**Alexander G. Ramm ^{1*}**

^{1 }Department of Mathematics, Kansas State University, Manhattan, KS 66506-2602, USA.

** *Corresponding Author:**Alexander G. Ramm, Department of Mathematics, Kansas State University, Manhattan, KS 66506-2602, USA.

**Citation: **Alexander G. Ramm (2024) Wave Scattering by Many Small Impedance Particles and Applications. *Nano Technol & Nano Sci J 6*: 158.

**Received**: June 19, 2024; **Accepted**: June 26, 2024; **Published**: June 30, 2024.

**Copyright: **© 2024 Alexander G. Ramm, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Formulas are derived for solutions of many-body wave scattering problem by small impedance particles embedded in a homogeneous medium. The limiting case is considered, when the size α of small particles tends to zero while their number tends to infinity at a suitable rate. The basic physical assumption is α<<d<<λ, where d is the minimal distance between neighboring particles, λ is the wavelength, and the particles can be impedance balls Β(*xm,α *) with centers *xm* located on a grid. Equations for the limiting effective (self-consistent) field in the medium are derived. It is proved that one can create material with a desired refraction coefficient by embedding in a free space many small balls of radius *α* with prescribed boundary impedances. The small balls can be centered at the points located on a grid. A recipe for creating materials with a desired refraction coefficient is formulated. It is proved that materials with a desired radiation pattern, for example, wave-focusing materials, can be created.

There is a large literature on wave scattering by small bodies, starting from Rayleigh’s work (1871), [1, 2, 36]. 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 [5], 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 [3], 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 , and the characteristic size α of a small body (particle) is much smaller than *λ*. Theoretically and practically the assumptions *α<<λ*, *d>>λ* are the simplest and they allow to neglect multiple scattering. By *k* = 2π the wave number is denoted. In contrast, in our theory the basic assumption is *α<<d<<λ*, and the multiple scattering is of basic importance. We give references to our papers and monographs in which the theory of wave scattering by small bodies of arbitrary shapes was developed under the assumption *α<<d<<λ*, [4–34]. The novelty of the results in this paper is in the location of the small bodies: they are placed o

n a grid. This may be of practical interest. In [35] for the first time the scattering problem for 10 billions small particles is solved numerically and numerical results are presented. This paper is a presentation of the new results under simplifying assumptions: the small particles *D*_{m}=*B*(*x _{m}*,α), 1 ≤

1. Solution to many-body wave scattering problem by small impedance particles, embedded in a homogeneous medium, under the assumptions *α<<d<<λ*, where *d* is the minimal distance between neighboring particles and *λ* is the wavelength in this medium.

2. Derivation of the equations for the limiting effective (self-consistent) field in this medium, in which many small impedance particles are embedded, when *α* → 0 and the number *M = M (α)* of the small particles tends to infinity at an appropriate rate.

3. Derivation of linear algebraic systems (LAS) for solving many-body wave scattering problems. These systems are not obtained by a discretization of boundary integral equations, and they give an efficient numerical method for solving many- body wave scattering problems in the case of small scatterers under the assumption *α<<d<<λ*.

4. Formulation of a recipe for creating materials with a desired refraction coefficient.

5. Formulation of a method for creating materials with a desired radiation pattern.

Our methods give powerful numerical methods for solving many-body wave scattering problems in the case when the scatterers are small (see [31]).

Let us formulate the wave scattering problems we deal with. Let *D* be a bounded domain in R3 with a sufficiently smooth boundary. The scattering problem consists of finding the solution to the problem:

It will be clear from Section 3 that the function *h (x)* can be determined by choosing a suitable boundary impedance *ζ(x)*. When *α* → 0, the *ζ _{m} *and

The many-body scattering problem (1)–(4) has a solution and this solution is unique, see [31] In Section 2 a method for solving this problem is given. In Section 3 a recipe for creating materials with a desired refraction coefficient is given. In Section 4 a recipe for creating materials with a desired radiation pattern is given.

**SolutionofMany-BodyScatteringProblem**

We look for the solution of the form

where *σ _{m}* (

This estimate justifies our claim since* α « d.* It follows that asymptotically, as → 0, one has for |*x − x _{m}* | ≥

The order *M* = 0(*α ^{x}*

LEMMA 2. Eq. (14) has a solution, this solution is unique and it is a limiting value of the solution to the scattering problem (1)– (4).

