- Email: [email protected]
Phase order and energy localization in acoustic propagation in random bubbly liquids
23 October 2000
Physics Letters A 275 Ž2000. 452–458 www.elsevier.nlrlocaterpla
Phase order and energy localization in acoustic propagation in random bubbly liquids Zhen Ye a,) , Haoren Hsu a , Emile Hoskinson a,b a
WaÕe Phenomena Laboratory, Department of Physics, National Central UniÕersity, Chungli, Taiwan, ROC b 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-consistent approach. It is shown that under proper conditions, multiple scattering leads to a peculiar phase transition in acoustic propagation. When the phase transition occurs, not only the acoustic waves are confined in the neighborhood of the transmitting source, but a previously unsuspected collective behavior of the air bubbles appears, responsible for effective cancellation of the propagating wave. A novel phase diagram method is used to depict the phase transition. q 2000 Elsevier Science B.V. All rights reserved. PACS: 02.60.-x; 45.05.-x; 43.20.q g
When propagating through media containing many scatterers, wave will be scattered by each scatterer. The scattered wave will be scattered again by other scatterers. Such a process will be repeated to set up an infinite recursive pattern of multiple scattering. Multiple scattering of waves accounts for many interesting phenomena including such as scintillation w1x, random laser w2,3x, and band gaps in crystal structures w4–6x. Under appropriate conditions, multiple scattering of waves leads to the ubiquitous phenomenon of wave localization, which has been and continues to be a subject of substantial research Žsee, e.g., Refs. w7–10x.. Wave localization refers to
)
Corresponding author. E-mail address: [email protected] ŽZ. Ye..
any situation in which waves in a scattering medium are trapped in space and will remain confined in the initial transmitting site until dissipated. In a recent Letter w11x, we have shown that localization can also be achieved for acoustic waves propagating in liquids with even a very small fraction of air-filled spherical bubbles, i.e. bubbly liquids, supporting some of the previous conjecture w12x. It is shown that the localization appears within a region of frequency slightly above the natural resonance of the individual air bubbles. Outside this region, wave propagation remains extended. In this communication, we present a further numerical investigation of acoustic localization in bubbly liquids. Unlike most previous approaches which derive approximately a diffusion equation for the ensemble averaged energy, our method is to solve rigorously
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 6 1 6 - 2
Z. Ye et al.r Physics Letters A 275 (2000) 452–458
the wave propagation from the fundamental wave equation. In particular, for the first time we show that when localization occurs, an amazing collective behavior of the spherical air bubbles emerges. In the context of field theory w13x, such a coherent behavior may be an indication of a global behavior of the system and may imply a symmetry breaking and appearance of a certain kind of Goldstone bosons. A novel phase diagram is used to illustrate different phase states in connection with the extended and localized transmissions. Considerable efforts have been devoted to both theoretical Žsee, e.g., Refs. w14–18x. and experimental w19,20x studies of acoustic propagation in bubbly liquids. The theoretical approaches include, for instance, the mean-field approach w21x, the fluid mechanic approach w18,22x, and the high-order perturbation theory w23x. A review of the subject can be referred to the recent textbook by Medwin and Clay w24x. The research in the circumstance of wave localization in bubbly liquids, however, is relatively scarce. Following Ref. w11x, we consider sound emission from a unit acoustic source located at the center of a bubble cloud. The source is transmitting a monochromatic wave of angular frequency v . For simplicity while without losing generality, the shape of the cloud is taken as spherical; such a model eliminates irrelevant effects due to an irregular edge. Total number N bubbles of the same radius a are randomly distributed in space within the cloud and their space coordinates are denoted by r i with i running from 1 to N. The bubble volume fraction, the space occupied by bubbles per unit volume, is taken as b . Adaptation of such a model for other geometries and situations is straightforward. The wave transmitted from the source propagates through the bubble layer, where multiple scattering incurs, and then it reaches a receiver located at some distance from the cloud. The multiple scattering in the bubbly layer is described by a set of self-consistent equations w21x. The energy transmission can be solved rigorously w11x. It is found that the localization of acoustic waves is reached when the bubble volume fraction b is greater than a threshold value of 10y5 w12,25x. Also the localization behavior is insensitive to the bubble size when the bubble radius is larger than 20 mm
453
w25x. Note that for smaller bubbles, the thermal and viscosity effects are significant and the wave localization seems less evident. In the simulation, we take b s 10y3 , and the total number of bubbles is varied from 100 to 2500, large enough to eliminate possible effects on localized states due to the finite sample size. The radius of the bubbles ranges from 20 m to 2 cm. We restrict our attention to the frequency range considered in Ref. w11x. The medium is water. We find that in this wide range of parameters all results are similar. The wave transmitted from the source is subject to scattering by individual bubbles. The scattered wave from each bubble is a linear response to the direct incident wave p 0 from the source and also all the scattered waves from other scatterers, and is written as w11x N
ž
ps Ž r ,r i . s f i p 0 Ž r i . q
Ý js1, j/i
=G 0 Ž r y r i . ,
ps Ž r i ,r j .
