for quenching SL and, also, the chem-istry of the dissociation product plays an important role inthe bubble stability [2].

(15) Recent experimental results of SL in high magneticfields [15] show a strange interplay between acoustic andmagnetic fields, with an increased region of stability underthe magnetic field.

3. A short review of early theories

There are many theories available in the literature thatattempt to deal with the mechanisms of SL, but there areno theories so far that do not suffer from some contradictionwith experimental facts and can completely account for allthe main experimental properties of SL as summarized above.Many theories, developed on the basis of the experimentalresults available at the time, later became questionable as newexperimental results became available.

For example, a theory considering a shock wave focusinga large amount of energy at the bubble center then generatingblack-body radiation as the cause of SL does not appearany more to be a correct candidate for SL. These shock

wave/plasma Bremsstrahlung theories were motivated by themeasurement of Barber et al. [16], which indicated that thewidth of the SBSL was shorter than 50 ps, as shock focussingprovides a natural mechanism for producing both extremelyhigh temperature and small pulse width. Actually, we shallshow in Appendix C, that due to high nonlinearities, shockwave focusing is stopped when the fluids approach their Vande Waals volumes.

Later, more accurate measurement of the pulse width wentagainst this shock wave theory of light emission. It wasdiscovered by Gompf et al. [6] that the measured width of thelight pulse is actually of the order of a few hundred picoseconds,much longer than the 50 ps upper bound measured by Barberet al. [16]. The much longer duration of the light pulse, asdiscovered by Gompf et al. [6], was later verified by Moranet al. [17] and Hiller et al. [18]. The same set of experiments [6,18] also proved that the black-body radiation models of SLare also questionable. Gompf et al. [6] and Hiller et al. [18]discovered that there is hardly any variation in the pulse widthwith the wavelength of the emitted radiation. This contradictedthermal models favoring black-body radiation, which predict alarge wavelength dependence of the pulse width (the red pulseduration is more than twice that of the UV pulse).

Thermal Bremsstrahlung theories also require an assumptionthat the temperature inside the bubble is extremely high∼106 K [19]. But neither an imploding shock nor a plasmaor such a high assumed temperature inside the bubble has yetbeen detected in any SL experiments. Very recently, however,there was a report [20] of observations of SL light emissionoriginating from collisions with high-energy electrons, ionsor particles from a hot plasma core. However, the measuredtemperature from the emission spectra is found to be very low(4000–15 000 K for Ar emission) and therefore inconsistentwith any such thermal process. The recent demonstration ofthe existence of chemical reactions within a single bubble [21]shows that the temperature reached inside the bubble duringcavitation may be substantially limited by the endothermicchemical reactions of the molecules inside the bubble. Also,outgoing pulses have been observed [5], but they would beemitted whether or not there is an imploding shock. Even if ashock wave strong enough for generating plasma could developfor large enough forcing, it would be limited to a tiny regionaround the bubble center and would last too short a time.

Another model ( jet theory) invoked an electrical mechanismfor light emission (fractoluminescence). This idea [22] requiresan asymmetric collapse of the bubble, during which jetsof water enter inside the bubble at very high speed.Sonoluminescence is produced by the collision of the jetwith the bubble wall, which would initiate fractoluminescence.However, it is now well confirmed experimentally that thebest is the bubble spherical symmetry, which favors SL themost. Otherwise, this model relies on specific properties ofwater and did not find favor with later experiments [13], whichshowed that bright SBSL is possible even in non-aqueousliquids.

Yet another line of research that involved the dynamicalCasimir effect as a potential photon-emission process [23,24]


had to be abandoned later when a calculation based on theactual model showed that the Casimir energy is of the orderof a magnitude that is too small to be relevant to SL [25]; inorder to match the observed light intensities, the bubble wallspeed would exceed the speed of light [26].

Other light emitting mechanism models include radiationfrom quantum tunneling [27] and the collision-inducedemission process [28]. However, the parameter values used forestimating the light emitted in the collision induced model [28](for example, the assumption that the life time of the radiation is1 ps) do not agree with the more recent experimental values [6].

It may also be appropriate here to mention other models thatplace the location of light emission in the surrounding fluid,with mechanisms different to our’s for SL. These are the proton-tunneling radiation model [27] and the electrical breakdownmodel of Garcia [29].

Thus, it is clear from these examples that the assump-tions/predictions of the existing theories of SL, which were for-mulated on the basis of experimental results available at thetime, do not agree with the new measurement results and thatnew suggestions would be welcome.

4. Preliminary discussion for a new approach

Our theory is also a shock wave theory, but instead ofassuming that it focuses a large energy density near the bubblecenter, generating a high-temperature plasma, we show thatthis shock wave generates a high-pressure pulse that last muchlonger and concerns a much larger volume, including thebubble and the surrounding fluid. The fluids inside this radiantsphere do not become a plasma, but nevertheless becomesonoluminescent.

4.1. Some very rough physical estimations

Let us first rexamine the order of magnitude of the physicalquantities (energies, temperature, radial bubble velocities etc.)which could be involved in the real experiments on SL.

Let us first consider the bubble temperature. According tothe experimental situation, the minimum diameter of the bubbleis about 10 times smaller than its diameter at rest, whichroughly corresponds to the minimum volume of the gas beforeit becomes hard core and almost incompressible. Then weassume the most favorable conditions for heating the gas insidethe bubble, that is:

• the gas in the bubble is a monoatomic gas and is perfectduring the whole compression;

• there is no heat diffusion in the fluid (the compression isperfectly adiabatic);

• the energy absorbed by the dissolution of the high-pressuregas inside the water is negligible (or, equivalently, the gas inthe bubble is hardly soluble in water).

The ratio Tm/Tr of the bubble temperature Tm at theminimum radius Rm with the temperature Tr ≈ 300 K atequilibrium radius Rr fulfills

Tm

Tr=

(Vm

Vr

)1−γ

=

(Rm

Rr

)3(1−γ )

where γ =53 for a monoatomic gas, and we get Tm ∼

30 000 K ∼ 3 eV. This result is a strict upper bound onthe final temperature inside the bubble, assuming that it hasuniform temperature. This overestimates the heating of thebubble, because the above assumptions are likely to be notfulfilled very well. Some real experiments and more accuratetheoretical calculations, which suggest that the maximumbubble temperature may not exceed 10 000–20 000 K [19,22],are consistent with our estimation.

Since the ionization energy of the rare gas (12.13 eV forfree xenon atoms and higher for other rare gas) inside thebubble is quite a lot higher than this thermal energy, the gasinside the bubble should not be substantially ionized withoutan extra mechanism for energy focusing. It cannot become aplasma, as suggested in some papers. Actually, experimentsshow that a substantial part of the energy injected by thepressure force is not used for heating the inside of the bubble,but is transiently converted into kinetic energy for the fluid (seeAppendix A). The order of magnitude of the energy W injectedby the external applied pressure during bubble collapse can beroughly estimated. We have W = P1V , where the pressureP ∼ 105 Pa is of the order of the atmospheric pressure, and thevolume variation 1V =

4π3 (R3

M − R3m) ≈

4π3 R3

M , where RM(resp. Rm) is the maximum (resp. minimum) bubble diameter.According to experiments, we may choose RM = 25 × 10−6 mand Rm = 25 × 10−8 m ∼ 0, which is W ∼ 6.25 × 10−9 J ∼

40 GeV. This is roughly the scale of the whole energy availablefor the SL process. In good experimental conditions, the kineticenergy of the fluid may reach an important fraction (typicallya half) of this energy. Since the kinetic energy of the wateris 2πρ R2 R3 as a function of bubble radius R and velocity R(see Appendix A), we obtain that, when this radius is almostminimum (R ≈ Rm = 25 × 10−8 m), R ∼ 6 km/s, which iscomparable to the velocities observed in real experiments [30].Most of this kinetic energy will be released in the shock wavegenerated by the supersonic impact of the fluid on the hard corebubble. The order of magnitude of the pressure peak due to thisshock wave, which is comparatively very high, is estimated inAppendices B and C.

Some authors speculatively suggested that the possibilityof nuclear fusion [31] could be generated by such implodingbubbles. Such a phenomena would require that a substantialamount of the available energy (∼40 GeV) generated frombubble collapse is focused over quite a small number of atoms.Although, for nearly linear theories and for a perfectly sphericalimpact, temperature and pressure near the bubble center shoulddiverge, it looks unlikely that energy could be sufficientlywell focused (at the atomic scale!) for nuclear fusion tooccur. Actually, most the available energy is involved in thecompression of a relatively large volume of fluid spreadingroughly uniformly much beyond the bubble core and thus doesnot focus at the center.

Our basic argument is to prove, by purely kinematicconsiderations, that this tremendous pressure peak mustbe associated with a tremendous hardening of the atomicinteractions which may raise the sound velocity in thecompressed material by several orders of magnitude (this


feature holds whatever the temperature and the state of thehighly compressed material could become: solid, liquid, orplasma). This only considers that mass is conserved and thatthe volume of a fluid element cannot become negative. Noprecise knowledge of the equation of state of the involved fluidsis required. We also assume, for simplicity, that this volumedecays monotonically as a function of pressure (that is, phaseseparation is absent). To be more realistic, assuming that avolume element cannot be reduced below a certain Van derWaals minimum (positive) volume will yield more tightenedbounds, favoring the occurrence of SL at impacts.

Although still at the qualitative level, the theory that wepropose is consistent with all the many different experimentson SL undertaken to date. Our theory simply relates SL to theadiabatic compression of standard matter at tremendously highpressure. SL involves the bubble, but also mostly a relativelylarge volume of the surrounding fluid, which is assumed toremain fluid.

4.2. Optimized conditions for sonoluminescence within ourtheory

We now have to consider the order of magnitude ofsome essential physical parameters in our theory. We set theassumptions required for our model precisely. We assume that,at the end of bubble collapse, when the bubble radius velocityis a maximum:

• The bubble radius shrinks at a very large velocity that ishighly supersonic or of the order of the sound velocity inthe fluid.

• The gas inside the bubble becomes a dense liquid nearmolecular or atomic close packing, with a compressibilitycomparable to the compressibility of the surrounding liquid.

Then, there is an impact between the bubble hard core andthe fluid, which generates a shock wave. We shall show that,very generally, any supersonic impact systematically generatesluminescent shock waves. According to our scenario, thebrightest SL requires the largest pressure pulse. The largestpressure pulse requires the largest impact velocity and, at thetime of impact, the fluids are the most incompressible they canpossibly be.

For the largest impact velocity, the maximum work providedby the driving pressure during the change of bubble volumeshould be converted into kinetic energy for the collapsingfluid. This is a problem of fluid mechanics, which involvesthe intrinsic properties of the fluid and of the gas inside thebubble. It also involves the frequency and amplitude of theapplied force. Experimentalists have already found empiricallythe conditions that realize sharp supersonic impacts.

Among the important issue is that the energy losses due tofluid viscosity should be minimized for better impact. In orderto minimize the friction forces in the flowing fluid, good bubblesphericity is highly favorable, since the standard dissipativeterms (A.3) in the Navier–Stokes equation vanish (at the lowestorder) for an incompressible fluid when the flow has perfectspherical symmetry. Breaking sphericity should diminish the

bubble velocity at impact and consequently SL. Thus, moreintense SL is favored when spherical bubbles are the moststable. In that respect, small bubbles are more favorable for SL,because their sphericity is better stabilized by surface tension.

