Waves in fluids and solids Part 13 pptx

Conclusions In this chapter, we first consider the acoustic propagation in a finite sample of bubbly soft elastic medium and solve the wave field rigorously by incorporating all multipl

Acoustic Waves in Bubbly Soft Media 289

Fig 5.2 Number densities of large (a) and small (b) bubbles in the bubbly silicone with optimal acoustic attenuation

Fig 5.3 Comparison of acoustic attenuations versus frequency for the four different cases

Waves in Fluids and Solids

290

6 Conclusions

In this chapter, we first consider the acoustic propagation in a finite sample of bubbly soft elastic medium and solve the wave field rigorously by incorporating all multiple scattering effects The energy converted into shear wave is numerically proved negligible as the longitudinal wave is scattered by the bubbles Under proper conditions, the acoustic localization can be achieved in such a class of media in a range of frequency slightly above the resonance frequency Based on the analysis of the spatial correlation characteristic of the wave field, we present a method that helps to discern the phenomenon of localization in a unique manner Then we taken into consideration the effect of viscosity of the soft medium and investigate the localization in a bubbly soft medium by inspecting the oscillation phases

of the bubble The proper analysis of the oscillation phases of bubbles is proved to be a valid approach to identify the existence of acoustic localization in such a medium in the presence

of viscosity, which reveals the existence of the significant phenomenon of phase transition characterized by an unusual collective behavior of the phases

For infinite sample of bubbly soft medium, we present an EMM which enables the investigation of the strong nonlinearity of such a medium and accounts for the effects of weak compressibility, viscosity, surrounding pressure, surface tension, and encapsulating shells Based on the modified equation of bubble oscillation, the linear and the nonlinear wave equations are derived and solved for a simplified 1-D case Based on the EMM which can be used to conveniently obtain the acoustic parameters of bubbly soft media with arbitrary structural parameters, we present an optimization method for enhancing the acoustic attenuation of such media in an optimal manner, by applying FL and GA together

A numerical simulation is presented to manifest the necessity and efficiency of the optimization method This optimization method is of potential application to a variety of situations once the objective function and optimizer are adjusted accordingly

7 References

[1] E Meyer, K Brendel, and K Tamm, J Acoust Soc Am 30, 1116 (1958)

[2] A C Eringen and E S Suhubi, Elastodynamics (Academic, New York, 1974)

[3] L A Ostrovsky, Sov Phys Acoust 34, 523 (1988)

[4] L A Ostrovsky, J Acoust Soc Am 90, 3332 (1991)

[5] S Y Emelianov, M F Hamilton, Yu A Ilinskii, and E A Zabolotskaya, J Acoust Soc

Am 115, 581 (2004)

[6] E A Zabolotskaya, Yu A Ilinskii, G D Meegan, and M F Hamilton, J Acoust Soc

Am 118, 2173 (2005)

[7] L D Landau and E M Lifshits, Theory of Elasticity (Pergamon, Oxford, 1986)

[8] B Liang and J C Cheng, Phys Rev E 75, 016605 (2007)

[9] B Liang, Z M Zhu, and J C Cheng, Chin Phys Lett 23, 871 (2006)

[10] B Liang, X Y Zou, and J C Cheng, Chin Phys Lett 26, 024301 (2009)

[11] B Liang, X Y Zou, and J C Cheng, Chin Phys B 19, 094301 (2010)

[12] B Liang, X Y Zou, and J C Cheng, J Acoust Soc Am 124, 1419 (2008)

[13] G C Gaunaurd and J Barlow, J Acoust Soc Am 75, 23 (1984)

[14] B Liang, X Y Zou, and J C Cheng, Chin Phys Lett 24, 1607 (2007)

Acoustic Waves in Bubbly Soft Media 291 [15] K X Wang and Z Ye, Phys Rev E 64, 056607 (2001)

[16] C F Ying and R Truell, J Appl Phys 27, 1086 (1956)

[17] G C Gaunaurd, K P Scharnhorst and H Überall, J Acoust Soc Am 65, 573

(1979)

[18] D M Egle, J Acoust Soc Am 70, 476 (1981)

[19] R L Weaver, J Acoust Soc Am 71, 1608 (1982)

[20] K Busch, C M Soukoulis, and E N Economou, Phys Rev B 50, 93 (1994)

