15 February 1999

PHYSICS LETTERS A

ELSEVIER Physics Letters A 252 (1999) 53-57

Localization transition in acoustic propagation Albert0 Alvarez *, Zhen Ye

Department of Physics and Center for Complex Systems, National Central University Chung-li. Taiwan, ROC

in bubbly liquids

Received 5 October 1998; accepted for publication 4 December 1998

Communicated by J. Flouquet

Abstract

In the past, it was suggested that localization can be achieved for acoustic wave propagation in liquids with even a very small concentration of air-filled bubbles. In this paper we present an ab initio numerical investigation of acoustic localization in bubbly liquids. The results show that the localization of acoustic waves is reached when the bubble volume fraction is greater than a threshold value about lo-‘. This is in good agreement with the previous prediction [Somette, Souillard, Europhys. Lett. 7 ( 1988) 1691. @ 1999 Elsevier Science B.V.

PAC.? 71.55.J; 43.20; 03.40.K

ICqvwordsr Acoustic localization; Wave propagation

Wave localization is the peculiar property of a scat- tering medium that completely blocks wave propaga- tion, thus inducing a surprising phase transition, for example in optical or acoustic propagation or electric conductivity. When localized, waves remained trapped in a spatial domain until dissipated. Such a localiza- tion may be realized in a variety of situations. In dis- ordered solids, electron localization is common [ 11. Water waves may be localized by underwater topogra- phy [ 21. The localization phenomenon has also been reported for microwaves [3,4] and recently for light in disordered media [S]. A systematic approach to the localization of light waves in three-dimensional dielectric media has recently developed by Rusek et al. [ 61. Likewise, acoustic localization has also been studied experimentally for two-dimensional inhomo- geneous systems [7] and theoretically for systems having internal resonances [g-lo]. Research has sug-

* Corresponding author. E-mail: aIberto@jouIe.phy.ncu.edu.tw

gested that acoustic localization may be observed in bubbly liquids. It was conjectured that acoustic local- ization occurs when the bubble volume fraction ex- ceeds about 10e5 [ lo].

Recently, we further considered acoustic localiza- tion in bubbly liquids in a rigorous numerical method. The preliminary numerical results confirmed that for a range of frequencies the wave localization is indeed possible in bubbly liquids [ 111. A few advantages of using bubbly liquids for the study of wave localiza- tion have also been identified. In this Letter, we con- tinue the study of acoustic localization in bubbly liq- uids. We attempt to explore the transition from the non-localization regime to the localization regime. In particular, we investigate how the wave propagation is affected as the bubble volume fraction varies. This study allows us to directly compare our numerical re- sults with the previous analytic prediction [ lo], there- fore providing a definite verification.

First consider acoustic scattering by a single bubble

0375-9601/99/$ - see front matter @ 1999 Elsevier Science B.V. All rights reserved PII SO375-9601(98)00926-8

54 A. Alvarez, 2. Ye/Physics Letters A 252 (1999) 53-57

lo2

(a) a=lOpm

loo m =_ -

1o‘2 F

0 0.1 0.2 0.3 0.4 ka

11. (b) a=2cm

I 0 0.1 0.2 0.3 0.4

ka

Fig. 1. Scattering function of a single bubble as a function of frequency in terms of ka.

of radius a in a liquid. Studies show that a strong res- onant scattering occurs for low frequencies, i.e. ka < 0.4 with k being the usual wavenumber, to which we

will restrict our attention. In this low frequency region, the scattering function is isotropic and can be readily

computed as [ 121

f= a

ws/02- 1 -ika-is’ (1)

where wg is the natural frequency of the bubble and S

is the damping factor that includes the thermal and vis- cosity effects causing acoustic absorption. However, the damping factor can be adjusted manually in the present study to inspect the sensibility of the results to the absorption.

Fig. 1 plots the scattering function versus frequency

in terms of ka for two bubble sizes. It shows that (i) the strongest scattering occurs at the resonance fre- quency 00, about ka = 0.0136; (ii) the acoustic ab- sorption is more pronounced for small bubbles, shown by the lower peak amplitude in the smaller bubble; (iii) except for the frequencies very close to the res- onance frequency, the scattering properties are almost identical for the different bubble sizes. Fig. 1 suggests that small bubbles may not be suitable for observing

localization, as they cause a significant acoustic ab- sorption.

Now consider wave propagation in liquids contain- ing many bubbles. Assume a unit point source in the

liquids. It transmits a monochromatic acoustic wave of angular frequency w. The source is assumed to be located at the origin. There are N spherical bubbles locatedatri (i= 1,2,... , N), surrounding the emis- sion point. For simplicity, all the bubbles are taken to

be the same size and are randomly distributed within a sphere. No two bubbles can occupy the same location, i.e. the hard sphere approximation. In other words, the point source is placed at the center of a spherical bub- ble cloud. The bubble radius is a. The volume frac- tion, the fraction of volume occupied by the bubbles

per unit volume, is denoted by p. Therefore the nu- merical density of the bubbles is n = 3P/4n-a3 and

the radius of the bubble cloud is R = (N/P) ‘13u. The emitted wave from the source is subject to multiple