Proof: Apply the operator ∇2+*k*2 to equation (14) and get (∇2+*k*2) *u* = 4*πh(x)u(x.)*

This is a Schrödinger equation with potential *q(x*):= 4*πh(x); *equations (2)– (3) hold. We assumed lm*h* ≤ 0. Therefore (15) has at most one solution. It is a Fredholm-type equation, so it has a solution. Lemma 2 is proved.

It follows from Lemma 2 that the LAS (13) for *u p* is solvable and its solution is unique. Let us write Eq. (15) as

∇^{2}*u+k ^{2}n^{2}(x)u = 0, n^{2}(x): = 1- 4πk^{2}h(x)*

Embedding small impedance balls *B (xm, α)* in *D* results in creating in *D *a new material with the refraction coefficient *n(x)* = (1- *4πk-2 h(x))1\2. *If one wants to have a material with the refraction coefficient *n(x)*, then one chooses by (17) the function *h(x)*. If *h(x)* is chosen, then one knows the boundary impedance ζ (*x*) which generates the desired *h(x)*. The practical problem is to prepare small particles with the desired boundary impedance.

**Recipe for Creating Materials with A Desired Refraction Coefficient**

Let us formulate a recipe for creating materials with a desired refraction coefficient. Formula (17) shows that if *h(x) *is chosen properly, then any *n(x) *can be obtained in *D*.

Recipe for creating materials with a desired refraction coefficient:

a) Calculate by formula (17) the function *h(x)*;

b) Distribute small impedance balls in the domain *D* by the distribution law (6).

The boundary impedances of these balls are defined by the function *h(x)*.

Theorem 1. The refraction coefficient of the resulting medium tends to the desired coefficient *n(x)* as *α* → 0.

Let us show that a practically negative refraction coefficient *n(x)* can be obtained by the above recipe. Denote b:= *4πk-2 > 0* and write (17) as *n(x)* = (1-*bh(x)*)1/2 = |1-*bh(x*)| 1/2 *eØ*/2, where *Ø* is the argument of 1-*bh(x)*. Since the operator in (14) is Fredholm, it remains Fredholm under small perturbations. Therefore, one can take *h - i∈* , where *∈*> 0 is sufficiently small and equation (14) will still have a unique solution.

By choosing *h *so that Re (1-*bh) > 0* and lm(1-

**Creating Materials with A Desired Radiation Pattern**

Let us define what we mean by radiation pattern. Consider the scattering problem for Eq. (15)

where ???? satisfies the radiation condition. Assume that *k > 0 *and α

We call *A(β) *the radiation pattern. Consider an inverse problem (IP): Given an arbitrary f *(β) ∈ L2 (S2) *and an arbitrary small

Theorem 2. For any f *(β) ∈ L2 (S2) *and an arbitrary small

Since small perturbations of *q* result in small perturbations of *A(β)*, there are infinitely many potentials *q* for which inequality (20) holds.

The conclusion of Theorem 2 follows from Lemmas 3 and 4.

Lemma 3. The set {∫*De - ik β.xh(x)dx} ∇h∈L 2 (D)* is dense in L 2 (S 2).

Corollary 1. Given f * ∈ L2 (S2) and ∈ > 0, *one can find

Lemma 4. The set {*q(x)u (x, α)*} ∀ *q ∈L 2 (D).*

Corollary 2. Given* h ∈L 2 (D)* and

Since the scattering amplitude depends continuously on *h*, the inverse problem IP is solved by Lemmas 3 and 4.

Proof of Lemma 3: Assume the contrary. Then ∃ψ* ∈L 2 (S 2) *such that

If *q ∈L 2 (D), *then this

The condition |∇*uj* |*l ≥ c > 0, j = 1, 2, *implies that a tubular neighborhood of the line *l, Nδ = {x: √ |μ*1* |*2 +

Here ξ∈*l, x *is a point on a plane passing through ???? and orthogonal to *l, p* = * |x - ξ|, *and

Claim 2, and, therefore, Lemma 4 are proved.

Therefore, Theorem 2 is proved.

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