Distribution of particles which produces a "smart" material

A. G. Ramm

doi:10.1007/s10955-007-9303-3

paper

Distribution of particles which produces a "smart" material

arXiv:math-ph/0606023 · doi:10.1007/s10955-007-9303-3

Abstract

If $A_{q} (β, α, k)$ is the scattering amplitude, corresponding to a potential $q \in L^{2} (D)$ , where $D \subset R^{3}$ is a bounded domain, and $e^{ik α \cdot x}$ is the incident plane wave, then we call the radiation pattern the function $A (β) := A_{q} (β, α, k)$ , where the unit vector $α$ , the incident direction, is fixed, and $k > 0$ , the wavenumber, is fixed. It is shown that any function $f (β) \in L^{2} (S^{2})$ , where $S^{2}$ is the unit sphere in $R^{3}$ , can be approximated with any desired accuracy by a radiation pattern: $∣∣ f (β) - A (β) ∣ ∣_{L^{2} (S^{2})} < ε$ , where $ε > 0$ is an arbitrary small fixed number. The potential $q$ , corresponding to $A (β)$ , depends on $f$ and $ε$ , 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 $D_{m} \subset D$ , $1 \leq m \leq M$ , distributed in an a priori given bounded domain $D \subset R^{3}$ . 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 $A (α^{'}, α)$ , $α^{'}, α \in S^{2}$ , at a fixed $k > 0$ , arbitrarily close in the norm of $L^{2} (S^{2} \times S^{2})$ to an arbitrary given scattering amplitude $f (α^{'}, α)$ , corresponding to a real-valued potential $q \in L^{2} (D)$ .

corrected typos

Distribution of particles which produces a "smart" material

Abstract

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