23 October 2000

Ž .Physics Letters A 275 2000 452–458www.elsevier.nlrlocaterpla

Phase order and energy localization in acoustic propagation inrandom bubbly liquids

Zhen Ye a,), Haoren Hsu a, Emile Hoskinson a,b

a WaÕe Phenomena Laboratory, Department of Physics, National Central UniÕersity, Chungli, Taiwan, ROCb Department of Physics, UniÕersity of California, Berkeley, CA 94720, USA

Received 8 June 2000; received in revised form 21 August 2000; accepted 24 August 2000

Communicated by V.M. Agranovich

Abstract

Propagation of acoustic waves in liquid media containing many air-filled bubbles is studied using a self-consistentapproach. It is shown that under proper conditions, multiple scattering leads to a peculiar phase transition in acousticpropagation. When the phase transition occurs, not only the acoustic waves are confined in the neighborhood of thetransmitting source, but a previously unsuspected collective behavior of the air bubbles appears, responsible for effectivecancellation of the propagating wave. A novel phase diagram method is used to depict the phase transition. q 2000 ElsevierScience B.V. All rights reserved.

PACS: 02.60.-x; 45.05.-x; 43.20.qg

When propagating through media containing manyscatterers, wave will be scattered by each scatterer.The scattered wave will be scattered again by otherscatterers. Such a process will be repeated to set upan infinite recursive pattern of multiple scattering.Multiple scattering of waves accounts for many in-teresting phenomena including such as scintillationw x w x1 , random laser 2,3 , and band gaps in crystal

w xstructures 4–6 . Under appropriate conditions, mul-tiple scattering of waves leads to the ubiquitousphenomenon of wave localization, which has beenand continues to be a subject of substantial researchŽ w x.see, e.g., Refs. 7–10 . Wave localization refers to

) Corresponding author.Ž .E-mail address: zhen@phy.ncu.edu.tw Z. Ye .

any situation in which waves in a scattering mediumare trapped in space and will remain confined in theinitial transmitting site until dissipated.

w xIn a recent Letter 11 , we have shown that local-ization can also be achieved for acoustic wavespropagating in liquids with even a very small frac-tion of air-filled spherical bubbles, i.e. bubbly liq-uids, supporting some of the previous conjecturew x12 . It is shown that the localization appears withina region of frequency slightly above the naturalresonance of the individual air bubbles. Outside thisregion, wave propagation remains extended. In thiscommunication, we present a further numerical in-vestigation of acoustic localization in bubbly liquids.Unlike most previous approaches which derive ap-proximately a diffusion equation for the ensembleaveraged energy, our method is to solve rigorously

0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0375-9601 00 00616-2

( )Z. Ye et al.rPhysics Letters A 275 2000 452–458 453

the wave propagation from the fundamental waveequation. In particular, for the first time we showthat when localization occurs, an amazing collectivebehavior of the spherical air bubbles emerges. In the

w xcontext of field theory 13 , such a coherent behaviormay be an indication of a global behavior of thesystem and may imply a symmetry breaking andappearance of a certain kind of Goldstone bosons. Anovel phase diagram is used to illustrate differentphase states in connection with the extended andlocalized transmissions.

Considerable efforts have been devoted to bothŽ w x.theoretical see, e.g., Refs. 14–18 and experimen-

w xtal 19,20 studies of acoustic propagation in bubblyliquids. The theoretical approaches include, for in-

w xstance, the mean-field approach 21 , the fluid me-w xchanic approach 18,22 , and the high-order perturba-

w xtion theory 23 . A review of the subject can bereferred to the recent textbook by Medwin and Clayw x24 . The research in the circumstance of wave local-ization in bubbly liquids, however, is relativelyscarce.

w xFollowing Ref. 11 , we consider sound emissionfrom a unit acoustic source located at the center of abubble cloud. The source is transmitting amonochromatic wave of angular frequency v. Forsimplicity while without losing generality, the shapeof the cloud is taken as spherical; such a modeleliminates irrelevant effects due to an irregular edge.Total number N bubbles of the same radius a arerandomly distributed in space within the cloud andtheir space coordinates are denoted by r with ii

running from 1 to N. The bubble volume fraction,the space occupied by bubbles per unit volume, istaken as b. Adaptation of such a model for othergeometries and situations is straightforward. Thewave transmitted from the source propagates throughthe bubble layer, where multiple scattering incurs,and then it reaches a receiver located at some dis-tance from the cloud. The multiple scattering in thebubbly layer is described by a set of self-consistent

w xequations 21 . The energy transmission can be solvedw xrigorously 11 .