/ Ž 1.
where f i is the scattering function of a single bubble, and G 0 Ž r . s expŽ ikr .rr is the usual 3D Green’s function. The scattering function f i can be readily computed and the solution can be written in the form of a modal series w11,26x, representing various vibrational modes of each spherical bubble. In the frequency range considered, the pulsating mode dominates the scattering and the scattering function is simplified as fi s
ai 2 2 v 0,irv y 1 y ika i
,
Ž 2.
where v 0, i is the natural frequency of the ith bubble. The imaginary term ka i in the denominator represents the radiation effect, responsible for the energy exchange between scattered waves and the oscillation of the bubbles. In the present case, all the bubbles are assumed identical. To solve for psŽ r j ,r i . and subsequently pŽ r,r i ., we set r in Eq. Ž1. to any scatterer other than the ith. Then Eq. Ž1. becomes a set of closed self-consistent equations which can be solved exactly by
Z. Ye et al.r Physics Letters A 275 (2000) 452–458
454
matrix inversion. Once ps is determined, the total wave at any space point is given by
ps Ž r ,r i . s A i G 0 Ž r y r i . .
N
p Ž r . s p0 Ž r . q
Ý ps Ž r ,r i . .
from the ith bubble Ž i s 1,2,3, . . . , N . is regarded as the radiated wave and is rewritten as
Ž 4.
Ž 3.
is1
In line with w11x, we define I s ²< p < 2 : to represent the squared modulus of the total waves, corresponding to the total energy. Here the brackets refer to an average over either the ensembles or the directions. According to the erdogic hypothesis, the two averages are expected to yield the same results. In this paper, the spatial distribution of energy is averaged over all directions. Also in the computation, the uninteresting geometrical spreading factor is removed. The frequency dependence of transmission shows that the waves are localized within a region of frequency from about ka s 0.014 to 0.09 w11x; where k is the usual wavenumber and a is the bubble radius. Note that in the simulation, all parameters are non-dimensionalized, and functions are dependent only on the dimensionless parameter ka rather than k and a separately. We assign three regions: Region I for ka - 0.014, Region II for 0.014 - ka - 0.09 and Region III for ka ) 0.09. Regions I and III are for non-localized states and Region II is the localization regime. Before going any further, a general discussion on wave propagation is appropriate. When waves propagate through media with many scatterers, multiple scattering of waves is established by an infinite recursive pattern of rescattering. In terms of wave fields, the energy flow in the system is calculated from J ; Rew i Ž p ) Ž r .=p Ž r .x. Writing p Ž r . s < AŽ r .< e i u Ž r . with < A < and u being the amplitude and phase respectively, the energy flow becomes J ; < A < 2 =u . Obviously, the energy flow will come to a complete halt and the waves could be localized in space when phase u is constant and < A < does not equal zero at the same time. In other words, the energy can be stored in the medium. Such a phase transition implies a condensation of modes in the real space. This perception is useful to our later discussion. Upon incidence, each air-bubble acts effectively as a secondary pulsating source. The scattered wave
The complex coefficient A i refers to the effective strength of the secondary source and is computed incorporating all multiple scattering effects. The total wave at any space point is the addition of the direct wave from the transmitting source and the radiated waves from all bubbles. We express A i as < A i
Z. Ye et al.r Physics 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; the distance is scaled by the cloud radius R. wOriginal in colour.x
qualitative properties for the phase behavior and the energy behavior are more or less the same for different ka in each region. Specifically, we observe that for frequencies below a certain value, roughly below ka s 0.014, there is no obvious ordering for the directions of the phase vectors, nor for the energy distribution. The phase vectors point to various direc-
tions. In this case, no energy localization appears, corresponding to the extended state discussed in Ref. w11x. These features are shown by the example of ka s 0.01254 in Fig. 1. The random behavior in the directions is attributed to the boundary effect of a finite number of the scatterers. In effect, as the wave is not localized, it can propagate through and is
456
Z. Ye et al.r Physics Letters A 275 (2000) 452–458