Bubbles of rare gas are also more favorable. Because theatoms of a rare gas are chemically inert, they interact weaklywith water molecules, which make them barely soluble in waterwith large surface tension. Moreover, since they are barelysoluble in water, there is almost no extra energy dissipationof the bubble kinetic energy due to the partial dissolution ofthe gas in water which could occur during compression. Sincesolubility increases with temperature, it is also preferable towork with water at the lowest possible temperature, close to0 ◦C.

It is interesting to note that energy dissipation during bubblecollapse has been increased artificially by applying a strongmagnetic field to water. It was found that the fast motion of thedipolar water molecules generate a torque on these molecules,which induces energy dissipation. Then, SL may be reducedor suppressed [32]. As mentioned in Section 2 above, all theseempirical features seems to be confirmed experimentally in theexperiments producing SL.

The second condition is again favored by rare gases, whichare monoatomic with complete electronic shells. Electronicoverlaps between nearest atoms cost a very large amount ofenergy, which makes these materials barely compressible in thesolid or liquid state when the atoms pack near to maximumcompactness. The hardness grows for rare gases with increasingmass, i.e. He, Ne, Kr, Xe.

However, the situation for water is special, because it isa relatively compressible liquid. The reason for this is thatit consists of a soft hydrogen bond network between oxygenatoms, leaving many empty spaces. Indeed, phase transitionshave been induced at low temperature from low-density ice toamorphous high-density ice with a density 1.19 at pressures ofabout 6 kbar ∼ 6 × 108 Pa [33,34], which corresponds to avolume contraction compared to normal water of about 20%.Despite the absence of first order transition in the liquid phase,we may reasonably expect, at the yet more gigantic pressuresinvolved in the shock wave, that the volume of water may shrinkby up to 20% (hW = 0.2), corresponding to the minimum Vander Waals volume.

Thus, there is an inversion point at relatively low pressure,where the rare gas suddenly becomes much less compressiblethan water. If this point is crossed during bubble dynamics,when water has already accumulated a large amount of kineticenergy, one gets an impact of water on the hard anvil formed bythe compacted rare gas.

In summary, within our approach of SL, there are threedifferent problems to analyze with more details for properlydescribing SL.

• The first problem is related to hydrodynamics. How do weproduce stable bubbles that reach radial velocities near tothe minimum radius which are supersonic and as large aspossible?

• What is the order of magnitude of the amplitude and thewidth of the pressure peak generated by the supersonic


impact of the liquid on the hard core gas in the bubble? Howdoes this pressure pulse propagate and how does it decay?

• Why does a high-pressure pulse produce SL?

We discuss these three points in the following, although ouranalysis will only be partial. We essentially focus on qualitativeaspects and the physical order of magnitudes.

5. Modelling bubble dynamics

The general behavior of the dynamics of a spherical bubbleis quite well described within a Rayleigh–Plesset (RP) model,which is studied in details in [35]. We show in Appendix Athat this model is nothing but an anharmonic oscillator whichis parametrically driven by the external pressure (Duffingoscillator).

When the amplitude of the driving pressure is not too large,the bubble radius oscillates smoothly at the driving frequency.The compressibility of the gas in the bubble always remainslarger than that of the surrounding fluid. Although the drivenbubble oscillator is highly nonlinear, with a motion that is notsine like and may become complex, it does not generate anysubstantial impact (and shock waves) at the minimum radius.The RP model accurately fits the experimental observationsquite well, but then there is no SL.

Beyond a certain critical amplitude of the driving pressure,there is a quite abrupt change in the bubble dynamics, becausethe compressibility of the bubble core becomes of the sameorder or smaller than those of the surrounding fluid. In thatcase, the fluid impacts the bubble at the minimum radiusand bounces back several times, with smaller and smallerimpacts. In that regime, the experimental observations [8] of thebubble dynamics, and especially the bubble radius bouncing,are not well fitted by the RP model. The main reason for thediscrepancy between the observations and the RP model islikely to be due to the extra energy dissipation by the shockwaves which are generated at each impact and carry out someenergy from the bubble oscillator. After these fast oscillationsdamps, the bubble radius grows smoothly again, while thebubble’s hard core returns to gas. The expansion of the bubbleis again well described by an RP model.

When the bubble is at maximum radius, the pressures arestill rather low everywhere in the fluid (and of the order ofthe ambient pressure, at most). At this stage, the gas insidethe bubble is highly compressible, while the fluid (water)outside the bubble is practically incompressible. The pressureinside the bubble is much lower than the increasing drivingpressure, so that the bubble starts to collapse, almost as for avacuum bubble. The radial velocity of the fluid near the bubbleaccelerates to very large velocities. As mentioned above, itmay reach up to four times the sound velocity in the fluid justbefore the minimum bubble radius. When the radial velocitybecomes sufficiently large, the bubble’s collapse ends with asharp impact on the hard core gas. The impact produces theshock wave that propagates in the fluid, as currently observedin experiments [36]. The pressure pulse generated at the firstimpact has been observed travelling radially in the water (orthe fluid) with an initial velocity well above supersonic [36].

SL is observed during the first few hundreds of picosecondsafter its creation. The amplitude of this pressure pulse decaysrapidly, not only as a function of the radial distance but alsobecause it spreads out. The SL flash rapidly cools down to lowerfrequencies, while its brightness disappear. The amplitude ofthe next bounce and the corresponding impacts are muchweaker because of the energy dissipation due the shock wavesof the previous impacts. No substantial SL is generated. This isqualitatively consistent with our theory, since the first impact isthe sharpest and should generate the most SL.

The impact occurs when the gas inside the bubble becomesless compressible than the fluid outside the bubble, whichis close to its Van der Waals radius. The compressibility ofthe fluid (initially assumed to be incompressible in the RPmodel) must be taken into account to describe the bubbleimpact and the travelling pressure pulse in the surroundingfluid. Because of the spherical symmetry, this pressure pulsepropagates radially from the bubble center. Its amplitude decayswhile it also spreads out. In Appendices B and C we model theimpacts in 1D and 3D spherical models, respectively.

When the gas inside the bubble becomes compact andless compressible than the surrounding fluid, heating of thebubble slows down sharply because most of the energy injectedfor compressing compact materials (solid or liquid) thenbecomes essentially potential energy and produces relativelylittle heating. The correct model to be treated to describe theimpact should be an improved RP model, where both thegas inside the bubble and the fluid around the bubble areconsidered to be compressible. Such a complex model shouldbe investigated numerically, but we may, however, draw strongconclusions with simple analytical modeling and arguments.

We first investigate the generation of a pressure pulseby a supersonic impact in a simplified situation where twoidentical compressible fluids in a one-dimensional pipe areinitially launched one against each other at supersonic velocity(the impact of water on water). This is done just by fixingan initial velocity field to an homogeneous fluid at constantpressure P0. This problem is treated in Appendix B. We havechosen a simple initial velocity profile where we have an exactsolution describing the impact, but nevertheless we think thatthe essential result should not depend much on the details ofthe initial conditions. The important physical feature for SLis not the amplitude of the pressure itself but the shift of thesound velocity in the fluid adiabatically submitted to this high-pressure pulse.

The volume of a fluid element of unit volume at pressure P0necessarily decays as a function of pressure, with an asymptoteat 1 − hW . Since it cannot be negative, we have 0 < hW < 1(the maximum Van der Waals contraction rate). We assume,for simplicity, that the sound velocity increases monotonicallyas a function of pressure. It must diverge at infinite pressure.Then, we proved that the sound velocity si in the region ofhigh pressure generated by the impact fulfills the inequalitysi > Vi/hW (see Eq. (B.19)). This inequality does not requireany precise knowledge of the equation of state of the fluid(s).

Actually, this minimum bound for the sound velocity si inthe high-pressure region generated at impact may be widely


Fig. 1. Scheme of the conjectured evolution of a spherical bubble impactdescribed by Eq. (C.7). The grey annulus represents the high-pressure zoneP > PW when it emits light. The dark zone represents the zone of the pressurepeak P < PW which does not emit light. Top left: Just after impact, a regionof high and roughly uniform pressure Pi appears, delimited by two frontsthat propagate at highly supersonic velocities in opposite directions. WeakUV–visible light starts to be emitted. Top right: The inner front reaches thecenter. Pressure is much larger than PW in an expanding sphere delimited bythe outer front moving at highly supersonic velocity. The whole fluid insidethat sphere is adiabatically compressed near to the Van der Waals minimumvolume. Intense UV–visible light is emitted. Bottom left: When the radius ofthe fluid compacted sphere reaches a minimum of a few bubble radii RSL, itsexpansion slows down. The compressed fluid starts to expand at the center,where pressure P ′

0 decreases. UV–visible light decreases. Bottom right (largerscale): When the sphere radius increases beyond RSL, pressure drops belowPW and, especially at the center, P ′

0 ∼ P0. Light emission stops. A high-pressure pulse forms, moves radially and becomes quasi-harmonic when itsvelocity decays to the normal sound velocity, while its profile ceases to spreadand becomes constant. The amplitude of the pressure peak decays as 1/r .

underestimated by this inequality (which is often the case forrigorous bounds). Only in the case of highly compressiblediluted gas, where hW ≈ 1, do we have both si > Vi andsi ∼ Vi . In contrast, dense fluids are barely compressible, sothat the maximum Van de Waals contraction, hw, is small. Inthe case of highly supersonic impacts, the fluid compressionrate approaches the maximum hW , at which the sound velocitydiverges (see Fig. 4). Then the sound velocity si in the high-pressure region generated at impact should reasonably beexpected to be much larger than this rigorous bound. It isthen rather obvious that highly supersonic impacts may shiftthe sound velocity in the compressed region to values whichare several order of magnitudes larger than the normal soundvelocity, s0, and which cannot be realized today in any staticexperiments.

The exact solution considered in Appendix B for provingthis inequality has the flaw of being one dimensional. It does notcorrespond to a pressure pulse but to a high-pressure region thatextends as a function of time by two fronts moving in oppositedirections. Moreover, our initial condition involves an infinite

kinetic energy, which is not realistic for describing a 3D bubbleimpact where the total kinetic energy of the fluid is finite.In 3D, the high-pressure region cannot extend indefinitely butbecomes a pulse. Nevertheless, one may consider that the 1Dimpact model could describe the very first stage of impact,when the high-pressure region is confined to a thin shell aroundthe sphere of impact. Thus, the 1D bound yields a lower boundfor the sound velocity at the impact region and at the time ofimpact (see Fig. 1).

We show in Appendix C that the equation describing animpact within a spherical geometry, considered for the variableu(r, t) = r P(r, t), is formally equivalent to the 1D equation forthe pressure P(r, t) but with inhomogeneous nonlinearities thatbecome stronger and stronger near the center. In the oppositecase, the effect of the nonlinearities disappears at great radialdistance. Unlike the 1D case, a discontinuous pressure step isnot an exact solution anymore.

We should expect that the shape of an initially sharp frontshould change as a function of time mostly close to the bubblecenter. We have not been able to perform an accurate analyticalinvestigation of the evolution of the pressure pulse generatedby a radial impact of the fluid on a sphere. A numericalinvestigation should be performed later.

