Distribution of Particles Which Produces a “Smart” Material

Abstract

If A _q(β, α, k) is the scattering amplitude, corresponding to a potential $q\in L^2(D)$, where D⊂ℝ³ is a bounded domain, and $e^{ik\alpha \cdot x}$ is the incident plane wave, then we call the radiation pattern the function $A(\beta):=A_q(\beta, \alpha, k)$, where the unit vector α, the incident direction, is fixed, β is the unit vector in the direction of the scattered wave, and k>0, the wavenumber, is fixed. It is shown that any function $f(\beta)\in L^2(S^2)$, where S ² is the unit sphere in ℝ³, can be approximated with any desired accuracy by a radiation pattern: $||f(\beta)-A(\beta)||_{L^2(S^2)}<\epsilon$, where ∊ >0 is an arbitrary small fixed number. The potential q, corresponding to A(β), depends on f and ∊, and can be calculated analytically. There is a one-to-one correspondence between the above potential and the density of the number of small acoustically soft particles D _m⊂ D, 1≤ m≤ M, distributed in an a priori given bounded domain D⊂ℝ³. The geometrical shape of a small particle D _m is arbitrary, the boundary S _m of D _m is Lipschitz uniformly with respect to m. The wave number k and the direction α of the incident upon D plane wave are fixed. It is shown that a suitable distribution of the above particles in D can produce the scattering amplitude $A(\alpha',\alpha)$, $\alpha',\alpha\in S^2$, at a fixed k>0, arbitrarily close in the norm of L ²(S ²× S ²) to an arbitrary given scattering amplitude f(α ', α), corresponding to a real-valued potential q∊ L ²(D), i.e., corresponding to an arbitrary refraction coefficient in D.