reflected by the asymmetric border; all bubbles can experience the effect via strong multiple scattering. As the frequency increases, moving inside the localization regime, the energy localization and an ordering of the phase vectors become evident. The case with ka s 0.0173 clearly shows that the energy is localized near the source. In the meantime, all bubbles oscillate completely in phase, but exactly out of phase with the transmitting source; all phase vectors point to the negative x-axis, in parallel to the x y y plane. Such a collective behavior allows for efficient cancellation of propagating waves. The energy decays nearly exponentially along the distance of propagation inside the cloud, setting the localization length Ž l . to be about 6 a w11,25x. At this length, kl f 0.088; therefore the Ioffe–Regel criterion for localization is satisfied. Outside the cloud, the transmission remains constant with distance, as expected when the geometric spreading factor has been removed. The energy localization and the order in the phase vectors are independent of the outer boundary and they always appear for sufficiently large b and N. In this case, as shown, all bubbles oscillate completely in phase, leading to the constant pressure
phases inside the medium and showing a previously unsuspected collective behaviour. Meanwhile, the energy gradually decreases as moving away from the transmitting source. These features are in accordance with the aforementioned perception about energy storage in random media. When the frequency increases further, moving outside the energy localization region, the in-phase order disappears. Meanwhile, the wave becomes non-localized again. This is illustrated by the case of ka s 0.1. In this case, the phase vectors again point to various directions Žleft column. and the energy distributes roughly uniformly in space Žright column.. From the above, we see that ka s 0.01254, ka s 0.0173 and ka s 0.1 respectively belong to different phase states. We may conclude as follows. Ž1. In Regions I and III, wave is not localized and the phase vectors point to various directions. Ž2. In Region II, the energy decays exponentially on average along the distance traveled, and all the phase vectors point more or less to the same direction against the source; in this region, states with different ka may have different localization length, but for almost all ka the phase vectors prevail a nearly
Fig. 2. Order parameter as a function of ka.
Z. Ye et al.r Physics Letters A 275 (2000) 452–458
perfect ordering. Implied by the phase diagram in the left column of Fig. 1, there must be a phase transition from Region I to II and from Region II to III. When waves are localized, the order parameter is significantly larger than that for extended waves. In Fig. 2, we plot the order parameter as a function of the non-dimensional parameter ka for one random realization of the bubble cloud. The result shows that as the frequency is increased from the low frequency end, the order parameter rises to about unity rapidly. Then it decreases as the frequency moves out of the localized regime. Due to the finite sample size, the order parameter does not vanish completely outside the localized regime w27x. Further numerical investigation shows that the patterns depicted by Fig. 1 hold for a wide range of bubble size and for any random distribution of bubbles. And the features are always valid for sufficiently large bubble volume fractions. The localization range depends crucially on the bubble volume fraction and are relatively insensitive to the bubble size. The collective phenomenon is caused by multiple scattering. As the multiple scattering is removed from Eq. Ž1., the collective behavior disappears completely. By varying the bubble numbers while keeping the volume fraction b constant, it can be shown that the localization and non-localization behavior are qualitatively unchanged, and thus the behavior is not caused by the boundary of the bubble arrays. The appearance of the collective behavior for localized waves implies existence of a kind of Goldstone bosons incurred in the phase transition. A further study also reveals the unexpected result that localization is relatively independent of the precise location or organization of the scatterers. The results for acoustic propagation in regular arrays and the connection to the band gap effect will be published elsewhere w28,29x. An intuitive understanding about the acoustic localization in the bubbly liquids may be helped from the following consensus. Imagine that two persons hold a thread. If one person pushes, while the other pulls, the two persons would act completely out of phase. No energy can be transferred from one to the other. In the bubbly liquid case, the state of localized waves shows such a completely out-of-phase behavior, which effectively prevents waves from propagating.
