Discussion on various model of multiple scattering in acoustics and elastic heterogeneous media

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Discussion on various model of multiple scattering in acoustics and elastic heterogeneous media

Loïc Le Marrec, Grégory Futhazar

To cite this version:

Loïc Le Marrec, Grégory Futhazar. Discussion on various model of multiple scattering in acoustics

and elastic heterogeneous media. Acoustics 2012, Apr 2012, Nantes, France. �hal-00810698�

Discussion on various model of multiple scattering in acoustics and elastic heterogeneous media

L. Le Marrec and G. Futhazar

IRMAR - Universit´ e Rennes 1, Campus Beaulieu, 35042 Rennes, France

loic.lemarrec@univ-rennes1.fr

A new formulation of the effective wave through a heterogeneous medium is the starting point of this paper. The coherent scatteredfield from afixed inclusion is supposed to propagate with a complex effective wave number K. The objective is tofind the expression ofK(ω) according to the concentration of inclusion. This is performed infirst order and traditional Foldy expression is obtained. Moreover the result at a second order differs from classical references as Linton-Martin formula. Each step of the method is detailed in order to discuss on the various consequences of proposed formulation. In particular, the solution for two coupled inclusions is detailed in order to highlight the consequences of hypotheses on the coupling regime.

1 Introduction

Multiple scattering methods [1] is applied to obtained some specifications of wave propagation through a random distribution of scatterer. This is performed by considering the coherent wave obtained by averaging thefields over all possible configurations of disorder [3, 4]. One of the main objective is to characterize an effective wave number. This later is generally defined as the wave number of thefield ex- citing a representative scatterer. Several traditional steps are used to obtain such results some are statistical [2] some other are more based on the physical process. We propose here an original formulation that does not give new fundamental re- sults but can be useful for specific problem typically when the source is no more a plane wave and when the physical contrast between matrix and inclusion is high.

The presentation is based on 2D-acoustical problem, where a random distribution of identical circular cylinders are con- fined in a specific domain.

2 Scattering formulation

We consider time-harmonic acoustical waves with angu- lar frequencyω. Displacementuand pressurepin a medium of densityρand celeritycare expressed thanks acoustical po- tentialψ:u=∇ψandp=−iωρψ. In homogeneous medium, the potential ψ satisfy the Helmholtz equation with wave numberk=ω/cand is a linear combination of 2D harmonic eigen functionφn(kr) = Jn(kr)e^inθand ˆφn(kr) = Hn(kr)e^inθ, where Hn ≡ Hn⁽¹⁾, r andθare respectively the length and angle of vector positionraccording to a given frame.

2.1 Scattering by one inclusion

Consider a single inclusion with radiusa. Material prop- erties of inclusions are with the indices 1 whereas no indices are placed for the matrix case. Incoming, scattered and trans- mittedfields have to satisfy some restrictions. First, the in- comingfieldψⁱ is supposed to be regular in the vicinity of the origin of the frame. Second the scatteredfieldψ^s must satisfy the Sommerfeld radiation condition. Third the trans- mittedfield must be defined in the inclusion. According to these limitations, these fields can be expressed in a frame centered on the circular inclusion byψi(r) =

ncnφn(kr), ψs(r)=

nanφˆn(kr) andψt(r)=

nbnφn(k1r) where the co- efficientscn,an,bnare the modal amplitude according to the specified frame. For perfect boundary condition continuity of displacement and pressure is imposed at the boundary, in term of potential:∂rψ0 =∂rψtandρ0ψ0=ρ1ψtat the bound- ary andψ0≡ψi+ψs. Then we have to solve:

n

an∂rφˆn(kr)+cn∂rφn(kr)

=

n

bn∂rφn(k1r) ρ

n

anφˆn(kr)+cnφn(kr)

=ρ1

n

bnφn(k1r) , r=a

We have∂rφˆn(kr) =k H_n(ka)e^inθet∂rφn(kr)= k J_n(ka)e^inθ. Because

e^inθdθ = 2πδn, the integration over θleads to a linear system where solutions for each modes are uncoupled:

anH_n(ka)− k₁

kbnJ_n(k1a)=−cnJ_n(ka) anHn(ka)− ρ1

ρbnJn(k1a)=−cnJn(k1a) (1) Introducing the coefficient

Δn = ρc

ρ1c1Hn(ka)J_n(k1a)−H_n(ka)Jn(k1a),

the scattering coefficients arean=zncnwithzn=−(Δn)/Δn. For further development we introduce infinite vectora :=