Nevertheless, using the same strong argument of fluidmass conservation as in the 1D case, we can make empiricalpredictions on the size evolution of the highly compressedregion (beyond the Van der Waals pressure) before it spreadsout and decays. We found that in the experimental conditions,a highly compacted sphere involving the bubble core and thesurrounding fluid near Van der Waals volume should form justafter bubble collapse. Its pressure is roughly uniform and nondiverging at bubble center. There is emission of light till theradius of this compacted sphere grows up to a maximum sizewhich corresponds to a sphere called radiance sphere withradius RSL which extends over few minimum bubble radiusRm . This result implies that most SL should be generated bythe fluid surrounding the bubble inside this radiance sphere.Pressures larger than the Van der Waals pressure will persistin that region only for a few hundred picoseconds. Accordingto our theory, when the pressure in the fluid is larger than itsVan der Waals pressure, the fluid should emit a broad band oflight with characteristic frequencies that drop from UV to IR(when the pressure decays). Beyond that distance, the pressurepeak becomes of the order of, or smaller than, the Van der Waalspressure. Then, this compacted sphere ceases to emit light andbegins to behave quasilinearly. First, its pressure drops at thecenter and the resulting spherical pressure pulse propagatesand decays as the inverse distance from bubble center, whileit velocity drops to the normal sound velocity in the fluid. Thisevolution is pictured schematically in Fig. 1.

6. Adiabatic high-pressure pulse generates light emission

The previous section, together with the three appendices,was devoted to the demonstration that

• Supersonic impacts can be generated by bubble collapse.


• Then, the pressure pulse (shock wave) that is generatedpropagates radially, initially at highly supersonic velocityin the fluid surrounding the bubble. It decays abruptly andslows down to the normal sound velocity within a few bubbleradii.

• The Gruneisen shift of the sound velocity in the fluid at thepressure pulse is very large within a few bubble radii, butdrops to one beyond this critical distance.

The question to be discussed here is to explain why a largesound velocity shift should produce SL.

First, the fluid compression should be considered to bepractically adiabatic, considering the scale in time and spaceof the pressure pulse that emits the light flash. In order to firmup the ideas, an exact spherical solution of the Fourier heat

equation yields the temperature T (t) =T0

(t+t0)3/2 exp −|r|2

4D(t+t0)of a hot spot as a function of time t and radius r , where Dis the heat diffusion constant and t0 is an arbitrary constant.At t = 0, the radius of the hot spot is r0 = 2

√Dt0 and the

temperature at r = 0 is T (0) =T0

t3/20

. It is divided by two after

a time τ(r0) = (22/3− 1)t0 = (22/3

− 1)r2

04D . Then, for a hot

spot with an initial radius of r0 = 0.25 µm in water, whereD ≈ 1.4 × 10−7 m2/s, its characteristic time τ to cool downto half the temperature is τ ≈ 1.6 × 10−7 s, which is threeorders of magnitude longer than the characteristic time for theSL flash. Thus we may neglect heat diffusion during the bubbleimpact.

6.1. Discussion about the concept of adiabaticity

When an element of fluid is submitted to a travellingpressure pulse, the molecules in it are suddenly compressedand then relaxed within a few hundred picoseconds. Although,for simplicity, we may model the pulse as being delimitedby two fronts that are a pure discontinuity (which is not anexact solution of the 3D spherical model), the pressure pulseshould correspond at the microscopic level to a slow pressurevariation, with a characteristic time much longer than the fastIR vibrations of the molecules, which means that they aresubmitted to an adiabatic compression.

We assume that the energies that are involved in bubblecollapse are insufficient for substantial ionization of the fluid(except perhaps within a negligible volume near the bubblecenter, which we may discard). This assumption requires alarge gap between the electronic excitations and the otheratomic modes, that is, the fluids involved in bubble collapseare good insulators. We consider that the fluid remainsstandard insulating condensed matter, where the atoms interactwith effective Born–Oppenheimer potentials and where theelectronic excitations are not directly involved.

During an adiabatic compression, a part of the energyprovided to the material becomes potential energy for theatomic interaction, and another part becomes kinetic energyfor the atoms. These are very opposite situations. For example,the energy released during the adiabatic compression of aperfect monoatomic gas is totally converted into the atomic

kinetic energy, since there is no atomic potential. This increasein the kinetic energy is directly related to the increase intemperature, which is thus a maximum. For a polyatomicgas, a given compression energy produces less heating, sincethere are intramolecular potentials that generate intramolecularmodes and capture part of the injected energy. The oppositesituation is obtained for a solid or any dense material. In thelimiting case, where this material is at 0 K, all the compressionenergy essentially goes into the potential energy of the atomicinteractions (elastic interactions) with no heating at all.

From the thermodynamical point of view, any material maybe described very generally by a collection of quantum levelsα at energy Eα , which completely defines its partition functionZ(β) =

∑α e−βEα where the temperature is T = 1/(kBβ). At

finite temperature, each quantum level α is occupied with theprobability fα = e−βEα/Z .

These energy levels Ei obviously depend on the atomicinteractions and consequently on the adiabatic change involumes. However, we have to refine the concept ofadiabaticity.

In thermodynamics, a compression is said to be adiabatic:

• when there is no exchange of heat;• when the volume variation is sufficiently slow, in order

that the system can be considered to always be at thermalequilibrium — its temperature varies as a function ofvolume.

We call this concept thermodynamical adiabaticity, whichshould be distinguished from the concept of dynamicaladiabaticity, well known in the theory of dynamical systems,which we recall now.

If we consider our system as only a large classical dynamicalsystem with external parameters (in our case, the volume v),and if we assume for simplicity that this system is integrable,1 itcan be described by a set of action-angle variables {Ii , θi } with aHamiltonian H({Ii }, v) which only depends on the actions andthe external parameters. Then, any trajectory is a quasiperiodicorbit involving a set of frequencies ωi ({Ii }, v) = ∂ H/∂ Ii .When model parameters vary slowly as a function of time(referring to those periods of oscillations), it is well knownthat the quasiperiodic orbits evolve slowly at constant actionsIi so that their frequencies ωi ({Ii }, v) change. If Ii are initiallychosen at thermal equilibrium, this system may become out ofequilibrium after a dynamically adiabatic transformation.

Actually, this action conservation law has a simple analogin the quantum case. When the system is initially in thequantum state α with energy Eα , it remains in that state whenthe external parameters vary slowly.2 The consequence is thatthe occupation number of each quantum state is unchangedduring a dynamically adiabatic transformation. Assuming thatthis quantum system is initially at thermal equilibrium at

1 For example, it is commonly assumed that the small amplitude dynamicsof systems close to the ground state can be described by harmonic phonons.

2 Within the standard semiclassical quantization, which consists ofquantizing the actions Ii = pi h as integer multiple pi of h, the conservation ofactions Ii means the conservation of the quantum numbers α = {pi }.


temperature T = 1/(kBβ), each quantum state α is occupiedwith probability fα . When the volume varies slowly, theoccupation probability fα of each quantum state must remainconstant, despite the energy level Eα changing into E ′

α . Butthen the system is a priori no longer at thermal equilibriumat some new temperature T ′, because these probabilities fαshould be determined from the new distribution of quantumlevels {E ′

α} instead of the old distribution.However, if there exists β ′ such that fα = e−βEα/Z =

e−β ′ E ′α/Z , this system is again at thermal equilibrium at the

new temperature T ′= 1/(kBβ ′). This condition requires

E ′α

Eα

=T ′

T= γ (1)

i.e. that the energy changes of all the excitations scale with thesame factor γ .

In principle, we have dynamical adiabaticity for veryfast volume variation (but not too fast, at the atomicscale) when thermalization does not occur locally. We havethermodynamical adiabaticy for medium-fast volume variation.The transformation is isothermal when the volume variation isextremely slow. All the intermediate situations are possible. SLshould occur only when volume variation is fast enough to haveeither dynamical or thermodynamical adiabaticity. We ignorewhich regime applies at the bubble impact that generates SL,but this question is not essential at the present stage.

In any case, no SL should be expected in the isothermalsituation when a tremendously large (quasi-static) pressure isapplied to the material.

6.2. Spectral rescaling and light emission

The variation of a quantum energy level Eα with respect tovolume v is usually characterized by a Gruneisen coefficientwhich a priori depends on the quantum excited state α:

γα = −d ln Eα

d ln v(2)

which yields, for a volume change from v to v′,

lnE ′

α

Eα

= −

∫ v′

v

γα(ζ )

ζdζ. (3)

With the assumption that γα is independent of α, the secondmember of Eq. (3) is independent of α, and thus condition (1)is fulfilled. Then, dynamical adiabaticity and thermodynamicaladiabaticity are strictly equivalent concepts only when theGruneisen coefficient γα is independent of the consideredmodes.

There are many simple physical models where the energiesof the excited states scale perfectly as a function of volume.For example, this property is found to be exact for a perfectmonoatomic gas viewed as non-interacting quantum particles ina box. In that case, the quantum energies Eα vary proportionallyto v−2/3. Another example concerns periodic monoatomiccrystals, where the phonon spectrum depends only of oneatomic parameter supposed to depend on volume. Then the

whole phonon spectrum is just rescaled with a constant factorwhen volume varies.

However, there are minor exceptions. Some vibrationmodes in ice (or water?) are known to exhibit anomaliesin their Gruneisen coefficients over a pressure range that isrelatively low on the scale of pressures involved in supersonicimpacts [34]. This is due to the presence of soft modesassociated with structural phase transitions concerning thehydrogen bond network. However, it is worthwhile recallingthat the kinetic energy carried by a single molecule at impact isof the order of several eV and is thus quite a lot larger than thehydrogen bond energy (a fraction of an eV). In a first approach,we may thus discard the role of possible phase transitions andthe small associated anomalies of some Gruneisen coefficients.In any case, this remark does not concern other liquids withoutphase transitions and where SL nevertheless also exists.

If the Gruneisen coefficients are not mode independentand if the pressure variation occurring at the passage of thepressure pulse is relatively fast, we should only have dynamicaladiabaticity. The pressure variation should be sufficientlyslow to have thermodynamical adiabaticity in order thatthermalization (independently) occurs in each fluid element. Inany case, we should expect SL, although the spectral emissioncould be slightly modified according to the assumption whichis made.

The supersonic impact of the bubble brings each element ofthe fluid up to a pressure beyond the Van der Waals pressureinside a sphere with radius RSL extending over a few bubbleradii (see Appendix C and Fig. 1). The pressure peak dependson the distance from the bubble center and lasts a time tSL onthe order of a few hundred picoseconds. As a consequence, theenergies of the quantum states of the fluid elements will alsovary.

We have seen above that the assumption that the Gruneisencoefficient γα does not depend on the mode α is not a severeapproximation that could change our global predictions aboutSL qualitatively. Then, the order of magnitude of the Gruneisenshift γ can simply be obtained from the frequency shift of theacoustic modes or, equivalently, the sound velocities γ ∼ si/s0.We have proved above that the sound velocity si at the pressurepeak created at supersonic bubble impact is necessarily larger,by one or several orders of magnitude, than the normal soundvelocity s0. Consequently, γ should range from at least 10 toperhaps 100 or more, depending on the strength of the impact.The important fact is that the whole frequency spectrum of thisfluid element is roughly multiplied by the same factor, as wellas its temperature.