457
The behavior of the present system is a result of the interplay between the single bubble scattering and multiple scattering among bubbles. We take the view that the coherent behavior allows for efficient cancellation of source wave, thus giving rise to wave localization. Exactly how the interplay between the single bubble response and the multiple scattering induces the localization and the global coherent behavior remains an open question and may be further understood by an analytic analysis which is under investigation. In summary, we have demonstrated a new phase transition for acoustic propagation in a bubbly liquid. The numerical results show that as the phase transition occurs, not only the acoustic waves are confined in the neighborhood of the transmitting source, but an amazing collective behavior of the air bubbles appears. A novel diagram method is proposed to describe the phase transition of the phase vectors associated with the bubbles. We suggest that such a coherence behavior may be crucial to differentiate the localization effect from the residual absorption effect. The ambiguity between the two effects has caused much debate in the literature w30,31x. The phase transition in the phase vectors for the present case draws remarkable similarities to the case of the magnetic dipole directions in magnetic materials, responsible for various magnetic phase states. Although these localization properties are only demonstrated for the special resonant air-bubbles, they are expected to hold true for other resonant scatterers as well.
Acknowledgements The work received support from the National Science Council.
References w1x A. Ishimaru, Wave Propagation and Scattering in Random Media, Academic Press, New York, 1978. w2x N.M. Lawandy, R.M. Balachandran, A.S.L. Gomes, E. Sauvain, Nature ŽLondon. 368 Ž1994. 436. w3x H. Cao et al., Phys. Rev. Lett. 82 Ž1999. 2278. w4x W.M. Robertson et al., J. Opt. Soc. Am. B 10 Ž1993. 322. w5x J.V. Sanchez et al., Phys. Rev. Lett. 80 Ž1998. 5325.
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Z. Ye et al.r Physics Letters A 275 (2000) 452–458
w6x M. Torres, J.P. Adrados, F.R. Montero de Espinosa, Nature ŽLondon. 398 Ž1998. 114. w7x P.W. Anderson, Phys. Rev. 109 Ž1958. 1492. w8x C.H. Hodges, J. Woodhouse, J. Acoust. Soc. Am. 74 Ž1983. 894. w9x R. Dalichouch, J.P. Amstrong, S. Shultz, P.M. Platzman, S.L. McCall, Nature ŽLondon. 354 Ž1991. 53. w10x J.H. Page et al., Science 271 Ž1996. 634. w11x Z. Ye, A. Alvarez, Phys. Rev. Lett. 80 Ž1998. 3503. w12x D. Sornette, B. Souillard, Europhys. Lett. 7 Ž1988. 269. w13x H. Umezawa, Advanced Field Theory, AIP, New York, 1995. w14x T.R. Kirkpatrick, Phys. Rev. B 31 Ž1985. 5746. w15x C.A. Condat, J. Acoust. Soc. Am. 83 Ž1988. 441. w16x Z. Ye, Phys. Rev. E 56 Ž1997. 2318. w17x Z. Ye, C. Feuillade, J. Acoust. Soc. Am. 102 Ž1997. 798. w18x S. Temkin, J. Fluid Mech. 380 Ž1999. 1. w19x E. Silberman, J. Acoust. Soc. Am. 29 Ž1957. 925.
w20x w21x w22x w23x w24x w25x w26x w27x w28x w29x w30x w31x
H. Medwin, J. Acoust. Soc. Am. 62 Ž1977. 1041. L.L. Foldy, Phys. Rev. 67 Ž1945. 107. A.S. Sangani, J. Fluid Mech. 232 Ž1991. 221. Z. Ye, L. Ding, J. Acoust. Soc. Am. 98 Ž1995. 1629. H. Medwin, C.S. Clay, Acoustical Oceanography, Academic Press, New York, 1998. A. Alvarez, Z. Ye, Phys. Lett. A 252 Ž1999. 53. P.M. Morse, H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, 1953. For extremely low or high frequencies Ž ka) 0.4., our model may not be applicable, but this is not our interest. Z. Ye, H. Hsu, preprint, 2000. P.-G. Luan, Z. Ye, preprint, 2000. F. Scheffold, R. Lenke, R. Tweer, G. Maret, Nature 398 Ž1999. 206. A.A. Chabanov, M. Stoytchev, A.Z. Genack, Nature 404 Ž2000. 850.