(. . . ,an, . . .) and scattering matrix Z: (Z)nm := znδnm. Ac- cording to these conventions we havea=Z c. In the special case of rigid inclusion∂nψ0 = 0 and for a voidψ0 = 0 ; for such impenetrable scatterer the same methodology gives zn = −J_n(ka)/H_n(ka) andzn = −Jn(ka)/Hn(ka) respectively.

Notice that these results have sense only forfields expressed in a frame centered on the cylinder.

2.2 Scattering by N inclusions

Let now considerNinclusions labeledsj(j=1, . . . ,N).

For each inclusionsja centered frame is defined. The posi- tion of a pointM is then given by the local position vector

rj. The vector formsitosjisri j. The transmitted and scat- teredfield of each inclusion expressed in its local frame is ψt^j =

nbn^j(sj)φn(k₁rj) andψs^j =

nan^j(sj) ˆφn(k₁rj). In order to solve the boundary condition for a given inclusion (say the first) all thefield at its boundary must be given in the frame ofs1. According to the Graf’s addition theorem [5] thefield scattered bysjatMis, in the frame ofs₁:

ψs^j(M)=

n

a¹_n(sj)φn(kr₁) , a¹(sj)=S^1ja^j(sj) ifr₁ < r₁j. This later restriction is of course fulfilled at the boundary ofs₁because inclusions do not overlap. The trans- formation matrixS^1jfrom the frame jto the frame 1 is de- fined by (S^1j)nm=φˆn−m(kr₁j). Using the same methodology than for a single scatterer one obtain a system ofNequations:

aⁱ(si)=Z

⎛⎜⎜⎜⎜⎜

⎜⎝cⁱ+

1j

S^{i j}a^j(sj)

⎞⎟⎟⎟⎟⎟

⎟⎠ , i=1, . . . ,N (2) wherecⁱis the modal amplitude of the incoming wave in the si-frame. This system is well-posed if the position of each inclusion is known.

3 Multiple scattering problems

3.1 Coherent Potential Approximation

The coherent wave can be obtained by averaging thefield over all possible configuration of scatterers. If the surface

concentration isϕ, the meanfield is thenψ=ϕψ1+(1− ϕ)ψ0 whereψ0 and ψ1 are the mean of external and internalfield. Coherent Potential Approximation (CPA) [1]

supposes that the coherence is lost far from the sources and the boundary∂Bof the heterogeneous mediumB; in other word the CPA impliesψ0=0 orψ1=0 equivalently. The incoming field is independent of the position of inclusion, thereforeψ0=ψi+ψsand the CPA implies:

ψs=−ψi (3) If an inclusion is placed atM,ψs+ψiis nothing else than the excitingfield used in literatures [6].

3.2 Ensemble average

A configuration is defined by the set of all positions of inclusionΛN =

r₀₁, r₀₂, . . . , r_0Naccording to afixed frame.

The configuration is governed by the probability densityp(ΛN) and the ensemble average is defined by

ψs=

...

ψs(ΛN)p(ΛN) dr_0N. . .dr₀₂dr₀₁ where the integrations are performed in all the domain except in a circular discCMof radiusacentered at the observation pointM. Otherwise thefield would be an internalfield.

The total scatteredfield isψs =

jψs^j. After ensemble av- erage, the contributions of an inclusion is indistinguishable from the others. Therefore

ψ¹s

= ψ^ks¹

andψs=N ψ¹s

ifs1is chosen as a representative inclusion.

As presented by [7], p(ΛN) = p(r02. . . r0N|r01)p(r01) where p(r₀₁) is the probability tofinds₁arr₀₁andp(r₀₂. . . r_0N|r₀₁) is the density probability tofind the other inclusionssjatr₀j

(j1) if the position of s₁is known. This later probability density can be decomposed as

p(r₀₂. . . r_0N|r₀₁)= p(r₀₃. . . r_0N|r₀₂, r₀₁)p(r₀₂|r₀₁) wherep(r02|r01) is the conditional probability tofinds2atr02

knowing the position ofs1. It is then useful to describe the following hierarchy

ψ¹s

= ψ¹s

1p(r01) dr01

ψ¹s

1= ψ¹s

12p(r02|r01) dr02

ψ¹s

12=

...