It is found that the localization of acoustic wavesis reached when the bubble volume fraction b is

y5 w xgreater than a threshold value of 10 12,25 . Alsothe localization behavior is insensitive to the bubblesize when the bubble radius is larger than 20 mm

w x25 . Note that for smaller bubbles, the thermal andviscosity effects are significant and the wave local-ization seems less evident. In the simulation, we takebs10y3, and the total number of bubbles is variedfrom 100 to 2500, large enough to eliminate possibleeffects on localized states due to the finite samplesize. The radius of the bubbles ranges from 20 m to2 cm. We restrict our attention to the frequency

w xrange considered in Ref. 11 . The medium is water.We find that in this wide range of parameters allresults are similar.

The wave transmitted from the source is subjectto scattering by individual bubbles. The scatteredwave from each bubble is a linear response to thedirect incident wave p from the source and also all0

the scattered waves from other scatterers, and isw xwritten as 11

N

p r ,r s f p r q p r ,rŽ . Ž . Ž .Ýs i i 0 i s i jž /js1, j/i

=G ryr , 1Ž . Ž .0 i

where f is the scattering function of a single bubble,iŽ . Ž .and G r sexp ikr rr is the usual 3D Green’s0

function. The scattering function f can be readilyi

computed and the solution can be written in the formw xof a modal series 11,26 , representing various vibra-

tional modes of each spherical bubble. In the fre-quency range considered, the pulsating mode domi-nates the scattering and the scattering function issimplified as

aif s , 2Ž .i 2 2v rv y1y ika0, i i

where v is the natural frequency of the ith bubble.0, i

The imaginary term ka in the denominator repre-i

sents the radiation effect, responsible for the energyexchange between scattered waves and the oscilla-tion of the bubbles. In the present case, all thebubbles are assumed identical.

Ž . Ž .To solve for p r ,r and subsequently p r,r ,s j i iŽ .we set r in Eq. 1 to any scatterer other than the

Ž .ith. Then Eq. 1 becomes a set of closed self-con-sistent equations which can be solved exactly by

( )Z. Ye et al.rPhysics Letters A 275 2000 452–458454

matrix inversion. Once p is determined, the totals

wave at any space point is given by

N

p r sp r q p r ,r . 3Ž . Ž . Ž . Ž .Ý0 s iis1

w x ² < < 2:In line with 11 , we define Is p to repre-sent the squared modulus of the total waves, corre-sponding to the total energy. Here the brackets referto an average over either the ensembles or the direc-tions. According to the erdogic hypothesis, the twoaverages are expected to yield the same results. Inthis paper, the spatial distribution of energy is aver-aged over all directions. Also in the computation, theuninteresting geometrical spreading factor is re-moved.

The frequency dependence of transmission showsthat the waves are localized within a region of

w xfrequency from about kas0.014 to 0.09 11 ; wherek is the usual wavenumber and a is the bubbleradius. Note that in the simulation, all parameters arenon-dimensionalized, and functions are dependentonly on the dimensionless parameter ka rather than kand a separately. We assign three regions: Region Ifor ka-0.014, Region II for 0.014-ka-0.09 andRegion III for ka)0.09. Regions I and III are fornon-localized states and Region II is the localizationregime.

Before going any further, a general discussion onwave propagation is appropriate. When waves propa-gate through media with many scatterers, multiplescattering of waves is established by an infiniterecursive pattern of rescattering. In terms of wavefields, the energy flow in the system is calculated

w Ž ) Ž . Ž .x Ž .from J ; Re i p r =p r . Writing p r s< Ž . < iu Ž r . < <A r e with A and u being the amplitude andphase respectively, the energy flow becomes J;< < 2A =u . Obviously, the energy flow will come to acomplete halt and the waves could be localized in

< <space when phase u is constant and A does notequal zero at the same time. In other words, theenergy can be stored in the medium. Such a phasetransition implies a condensation of modes in thereal space. This perception is useful to our laterdiscussion.