According to the standard laws of black-body radiation, thetypical frequency of its black-body spectrum is extended by afactor γ (Wien’s law) and its global intensity is increased byγ 4 (Stefan’s law). However, the Planck model for black-bodyradiation assumes a simple continuous frequency spectrumextending up to infinity, which does not take into accountthe real frequency spectrum of the material. It is well knownthat the vibrational spectrum of any insulating material atnormal atmospheric pressure is limited up to IR frequenciesnot exceeding ∼0.5 eV. Above that threshold, there is a large


Fig. 2. Scheme of intensity versus frequency, showing the change in lightemission of the fluid under adiabatic pressure. Before impact, the thermalemission (dark grey area) of the cold fluid is mostly due to the vibrationalspectrum in the IR range. There is also a negligible electronic contributionin the UV range. The fluid is transparent in a visible frequency gap betweenthese two contributions. At impact, the pressure and temperature of the fluidelements near the bubble center simultaneously increase during a short instant(a fraction of a nanosecond) and drop down. During that instant, the thermalIR spectrum of the fluid (light grey) is dilated by a large Gruneisen coefficient∗γ up to UV frequencies, while its intensity is drastically magnified by a factor∗γ 4. The emitted light is screened by the surrounding cold fluid, except forthe frequencies in the transparency gap. The electronic contribution remainsnegligible, since practically no electronic excitations were initially present inthe cold fluid.

frequency gap between the vibrational IR frequencies and themuch higher UV electronic excitation energies. In that gap, theinsulating fluid at normal pressure is transparent and cannotemit any black-body radiation. When the fluid element isbrought simultaneously to high pressure and high temperature,its IR spectrum is dilated upwards and covers the transparencygap (see the scheme in Fig. 2). Then, the compressed fluidemits in the frequency gap where the fluid at normal pressure istransparent, which allows the emitted light not to be absorbedby the surrounding fluid at normal pressure and to be observablefar from the bubble center.

Since it is not absolutely granted that the fluid submitted tothe pressure pulse is at thermal equilibrium, we note, to be morerigorous, that if the compression is only dynamically adiabatic,the radiation power emitted by each mode α at frequency ωα

is nevertheless proportional both to its occupation probabilityfα and to its frequency at power 4 ω4

α , according to Fermi’sgolden rule [37]. If we assume that the dipole of this mode isnot substantially changed by the compression, fα should not bechanged, while its frequency is multiplied by γα . As a result, thepower emitted by this mode is multiplied by γ 4

α and is shiftedat frequency γαωα .

In the case γα = γ independent of α, the global powerof the emitted spectrum is multiplied by γ 4, which is thesame result as those given by Stefan’s law for black-bodyradiation. The spectrum of emission and its intensity coulddiffer substantially from those due to purely thermal radiationwhen γα is not independent of α, but nevertheless the results

will be qualitatively the same concerning an intense emissionof light.

To explain the (almost) frequency independence of thepulse width, various early theories have been proposed,such as temperature-dependent photon absorption of thegas [38], and weak dependence of absorbtion and emissioncoefficients on wavelength but strong (exponential) dependenceon temperature [39] etc.

Our explanation of the frequency independence of the pulsewidth is simply related to the sharp pressure dependence ofthe Gruneisen parameter γ as a function of pressure due to thefact the potential V(v) should reasonably be assumed to divergevery sharply at v = −hW when the fluid is very close to its Vander Waals volume. When the pressure increases beyond PW , thecurve F(−P) becomes very flat and close to its asymptote (seeFig. 4), which implies that the ratio between the sound velocityin the high-pressure region and the corresponding front velocitydeparts from unity and diverges sharply (take, as an example,atomic interactions described at short distance by a Lennard-Jones potential). Then, although the space-time variation frontvelocities and the associated pressures may be relatively smooth(at the scale of the experiment), the Gruneisen coefficient γ

that simultaneously controls the change of temperature and theemission spectrum of the fluid element falls abruptly to unity atthe crossover pressure PW . Consequently, the light emission ofthis fluid element switches off abruptly almost simultaneouslyfor all observable frequencies as soon its pressure decaysbelow PW . However, this result is only valid for observablefrequencies that are not too large (and belonging to thetransparency gap of the surrounding normal fluid).

Then, when the radius of the sphere of the compactedfluid expands beyond RSL, the highly compacted fluid starts torelax from the Van der Waals volume and the light emissionprogressively switches off shell by shell, but nevertheless thelight emission decays quasi-identically for all frequencies.However, our model may be refined, because light emissiondoes not really switch off instantaneously. The consequence isthat the lower frequencies should exhibit a slightly prolongedtail, as mentioned in certain experiments [17].

In our model, we assume that the characteristic time of thepressure pulse tSL is sufficiently short so that each fluid elementcannot emit enough thermal radiation (the SL light flash) tocool down by itself. Its temperature is essentially driven by itspressure, which depends on time and distance from the bubblecenter. As a result, the emitted SL light is a superposition of themany different emission spectra of the fluid elements (whichdepends on their pressure profile versus time) in the rather well-defined radiance sphere with radius RSL.

This energy for light emission is borrowed from part of thecompression energy, a part being used for shifting to higherenergy the occupied quantum excitations of the initial fluid atnormal pressure and temperature.

7. Concluding remarks

Here we have not proposed a complete quantitative theoryof SL but suggested the basic principles that should be takeninto account for a new approach to this puzzling problem.


Our basic argument is that we proved first that the Gruneisenshift involved in supersonic impacts must be very large.Although the models that we investigated for supporting ourtheory are incompletely solved (especially concerning theirnumerical analysis), or simplified (homogeneous fluid), theyshould easily be extendable and should yield similar results.

We claim, more generally, that light flash emission isan ubiquitous phenomenon for shock waves when they aresufficiently intense (like those that are generated by supersonicor close-to-supersonic impacts). The important parameter thatcontrols the light emission is the Gruneisen parameter whichis determined by the instantaneous pressure. The mechanismof light production is that the excitation spectrum of a fluidelement adiabatically submitted to a very intense and briefpressure pulse is globally shifted to higher frequencies by alarge factor (the Gruneisen coefficient), while its temperatureis simultaneously increased by the same factor. Then, a briefand hot thermal light flash is emitted simultaneously.

Concerning bubble SL, unlike other existing models inthe literature which place the location of light emissionessentially inside the bubble, we propose that the light is mostlyemitted from the liquid surrounding the collapsing bubble. Wediscarded the effect of the formal divergence of pressure atthe bubble center, which could generate a plasma for severalreasons. The first reason is that it requires almost perfect bubblesphericity, which is unlikely to ever be realized in experiment.The second reason is that, even assuming perfect enoughsphericity, the focusing time of this pressure peak calculatedby some authors [40] ranges on the scale of just picoseconds,which is shorter by at least two orders of magnitude than thewidth of the observed light flash of SL. A third reason is that ourinvestigation in Appendix C, suggests the absence of pressuredivergency near bubble center because of the tremendously highnonlinearities which tend to uniformize the pressure when thefluid approaches its Van der Waals volume.

SL emitted by collapsing bubbles is the superposition ofthe thermal emission of many fluid elements with a spectrumand intensities that depend on their distance to the bubblecenter and on time. Thus, SL is not simple black-body radiationof hot matter at a well-defined temperature. We found that,schematically, SL is essentially emitted by a radiance spherewith a radius of a few minimum bubble radii, where the fluidsare highly compacted close to their minimum Van der Waalsvolume during a time tSL which we estimated to be compatiblewith the observations.

However, many questions remain to debate, but neverthelesswe have a clue for interpreting the most puzzling featuresunexplained to date.

Why is water the best fluid for producing intense SL? Wethink that this question is essentially a problem of bubblehydrodynamics. Since SL is generated by supersonic impacts,it requires that the collapsing bubble velocity reaches at leastclose to supersonic velocities (referred to the surrounding fluid).Then, an essential condition is the absence of energy dissipationwhich could slow down the bubble collapse and generate onlya soft impact. For that purpose, it is more favorable that thecollapsing bubble preserves its sphericity as well as possible

until the end of the collapse and does not develop any asphericalinstabilities (we noted in Appendix A that dissipation is smallerin that case). Our models were assumed to be Hamiltonianwithout energy dissipation, which corresponds to the idealsituation for SL. However, in real situations, there are manypossible sources of energy dissipation that could reduce thesharpness of the bubble impact and consequently suppress SL.For example, it should be avoided that the bubble containsgases or liquid vapor that are too soluble under pressure.3 Manyparameters concerning the surrounding fluid and the gas insidethe bubble are involved in producing sharp and stable bubblecollapses generating supersonic impacts and SL. At the presentstage, these parameters are not controllable in detail. In anycase, up to now, experiments have shown that cold degassedwater with rare gas bubbles is the most efficient system forproducing bright SL.

According to our theory, the brightness of SL is controlledby the maximum Gruneisen parameter γ which could begenerated at the bubble impact. It depends sharply on the impactvelocity but, moreover, there is a threshold impact velocitybelow which there is practically no SL.

Beyond this threshold but close to it, the coefficient γ isstill not very large and close to unity. The emission spectrumof the fluid is insufficiently dilated to overlap completely thetransparency gap of the normal fluid. Moreover, the temperatureis relatively low. This is the regime of dim SL. The emitted lightflash is weakly intense, with a frequency at maximum intensityshifted to the red. This is also a regime that is especiallysensitive to isotopic effect, which may sharply modify theemission spectrum of the normal fluid.

For example, the substitution of light water (H2O) by heavywater (D2O) reduces the cut-off IR frequency by a factor of√

2, because the highest frequency modes, which are protonor deuteron vibrations, are proportional to the inverse squareroots of their masses. The transparency gap is also extendedon the IR side, while the UV side does not change much. Theresult is that, for the same bubbles and the same impact velocity(neglecting the minor effect of different fluid densities), thewhole SL spectrum is roughly shifted toward the red in thesame proportion and with a lower intensity, as observed in realexperiments (see [7]).

For larger impact velocities, γ becomes larger, and theemission spectrum of the compressed fluid completely overlapsthe transparency gap of the normal fluid. The consequence isthat SL is brighter, with a maximum intensity at a frequencyshifted close to the upper UV edge of the absorption gap. Then,in equivalent conditions for both light and heavy water, thereis no important change in the SL spectrum, but neverthelesswith an important diminution of SL intensity. In contrast, thesame experiments [7] show only a small change due to isotopicsubstitution concerning the rare gas inside the bubble, whichconfirms that their contribution is minor for SL according toour theory.

3 We may note that there is systematically some energy dissipation of thebubble oscillator energy by the propagating shock wave at the bubble impact,but this does not affect the precursive light flash emission, which also yields aminor contribution to the dissipation of the bubble kinetic energy.


If the pressure peak of the shock wave could becomesufficiently large to make the fluid inside (or outside) thebubble metallic, then electronic excitations could be involvedfor generating SL (plasma?). Unfortunately, the experimentsperformed up to today have shown that bubble sphericitybecomes unstable when the amplitude of the driving pressureincreases too much [31]. As a consequence, the impact atbubble collapse is more complex and most likely not sharpenough for generating SL.

Acknowledgements

Both of us are indebted to J. Teixeira from Laboratoire LonBrillouin (LLB), CEA Saclay, France for useful discussionsabout the amazing properties of water. One of the authors (BD)would like to thank LLB for kind hospitality during his visitsthere. He also thanks the DST (India) for financial assistancethrough a research grant.

S.A. acknowledges T. Dauxois and all the organizers of this60th birthday symposium for the honor they gave him.

Appendix A. Rayleigh–Plesset model

We show that a spherical bubble of gas in an incompressiblefluid submitted to an external pressure is nothing but ananharmonic oscillator driven by an external force. This modelyields that the bubble radius reaches its maximum radialvelocity when the bubble pressure becomes equal to the appliedpressure and is still low. This model is valid up to this point, butcannot properly describe the following impact of the fluid onthe hard core gas.