[21] A A Asatryan, P A Robinson, R C McPhedran, L C Botten, C Martijin de Sterke, T

L Langtry, and N A Nicorovici, Phys Rev E 67, 036605 (2003)

[22] S Gerristsen and G E W Bauer, Phys Rev E 73, 016618 (2006)

[23] C A Condat, J Acoust Soc Am 83, 441 (1988)

[24] Z Ye and A Alvarez, Phys Rev Lett 80, 3503 (1998)

[25] S Catheline, J L Gennisson, and M Fink, J Acoust Soc Am 114, 3087 (2003)

[26] Z Fan, J Ma, B Liang, Z M Zhu, and J C Cheng, Acustica 92, 217 (2006)

[27] I Y Belyaeva and E M Timanin, Sov Phys Acoust 37, 533 (1991)

[28] Z Ye, H Hsu, and E Hoskinson, Phys Lett A 275, 452 (2000)

[29] A Alvarez, C C Wang, and Z Ye, J Comp Phys 154, 231 (1999)

[30] L L Foldy, Phys Rev 67, 107 (1945)

[31] A Ishimaru, Wave Propagation and Scattering in Random Media (Academic press, New

York, 1978)

[32] B C Gupta and Z Ye, Phys Rev E 67, 036606 (2003)

[33] M Rusek, A Orlowski, and J Mostowski, Phys Rev E 53, 4122 (1996)

[34] A Alvarez and Z Ye, Phys Lett A 252, 53 (1999)

[35] C C Church, J Acoust Soc Am 97, 1510 (1995)

[36] C H Kuo, K K Wang and Z Ye, Appl Phys Lett 83, 4247 (2003)

[37] H J Feng and F M Liu, Chin Phys B 18, 1574 (2009)

[38] M Mooney, J Appl Phys 11, 582 (1940)

[39] J Ma, J F Yu, Z M Zhu, X F Gong, and G H Du, J Acoust Soc Am 116, 186

(2004)

[40] G C Gaunaurd, H Überall, J Acoust Soc Am 71, 282 (1982)

[41] G C Gaunaurd and W Wertman, J Acoust Soc Am 85, 541 (1989)

[42] A M Baird, F H Kerr, and D J Townend, J Acoust Soc Am 105, 1527 (1999)

[43] D G Aggelis, S V Tsinopoulos, and D Polyzos, J Acoust Soc Am 116, 3343

(2004)

[44] B Qin, J J Chen, and J C Cheng, Acoust Phys 52, 490 (2006)

[45] B Liang B, Z M Zhu, and J C Cheng, Chin Phys 15, 412 (2006)

[46] D H Trivett, H Pincon, and P H Rogers, J Acoust Soc Am 119, 3610 (2006)

[47] A C Hennion, and J N Decarpigny, J Acoust Soc Am 90, 3356 (1991)

[48] L F Shen, Z Ye, and S He, Phys Rev B 68, 035109 (2003)

[49] H Li, and M Gupta, Fuzzy logic and intelligent systems (Boston, Kluwer Academic

Waves in Fluids and Solids

292

[52] Y Xu et al, Chin Phys Lett 22, 2557 (2005)

[53] Y C Chang, L J Yeh, and M C Chiu, Int J Numer Meth Engng 62, 317 (2005)

Inverse Scattering in the Low-Frequency Region

by Using Acoustic Point Sources

Nikolaos L Tsitsas

Department of Mathematics, School of Applied Mathematical and Physical Sciences,

National Technical University of Athens, Athens

Greece

1 Introduction

The interaction of a point-source spherical acoustic wave with a bounded obstacle possessesvarious attractive and useful properties in direct and inverse scattering theory More precisely,concerning the direct scattering problem, the far-field interaction of a point source with anobstacle is, under certain conditions, stronger compared to that of a plane wave On theother hand, in inverse scattering problems the distance of the point-source from the obstacleconstitutes a crucial parameter, which is encoded in the far-field pattern and is utilizedappropriately for the localization and reconstruction of the obstacle’s physical and geometricalcharacteristics