ψ¹s(ΛN)p(r03...r0N|r01,r02) dr03...drN

(4)

If the probability densities are supposed uniform:

p(r01)≈ 1

V, p(r02|r01)≈ 1

VH(r12−2a)=p(r12) (5) whereV = N/n₀ is the surface of the integration domain, H is the Heaviside step function andn₀ is the number of inclusion per unit area (or density of inclusion). In particular the CPA Eq.3 is at thefirst order of integration:

n₀ ψ¹s

1 dr₀₁=−ψi

4 Independent scattering

4.1 Discussion on its formulation

On the previous equation the objective is to give the more realistic sense to

ψ¹s

1. In the traditional form of Indepen- dent Scattering Approximation (ISA) proposed by Foldy and

other, one consider that the scattering patternf(θ)=

nzne^inθ of the inclusion is not perturbed by ensemble average. More- over the incomingfield is replaced by an excitingfield having an effective wave numberK. Consider the case of incoming plane wave propagating along thexdirection and probings₁ placed atx₀₁, the radiation far froms₁is

ψ¹s

1=e^{iK x01} f(θ)H0(kr1) (6) The radial dependence of the scattering is governed by k.

This is mainly motivated by the Sommerfeld radiation con- dition satisfied by ψ¹ before averaging over the position of all the inclusion excepts₁. Moreover this formulation can be taken into question. First, Foldy approach leads to searching an excitingfield solution of (Δ+K²)ψ=0, whereKis the ef- fective wave number. Second, it looks reasonable to suppose that the scattering must be modified along the radial direc- tion after this ensemble average: the global effect of other in- clusions placed betweens1 and the observation point would induce additional dispersion and attenuation (Im(K)≤0). In order to take into account this remarks, we propose to replace the previous equation by

ψ¹s

1 =e^ikx01 f(θ)H0(Kr1) (7) where the effect of translation of s1 induces simply a phase difference governed by k. According to this methods, the excitingfield is considered to be the incoming wave in the first order of approximation. This proposed version of ISA in Eq.(7) is called ISA’ in the following. If ISA’ consists to suppose that s₁ is insonified only by the incoming wave ψ¹s

12is consequently independent of the position ofs2and by construction:

ψ¹s

1= ψ¹s

12 p(r12)dr₀₂= ψ¹s

12

the quasi-crystalline approximation (QCA) [8] is verified.

The contribution of other inclusions will be taken into ac- count for methods using higher order whereψs₁₂ depends on the position ofs₂.

4.2 E ff ective wavenumber

Wefixe the reference frame at the pointMwhere the CPA has to be verified. In this frame, the incomingfield is simply c⁰₀becauseφn(k0)=δn. Suppose thats1is placed atr01in the reference frame. Its scattering measured atM and obtained after averaging all other inclusions is:

ψ¹s

1=

n

a¹_n

1φˆn(Kr₁₀) wherea_n^jdenotesa_n^j(sj) in the following.

If ISA’ is used, the excitingfield is the incomingfield:

a¹

1≡ Zc¹ = ZS¹⁰c⁰ where the transformation S¹⁰ is defined by (S¹⁰)nm=φn−m(kr₁₀)=φm−n(kr₀₁) according to Graf’s theo- rem [5]. Then, the CPA supposes thatKsatisfies:

n₀

n

(ZS¹⁰c⁰)nφˆn(Kr₁₀) dr₀₁=−c⁰₀ (8) We have ˆφn(Kr₁₀)=(−1)ⁿφˆn(Kr₀₁) and the integral is:

(−1)ⁿzn

m

c⁰_m

φm−n(kr₀₁) ˆφn(Kr₀₁) dr₀₁ (9)

Using Green Theorem, the surface integral can be replaced by a contour integration:

Bφm−n(kr) ˆφn(Kr) dr=

∂B

φˆn(Kr)∂nφm−n(kr)−φm−n(kr)∂nφˆn(Kr)

K²−k² dr

(10)

what generalize the case of ([7]-Eq.(67)) for an incoming plane wave.