A.1. Assumptions

We make the following assumptions:

• The flow is radial and the bubble is spherical. The dynamicsof the bubble are characterized by the time dependence of itsradius R(t).

• The fluid outside the bubble is incompressible and itspressure far from the bubble center is Pa(t), the externalpressure.

• The gas inside the bubble is compressible. It is assumedto remain in adiabatic equilibrium when the radius of thebubble varies. As a result, there is no energy dissipation.

• The energy of the gas inside the bubble, Φ(V ), is supposedto be a certain function of its volume V = (4π/3)R3,only. The pressure inside the bubble is Pi = −dΦ/dV .The bubble is in static equilibrium with the external pressurePa when Φ(V ) + Pa V is a minimum, which implies thatPi = Pa .

We may assume that the energy Φ(V ) is a function of thevolume, assuming that the gas compresses adiabatically with aVan der Waals radius h. Φ(V ) = K/(V − h)γ−1. The surfacetension energy σ S = 4πσ R2

= (36π)1/3σ V 2/3 can be addedto the bulk energy terms. Any more accurate potential canbe chosen. The physically important property is that potentialΦ(V ) diverges to +∞ for small V or V → h (Van der Waalsvolume).

A.2. Rayleigh–Plesset equation for the bubble dynamics

The field of radial velocity v(r) = v(r).r/r in theincompressible fluid outside the bubble fulfills the condition∇.v(r) = 0, which that implies v(r) = k/r2. Since v(R) = R,it comes out as

v(r) =R R2

r2 . (A.1)

The kinetic energy of the fluid outside the bubble is then

EK =

∫∞

R

12ρv2(r)4πr2dr = 2πρ R2 R4

∫∞

R

dr

r2

= 2πρ R2 R3. (A.2)

We neglect the kinetic energy of the gas inside the bubble.For a bubble with radius R = 0.5 × 10−6 m and velocity

R = 6 km/s (which is four times the sound velocity 1.5 km/sin water), the kinetic energy of the fluid is EK = 2π × 103

×

62× 106

× (0.5)3× 10−18 J ≈ 3 × 10−8 J ≈ 2 × 1011 eV.

Note that the kinetic energy of a single water molecule movingat 6 km/s is ≈3.4 eV.

The viscosity force

F = η∇2v + (η + λ)∇(∇.v) (A.3)

is zero, since ∇.v = 0, as the fluid outside the bubble isincompressible, and ∇

2v = 0, as the velocity field is radial.There might exist viscosity forces at highest order, but we shallneglect it.4

Thus, there is no energy dissipation and the total energy isconserved (assuming that the external pressure Pa is constant).We have

H = 2πρ R2 R3+ Φ

(4π

3R3)

+ Pa .4π

3R3. (A.4)

With the new variable

u(t) = V 5/6=

(4π

3

)5/6

R5/2 (A.5)

the total energy defines the Hamiltonian

H =12

Mu2+ W (u; Pa) (A.6)

of a particle with mass

M =1225

(3

4π

)2/3

ρ (A.7)

in the potential

W (u; Pa) = Φ(u6/5) + Pa .u6/5. (A.8)

This potential behaves as Pa .u6/5 for large u, while itbecomes very steep and diverges for u → uW , where uW =

4 If the velocity field is non-radial, it dissipates energy into heat in the water,which necessarily reduces the kinetic energy of the water at impact.


Fig. 3. Bubble potential W (u; Pa) versus u at different pressure Pa . During thepressure variation that generates SL, the minimum uc(Pa) of the curve variesby a factor 105 from the smallest to the largest bubble size.

h5/6 corresponds to the hard core (Van der Waals) volume of thegas inside the bubble (see Fig. 3). The minimum at u = uc(Pa)

of potential W (u; Pa) versus u yields the equilibrium radius ofthe bubble at this pressure. We have W ′(uc(Pa); Pa) = 0.

At small pressure, the minimum of this potential is obtainedfor large uc(Pa), while for large Pa the minimum is obtainedfor u = uc(Pa) → uW close to the hard core radius.

The equation for the bubble dynamics is simply describedby the equation

Mu + W ′(u; Pa) = 0. (A.9)

A.3. Pressure field

The Navier–Stokes equations are obtained from the Newtonlaw, which equates the force (−∇ P(r) + F(r) + F(r)) dr ona small volume element dr to the product of its mass and itsacceleration ρ (v + ∇v.v). F(r) is the friction force and F(r)is the force generated by an external field. In our case, there isno external field and the friction force is also zero for a radialbubble oscillation with an incompressible fluid. Then, using thespherical symmetry, we get

−∂ P

∂r= ρ

(v + v

∂v

∂r

)= ρ

(d(R R2)

dt

1

r2 − 2R2 R4

r5

).

This equation can be integrated, which yields

P(r, t) = ρ

(R R2

+ 2R R2

r−

12

R2 R4

r4

)+ Pa . (A.10)

The pressure at the bubble surface is, using Eqs. (A.9), (A.5)and (A.7),

P(R, t) = ρ

(R R +

32

R2)

+ Pa = Pa +2ρ

5√

R

d2 R5/2

dt2

= Pa +2ρ

5

(3

4π

)2/3 u

u1/5

= Pa −2512

W ′(u; Pa)

u1/5 . (A.11)

Consequently, when u(t) reaches the value uc(Pa) thatcorresponds to the minimum versus i of the potential W (u; Pa),we have W ′(uc(Pa); Pa) = u = 0, which implies that P(R) =

Pa . Next, the acceleration changes sign, so we conclude that thepressure in the bubble is equal to the external pressure when thevelocity u is negative with maximum modulus.

A.4. Bubble dynamics

When the external pressure Pa(t) varies as a function oftime, this anharmonic oscillator (A.6) is parametrically driven.Potential W (u; Pa) also depends on time, so that there is noenergy conservation. Then Eq. (A.9) becomes non-integrablewhen Pa(t) is time dependent, and consequently this undampedmodel should exhibit chaotic trajectories in addition to periodicand quasi-periodic trajectories corresponding to KAM tori. Asmall damping should favor the existence of attractors. It isobserved experimentally that, in good conditions for SL, theradial motion of the bubble is periodic at the frequency of thedriving ultrasound.

Although numerical simulation of Eq. (A.9) should beperformed for a more complete analysis, we can alreadymake some qualitative remarks concerning the possibility ofgenerating a sharp impact.

We initially consider the bubble at its maximum size whenthe external pressure Pa(t) is a minimum. If Pa(t) increasesvery slowly from its minimum value Pm to its maximumvalue PM , the motion of the bubble is adiabatic. Its radiusis practically the bubble equilibrium position uc(Pa(t)) at thecorresponding pressure. Then, the pressure remains practicallyuniform everywhere and equal to Pa(t). No impact would begenerated.

Actually, the characteristic frequency ωB(Pa) of the bubbleoscillator goes to zero at small pressure Pa and large radiusuc(Pa), and in contrast diverges at large pressure Pa and smallbubble radius. If the amplitude of the driving pressure becomestoo large, the bubble motion cannot be adiabatic anymore.

Indeed, if the maximum radius of the bubble is too large,when the bubble is at this maximum radius the variation inthe time-periodic driving pressure Pa(t) becomes too fast, sothat the response of the bubble oscillator u(t) cannot follow theadiabatic solution at that bubble size region. It will be delayed atu(t) > uc(Pa(t)) and then the bubble radius will be submittedto a (negative) acceleration, W ′(u; Pa)/M . It is then intuitivethat, from that point, the following conditions will favor thesharpest bubble impacts.

• The pressure Pa(t) increases very quickly up to itsmaximum value PM . Then, the acceleration of the bubbleoscillator becomes a maximum, and a maximum amount ofpotential energy can be transformed into kinetic energy atu = uc(PM ) (see Fig. 3).

• The radius uc(PM ) of the bubble at equilibrium at maximumpressure PM should be close to its Van der Waals radiusin order that potential W (u, PM ) becomes very steep foru < uc(PM ) and behaves like a hard wall. This conditionrequires that PM is large enough.


• After impact, the decay of the pressure from PM to Pmshould be as smooth as possible in order that the bubblecould reach its maximum radius almost adiabatically.

It turns out that the ideal shape of Pa(t) in a given periodand amplitude is more like a sawtooth periodic function of time.However, a sine time-periodic function is still efficient if a goodcompromise is chosen for its frequency.

First, it is clear that, for generating bubble impact, themaximum pressure PM should be large enough. Second, thefrequency of the driving pressure should have a range in thefrequency interval visited by the bubble frequency ωB(P) inorder to be non-adiabatic enough during the bubble collapseand adiabatic enough when it expands.

Note that, a priori, the maximum velocity for u does notcorrespond to those of the radial bubble velocity R because u ∝

R5/2. However, since R ≈ Rm is close to its minimum radiusRm , we may linearly expand u ∝ R5/2

m +52 R3/2

m (R − Rm)+· · ·,which yields that u = 0 and R = 0 occurs practically at thesame time ti .

In real experiments on SL, the maximum radial velocity Rhas been observed close to the minimum bubble radius Rm to beabout 6 km/s, which is four times larger than the sound velocityin water at rest [30]. At that point, the pressure field in the fluidis, from Eq. (A.10) (for r > R),

P(r, ti ) = ρR R2

2r

(1 −

R3

r3

)+ Pa .

This pressure field decays as a function of r but, sinceρ R2

≈ 103× 62

× 106= 36 GPa, it reaches the order of

magnitude of 100 kB (atmospheric pressure), where water isnot incompressible anymore.

Beyond this point at time ti , the RP model does not hold,because the pressure in the fluid close to the bubble willbecome tremendously large, so it cannot be assumed to beincompressible. The impact generates a pressure pulse in thefluid propagating radially from the bubble center (which alsodissipates energy away from the bubble center).

We now model the system after time ti by an impact modelwhere the fluid is compressible. It is initially at a uniformpressure, with a field of initial velocities roughly correspondingto those generated by the bubble collapse. We first consider thismodel in one dimension, which is simpler and already yieldsinteresting information.

Appendix B. Impact in one dimension

Our purpose is to obtain some order of magnitude forimportant physical parameters involved in the impact withoutaccurate calculations, from which we can argue that there issubstantial light emission. Actually, we do not estimate theamplitude of the pressure pulse that depends explicitly on theequation of state of the fluid, which is unknown in situations attremendous pressure.

After the bubble radius reaches maximum velocity at timeti , which we now consider as the origin of time, we model theimpact that occurs immediately afterwards by a different modelwhere the fluid is compressible.

Since, at this time, the fluid pressure is still relatively smallcompared with those involved in the pressure pulse, we assumea uniform pressure in the fluid. We simplify the model byassuming that the impact occurs from water on water. This isequivalent to saying that the highly compressed gas inside thebubble has the same density and the same compressibility aswater when impact occurs.

We first consider this model in one dimension, which issimpler and already yields essential information. We show thatwe can bound from below the sound velocity in the regionsubmitted to the pressure pulse from only the knowledge of theimpact velocity.