Physics Letters A 275 Ž2000. 452–458 www.elsevier.nlrlocaterpla
Phase order and energy localization in acoustic propagation in random bubbly liquids Zhen Ye a,) , Haoren Hsu a , Emile Hoskinson a,b a
WaÕe Phenomena Laboratory, Department of Physics, National Central UniÕersity, Chungli, Taiwan, ROC b 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-consistent approach. It is shown that under proper conditions, multiple scattering leads to a peculiar phase transition in acoustic propagation. When the phase transition occurs, not only the acoustic waves are confined in the neighborhood of the transmitting source, but a previously unsuspected collective behavior of the air bubbles appears, responsible for effective cancellation of the propagating wave. A novel phase diagram method is used to depict the phase transition. q 2000 Elsevier Science B.V. All rights reserved. PACS: 02.60.-x; 45.05.-x; 43.20.q g
When propagating through media containing many scatterers, wave will be scattered by each scatterer. The scattered wave will be scattered again by other scatterers. Such a process will be repeated to set up an infinite recursive pattern of multiple scattering. Multiple scattering of waves accounts for many interesting phenomena including such as scintillation w1x, random laser w2,3x, and band gaps in crystal structures w4–6x. Under appropriate conditions, multiple scattering of waves leads to the ubiquitous phenomenon of wave localization, which has been and continues to be a subject of substantial research Žsee, e.g., Refs. w7–10x.. Wave localization refers to
)
Corresponding author. E-mail address: [email protected] ŽZ. Ye..
any situation in which waves in a scattering medium are trapped in space and will remain confined in the initial transmitting site until dissipated. In a recent Letter w11x, we have shown that localization can also be achieved for acoustic waves propagating in liquids with even a very small fraction of air-filled spherical bubbles, i.e. bubbly liquids, supporting some of the previous conjecture w12x. It is shown that the localization appears within a region of frequency slightly above the natural resonance of the individual air bubbles. Outside this region, wave propagation remains extended. In this communication, we present a further numerical investigation of acoustic localization in bubbly liquids. Unlike most previous approaches which derive approximately a diffusion equation for the ensemble averaged energy, our method is to solve rigorously
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 6 1 6 - 2
Z. Ye et al.r Physics Letters A 275 (2000) 452–458
the wave propagation from the fundamental wave equation. In particular, for the first time we show that when localization occurs, an amazing collective behavior of the spherical air bubbles emerges. In the context of field theory w13x, such a coherent behavior may be an indication of a global behavior of the system and may imply a symmetry breaking and appearance of a certain kind of Goldstone bosons. A novel phase diagram is used to illustrate different phase states in connection with the extended and localized transmissions. Considerable efforts have been devoted to both theoretical Žsee, e.g., Refs. w14–18x. and experimental w19,20x studies of acoustic propagation in bubbly liquids. The theoretical approaches include, for instance, the mean-field approach w21x, the fluid mechanic approach w18,22x, and the high-order perturbation theory w23x. A review of the subject can be referred to the recent textbook by Medwin and Clay w24x. The research in the circumstance of wave localization in bubbly liquids, however, is relatively scarce. Following Ref. w11x, we consider sound emission from a unit acoustic source located at the center of a bubble cloud. The source is transmitting a monochromatic wave of angular frequency v . For simplicity while without losing generality, the shape of the cloud is taken as spherical; such a model eliminates irrelevant effects due to an irregular edge. Total number N bubbles of the same radius a are randomly distributed in space within the cloud and their space coordinates are denoted by r i with i running from 1 to N. The bubble volume fraction, the space occupied by bubbles per unit volume, is taken as b . Adaptation of such a model for other geometries and situations is straightforward. The wave transmitted from the source propagates through the bubble layer, where multiple scattering incurs, and then it reaches a receiver located at some distance from the cloud. The multiple scattering in the bubbly layer is described by a set of self-consistent equations w21x. The energy transmission can be solved rigorously w11x. It is found that the localization of acoustic waves is reached when the bubble volume fraction b is greater than a threshold value of 10y5 w12,25x. Also the localization behavior is insensitive to the bubble size when the bubble radius is larger than 20 mm
453
w25x. Note that for smaller bubbles, the thermal and viscosity effects are significant and the wave localization seems less evident. In the simulation, we take b s 10y3 , and the total number of bubbles is varied from 100 to 2500, large enough to eliminate possible effects on localized states due to the finite sample size. The radius of the bubbles ranges from 20 m to 2 cm. We restrict our attention to the frequency range considered in Ref. w11x. The medium is water. We find that in this wide range of parameters all results are similar. The wave transmitted from the source is subject to scattering by individual bubbles. The scattered wave from each bubble is a linear response to the direct incident wave p 0 from the source and also all the scattered waves from other scatterers, and is written as w11x N
ž
ps Ž r ,r i . s f i p 0 Ž r i . q
Ý js1, j/i
=G 0 Ž r y r i . ,
ps Ž r i ,r j .