HereBis defined byr>aas mentioned in section 3.2. The contribution from infinity is null due to the radial attenuation imposed by ˆφn(Kr). Therefor∂Bis mainly∂CMthe contour of circleCMwith radiusa. In the reference frame the external normal isn = −er. Using∂nφn(kr) = −kJ_n(kr)e^inθatr = a and the orthogonality relation

e^imθdθ= 2πδmthe previous equation gives after simplification

Bφm−n(kr01) ˆφn(Kr01) dr01 =2πa²(−1)ⁿNn(k,K)δm

with:

Nⁿ(k,K)= kaJ_n(ka)Hn(Ka)−KaJn(ka)H_n(Ka) (ka)²−(Ka)²

Rememberingϕ=n₀πa²and simplify byc⁰₀, the CPA is:

2ϕ

n

znNn(k,K)=−1 (11) The effective wavenumber must satisfy this equation. An ap- proximate form can be found ifKais close enough fromka:

Ka=ka+with|| 1. Taylor expansion to thefirst order of the functionHn(Ka) andH_n(Ka) in the numerator ofN gives:

Nn(k,K)≈ −2i π

1 (ka)²−(Ka)² and then the traditional form of the ISA is obtained:

K²=k²−4in0

n

zn

Finally the new formulation of ISA’ does not introduce par- ticular effect and is efficient with the CPA if the hole correc- tion formulated in section 3.2 is respected. The method has been developed without care on the form and directivity of the incoming wave, the general expression of the effective wave number is independent of source.

5 Second order

5.1 Scattering by two inclusions

First consider the scattering problem Eq.(2) for two in- clusions. It reduces to solve the system:

a¹=Z

c¹+S¹²a² a²=Z

c²+S²¹a¹ (12) Hencea¹is solution of:

(I−ZS¹²ZS²¹)a¹=Z(I+S¹²ZS²¹)c¹=Zc¹+ZS¹²Zc² (13) The contribution of each terms can be highlighted by consid- ering the farfield interaction wherekr₁₂ 1. In this case,

the asymptotic form of Bessel functions for large argument can be used:

Hn(z)≈H₀(z)e^−inπ2, Jn(z)≈H₀(z)e^−inπ2+e^inπ2

2 , z1

Introducing these hypotheses gives the following forms for the two main matrices:

(ZS¹²ZS²¹)nm=H₀²(kr12)zne^{in(−π2−θ12)}e^{im(−π2+θ12)}f(π) (ZS¹²Z)nm=H₀(kr₁₂)zne^{−in(π2+θ12)}zme^im(π2+θ12) where f(π) =

p(−1)^pzp is the backscattering contribution of the farfield pattern.

Consider an incoming plane wave with directionθi. The in- comingfield at any pointMcan be decomposed:

ψi(M)=ψ¹ie^{ikr1cos (θ1−θi))}=ψ¹i

n

iⁿJn(kr)e^{in(θ1−θi)} whereψ¹i is the incomingfield at the center ofs₁. In other word we havec¹_n=ψ¹ie^{in(π2−θi)}andc²_m=ψ²ie^{im(π2−θi)}. It is then possible to express each vectors:

ZS¹²ZS²¹a¹

n= H₀²(kr12)f(π)zne^{in(−π2−θ12)}

m

a¹_me^{im(−π2+θ12)} Zc¹

n= ψ¹izne^{in(π2−θi)} ZS¹²Zc²

n= H₀(kr12)zne^{−in(π2+θ12)}ψ²i

m

zme^{im(π+θ12−θi)}

The farfield contribution of the inclusions1is ψ¹s∼H0(kr1)F(θ1) with F(θ1)=

n

ane^{in(θ1−π2)} Strictly speaking,F is not the scattering pattern f as it de- pends on the probing wave. Moreover it gives the same in- formations.