B.1. Dynamical equation

We consider a fluid moving in a rigid one-dimensional pipewith a strictly constant section (assumed to be the unit surface).The fluid is initially at uniform pressure P0. When the fluid ismoving, the element of fluid initially at x is at x + ε(x, t) attime t . If ρ is the mass of a unit volume of fluid at the initialpressure P0, the kinetic energy of the fluid is then

K =

∫12ρε2dx . (B.1)

V(v) is the potential energy of an element of fluid as afunction of its volume 1 + v (supposed to be the unit volume)in the initial state (v = 0) at pressure P0 (we have V ′

= −P0).We assume that V(v) is defined for v > −hW and becomes

infinite for v → −hW , where 1 − hW is the minimum volume(Van der Waals hard core). Moreover, we assume that V(v) isa convex function of v and that its second derivative V ′′(v)

(hardness) is a monotonically decreasing function of v.The relative change in volume of a small fluid element

initially at x is v(x, t) = ∇ε(x, t) =∂ε∂x . Thus, the potential

energy of the fluid is

W =

∫V(

∂ε

∂x

)dx . (B.2)

The action of this Hamiltonian system is (dissipation isneglected)

A = K − W =

∫ ∫ (12ρε2

− V(

∂ε

∂x

))dtdx . (B.3)

Then, with vanishing variation,

δA = 0

yields the nonlinear dynamical equation

ρε −∂.

∂xV ′

(∂ε

∂x

)= 0. (B.4)

It is convenient to consider the pressure P(x, t) = −V ′( ∂ε∂x )

as variable instead of ε(x, t) and to define the inverse functionF(ξ) of V ′ which fulfills

ξ = V ′(F(ξ)). (B.5)


Fig. 4. Sketch of the variation of F(−P) versus −P . F(−P) and its derivativeF ′(−P) are monotonically increasing functions of −P with an asymptote at−hW infinite pressure P . The slopes at −P0 and −Pi are proportional to theinverse square of the sound velocities s2

0 and s2i , respectively. The slope of the

cord between −P0 and −Pi is proportional to the inverse square velocity c2i of

a front between two regions at the corresponding pressures. PW is the cross-over pressure between the linear regime and the Van der Waals regime. ForPi � PW , this figure shows that s0 � ci � si .

−F(−P) is the static contraction rate of the fluid at pressure−P referred to the fluid at pressure P0 (a unit volume offluid initially at pressure P0 becomes 1 + F(−P) at pressureP). Because of the assumptions made on potential V , F(ξ)

is a convex monotonically increasing function of ξ with anasymptote F(−∞) = −hW < 0, where 1−hW is the minimumVan der Waals volume of an initial unit volume. A typicalvariation of F(ξ) is shown in Fig. 4.

Then ∂ε∂x = F(−P). We note that F(−P0) = 0, since

V ′= −P0. Eq. (B.4) becomes

ρ∂2 F(−P)

∂t2 +∂2 P

∂x2 = 0 (B.6)

where function F(ξ) is some monotonically increasing functionrelated to the equation of state of the fluid. It has an asymptoteat −∞: F(−∞) = −hW , as shown in Fig. 4.

Eq. (B.6) has trivial solutions with arbitrary uniformpressure P(x, t) = Pa . Small pressure fluctuations Pa +

δP(x, t) fulfill the linearized equation

−ρF ′(−Pa)∂2δP

∂t2 +∂2δP

∂x2 = 0 (B.7)

which has general solutions with the form δP(x, t) = f (x −

s(Pa)t)+ g(x + s(Pa)t), where f and g are arbitrary functionsand the sound velocity is defined as

s(P) =1

√ρF ′(−P)

. (B.8)

It is convenient to define a cross-over pressure PW that wecall the Van der Waals pressure, at which the sound velocitystarts to diverge. This situation is obtained when F(−P) bendsto its asymptote. Strong nonlinear effects will be manifestedwhen the pressure goes beyond this Van der Waals pressurePW while, for smaller pressures, the harmonic features remains

Fig. 5. Scheme of the evolution of a Hugoniot solution of Eq. (B.6). Anarbitrary initial pulse-shaped pressure profile P(x, t) versus x propagates tothe right side. Each isobar propagates toward the right at a velocity s(P) Eq.(B.8) which is a monotonically increasing function of pressure P . The frontside becomes steeper, while the backside flattens. Since P(x, t) must remainunivalued, a pressure discontinuity must appear within a finite time. Actually,the solution should leave the Hugoniot class, because the pressure profile is nota univalued function anymore. Its further evolution is unknown in general.

qualitatively good. We may choose (see Fig. 4)

PW − P0 = hW /F ′(−P0) = ρhW s20 (B.9)

where s0 = s(P0). For water, where s0 ≈ 1.5 × 103 m/s,ρ = 103 kg/m3 and choosing hW = 0.2, we obtain PW − P0 ∼

0.45 GPa ∼ 4.5 kbar.

B.2. Rankine characteristics

For this equation in 1D, it is convenient to consider, insteadof the function P(x, t) of space x and time t , the motion of theisobars defined as the spatial location x(P, t) at time t of thepoint where the pressure is equal to P . x(P, t) is defined by theimplicit equation P(x(P, t), t) = P . Then, Eq. (B.6) becomes,after some tedious calculations,

ρF ′′(−P)∂x

∂ P

(∂x

∂t

)2

−∂2x

∂ P2

+ ρF ′(−P)

((∂x

∂t

)2∂2x

∂ P2 − 2∂x

∂t

∂x

∂ P

∂2x

∂ P∂t

+

(∂x

∂ P

)2∂2x

∂t2

)= 0. (B.10)

This equation has special travelling solutions that areobtained from the ansatz

∂x

∂t(P, t) = σ s(P) (B.11)

where σ = ±1 determines the direction of propagation ofthe pressure pulse and s(P) is the sound velocity (B.8) in thefluid at uniform pressure P . Then, Eq. (B.11) yields x(P, t) =

σ s(P)t + x(P, 0), which are special solutions of Eq. (B.10).These solutions are called Hugoniot solutions.

Consequently, if we consider a travelling pressure profilewhich is initially P(x, 0) (see Fig. 5), each isobar at pressureP that is located at x(P, t) moves uniformly either to theleft side or the right right at a velocity s(P), which is only afunction of P . Since V(v) has been assumed to be a convexfunction of v with a Van der Waals hard core at −hW , V ′(v)

is monotonically increasing as well as its inverse functionF(ξ), which fulfils F(−∞) = −hW . Consequently, the sound


Fig. 6. Scheme of a one-dimensional impact between two fluids moving inopposite directions initially at pressure P0 and velocity Vi . A region of highpressure Pi grows from the impact line. The two fronts between the low-pressure and high-pressure regions move at velocity ci in opposite directions.The sound velocities in those regions are s0 and si , functions of the pressure P0and Pi , respectively.

velocity s(P) monotonically increases as a function of thepressure and diverges at infinite pressure.

Since an isobar at larger pressure moves faster, it is clearthat any travelling pressure profile has to develop a singularitywithin a finite time; see Fig. 5. Since for a given x only onepressure can be defined, this singularity should appear as apressure discontinuity.

B.3. Propagation of a front

Actually, Eq. (B.6) has exact discontinuous step solutions(considering the time and space derivatives as generalizedfunctions) with the form

Pd(x, t) = P− + (P+ − P−)Y (x − xd(t)) (B.12)

where xd(t) = ±c(P−, P+)t + xd(0). Y (x) is theHeaviside function (Y (x) = 0 for x < 0 and Y (x) = 1 forx ≥ 0) and the velocity is

c(P−; P+) =

√P+ − P−

−ρ(F(−P+) − F(−P−)). (B.13)

It corresponds to a travelling pressure discontinuity at xd(t) =

±c(P+; P−)t + xd(0) between two regions with constantpressure P− for x < xd(t) and P+ for x > xd(t). Sincefunction F(ξ) is monotonically increasing (see Fig. 4), we havethe important inequality for P+ < P−:

s(P+) < c(P+; P−) < s(P−). (B.14)

Assuming that P+ < P− in order to fix the ideas, thissolution is stable when xd(t) = +c(P+; P−) and unstablewhen xd(t) = +c(P+; P−). In the stable case, if we split thisdiscontinuity into two nearby discontinuities, according to thescheme shown Fig. 5, the second discontinuity travels fasterthan the first one and merges with it within a finite time. Inthe unstable case, the distance between these discontinuitiesalways increases. Actually, the width of the discontinuityfront spontaneously increases under perturbations. We get theimportant conclusion that the front is stable only when itpropagates from a high-pressure region to a lower-pressureregion. A front propagating backwards is unstable andspontaneously thickens.

More generally, if P(x, 0) is an initial pressure profile that isdiscontinuous at time 0 at xd(0), it generates a solution P(x, t)

of Eq. (B.6) which is discontinuous with respect to x at xd(t).The limit pressures at x → xd(0) are P+(t) for x > xd(0)

and P−(t) for x < xd(0). Vanishing (only) the components ofthe Dirac derivatives in Eq. (B.6) yields the instantaneous frontvelocity

x2d(t) =

P+ − P−

ρ(F(−P+) − F(−P−))= c2(P+(t); P−(t)). (B.15)

The velocity of the discontinuity is determined by the pressuresP+(t) and P−(t) on both sides, which are also time dependent.Consequently, in general, the pressures at both sides of thediscontinuity are not constant, so its velocity is not uniform.

B.4. Impact solution

We choose initial conditions where the fluid is initially atuniform pressure P0 and moves at velocity −Vi for x > 0 andthe opposite velocity Vi for x < 0 (see Fig. 6). Vi is the impactvelocity. Thus, we have ε(x, 0) = 0 and ε(x, 0) = −Vi sign(x).An exact continuous solution of Eq. (B.4) is

ε(x, t) = F(−Pi )((ci t + x)Y (ci t + x)

− (ci t − x)Y (ci t − x) − x) = −ε(−x, t) (B.16)

where Pi is the pressure generated at the impact to bedetermined and ci = c(Pi , P0) is the front velocity betweenregions at pressures Pi and P0, respectively.5 We haveε(x, 0) = 0 and ε(x, t) = F(−Pi )ci (Y (ci t + x)− Y (ci t − x)).which fulfills the required initial condition at t = 0 when

Vi = −F(−Pi )ci . (B.17)

Since we have ∂ε∂x = F(−Pi ) (Y (ci t + x) + Y (ci t − x) − 1),

it comes out that ∂ε∂x = F(−Pi ) and P(x, t) = Pi for −ci t <

x < ci t and ∂ε∂x = 0 and P(x, t) = 0 elsewhere. We have

P(x, t) = P0 + (Pi − P0)(Y (ci t + x) + Y (ci t − x) − 1)

(B.18)

which yields the pressure evolution versus time and space fora symmetric impact of the fluid at the origin at time 0 and atvelocity Vi .

Since ci = c(Pi , P0) is also a function of Pi , Eq. (B.17)determines the amplitude of the pressure Pi generated by theimpact as a function of the impact velocity Vi . Actually, Eq.(B.17) is nothing but a kinematic equation that expresses themass conservation of the fluid. The volume of a large mass offluid defined by −L < x < L , with L large, shrinks per unittime by 2Vi . Considering that a unit volume element initiallyat pressure P0 shrinks its volume to 1 + F(−Pi ) when itspressure becomes Pi , this volume also shrinks per unit time by−2ci F(−Pi ), where ci is the front velocity. These two resultshave to be equal, which yields Eq. (B.17).

5 However, let us note that the apparent location of the front is not xF (t) =

±ci t but xF +ε(xF , t) = ±(ci −Vi )t . The front velocity ci −Vi > 0 appearingin a real experiment would be smaller, because it moves with respect to the fluidmoving in the opposite direction at velocity Vi .