Upon incidence, each air-bubble acts effectivelyas a secondary pulsating source. The scattered wave

Ž .from the ith bubble is1,2,3, . . . , N is regarded asthe radiated wave and is rewritten as

p r ,r sA G ryr . 4Ž . Ž . Ž .s i i 0 i

The complex coefficient A refers to the effectivei

strength of the secondary source and is computedincorporating all multiple scattering effects. Thetotal wave at any space point is the addition of thedirect wave from the transmitting source and theradiated waves from all bubbles.

'< < Ž .We express A as A exp ju with js y1i i i

here; the modulus A represents the strength, whereasi

u the phase of the secondary source, physicallyi

corresponding to the oscillation phase of the bubble.We assign a two dimensional unit vector u , here-i

after termed phase vector, to each phase u , andi

these vectors are represented on a phase diagramparallel to the xyy plane. That is, the starting pointof each phase vector is positioned at the center ofindividual scatterers with an angle with respect to thepositive x-axis equal to the phase, u scosu xqˆi i

sinu y. Letting the phase of the initiative emittingˆi

source be zero, numerical experiments are carriedout to study the behavior of the phases of the bub-bles and the spatial distribution of the acoustic en-ergy. The normalized magnitude of the summation ofall phase vectors may represent an order parameter.We note that this order parameter is only tentative; itis meant to characterize the phase behavior of thesystem.

Fig. 1 shows the 3D phase diagrams of the phasevectors and the averaged energy distribution as afunction of distance from the source. We have cho-sen three particular frequencies below, within andabove the localization regime, according to the re-

w xsults of Ref. 11 : kas0.01254,0.0173, and 0.1.They represent the phase states in the three afore-mentioned regions respectively. The dots refer to thethree-dimensional positions of the air bubbles at

Ž .r s x , y , z , with is1,2, . . . , N. The arrows referi i i i

to the phase vectors which are located at the bubblesites and are parallel to the xyy plane. For thepurpose of demonstrating the physical picture in itsmost explicit way, we have only shown here thephase vectors for 136 bubbles.

In general, we find that although quantitativeresults for the energy spatial variation may differ, the

( )Z. Ye et al.rPhysics Letters A 275 2000 452–458 455

Fig. 1. Left column: The phase diagram for the two dimensional phase vectors defined in the text for one realization of the bubble cloud.Right column: The acoustic energy distribution or transmission, averaged for all directions, as a function of distance away from the source;

w xthe distance is scaled by the cloud radius R. Original in colour.

qualitative properties for the phase behavior and theenergy behavior are more or less the same for differ-ent ka in each region. Specifically, we observe thatfor frequencies below a certain value, roughly belowkas0.014, there is no obvious ordering for thedirections of the phase vectors, nor for the energydistribution. The phase vectors point to various direc-

tions. In this case, no energy localization appears,corresponding to the extended state discussed in Ref.w x11 . These features are shown by the example ofkas0.01254 in Fig. 1. The random behavior in thedirections is attributed to the boundary effect of afinite number of the scatterers. In effect, as the waveis not localized, it can propagate through and is

( )Z. Ye et al.rPhysics Letters A 275 2000 452–458456

reflected by the asymmetric border; all bubbles canexperience the effect via strong multiple scattering.

As the frequency increases, moving inside thelocalization regime, the energy localization and anordering of the phase vectors become evident. Thecase with kas0.0173 clearly shows that the energyis localized near the source. In the meantime, allbubbles oscillate completely in phase, but exactly outof phase with the transmitting source; all phasevectors point to the negative x-axis, in parallel to thexyy plane. Such a collective behavior allows forefficient cancellation of propagating waves. The en-ergy decays nearly exponentially along the distanceof propagation inside the cloud, setting the localiza-

Ž . w xtion length l to be about 6a 11,25 . At this length,klf0.088; therefore the Ioffe–Regel criterion forlocalization is satisfied. Outside the cloud, the trans-mission remains constant with distance, as expectedwhen the geometric spreading factor has been re-moved. The energy localization and the order in thephase vectors are independent of the outer boundaryand they always appear for sufficiently large b andN. In this case, as shown, all bubbles oscillate com-pletely in phase, leading to the constant pressure

phases inside the medium and showing a previouslyunsuspected collective behaviour. Meanwhile, theenergy gradually decreases as moving away from thetransmitting source. These features are in accordancewith the aforementioned perception about energystorage in random media.