The research of point-source scattering initiated in (1), dealing with analytical investigations

of the scattering problem by a circular disc The main results for point-source scattering bysimple homogeneous canonical shapes are collected in the classic books (2) and (3) Thetechniques of the low-frequency theory (4) in the point-source acoustic scattering by soft,hard, impedance surface, and penetrable obstacles were introduced in (5), (6), and (7), wherealso explicit results for the corresponding particular spherical homogeneous scatterers wereobtained Moreover, in (5), (6), and (7) simple far-field inverse scattering algorithms weredeveloped for the determination of the sphere’s center as well as of its radius On the otherhand, point-source near-field inverse scattering problems for a small soft or hard sphere werestudied in (8) For other implementations of near-field inverse problems see (9), and p 133 of(10); also we point out the point-source inverse scattering methods analyzed in (11)

In all the above investigations the incident wave is generated by a point-source located inthe exterior of the scatterer However, a variety of applications suggests the investigation

of excitation problems, where a layered obstacle is excited by an acoustic spherical wavegenerated by a point source located in its interior Representative applications concernscattering problems for the localization of an object, buried in a layered medium (e.g insidethe earth), (12) This is due to the fact that the Green’s function of the layered medium(corresponding to an interior point-source) is utilized as kernel of efficient integral equationformulations, where the integration domain is usually the support of an inhomogeneityexisting inside the layered medium Besides, the interior point-source excitation of a layeredsphere has significant medical applications, such as implantations inside the human head forhyperthermia or biotelemetry purposes (13), as well as excitation of the human brain by theneurons currents (see for example (14) and (15), as well as the references therein) Several

11

physical applications of layered media point-source excitation in seismic wave propagation,underwater acoustics, and biology are reported in (16) and (17) Further chemical, biologicaland physical applications motivating the investigations of interior and exterior scatteringproblems by layered spheres are discussed in (18) Additionally, we note that, concerning theexperimental realization and configuration testing for the related applications, a point-sourcefield is more easily realizable inside the limited space of a laboratory compared to a planewave field.

To the direction of modeling the above mentioned applications, direct and inverse acousticscattering and radiation problems for point source excitation of a piecewise homogeneoussphere were treated in (19)

This chapter is organized as follows: Section 2 contains the mathematical formulation of theexcitation problem of a layered scatterer by an interior point-source; the boundary interfaces

case where the boundary surfaces are spherical and deal with the direct and inverse acousticpoint-source scattering by a piecewise homogeneous (layered) sphere The point-source may

be located either in the exterior or in the interior of the sphere The layered sphere consists of

and the N-th layer (core) is soft, hard, resistive or penetrable More precisely, Section 3.1addresses the direct scattering problem for which an analytic method is developed for thedetermination of the exact acoustic Green’s function In particular, the Green’s function isdetermined analytically by solving the corresponding boundary value problem, by applying

a combination of Sommerfeld’s (20), (21) and T-matrix (22) methods Also, we give numericalresults on comparative far-field investigations of spherical and plane wave scattering, whichprovide quantitative criteria on how far the point-source should be placed from the sphere

in order to obtain the same results with plane wave incidence Next, in Section 3.2 thelow-frequency assumption is introduced and the related far-field patterns and scatteringcross-sections are derived In particular, we compute the low-frequency approximations of

to the corresponding ones due to plane wave incidence on a layered sphere and also recover

as special cases several classic results of the literature (contained e.g in (2), and (5)-(7)),concerning the exterior spherical wave excitation of homogeneous small spheres, subject

to various boundary conditions Also, we present numerical simulations concerning theconvergence of the low-frequency cross-sections to the exact ones Moreover, in Section 3.3certain low-frequency near-field results are briefly reported

Importantly, in Section 4 various far- and near-field inverse scattering algorithms for a smalllayered sphere are presented The main idea in the development of these algorithms is thatthe distance of the point source from the scatterer is an additional parameter, encoded in thecross-section, which plays a primary role for the localization and reconstruction of the sphere’scharacteristics First, in Section 4.1 the following three types of far-field inverse problems areexamined: (i) establish an algorithmic criterion for the determination of the point-source’slocation for given geometrical and physical parameters of the sphere by exploiting thedifferent cross-section characteristics of interior and exterior excitation, (ii) determine themass densities of the sphere’s layers for given geometrical characteristics by combining thecross-section measurements for both interior and exterior point-source excitation, (iii) recoverthe sphere’s location and the layers radii by measuring the total or differential cross-sectionfor various exterior point-source locations as well as for plane wave incidence Furthermore,

in Section 4.2 ideas on the potential use of point-source fields in the development of near-fieldinverse scattering algorithms for small layered spheres are pointed out.