For a weak impact, that is, when Vi � s0 (much smallerthan the sound velocity), Pi is also small and we get at lowestorder ci ≈ s0 and Pi − P0 ≈ V ′′(0)

Vis0

= ρs0Vi .When the impact is supersonic (Vi > s0), Pi becomes

tremendously large, as we show now. As represented in Fig. 4,function F(ξ) is asymptotic to −hW , which represents themaximum Van der Waals contraction of the fluid. Consequently,0 < −F(−Pi ) < hW < 1. If the fluid is barely compressible,that is, the fluid compactness is close to the Van der Waalsvolume, then F(−Pi ) is small. Then Eq. (B.17) implies thatthe front velocity ci is bounded as Vi/hW � ci . In any case,even assuming the fluid volume could shrink to zero, which isunrealistic (hW = 1), the front velocity is always larger thanthe impact velocity Vi < ci . In the limit of large pressure Pi ,−F(−Pi ) ≈ hW , this inequality becomes an equality. We haveci = Vi/hW . Since ci =

√(Pi − P0)/(−ρF(−Pi )), we get

Pi −P0 = hW ρc2i = ρV 2

i /hW . We note for consistency that weshould have Pi � PW , where PW is the Van der Waals pressure(B.9). This situation is fulfilled when Vi � hW s0. Thus, forbarely compressible materials where hW is small, we alreadyget supersonic fronts for subsonic impacts.

For understanding SL, it is essential to get information aboutthe sound velocity si in the compressed region at pressurePi defined by Eq. (B.8). We have the following sequence ofinequalities (see Fig. 4), which is always strictly fulfilled:

Vi <Vi

hW∼< ci < si . (B.19)

When the pressure Pi goes beyond the Van der Waalspressure, it is clear (see Fig. 4) that ci � si . The sound velocityin the high-pressure region si may become larger than ci by one(or several) orders of magnitude. For highly supersonic impactvelocity Vi � s0, the sound velocity in the high-pressure regionshould be shifted up to many more orders of magnitude abovenormal sound velocity.

For example, if we assume an impact velocity Vi = 2 s0only twice the sound velocity s0 and a maximum Van der Waalscontraction hW = 0.2 (−20%), then the front velocity ci willbe strictly larger than 10 s0, that is, one order of magnitudeabove the normal sound velocity. The sound velocity in thecompressed region is strictly larger than this front velocity.Since the compression at impact brings the fluid very closeto its minimum volume (F(−Pi ) ≈ −hW ), very close to theasymptote, the slope of F(−Pi ) at Pi is almost horizontal (seeFig. 4), which suggests that si could be larger than ci by one (orseveral) orders of magnitude. Thus, assuming si ∼ 100s0 is notphysically unreasonable.

The impact pressure Pi is given by Eq. (B.13). For highlysupersonic impacts, −F(−Pi ) ∼ hW which yields ci =√

(Pi − P0)/(ρhW ). Since hW ci ∼ Vi , we obtain

Pi ∼ρ

hWV 2

i (B.20)

for highly nonlinear impacts. For example, for a supersonicimpact with water at velocity Vi = 2 s0 ∼ 3 × 103 m/s,hw = 0.2 and ρ = 103 kg/m3, we could obtain Pi ∼

0.45 × 1011 Pa ∼ 0.45 Mbar (and then Pi � PW ∼ 4.5 kbar).

Fig. 7. One-dimensional scheme showing an incoming front at pressure Pi andvelocity ci (top) crossing an interface (dashed line) between two different media(top). An intermediate pressure region at pressure Pm appears delimited by twofronts that propagate in opposite directions from the interface at velocity cr forthe reflected front and ct for the the transmitted front. The fluid contractionrate is hi = −F−(Pi ), hr = −F−(−Pm ) for the regions in the left medium atpressure Pi and Pm , respectively. It is ht = −F+(−Pm ) in the region in the leftmedium at pressure Pm . The middle scheme corresponds to a situation wherethe right medium is less compressible (‘harder’) than the left medium (thenPm > Pi ). The bottom scheme corresponds to the opposite situation where theright medium is softer (Pm < Pi ). Then the front moving backwards is unstableand flattens.

B.5. Inhomogeneous media

When the medium is not uniformly homogeneous, thepropagation of pressure fronts may be sharply affected. It ispedagogical to consider the simple example of inhomogeneitywhere the nonlinear equation Eq. (B.6) is modified fordescribing front propagation through the interface at x = 0(see Fig. 7) between two different media at the left and atthe right. This situation may model the behavior of a pressurefront penetrating inside the compacted rare gas bubble, butshould also give some intuition concerning the propagation ofa spherical 3D front, which we will show to be described by aninhomogeneous 1D model.

An exact solution can be found analytically for that modelwhich describes the scattering of a single front (with the form(B.12) at negative x) at the interface. This front splits into twofronts propagating in opposite directions in the different media,as shown Fig. 7.

One considers two different contraction rates and densitiesF−(−P) and ρ− for x < 0 in the left medium and F+(−P)

and ρ+ for x < 0 in the right medium.


Since the global rate of contraction is determined by thefluid velocity at infinity, which is unchanged because ofthe scattering at the interface, it is invariant before and af-ter scattering, which directly yields the equation hi ci =

(hr − hi )cr + ht ct where hi = −F−(−Pi ), hr =

−F−(−Pi ) and ht = −F+(−Pm) and the front veloci-ties are from Eq. (B.13), ci =

√(Pi − P0)/(−ρ−F−(−Pi )),

cr =√

(Pm − Pi )/(−ρ−(F−(−Pm)) − F−(−Pi )) and ct =√(Pm − P0)/(−ρ+F+(−Pm)). Actually, this equation deter-

mines the pressure Pm as a function of Pi , since the pa-rameters are only functions of the involved pressures Pi ,Pm and P0. Since we have hi ci =

1√

ρ−

√(Pi − P0)hi and,

with σm = sign(Pm − Pi ), we have (hr − hi )cr =

σm1

√ρ−

√(Pm − Pi )(hr − hi ) and ht ct =

1√

ρ+

√(Pm − P0)ht ,

then this equation becomes√(Pi − P0)hi − σm

√(Pm − Pi )(hr − hi )

=

√ρ−

ρ+

√(Pm − P0)ht . (B.21)

If one assume two identical media, it comes out readily thatPm = Pi is the expected solution.

By definition, we say that the left fluid is less compressiblethan the right fluid at pressure P when

−ρ+F+(−P) < −ρ−F−(−P) (B.22)

is fulfilled. We assume for simplicity that this inequality isfulfilled for the whole range of pressures P in the interval[Pi , Pm].

Then, it can be proved readily that an overpressure (Pm > Piand σm = +1) is generated at the interface when the pressurefront comes from the left fluid, which is more compressible thanthe right fluid. Then the front propagation are both stable.

In the opposite case, when the left fluid is less compressiblethan that on the right side, we have a depression (Pm < Piand σm = −1) at the interface. However, in that case, the frontpropagating backwards to the left side is unstable and shouldsmooth out (see Fig. 5), while that propagating forwards to theright side remains stable.

For example, if we assume that the right medium is totallyincompressible (ht = 0), we obtain Pm − P0 = (Pi − P0)

hrhr −hi

> Pi − P0 and cr =hi

hr −hici > ci , since hi < hr < h−W .

If, moreover, we assume that the pressure front in the leftmedium involves a contraction rate hi close to its maximumVan der Waals contraction h+,W , then hi

hr −himust be very large,

which implies that the relative increase in pressure Pm and alsoin the front velocity cr can be very large.

In conclusion, when a pressure front propagates throughan interface towards a less compressible medium, it developsa region with a larger pressure at the interface. This regionis delimited by two fronts that propagate faster than theinitial front in both directions. When the medium is morecompressible, it is the opposite; the pressure is smaller atthe interface. The front penetrates the soft medium with asmaller amplitude and propagates slower. On the other side, thebackward front smooths out.

Appendix C. Spherical bubble impact

For simplicity, we again study the simpler situation of animpact of water on water, which mimics the impact of the fluidonto the spherical hard core bubble.

We assume, as in 1D, that the pressure P0 is initially uniformin the fluid but that the fluid outside the sphere with radius r >

R is initially moving radially with a large velocity toward thebubble center, according to Eq. (A.1), and while the fluid insidethe sphere is at rest inside. We neglect the fluid viscosity andthus assume that the system is Hamiltonian. We also assumethat the solution with perfect spherical symmetry is stable undernon-spherical perturbations.

C.1. Dynamical equation

An element of fluid at the radial distance 0 < r in thesteady fluid at pressure P0 moves radially at r + ε(r, t) at time t(Euler coordinates). The volume contraction rate v(r, t) fulfillsthe inequality −1 < v(r, t) =

∂ε∂r , since the volume 1 + v of a

unit volume of fluid at pressure P0 is never negative. If ρ is themass of this unit volume of fluid, the total kinetic energy of thefluid is then different to the 1D case (B.1):

K =

∫∞

0

12ρε24πr2dr. (C.1)

Potential V(v) is the potential energy of a unit volume offluid initially at pressure P0 as a function of its volume 1 + v. Itis assumed to have the same properties as in the 1D case; V(v)

becomes infinite for v → −1 or v → −hW (maximum Van derWaals contraction rate). Again we assume that V(v) is a convexfunction of v with monotone decreasing second derivatives.

The volume contraction rate of a small element initially atradial distance r is v(r, t) = ∇ε(r, t) = r2 ∂r2ε

∂r . Then, thepotential energy of the fluid becomes formally different fromthe 1D case (B.2):

W =

∫∞

0V(

1

r2

∂r2ε

∂r

)4πr2dr. (C.2)

The action of this Hamiltonian system is

A = K − W =

∫dt∫

∞

0

(12ρε2

− V(

1

r2

∂r2ε

∂r

))4πr2dr.

(C.3)

Vanishing the variation δA yields the corresponding nonlineardynamical equation of the fluid

ρε −∂.

∂rV ′

(1

r2

∂r2ε

∂r

)= 0. (C.4)

As in 1D, it may be more convenient to write an equation for thepressure instead of the displacements, P(r, t) = −V ′(v(r, t))(we have P0 = −V ′(0)) or

P(r, t) = −V ′

(1

r2

∂r2ε

∂r

). (C.5)


Function F(ξ) is defined, as in the 1D case, as the inversefunction of V ′ (see Fig. 4). Then, multiplying both membersof Eq. (C.4) by r2 and next differentiating with respect to r , weobtain

ρ∂2.

∂t2 F(−P(r, t)) +1r

∂2.

∂r2 (r P(r, t)) = 0 (C.6)

which is defined for r ≥ 0. It is now more convenient to definethe new variable u(r, t) = r P(r, t) which fulfills

ρ∂2.

∂t2

[r F

(−

u

r

)]+

∂2u

∂r2 = 0. (C.7)

Since the pressure is a radial function P(r, t) = P(−r, t),we have to search only for antisymmetric solutions u(r, t) =

−u(−r, t) of Eq. (C.7). The advantage of this equation is thatit is formally similar to Eq. (B.6), except that the uniformnonlinear function F(ξ) is replaced by a spatially dependentnonlinear function G(ξ, r) = r F(

ξr ). Actually, this change only

affects the nonlinear part, since G(ξ, r) = κξ is independent ofr in the case F(ξ) = κξ is linear.