/ Ž 1.
where f i is the scattering function of a single bubble, and G 0 Ž r . s expŽ ikr .rr is the usual 3D Green’s function. The scattering function f i can be readily computed and the solution can be written in the form of a modal series w11,26x, representing various vibrational modes of each spherical bubble. In the frequency range considered, the pulsating mode dominates the scattering and the scattering function is simplified as fi s
ai 2 2 v 0,irv y 1 y ika i
,
Ž 2.
where v 0, i is the natural frequency of the ith bubble. The imaginary term ka i in the denominator represents the radiation effect, responsible for the energy exchange between scattered waves and the oscillation of the bubbles. In the present case, all the bubbles are assumed identical. To solve for psŽ r j ,r i . and subsequently pŽ r,r i ., we set r in Eq. Ž1. to any scatterer other than the ith. Then Eq. Ž1. becomes a set of closed self-consistent equations which can be solved exactly by
Z. Ye et al.r Physics Letters A 275 (2000) 452–458
454
matrix inversion. Once ps is determined, the total wave at any space point is given by
ps Ž r ,r i . s A i G 0 Ž r y r i . .
N
p Ž r . s p0 Ž r . q
Ý ps Ž r ,r i . .
from the ith bubble Ž i s 1,2,3, . . . , N . is regarded as the radiated wave and is rewritten as
Ž 4.
Ž 3.
is1
In line with w11x, we define I s ²< p < 2 : to represent the squared modulus of the total waves, corresponding to the total energy. Here the brackets refer to an average over either the ensembles or the directions. According to the erdogic hypothesis, the two averages are expected to yield the same results. In this paper, the spatial distribution of energy is averaged over all directions. Also in the computation, the uninteresting geometrical spreading factor is removed. The frequency dependence of transmission shows that the waves are localized within a region of frequency from about ka s 0.014 to 0.09 w11x; where k is the usual wavenumber and a is the bubble radius. Note that in the simulation, all parameters are non-dimensionalized, and functions are dependent only on the dimensionless parameter ka rather than k and a separately. We assign three regions: Region I for ka - 0.014, Region II for 0.014 - ka - 0.09 and Region III for ka ) 0.09. Regions I and III are for non-localized states and Region II is the localization regime. Before going any further, a general discussion on wave propagation is appropriate. When waves propagate through media with many scatterers, multiple scattering of waves is established by an infinite recursive pattern of rescattering. In terms of wave fields, the energy flow in the system is calculated from J ; Rew i Ž p ) Ž r .=p Ž r .x. Writing p Ž r . s < AŽ r .< e i u Ž r . with < A < and u being the amplitude and phase respectively, the energy flow becomes J ; < A < 2 =u . Obviously, the energy flow will come to a complete halt and the waves could be localized in space when phase u is constant and < A < does not equal zero at the same time. In other words, the energy can be stored in the medium. Such a phase transition implies a condensation of modes in the real space. This perception is useful to our later discussion. Upon incidence, each air-bubble acts effectively as a secondary pulsating source. The scattered wave
The complex coefficient A i refers to the effective strength of the secondary source and is computed incorporating all multiple scattering effects. The total wave at any space point is the addition of the direct wave from the transmitting source and the radiated waves from all bubbles. We express A i as < A i
Z. Ye et al.r Physics 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; the distance is scaled by the cloud radius R. wOriginal in colour.x
qualitative properties for the phase behavior and the energy behavior are more or less the same for different ka in each region. Specifically, we observe that for frequencies below a certain value, roughly below ka s 0.014, there is no obvious ordering for the directions of the phase vectors, nor for the energy distribution. The phase vectors point to various direc-
tions. In this case, no energy localization appears, corresponding to the extended state discussed in Ref. w11x. These features are shown by the example of ka s 0.01254 in Fig. 1. The random behavior in the directions is attributed to the boundary effect of a finite number of the scatterers. In effect, as the wave is not localized, it can propagate through and is
456
Z. Ye et al.r Physics Letters A 275 (2000) 452–458