When the frequency increases further, movingoutside the energy localization region, the in-phaseorder disappears. Meanwhile, the wave becomesnon-localized again. This is illustrated by the case ofkas0.1. In this case, the phase vectors again point

Ž .to various directions left column and the energyŽ .distributes roughly uniformly in space right column .

From the above, we see that kas0.01254, kas0.0173 and kas0.1 respectively belong to different

Ž .phase states. We may conclude as follows. 1 InRegions I and III, wave is not localized and the

Ž .phase vectors point to various directions. 2 InRegion II, the energy decays exponentially on aver-age along the distance traveled, and all the phasevectors point more or less to the same directionagainst the source; in this region, states with differ-ent ka may have different localization length, but foralmost all ka the phase vectors prevail a nearly

Fig. 2. Order parameter as a function of ka.

( )Z. Ye et al.rPhysics Letters A 275 2000 452–458 457

perfect ordering. Implied by the phase diagram in theleft column of Fig. 1, there must be a phase transi-tion from Region I to II and from Region II to III.

When waves are localized, the order parameter issignificantly larger than that for extended waves. InFig. 2, we plot the order parameter as a function ofthe non-dimensional parameter ka for one randomrealization of the bubble cloud. The result shows thatas the frequency is increased from the low frequencyend, the order parameter rises to about unity rapidly.Then it decreases as the frequency moves out of thelocalized regime. Due to the finite sample size, theorder parameter does not vanish completely outside

w xthe localized regime 27 .Further numerical investigation shows that the

patterns depicted by Fig. 1 hold for a wide range ofbubble size and for any random distribution of bub-bles. And the features are always valid for suffi-ciently large bubble volume fractions. The localiza-tion range depends crucially on the bubble volumefraction and are relatively insensitive to the bubblesize. The collective phenomenon is caused by multi-ple scattering. As the multiple scattering is removed

Ž .from Eq. 1 , the collective behavior disappears com-pletely. By varying the bubble numbers while keep-ing the volume fraction b constant, it can be shownthat the localization and non-localization behaviorare qualitatively unchanged, and thus the behavior isnot caused by the boundary of the bubble arrays. Theappearance of the collective behavior for localizedwaves implies existence of a kind of Goldstonebosons incurred in the phase transition. A furtherstudy also reveals the unexpected result that localiza-tion is relatively independent of the precise locationor organization of the scatterers. The results foracoustic propagation in regular arrays and the con-nection to the band gap effect will be published

w xelsewhere 28,29 .An intuitive understanding about the acoustic lo-

calization in the bubbly liquids may be helped fromthe following consensus. Imagine that two personshold a thread. If one person pushes, while the otherpulls, the two persons would act completely out ofphase. No energy can be transferred from one to theother. In the bubbly liquid case, the state of localizedwaves shows such a completely out-of-phase behav-ior, which effectively prevents waves from propagat-ing.

The behavior of the present system is a result ofthe interplay between the single bubble scatteringand multiple scattering among bubbles. We take theview that the coherent behavior allows for efficientcancellation of source wave, thus giving rise to wavelocalization. Exactly how the interplay between thesingle bubble response and the multiple scatteringinduces the localization and the global coherent be-havior remains an open question and may be furtherunderstood by an analytic analysis which is underinvestigation.

In summary, we have demonstrated a new phasetransition for acoustic propagation in a bubbly liquid.The numerical results show that as the phase transi-tion occurs, not only the acoustic waves are confinedin the neighborhood of the transmitting source, butan amazing collective behavior of the air bubblesappears. A novel diagram method is proposed todescribe the phase transition of the phase vectorsassociated with the bubbles. We suggest that such acoherence behavior may be crucial to differentiatethe localization effect from the residual absorptioneffect. The ambiguity between the two effects has

w xcaused much debate in the literature 30,31 . Thephase transition in the phase vectors for the presentcase draws remarkable similarities to the case of themagnetic dipole directions in magnetic materials,responsible for various magnetic phase states. Al-though these localization properties are only demon-strated for the special resonant air-bubbles, they areexpected to hold true for other resonant scatterers aswell.

Acknowledgements

The work received support from the NationalScience Council.

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