If we assume u(r, t) bounded for large r , then ur goes to

zero when r → +∞. Consequently, F(ξ) ≈ ξ/(ρs20) can be

linearized for large r , which yields the planewave equation

−∂2u

∂t2 + s20∂2u

∂r2 = 0. (C.8)

Since u(r, t) = −u(−r, t), this linear equation (C.8) has exactsolutions with the general form u(r, t) = f (s0t−r)− f (s0t+r),where f (x) is an arbitrary function. If f (x) is pulse shaped witha single peak, it comes out that this pulse propagates radiallyfar from bubble center at the sound velocity s0 of the fluid atrest. It keeps its shape over time, although the correspondingpressure field P(r, t) =

f (s0t−r)r decays with inverse distance

from bubble center.The role of the nonlinearities necessarily becomes essential

for the pulse propagation near the bubble center. Theequation that yields the Hugoniot characteristics analogous toEq. (B.10) can be written formally but, because of the spatialinhomogeneity of the nonlinearities, it does not have simplesolutions with the form (B.11).

Another consequence is that discontinuous front solutionssimilar to (B.12) are no more exact solutions of Eq. (C.7). Thelast section of the previous Appendix B has shown that a spatialinhomogeneity in a 1D model at a discontinuous interfacebreaks a single front in two fronts. In the 3D model, wherethe spatial inhomogeneity varies continuously, the situation issurely more complex. Some numerical investigations wouldhelp, but have not been performed up to now.

Nevertheless, we may obtain some analytical informationabout the initial evolution of some given u profiles. We considerfor example, an initial profile u(r, 0) that is discontinuous atr = rd(0). The corresponding solution u(r, t) is assumed to bediscontinuous at r = rd(t), and there are two different limits foru(r, t) when r → rd(t) which are u+(t) = rd(t)P+(t) for r >

rd(t) and u−(t) = rd(t)P−(t) for r < rd(0). By substitution ofsuch a form in Eq. (C.7), one gets different components which

are Dirac derivatives, Dirac functions, and standard functions orr and t , which should be vanished separately. Then, vanishingonly the components of the Dirac derivatives yields that thepressure discontinuity necessarily travels at velocity

r2d =

u+ − u−

−ρrd

(F(−

u+

rd) − F(−

u−

rd))

=P+ − P−

−ρ (F(−P+) − F(−P−))= c2(P+, P−) (C.9)

which may be either positive or negative and is the samevelocity as in the 1D case (B.15). However, this velocitydepends on the values u−(t) and u+(t) of u(r, t) at thediscontinuity for which the evolution is unknown.

We now propose an empirical scenario for a spherical impactwhich, however, should require some numerical confirmations.

The initial conditions that we propose for our impact modelare obtained from the RP velocity field (A.1), which yields

ε(r, 0) = 0 and ε(r, 0) = −Vr R2

mr2 sign(r) for |r | > Rm , where

Vr = R(0) is the velocity of the bubble radius at the time ofimpact. We assume that the bubble core is initially immobile,ε(r, 0) = 0 for |r | < Rm . Then we have initially P(r, 0) = P0and u(r, 0) = P0r . For simplicity, we assume that P0 = 0 andthus u(r, 0) = 0.6 Other initial conditions could be chosen tobe more realistic, but we think that the general scenario wouldbe similar, except for the very short initial stage of the impact.

C.2. Scenario of bubble collapse

We may take the risk to draw some conjectures about thepulse shape evolution (see Fig. 1) according to Eq. (C.7)without performing any numerical tests, which are left for laterworks. We propose only a qualitative scheme about the coarseevolution of the pressure in the fluid neglecting fluctuations.

During a short time after the impact of the fluid on thesphere, this impact is planar and may be described by the one-dimensional model described in the previous Appendix B. Thenwe should consider Vi = Vr/2, since only the fluid on one sideof the impact surface is moving. This impact generates a pulseinitially at pressure Pi ∼

ρhW

V 2i (B.20).

Just after impact (see Fig. 1 top left), the solution ofEq. (C.7) is such that u(r, t) = 0 for all r except in a narrowshell delimited by two fronts at the inner radius rI (t) and theouter radius rO(t), where u(r, t) = ui is constant. We initiallyhave rI (0) = rO(0) = Rm at time zero and −rI (0) =

rO(0) = ci , where ci is defined from Eq. (B.17) by Vi =

Vr/2 = −F(−Pi )ci at time 0. The two u-fronts appear andpropagate in opposite directions at velocities cI = −rI (t) andcO = rO(t). These two fronts propagate from high-pressureto low-pressure regions so that, according to the 1D model,they may be considered as initially stable. However, the spatialnonlinear inhomogeneity drastically modifies the propagationof the pressure fronts which should not remain square shaped.

6 There is no lack of generality in assuming P0 = 0, which can be obtainedby shifting the origin of pressures at P0. The equations are the same, exceptthat potential Vs (ξ) = V(ξ) − V ′(0)ξ replaces potential V(ξ) everywhere.


We have ∂2 f∂r2 =

ξ2

r3 F ′′(ξr ) > 0. Since F(ξ) has been

assumed to be a convex function of ξ , function G(ξ, r) isalso a convex function of r for arbitrary fixed ξ . Then ∂ f

∂r =

F(ξr )− (

ξr )F ′(

ξr ) is monotonically increasing with respect to r .

Since ∂ f∂r (ξ, +∞) = 0, this derivative is strictly negative, which

implies that, for fixed ξ , G(ξ, r) is a monotonically decreasingfunction of r . Then, when r1 < r2, we have the inequalityG(ξ, r1) > G(ξ, r2), which is valid for any ξ . We concludethat this inhomogeneous medium becomes less compressible atsmaller r referred to variable u, according to definition (B.22)(in our case, mass density ρ of the medium at rest is constant).We say that the system is less u-compressible at smaller r .

According to Appendix B.5 which shows some frontbehaviors in 1D when the compressibility is spatiallydiscontinuous, we expect that the u-pulse should not remainsquare-shaped. Complex pressure oscillations should begenerated in the high pressure region behind the two fronts inthe highly compressed region. In any case, an extremely shorttime is needed for the pressure pulse to reach the bubble center.If we assume, for example, average front velocities cI of theorder of 10–20 s0 ∼ 25 × 103 m/s, then a typical time forthe inner front to reach the center is about 10 ps (for a 0.5 µmdiameter bubble); see Fig. 1 top right. At this stage, the fluid iscompressed close to its Van der Waals volume in a full sphereat a pressure that is much beyond PW .

Since u(0, t) = 0 at the center, a naive scheme would bethat the inner front simply reflects at the center and movesbackwards, while its amplitude and velocity decays again.However, this scheme is surely wrong, because we have shownin the 1D case that a front moving backwards is unstable andthickens (see Fig. 5). Another effect is that, when the directionof propagation of this front reverses, it should interact with itsown tail generated beforehand (see Fig. 7 middle).

The pressure evolution in the highly compressed regionshould be highly chaotic. Actually, in the Van der Waalsregime when the pressure becomes tremendously large, thesound velocity becomes larger than the front velocities by manyorder of magnitudes. Thus, we should expect that the pressurefluctuations in that highly compressed region delimited by thisrelatively immobile fronts, tend to dissipate extremely fast intoheat so that the pressure should be roughly uniform in thatregion. This argument for fast pressure uniformization whichholds essentially in the Van der Waals regime, forbids anypressure divergency at bubble center.

We may describe this regime with a highly compacted sphereof fluid by assuming the existence of a rather well-definedsphere with radius rO(t), where the fluid is compacted close toits Van der Waals volume, which persists (see Fig. 1 bottom left)for some time (this assumption does not mean that the pressureinside is uniform but only that it is larger than PW ) and suchthat the fluid outside is close to the normal pressure P0.

This compact sphere regime lasts a certain time tSL, whichwe can simply estimate by using the same argument of massconservation as in the 1D case (Appendix B). The volumeof fluid (at rest) per unit time entering from far away in alarge sphere with radius r is 4π r2

R2m

Vr , and Vr are the bubble

radius and velocities at the time 0 of impact. The volumeof fluid (at rest) that is compacted per unit time is 4πr2

O rO .Since the contraction rate is close to the maximum Van derWaals contraction hW , we should have 4π r2

R2m

Vr ≈ hW 4πr2O rO ,

which yields

rO = cO =Vr

hw

R2m

r2O

(C.10)

which yields the time dependence r3O(t) ≈ R3

m(1+3Vr

Rm hwt). The

outer front velocity rO decays as a function of its distance rOfrom the bubble center as 1/r2

O ! Since this velocity is relatedto the pressure discontinuity from P ′

i to P0 according to Eq.

(B.13), we get rO =

√Pi −P0

ρF(−Pi )≈

√Pi −P0ρhW

. Consequently, forsmall rO , while the whole fluid in the sphere is compacted at itsVan der Waals volume, the pressure discontinuity behaves as

P ′

i − P0 ∝1

r4O

. (C.11)

Our assumption of a compacted sphere is consistent whilethe front velocity rO is larger than the sound velocity s0 (whichis the minimum velocity of a pressure front), which yields

rO < RSL = Rm

√Vr

s0hw. Thus, if we choose a Van der Waals

contraction rate hW = 0.2 and an impact bubble velocity Vr =

4s0, it comes out that RSL ≈ 4.5Rm , where Rm is the minimumbubble radius at impact. We call the sphere with radius RSL theradiance sphere, because most light will be emitted from thatregion. The consequence is that, under realistic experimentalconditions, a substantial volume of the fluid around the bubble,and not only the inner part, is highly compressed up to its to Vander Waals volume and emits light according to the explanationdeveloped in Section 6.

The time duration tSL where this outer front is in thesupersonic regime, that is, the life time of the compacted sphere,can be estimated. Integrating Eq. (C.10) with respect to time t

yieldsr3

OR3 = 1+3 Vr

Rm hwt . Thus, for having rO = RSL ≈ 4.5Rm ,

where the impact radius is Rm = 0.25 × 10−6 m and Vr =

4s0 ≈ 6×103 m, it comes out that tSL ≈ 0.5×10−9 s ∼ 500 ps,which is typically the order of duration of the SL light flash.

When rO becomes larger RSL, the outer front velocity rO isbounded from below by the sound velocity s0. Then, the fluidin the compacted sphere starts to relax to a larger volume. Thevolume of the fluid most likely expands first from the bubblecenter, where the pressure decays to smaller pressures P ′

0smaller than PW (see Fig. 1 bottom left). A pressure pulse startsto form that is initially more extended spatially (∼RSL) thanthe bubble size Rm at impact. The effect of nonlinearities andtheir inhomogeneity still spreads the pressure pulse to largerwidth. At this point, the pressure drops in the compacted sphereto smaller values than the Van der Waals pressure PW definedin (B.9). Next, the pressure pulse move radially away fromthe bubble center almost at sound velocity s0 with an almostconstant profile (see Fig. 1 bottom right). The amplitude of thepressure peak now decays as 1/r , as expected for harmonicwaves. Measurements of the velocity and the relatively large


spatial width of the acoustic pulse observed far from the bubblecenter qualitatively confirms this scenario [36].

Emission of light (SL) will occur from the fluid compressedto pressures much larger than PW (see Section 6), that is, onlyat the initial stage of the pressure pulse formation in the highlycompressed sphere.

It is worthwhile noting now that it has been observed that thepressure pulse velocity is much larger than the normal soundvelocity when it is observed just after impact. Pulse velocitiesup to 4 km/s were observed in water [2,36] with a spatial widthof several µm. This velocity was found to decrease rapidly tothe normal sound velocity away from the bubble center.

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New suggestion concerning the origin of sonoluminescence