reflected by the asymmetric border; all bubbles can experience the effect via strong multiple scattering. As the frequency increases, moving inside the localization regime, the energy localization and an ordering of the phase vectors become evident. The case with ka s 0.0173 clearly shows that the energy is localized near the source. In the meantime, all bubbles oscillate completely in phase, but exactly out of phase with the transmitting source; all phase vectors point to the negative x-axis, in parallel to the x y y plane. Such a collective behavior allows for efficient cancellation of propagating waves. The energy decays nearly exponentially along the distance of propagation inside the cloud, setting the localization length Ž l . to be about 6 a w11,25x. At this length, kl f 0.088; therefore the Ioffe–Regel criterion for localization is satisfied. Outside the cloud, the transmission remains constant with distance, as expected when the geometric spreading factor has been removed. The energy localization and the order in the phase vectors are independent of the outer boundary and they always appear for sufficiently large b and N. In this case, as shown, all bubbles oscillate completely in phase, leading to the constant pressure
phases inside the medium and showing a previously unsuspected collective behaviour. Meanwhile, the energy gradually decreases as moving away from the transmitting source. These features are in accordance with the aforementioned perception about energy storage in random media. When the frequency increases further, moving outside the energy localization region, the in-phase order disappears. Meanwhile, the wave becomes non-localized again. This is illustrated by the case of ka s 0.1. In this case, the phase vectors again point to various directions Žleft column. and the energy distributes roughly uniformly in space Žright column.. From the above, we see that ka s 0.01254, ka s 0.0173 and ka s 0.1 respectively belong to different phase states. We may conclude as follows. Ž1. In Regions I and III, wave is not localized and the phase vectors point to various directions. Ž2. In Region II, the energy decays exponentially on average along the distance traveled, and all the phase vectors point more or less to the same direction against the source; in this region, states with different ka may have different localization length, but for almost all ka the phase vectors prevail a nearly
Fig. 2. Order parameter as a function of ka.
Z. Ye et al.r Physics Letters A 275 (2000) 452–458
perfect ordering. Implied by the phase diagram in the left column of Fig. 1, there must be a phase transition from Region I to II and from Region II to III. When waves are localized, the order parameter is significantly larger than that for extended waves. In Fig. 2, we plot the order parameter as a function of the non-dimensional parameter ka for one random realization of the bubble cloud. The result shows that as the frequency is increased from the low frequency end, the order parameter rises to about unity rapidly. Then it decreases as the frequency moves out of the localized regime. Due to the finite sample size, the order parameter does not vanish completely outside the localized regime w27x. Further numerical investigation shows that the patterns depicted by Fig. 1 hold for a wide range of bubble size and for any random distribution of bubbles. And the features are always valid for sufficiently large bubble volume fractions. The localization range depends crucially on the bubble volume fraction and are relatively insensitive to the bubble size. The collective phenomenon is caused by multiple scattering. As the multiple scattering is removed from Eq. Ž1., the collective behavior disappears completely. By varying the bubble numbers while keeping the volume fraction b constant, it can be shown that the localization and non-localization behavior are qualitatively unchanged, and thus the behavior is not caused by the boundary of the bubble arrays. The appearance of the collective behavior for localized waves implies existence of a kind of Goldstone bosons incurred in the phase transition. A further study also reveals the unexpected result that localization is relatively independent of the precise location or organization of the scatterers. The results for acoustic propagation in regular arrays and the connection to the band gap effect will be published elsewhere w28,29x. An intuitive understanding about the acoustic localization in the bubbly liquids may be helped from the following consensus. Imagine that two persons hold a thread. If one person pushes, while the other pulls, the two persons would act completely out of phase. No energy can be transferred from one to the other. In the bubbly liquid case, the state of localized waves shows such a completely out-of-phase behavior, which effectively prevents waves from propagating.