2 Interior acoustic excitation of a layered scatterer: mathematical formulation

normal unit vector ˆn.

Applying Sommerfeld’s method (see for example (20), (21), (22)), the primary spherical field

urprq, radiated by this point-source, is expressed by

urprq(r) =rqexp(−ikqrq)exp(ikq|r−rq|)

q}, (1)

the primary field reduces to a plane wave with direction of propagation that of the unit vector

−ˆrq, when the point source recedes to infinity, i.e

lim

r q→∞u

pr

rq(r) =exp(−ikqr ˆq·r) (2)

Vj The respective secondary fields in Vj (j 6= q) and Vq are denoted by ujrq and usecrq By

and the secondary field

uqr

q(r) =urpr

q(r) +usecr

q (r), r∈Vq\ {rq} (3)Moreover, the total field in Vj(j6=q) coincides with the secondary field ujrq

The total field ujr

For a penetrable core (7) hold also for j=N On the other hand, for a soft, hard and resistive

the Dirichlet

Since scattering problems always involve an unbounded domain, a radiation condition for the

spherical acoustic wave defined by (1) satisfies the Sommerfeld radiation condition (11), whichclearly is not satisfied by an incident plane acoustic wave

Besides, the secondary usecr0 and the total field u0rq in V0have the asymptotic expressions

u0rq(r) =grq(ˆ r)h0(k0r) + O(r−2), r→∞ (q>0) (13)where h0(x)=exp(ix)/(ix)is the zero-th order spherical Hankel function of the first kind The

direction of observation ˆr of the far-field, due to the excitation by the particular primary field

urprq in layer Vq.Moreover, we define the q-excitation differential (or bistatic radar) cross-section

σrq( ) = 4π

which specifies the amount of the field’s power radiated in the direction ˆr of the far field Also,

we define the q-excitation total cross-section

σreq =σraq+σrq (17)

the other layers have been assumed lossless) and the latter the total power that the scatterer

rq =0 for

a soft, hard, or penetrable lossless core, and σraq≥0 for a resistive core

We note that scattering theorems for the interior acoustic excitation of a layered obstacle,subject to various boundary conditions, have been treated in (23) and (24)

3 Layered sphere: direct scattering problems

The solution of the direct scattering problem for the layered scatterer of Fig 1 cannot

be obtained analytically and thus generally requires the use of numerical methods; for anoverview of such methods treating inhomogeneous and partially homogeneous scatterers

analytically and the exact Green’s function can be obtained in the form of special functionsseries To this end, we focus hereafter to the case of the scatterer V being a layered sphere

3.1 Exact acoustic Green’s function

A classic scattering problem deals with the effects that a discontinuity of the medium ofpropagation has upon a known incident wave and that takes care of the case where theexcitation is located outside the scatterer When the source of illumination is located insidethe scatterer and we are looking at the field outside it, then we have a radiation and not

a scattering problem The investigation of spherical wave scattering problems by layeredspherical scatterers is usually based on the implementation of T-matrix (22) combined withSommerfeld’s methods (20), (21) The T-matrix method handles the effect of the sphere’slayers and the Sommerfeld’s method handles the singularity of the point-source and unifiesthe cases of interior and exterior excitation The combination of these two methods leads

to certain algorithms for the development of exact expressions for the fields in every layer.Here, we impose an appropriate combined Sommerfeld T-matrix method for the computation

of the exact acoustic Green’s function of a layered sphere More precisely, the primary andsecondary acoustic fields in every layer are expressed with respect to the basis of the sphericalwave functions The unknown coefficients in the secondary fields expansions are determinedanalytically by applying a T-matrix method

We select the spherical coordinate system (r,θ,φ) with the origin O at the centre of V, so that

Fig 2 Geometry of the layered spherical scatterer.

The secondary field urjqin Vj(j=1, ,N−1) is expanded as

point-source, from the layers above and below respectively.

the transformations (19)

By using the above method we recover (for q=0 and N=1) classic results of the literature,concerning the scattered field for the exterior point-source excitation of a homogeneous sphere(see for example (10.5) and (10.70) of (2) for a soft and a hard sphere)

of an efficient recursive algorithm (19)