Scattering of scalar waves by many small particles

Formulas are derived for solutions of many-body wave scattering problems by small particles in the case of acoustically soft, hard, and impedance particles embedded in an inhomogeneous medium. The limiting case is considered, when the size $a$ of small particles tends to zero while their number tends to infinity at a suitable rate. Equations for the limiting effective (self-consistent) field in the medium are derived.


Introduction
There is a large literature on wave scattering by small bodies, starting from Rayleigh's work (1871). For the problem of wave scattering by one body an analytical solution was found only for the bodies of special shapes, for example, for balls and ellipsoids. If the scatterer is small then the scattered field can be calculated analytically for bodies of arbitrary shapes, see [7], where this theory is presented.
The many-body wave scattering problem was discussed in the literature mostly numerically, if the number of scatterers is small, or under the assumption that the influence of the waves, scattered by other particles on a particular particle is negligible ( see [5], where one finds a large bibliography, 1386 entries). This corresponds to the case when the distance d between neighboring particles is much larger than the wavelength λ, and the characteristic size a of a small body (particle) is much smaller than λ. By k = 2π λ the wave number is denoted. The basic results of our paper consist of: i) Derivation of formulas for the scattering amplitude for the wave scattering problem by one small (ka ≪ 1) body of an arbitrary shape under the Dirichlet, impedance, or Neumann boundary condition (acoustically soft, impedance, or hard particle), ii) Solution to many-body wave scattering problem by many such particles under the assumptions a ≪ d and a ≪ λ, where d is the minimal distance between neighboring particles, iii) Derivation of the equations for the limiting effective (self-consistent) field in the medium when a → 0 and the number M = M (a) of the small particles tends to infinity at an appropriate rate, iv) Derivation of linear algebraic systems for solving many-body wave scattering problems; these system are not obtained by a discretization of boundary integral equations.
Let us formulate the wave scattering problems we deal with. First, let us consider a one-body scattering problem. Let D 1 be a bounded domain in R 3 with a sufficiently smooth boundary S 1 . The scattering problem consists of finding the solution to the problem: where S 2 is the unit sphere in R 3 , u 0 is the incident field, v is the scattered field satisfying the radiation condition Γu is the boundary condition (bc) of one of the following types where ζ 1 is a constant, N is the unit normal to S 1 , pointing out of D 1 , and It is well known (see, e.g., [6]) that problem (1)-(3) has a unique solution.
We now assume that a := 0.5 diamD 1 , ka ≪ 1, and look for the solution to problem (1)-(3) of the form where dt is the element of the surface area of S 1 . One can prove that the unique solution to the scattering problem (1)-(3) with any of the boundary conditions (6)- (8) can be found in the form (10), and the function σ 1 in equation (10) is uniquely defined from the boundary condition (2). The The equations for finding σ 1 are: respectively, for conditions (6)- (8). The operator A is defined as follows: Equations (12)- (14) are uniquely solvable, but there are no analytic formulas for their solutions for bodies of arbitrary shapes. However, if the body D 1 is small, ka ≪ 1, one can rewrite (10) as where and 0 ∈ D 1 is the origin. If ka ≪ 1, then we prove that Therefore, the scattered field is determined outside D 1 by a single number Q 1 . This number can be obtained analytically without solving equations (12)- (13). The case (14) requires a special approach by the reason discussed in detail later. Let us give the results for equations (12) and (13) first. For equation (12) one has where C is the electric capacitance of a perfect conductor with the shape D 1 . For equation (13) one has where |S 1 | is the surface area of S 1 . The scattering amplitude for problem (1)-(3) with Γ = Γ 1 (acoustically soft particle) is since u 0 (0) = e ikα·x | x=0 = 1.
Therefore, in this case the scattering is isotropic and of the order O(a), because the capacitance C = O(a). The scattering amplitude for problem (1)-(3) with Γ = Γ 2 (small impedance particles) is : since u 0 (0) = 1.
In this case the scattering is also isotropic, and of the order O(ζ|S 1 |).
The scattering amplitude for problem (1)-(3) with Γ = Γ 3 (acoustically hard particles) is Here and below summation is understood over the repeated indices, α q = α · e q , α · e q denotes the dot product of two vectors in R 3 , p, q = 1, 2, 3, {e p } is an orthonormal Cartesian basis of R 3 , |D 1 | is the volume of D 1 , β pq is the magnetic polarizability tensor defined as follows ( [7], p.62): σ 1q is the solution to the equation N q (s) = N (s) · e q , N = N (s) is the unit outer normal to S 1 at the point s, i.e., the normal pointing out of D 1 , and A 0 is the operator A at k = 0. For small bodies A − A 0 = o(ka).
If u 0 (x) is an arbitrary field satisfying equation (1), not necessarily the plane wave e ikα·x , then The above formulas are derived in Section 2. In Section 3 we develop a theory for many-body wave scattering problem and derive the equations for effective field in the medium, in which many small particles are embedded, as a → 0. The results, presented in this paper, are based on the earlier works of the author ([9]- [23]). Our presentation and some of the results are novel. These results and methods of their derivation differ much from those in the homogenization theory ( [2], [4]). The differences are: i) no periodic structure in the problems is assumed, ii) the operators in our problems are non-selfadjoint and have continuous spectrum, iii) the limiting medium is not homogeneous and its parameters are not periodic, iv) the technique for passing to the limit is different from one used in homogenization theory.