scattering by the surrounding bubbles before reaching a receiving point. The scattered wave from each bub- ble is a response to the incident wave composed of the direct incident wave from the source and the multi- ply scattered waves from other bubbles. The response function is given in Eq. ( 1). The total wave at any space point is the addition of the direct wave from the source and scattered wave from all the bubbles. Mul-

tiple scattering in such a system can be computed rig-

orously using the self-consistent method proposed by Foldy [ 133 and the matrix inversion scheme [ 111. We

note that the above approach is similar to what Rusek et al. have done in the study of light localization by random dielectric media [ 61. They show that although

for a finite system it is not possible to achieve perfect localization, the prediction is experimentally indistin- guishable from a complete localization. This is sup-

ported by our work. Indeed, the transmission in our systems, as will be shown, follows exponential decay

behavior, the localization can therefore only depend

on the boundary through an exponential tail, in agree- ment with the Thouless scaling argument [ 141

When localization occurs, the total energy is con- fined in the neighborhood of the transmission point. Therefore it would be sufficient to examine the spa- tial distribution of the total energy. For this purpose, we compute the averaged squared modulus of the to- tal wave denoted by I = (ip12), which can be decom- posed into two parts, that is, the coherent portion 1~ G

A. Alvarez, 2. Ye/Physics Letters A 252 (1999) 53-57 5s

ka

04

P @le-3

.$ loo p=ie-5

f

H 8 m

;_ 10-s-

0 0.1 0.2 0.3 0.4 ka

Fig. 2. Transmission (a) and backscattering (b) for an acoustic

wave propagation in bubbly liquids for two bubble concentrations.

Ik)12 d h d’ff an t e 1 usive portion 1, I (lp - (p) 12). we

have I = Ic + lo. To compare with other approaches

that use the information of the diffusive energy, we also compute the spatial distribution of the diffusive

energy. The backscattering strength is obtained by re- placing p with CE, ps (0; i), i.e. the summation of all the scattered waves at the source position. We note

that an analytic method for the study of the localiza-

tion may be to compute the Bethe-Salpeter equation for the Green’s function of the energy, for which the

diffusive approximation is invoked to obtain the diffu- sion coefficient and the transport mean free path in the

long time and range limits. In the present approach, the total wave is directly computed without resorting

to unnecessary approximations. A set of numerical simulations has been carried out

for various bubble volume fractions. The parameters used are the same as in Ref. [ 111. In the simulation, the number of bubbles has been set to 200, 500 up to 2500 respectively. The radius of the bubbles ranges from 20 Gum to 2 cm. We find that in this wide range of parameters all results are similar. Below we present some typical results.

Fig. 2 shows the transmission and backscattering as a function of frequency in terms of ka for volume

fractions /3 = 10e3 and low5 respectively. The bubble

number is 1000 and the radius 0.2 cm. In Fig. 2a, the hydrophone is placed at a distance away from the bub- ble cloud to receive the transmission. Here it is shown that the transmission is significantly reduced between ka = 0.01 and 0. I in the case of /3 = 10-j. The great-

est inhibition occurs at about ka = 0.017. Such an in-

hibition is not caused by the dissipation mechanisms mentioned earlier. To justify this, we manually make the absorption factor S zero and find that the results remain unchanged. The absorption effect becomes sig- nificant only when we use very small bubbles such

as a = I ,um. We also verified that the transmission inhibition is related to multiple scattering. When the

multiple scattering is turned off in our simulation, the inhibition disappears. This provides a constraint on

how many bubbles should be considered in the sim- ulation so that sufficient multiple scattering can be established. For reasons that will become clear later, such an inhibition is in line with wave localization. We

also verify that when localization occurs the intuitive Ioffe-Regel criterion is satisfied [ 111.

For /? = 10P5, no significant inhibition is found at any frequency with reference to the fl = low3 situa- tion, implying that a transition from the localization

to non-localization occurs as the bubble volume frac-

tion is lowered. Fig. 2b shows the backscattering sit- uation. We observe stronger backscattering enhance- ment for the /? = 10e3 case as compared to the case p = 10P5. Our results do not indicate a factor of 2 en-

hancement. It is our opinion that although connected with multiple scattering, the backscattering enhance- ment is not a direct indicator of localization of waves.

Even for non-localization and weakly multiple scat- tering situations, people can observe a factor of two enhancement in backscattering (see e.g. Ref. [ 151) .

Indeed, as pointed out by one of the referees, there is a wild discussion in the literature about backscat-

tering enhancement. Backscattering enhancement ap- pears as long as there is multiple scattering. However, only appropriate multiple scattering can induce the lo-

calization phase transition. In separate papers we will discuss this in further detail [ 16,171.