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The behavior of the present system is a result of the interplay between the single bubble scattering and multiple scattering among bubbles. We take the view that the coherent behavior allows for efficient cancellation of source wave, thus giving rise to wave localization. Exactly how the interplay between the single bubble response and the multiple scattering induces the localization and the global coherent behavior remains an open question and may be further understood by an analytic analysis which is under investigation. In summary, we have demonstrated a new phase transition for acoustic propagation in a bubbly liquid. The numerical results show that as the phase transition occurs, not only the acoustic waves are confined in the neighborhood of the transmitting source, but an amazing collective behavior of the air bubbles appears. A novel diagram method is proposed to describe the phase transition of the phase vectors associated with the bubbles. We suggest that such a coherence behavior may be crucial to differentiate the localization effect from the residual absorption effect. The ambiguity between the two effects has caused much debate in the literature w30,31x. The phase transition in the phase vectors for the present case draws remarkable similarities to the case of the magnetic dipole directions in magnetic materials, responsible for various magnetic phase states. Although these localization properties are only demonstrated for the special resonant air-bubbles, they are expected to hold true for other resonant scatterers as well.
Acknowledgements The work received support from the National Science Council.
References w1x A. Ishimaru, Wave Propagation and Scattering in Random Media, Academic Press, New York, 1978. w2x N.M. Lawandy, R.M. Balachandran, A.S.L. Gomes, E. Sauvain, Nature ŽLondon. 368 Ž1994. 436. w3x H. Cao et al., Phys. Rev. Lett. 82 Ž1999. 2278. w4x W.M. Robertson et al., J. Opt. Soc. Am. B 10 Ž1993. 322. w5x J.V. Sanchez et al., Phys. Rev. Lett. 80 Ž1998. 5325.
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w6x M. Torres, J.P. Adrados, F.R. Montero de Espinosa, Nature ŽLondon. 398 Ž1998. 114. w7x P.W. Anderson, Phys. Rev. 109 Ž1958. 1492. w8x C.H. Hodges, J. Woodhouse, J. Acoust. Soc. Am. 74 Ž1983. 894. w9x R. Dalichouch, J.P. Amstrong, S. Shultz, P.M. Platzman, S.L. McCall, Nature ŽLondon. 354 Ž1991. 53. w10x J.H. Page et al., Science 271 Ž1996. 634. w11x Z. Ye, A. Alvarez, Phys. Rev. Lett. 80 Ž1998. 3503. w12x D. Sornette, B. Souillard, Europhys. Lett. 7 Ž1988. 269. w13x H. Umezawa, Advanced Field Theory, AIP, New York, 1995. w14x T.R. Kirkpatrick, Phys. Rev. B 31 Ž1985. 5746. w15x C.A. Condat, J. Acoust. Soc. Am. 83 Ž1988. 441. w16x Z. Ye, Phys. Rev. E 56 Ž1997. 2318. w17x Z. Ye, C. Feuillade, J. Acoust. Soc. Am. 102 Ž1997. 798. w18x S. Temkin, J. Fluid Mech. 380 Ž1999. 1. w19x E. Silberman, J. Acoust. Soc. Am. 29 Ž1957. 925.
w20x w21x w22x w23x w24x w25x w26x w27x w28x w29x w30x w31x
H. Medwin, J. Acoust. Soc. Am. 62 Ž1977. 1041. L.L. Foldy, Phys. Rev. 67 Ž1945. 107. A.S. Sangani, J. Fluid Mech. 232 Ž1991. 221. Z. Ye, L. Ding, J. Acoust. Soc. Am. 98 Ž1995. 1629. H. Medwin, C.S. Clay, Acoustical Oceanography, Academic Press, New York, 1998. A. Alvarez, Z. Ye, Phys. Lett. A 252 Ž1999. 53. P.M. Morse, H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, 1953. For extremely low or high frequencies Ž ka) 0.4., our model may not be applicable, but this is not our interest. Z. Ye, H. Hsu, preprint, 2000. P.-G. Luan, Z. Ye, preprint, 2000. F. Scheffold, R. Lenke, R. Tweer, G. Maret, Nature 398 Ž1999. 206. A.A. Chabanov, M. Stoytchev, A.Z. Genack, Nature 404 Ž2000. 850.