Derivation of the formulas for one-body wave scattering problems
Let us recall the known result (see e.g., [6]) concerning the limiting value of the normal derivative of single-layer potential from outside. Let In this Section m = 1, and x m = 0 is the origin. We assume that ka ≪ 1, Then Let us derive estimate (19). Since |t| ≤ a on S 1 , one has where g 0 (s, t) = 1 4π|s−t| . Since u 0 (s) is a smooth function, one has |u 0 (s) − u 0 (0)| = O(a). Consequently, equation (12) can be considered as an equation for electrostatic charge distribution σ 1 (t) on the surface S 1 of a perfect conductor D 1 , charged to the constant potential −u 0 (0) (up to a small term of the order O(ka)). It is known that the total charge Q 1 = S 1 σ 1 (t)dt of this conductor is equal to where C is the electric capacitance of the perfect conductor with the shape D 1 . Analytic formulas for electric capacitance C of a perfect conductor of an arbitrary shape, which allow to calculate C with a desired accuracy, are derived in [7]. For example, the zeroth approximation formula is and we assume in (32) that ǫ 0 = 1, where ǫ 0 is the dielectric constant of the homogeneous medium in which the perfect conductor is placed. Formula (31) is formula (19). If u 0 (x) = e ikα·x , then u 0 (0) = 1, and Q 1 = −C(1 + O(ka)). In this case which is formula (21). Consider now wave scattering by an impedance particle.
One can see that the wave scattering by an impedance particle is isotropic, and the scattered field is of the order O( . Consider now wave scattering by an acoustically hard small particle, i.e., the problem with the Neumann boundary condition.
In this case we will prove that: i) The scattering is anisotropic, ii) It is defined not by a single number, as in the previous two cases, but by a tensor, and iii) The order of the scattered field is O(a 3 ) as a → 0, for a fixed k > 0, i.e., the scattered field is much smaller than in the previous two cases.
When one integrates over S 1 equation (13), one gets Thus, . Therefore, the contribution of the term e −ikx o ·t in formula (28) with x m = 0 will be also of the order O(a 3 ) and should be taken into account, in contrast to the previous two cases. Namely, One has where the terms of higher order of smallness are neglected and summation over index p is understood. The function σ 1 solves equation (14): if u 0 (x) = e ikα·x . Comparing (43) with (25), using (24), and taking into account that ka ≪ 1, one gets If u 0 (x) is an arbitrary function, satisfying equation (1), then ikα q in (43) is replaced by ∂u 0 ∂xq , and −k 2 u 0 = △u 0 , which yields formula (26). This completes the derivation of the formulas for the solution of scalar wave scattering problem by one small body on the boundary of which the Dirichlet, or the impedance, or the Neumann boundary condition is imposed.