Multiply each terms of Eq.(13) bye^{in(−π2+θ1)} and sum overn give:

n

a¹_ne^{in(θ1−π2)} =F(θ1)

n

ZS¹²ZS²¹a¹

ne^{in(θ1−π2)} =H₀²(kr12)f(π)f(θ1−θ21)F(θ12)

n

Z¹c¹

ne^{in(θ1−π2)} = f(θ1−θⁱ)ψ¹i

n

ZS¹²Zc²

ne^{in(θ1−π2)} =H₀(kr₁₂)f(θ1−θ21)f(θ21−θi)ψ²i

A simple form of the scattering pattern is then solution of:

F(θ1)−H²₀(kr12)f(π)f(θ1−θ21)F(θ12) = f(θ1−θi)ψ¹i +H0(kr12)f(θ1−θ21)f(θ21−θi)ψ²i

For observation alongs2directionθ1=θ12 =θ21+π:

F(θ12)= f(θ12−θi)ψ¹_i +H0(kr12)f(π)f(θ21−θi)ψ²_i 1−H₀²(kr12)f(π)²

And after rearrangement:

F(θ1)= f(θ1−θi)ψ¹i +. . .

H₀(kr₁₂)f(θ1−θ21)f(θ21−θi)ψ²_i +H0(kr12)f(π)f(θ12−θi)ψ¹_i 1−H₀²(kr12)f²(π)

Of course if theO(H²₀(kr12)) terms are neglected we observe a more intuitive from of the farfield contribution:

F(θ1)= f(θ1−θi)ψ¹i +H₀(kr₁₂)f(θ1−θ21)f(θ21−θi)ψ²i

The far field pattern is obtained by a weak coupling with the second inclusion: f andH0 acts as an angular and ra- dial transfer function. NeglectingO(H₀²(kr12)) is equivalent to neglectZS¹²ZS²¹ in Eq.(13). In other wordsZS¹²ZS²¹ is mainly a nearfield contribution. As shown infigure 1 the contribution ofZS¹²ZS²¹ looks far from negligible for the nearfield contribution.

0.5 1 1.5

2 2.5

30

210

60

240 90

270 120

300 150

330

180 0

f Fex Fapp

1 2 3 4 5

30

210

60

240 90

270 120

300 150

330

180 0

f Fex Fapp

Figure 1:F evaluated by solving Eq.(13). Exact form (Fex) and obtained by neglectingZS¹²ZS²¹(Fapp), exact form for

a single inclusion (f). Left: rigid inclusion; right: small contrast (ρ1,c₁)= ³₄(ρ,c). Incidence alongxandθ12=π/3.

Case withka=3 andkr₁₂=2ka

5.2 Ensemble average

ForNinclusions the scattering amplitude is given in Eq.(2).

Ensemble average on all inclusions excepts1gives:

a¹

1=Z

⎛⎜⎜⎜⎜⎜

⎜⎝c¹+

k1

S^1k a^k

1k p(r1k) dr_1k

⎞⎟⎟⎟⎟⎟

⎟⎠

HereS^{i j}is the same asS^{i j}but evaluated withKin place ofk because Graf’s Theorem is applied on:

ψ¹s

1=

n

a¹_n

1 φˆn(Kr₁)

The contribution ofsk1are indistinguishable and are associ- ated toN−1≈Nrepresentative inclusions₂. Hence:

a¹

1 =Z

c¹+N S¹²

a²

12 p(r12) dr₁₂

(14) or equivalently:

a¹

12=Z

c¹+NS¹² a²

12

(15)

5.3 Linton-Martin closure assumption

In order to solve the previous equation it is necessary to have a relation between

a¹

1and a²

12in Eq.(14) or a¹

12

and a²

12in Eq.(15).

The Linton-Martin approximation [7] is based on the QCA where we suppose

a²

12≈S²¹ a¹

1

according to the ISA. The objective is then to solve:

I−N

S¹²ZS²¹ p(r₁₂) dr₁₂ a¹

1=Zc¹ (16)

In fact the dispersion solution is found supposing no probing wave:c¹ ≡0 looking for non trivial solution of the previous equation. This can be made by definingK(ω) solution of

det

I−N

S¹²ZS²¹ p(r12) dr12

=0 (17)

5.4 Two inclusions - closure assumptions

Moreover, according to Eq.(2) it is possible to express the contribution of

a²

12: Z

⎛⎜⎜⎜⎜⎜

⎜⎝c²+S²¹ a¹

12+

k(1,2)