The conjectured transition is confirmed by exam- ining the spatial distribution of energy. Fig. 3 plots the transmission as a function of distance away from the source for different bubble volume fractions at ka = 0.0171, where a strong transmission inhibition

56 A. Alvarez. 2. Ye/Physics Letters A 252 (1999) 53-57

10" a) p=1e4

10-z

r F e104

1o-6 ,\

0 0.5 1 1.5 2 r/R

10"

4 p=3e-5

lo-*

N-

% r lo-'

lo4 r

Fig. 3. Total energy (solid lines) and its diffusive portion (dotted lines) as a function of distance Tom the acoustic source for various bubble volume fractions: (a) 10V4, (b) 3 x toes, (c) low5 and (d) 10m6. The distance is scaled by the radius of the bubble cloud R.

appeared. In the plots, both the total energy and its dif- fusive portion are shown. It is evident from plots (a) to (d) that the transition occurs when the bubble con- centration is about 3 x 10m5 to 10m5. At p = 10V4, the

wave is spatially trapped near the transmission point

and localized inside the bubble cloud, rending the fea- tures of wave localization. The transmission decays exponentially along the distance traveled by the wave. The slope sets the localization length to be around 5a, which is significantly smaller than the size of bubble

clouds due to the strong interaction and could not be obtained by an effective medium theory. We also note

that the attenuation length due to the acoustic absorp- tion is about 118~ which is much larger than the lo- calization length, excluding the acoustic absorption as a cause for the wave trapping. In the localized state, one can intuitively expect that the diffusive energy in- creases initially as more and more scattering happens, then it will be eventually stopped by the interference of the multiply scattered waves. This is clearly shown in Fig. 3a. Explicitly, this figure demonstrates how the diffusive portion increases at the beginning as a result of scattering. After some distance from the emission point, around r/R = 0.15, its transport is suddenly

blocked along with the total transmission. Such a be- havior in the diffusive wave cannot be explained by a classical diffusion model, in line with the experimen- tal observation by Weaver [ 71.

When the bubble volume fraction is lowered, the localization starts to disappear. At 3 x lo-‘, a local-

ization is still barely seen. For p around low5 and 10m6, the localization vanishes completely. Therefore we conclude that the localization transition occurs at about p = lo-‘. Further numerical experiments show

that this is true for other bubble sizes and frequencies at which localization may occur. This is in favorable

agreement with the previous theoretical estimate [ lo].

The robustness of the above results has been tested by performing further numerical simulations in which

experimental parameters are adjusted manually. Such a test makes certain that the localization transition is indeed caused by scattering rather than by the en- ergy dissipation. Finally, we note from Fig. 2 that the strongest inhibition does not occur at the resonance of the bubble. In fact, we found that the wave is not lo- calized at the resonance, rather at parameters slightly different from the internal resonance, indicating that mere resonance does not promise localization. This

A. Alvarez, Z. Ye/Physics Letters A 252 (1999) 53-57 51

supports the assertion of Rusek et al. [ 61. As will be

shown elsewhere, the acoustic localization in systems of internal resonances is in fact attributed to an inter- esting collective behavior, which allows for an effi- cient phase cancellation and induces the localization phase transition [ 171.

We have studied the localization transition in acous- tic propagation in bubbly liquids. The numerical re- sults clearly show that the localization appears when

the bubble volume fraction reaches around 10e5, sup- porting the previous prediction.

We thank one referee for useful comments/sug- gestions that helped improve the manuscript. He also

brought our attention to the important work of Weaver

in Ref. [ 71. We share many of his concerns with the ambiguous results in the literature. The work received support from the National Science Council of ROC. One of us (AA) also acknowledges the FPIE fellow- ship from the Spanish Ministry of Education and a post doctoral fellowship from the NSC of ROC.

References

[ 11 P.A. Lee. T.V. Ramakrishnan. Rev. Mod. Phys. 57 (1985)

287.

21 M. Belzons. E. Guazzelli, B. Souillard, Localization of

surface gravity waves on a random bottom, in: Scattering

and Localization of Classical Waves in Random Media. ed.

P. Sheng (World Scientific, Singapore. 1990).

3 I R. Dalichaouch et al., Nature 354 ( 1991) 53.

41 A.Z. Genack. N. Garcia, Phys. Rev. Len. 66 ( 199 1) 2064.

51 D.S. Wiersma et al., Nature 390 (1997) 671.

61 M. Rusek, A. Orlowski, J. Mostowski. Phys. Rev. E 53

(1996) 4122.

171 R.L. Weaver, Wave Motion 12 ( 1990) 129.

[81 T.R. Kirkpatrick, Phys. Rev. B 31 ( 1985) 5746.

191 C.A. Condat, J. Acoust. Sot. Am. 83 (1988) 441.

[lo] D. Somette. B. Souillard, Europhys. Lett. 7 ( 1988) 269.

[ 111 Z. Ye, A. Alvarez, Phys. Rev. Lett. 80 ( 1998) 3503.

[I21 C.S. Clay, H. Medwin. Acoustical Oceanography (Wiley,

New York, 1977).

[ 131 L.L. Foldy, Phys. Rev. 67 (1945) 107.

[ 141 P. Sheng, Introduction to Wave Scattering, Localization, and

Mesoscopic Phenomena (Academic Press, New York, 1995).

[ 151 D. Chu et al., J. Acoust. Sot. Am. 102 ( 1997) 806.

[ 161 Z. Ye et al., Chin. J. Phys. (to be published).

[ 171 Z. Ye. E. Hoskinson. Nature (submitted).