Many-body scattering problem
In this Section we assume that there are M = M (a) small bodies (particles) between neighboring bodies is much larger than a, d ≫ a, but we do not assume that d ≫ λ, so there may be many small particles on the distances of the order of the wavelength λ. This means that our medium with the embedded particles is not necessarily diluted.
We assume that the small bodies are embedded in an arbitrary large but finite domain D, By N we denote a unit normal to ∂Ω, pointing out of Ω, by |D m | the volume of the body D m is denoted.
The scattering problem consists of finding the solution to the following problem where u 0 is the incident field, satisfying equation (45) in R 3 , for example, u 0 = e ikα·x , α ∈ S 2 , and v is the scattered field, satisfying the radiation condition (5). The boundary condition (46) can be of the types (6)- (8).
In the case of impedance boundary condition (7) we assume that so the impedance may vary from one particle to another. We assume that where x m ∈ D m is a point in D m , and h(x), x ∈ D, is a given function, which we can choose as we wish, subject to the condition Imh(x) ≤ 0. For simplicity we assume that h(x) is a continuous function. Let us make the following assumption about the distribution of small particles: if ∆ ⊂ D is an arbitrary open subset of D, then the number N (∆) of small particles in ∆, assuming the impedance boundary condition, is: where N (x) ≥ 0 is a given function. If the Dirichlet boundary condition is assumed, then The case of the Neumann boundary condition will be considered later.
We look for the solution to problem (45)-(47) with the Dirichlet boundary condition of the form where σ m (t) are some functions to be determined from the boundary condition (46). It is proved in [9] that problem (45) rather than M functions σ m (t). This is possible to prove if the particles D m are small. We derive analytical formulas for Q m as a → 0.
Let us define the effective (self-consistent) field u e (x) = u (j) e (x), acting on the j−th particle, by the formula Physically this field acts on the j−th particle and is a sum of the incident field and the fields acting from all other particles: Let us rewrite (55) as follows: We want to prove that the last sum is negligible compared with the first one as a → 0. To prove this, let us give some estimates. One has |t − x m | ≤ a, One can also prove that where J 1 is the first sum in (56) and J 2 is the second sum in (56). Therefore, at any point x ∈ Ω ′ = R 3 \ Ω one has where the terms of higher order of smallness are omitted.

The case of acoustically soft particles
If (46) is the Dirichlet condition, then, as we have proved in Section 2 (see formula (31)), one has Thus, One has so the full field and effective field are practically the same. Let us write a linear algebraic system (LAS) for finding unknown quantities u e (x m ): If M is not very large, say M = O(10 3 ), then LAS (64) can be solved numerically, and formula (62) can be used for calculation of u e (x). Consider the limiting case, when a → 0. One can rewrite (64) as follows: where {∆ p } P p=1 is a union of cubes which forms a covering of D, By |∆ p | we denote the volume (measure) of ∆ p , and ξ p is the center of ∆ p , or a point x p in an arbitrary small body D p , located in ∆ p . Let us assume that there exists the limit For example, one may have for all m such that x m ∈ ∆ p , where c(x) is some function in D. If all D m are balls of radius a, then c(x) = 4π. We have so limit (67) exists, and From (65), (68)-(70) one gets Linear algebraic system (71) can be considered as the collocation method for solving integral equation It is proved in [24] that system (71)  Applying the operator L 0 := ∇ 2 + k 2 to (72), and using the formula L 0 g(x, y) = −δ(x − y), where δ(x) is the delta-function, one gets The physical conclusion is: If one embeds M (a) = O(1/a) small acoustically soft particles, which are distributed as in (51), then one creats, as a → 0, a limiting medium, which is inhomogeneous, and has a refraction coefficient n 2 (x) = 1 − k −2 q(x).
It is interesting from the physical point of view to note that the limit, as a → 0, of the total volume of the embedded particles is zero.
Indeed, the ). In this case, for 3γ > 1, the limit in (69) is C(ξ p ) = lim a→0 c p aO(a −3γ ) = ∞. Therefore, the product of this limit by u remains finite only if u = 0 in D. Physically this means that if the distances between neighboring perfectly soft particles are smaller than O(a 1/3 ), namely, they are O(a γ ) with any γ > 1 3 , then u = 0 in D. On the other hand, if γ < 1 3 , then the limit C(ξ p ) = 0, and u = u 0 in D, so that the embedded particles do not change, in the limit a → 0, properties of the medium.
This concludes our discussion of the scattering problem for many acoustically soft particles.