S^2k a^k

12kp(r0k|r01,r02)dr0k

⎞⎟⎟⎟⎟⎟

⎟⎠

again contribution ofsk(1,2)are indistinguishable and can be attributed toN−2≈Nrepresentative inclusions₃to obtain

a²

12=Z

c²+S²¹ a¹

12+N S²³

a³

123p(r₀₃|r₀₁,r₀₂)dr₀₃

If the contribution of a³

123is supposed null, the coupling between three inclusions is neglected, but the pair coupling betweens₂ands₁is fully represented by:

a²

12=Z

c²+S²¹ a¹

12

Introducing this relation in Eq.(14) gives a more complete form of the second order ensemble average, where the con- tribution of

a²

12has not been lowered by QCA:

(I−N ZS¹²ZS²¹) a¹

12=Z(I+NS¹²ZS²¹)c¹ If one considerc¹ ≡0, the dispersion relation is found to be solution of

det

I−N

S¹²ZS²¹ p(r12) dr12

=0 (18) An alternative is to compute

a¹

1from the previous equa- tion. Numerically, this can be done by left-multiply the right hand side by (I−N ZS¹²ZS²¹)⁻¹then integrating the obtained matrix. But this matrix is ill conditioned and integration is time consuming.

An alternative is to suppose that a¹

1is solution of:

(I−N ZB1) a¹

1=Z(I+ND1)c¹ (19) with B1=

S¹²ZS²¹p(r12)dr₂

D1=

S¹²ZS²¹p(r12)dr₂

This method supposes that a¹

12 is not affected by integra- tion. And QCA is replaced by the following closure assump-

tion:

a¹

12≈ a¹

1

without imposing a particular form of a¹

1or a²

1. The section 5.1 gives an other point of view for comparing the LM-formulation Eq.(16) and the proposed one Eq.(19):

with Eq.(19) the nearfield coupling looks more accurate.

5.5 Analytical integration

In order to give an explicit form of a¹

1,D1andB1

have to been evaluatedfirst.

5.5.1 Calculation ofD1

(D1)nm=

p

zp(−1)^p−m

B\C1

φˆn−p(Kr12)φp−m(kr12)dr12

According to Eq.(10) the integration is performed on∂C1of radius 2aand gives:

8πa²(−1)^p−nNn−p(2k,2K)δnm

The matrixD1is diagonal with a general term (D1)nn =8πa²

p

zpNn−p(2k,2K)

5.5.2 Calculation ofB1

(B1)nm=

p

zp(−1)^p−n

B\C¹

φˆn−p(Kr12) ˆφp−m(Kr12)dr₁₂ Green Therorem can’t be used because the argument of the Bessel functions are the same. Integration alongθ12 gives n=mand the matrix is diagonal withn^thterm:

(B1)nn=2π

p

zp

_∞

2a Hn−p(Kr12)Hn−p(Kr12)dr12

Using [9]-eq.10.22.5 the integral is 2a²Pn−p(2K) with Pn(K)=H_n²(Ka)−Hn−1(Ka)Hn+1(Ka) Hence

(B1)nn=4πa²

p

zpPn−p(2K)

5.5.3 Calculation of a¹

1and a¹

Because matrixB1andD1are diagonal a¹

1is sim-

ply:

a¹

1=GZc¹ whereGis the diagonal matrix:

(G)nn= 1+8ϕ

pzpNn−p(2k,2K) 1−4ϕ

pzpPn−p(2K)

AsGis independent of the position ofs1, the integration over all the positions of s₁ inB\CM gives similar result than in section 4.2. We obtain:

2ϕ

n

zn

1+8ϕ

pzpNn−p(2k,2K) 1−4ϕ

pzpPn−p(2K) Nn(k,K)=−1 (20) It is a proposition of implicit solution ofKfor a second order model.

6 Conclusion

The proposed formulation of the multiple scattering prob- lem is consistent with the Foldy approximation. At higher order this formulation induces a reformulation of the closure assumption. Among the various solutions the comparison with the 2-inclusions problem supposes that Linton-Martin formulation is based on weak coupling between two repre- sentative inclusions. Current numerical investigations are performed in order to give quantitative comparison.

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