3.2
Wave scattering by many impedance particles We assume now that (49) and (50) hold, use the exact boundary condition (46) with Γ = Γ 2 , that is, and integrate (75) over S m in order to derive an analytical asymptotic formula for Q m = Sm σ m (t)dt. We have This yields the formula for the approximate solution to the wave scattering problem for many impedance particles: The integral equation for the limiting effective field in the medium with embedded small particles, as a → 0, is where and we have assumed in (83) for simplicity that b m = b for all m, that is, all small particles are of the same shape and size. Applying operator L 0 = ∇ 2 + k 2 to equation (83), one finds the differential equation for the limiting effective field u(x): and u satisfies condition (47). The conclusion is: the limiting medium is inhomogeneous, and its properties are described by the function Since the choice of the functions N (x) ≥ 0 and h(x), Imh(x) ≤ 0, is at our disposal, we can create the medium with desired properties by embedding many small impedance particles, with suitable impedances, according to the distribution law (50) with a suitable N (x). The function is the refraction coefficient of the limiting medium. Given a desired refraction coefficient n 2 (x), Imn 2 (x) ≥ 0, one can find N (x) and h(x) so that (87) holds, that is, one can create a material with a desired refraction coefficient by embedding into a given material many small particles with suitable boundary impedances. This concludes our discussion of the wave scattering problem with many small impedance particles.

Wave scattering by many acoustically hard particles
Consider now the case of acoustically hard particles, i.e., the case of Neumann boundary condition. The exact boundary integral equation for the function σ m in this case is: Arguing as in Section 2, see formulas (40)-(44), one obtains (89) Here we took into account that the unit vector β in (44)  one sets x = x j in (89), takes the sum in (89) with m = j. This yields the first set of equations for finding these unknowns. Then one takes derivative of equation (89) with respect to (x) q , sets x = x j , and takes the sum in (89) with m = j. This yields the second set of equations for finding these unknowns. Finally, one takes Laplacian of equation (89), sets x = x j , and takes the sum in (89) with m = j. This yields the third set of linear algebraic equations for finding u e (x m ), ∂ue(xm) ∂(x)q , and ∆u e (x m ). Passing to the limit a → 0 in equation (89), yields the equation for the limiting field where ρ(y) and B pq (y) are defined below, see formulas (92)  Thus, there are many small particles D m in ∆ l . Let x l be a point in ∆ l .
One has M m=1 g(x, x m ) △u e (x m ) + ik ∂u e (x m ) Assume that the following limit exist: Then, the sum in (91) converges to D g(x, y) ρ(y)∇ 2 u(y) + ik ∂u(y) ∂y q x p − y p |x − y| B pq (y) dy.
Consequently, (89) yields in the limit a → 0 equation (90). Equation (90) cannot be reduced to a differential equation for u(x), because (90) is an integrodifferential equation whose integrand depends on x and y.

Scattering by small particles embedded in an inhomogeneous medium
Suppose that the operator ∇ 2 + k 2 in (1) and in (45) is replaced by the operator L 0 = ∇ 2 + k 2 n 2 0 (x), where n 2 0 (x) is a known function, Im n 2 0 (x) ≥ 0.
The function n 2 0 (x) is the refraction coefficient of an inhomogeneous medium in which many small particles are embedded. The results, presented in Section 1-3 remain valid if one replaces function g(x, y) by the Green's function G(x, y), [∇ 2 + k 2 n 2 0 (x)]G(x, y) = −δ(x − y), satisfying the radiation condition. We assume that The function G(x, y) is uniquely defined (see, e.g., [9]). The derivations of the results remain essentially the same because where g 0 (x, y) = 1 4π|x−y| . Estimates of G(x, y) as |x−y| → 0 and as |x−y| → ∞ are obtained in [9]. Smallness of particles in an inhomogeneous medium with refraction coefficient n 2 0 (x) is described by the relation kn 0 a ≪ 1, where n 0 := max x∈D |n 0 (x)|, and a = max 1≤m≤M diamD m .