Bodies of zero resistance and bodies invisible in one direction

Abstract

We consider a body in a parallel flow of non-interacting particles. The interaction of particles with the body is perfectly elastic. We introduce the notions of a body of zero resistance, a body that leaves no trace, and an invisible body, and prove that all such bodies do exist.

The problem of camouflaging via mirror reflections

“…In other words, is it possible to construct a sequence of domains with fixed volume and with the visibility index going to zero? 1 We do not know the answer to this question.…”

Acoustically invisible gateways

“…This problem was originally considered by Isaac Newton [15] in 1687. Very recently Aleksenko and Plakhov [1] have taken up the problem and introduced the notion of a body with zero resistance, a body that leaves no trace and an invisible body. They prove that such bodies exist and construct examples.…”

“…The paper is organised as follows: In section 2 we describe the results of Aleksenko and Plakhov [1] and, in particular, recall their examples of bodies which have zero resistance and are invisible. Section 3 is devoted to the formulation of our acoustic scattering problem, the Kirchoff approximation and the geometrical theory of diffraction.…”

“…Aleksenko and Plakhov [1] develop their arguments further by considering a 1parameter family of bodies having all the desired properties claimed in theorem 1.…”

“…Conclusions. Motivated by the work of Aleksenko and Plakhov [1] concerning the problem of a parallel flow of particles falling on a body with piecewise smooth boundary and leaving no trace we have considered the analogous acoustic scattering problem of a plane wave illuminating the same body which is viewed as a gateway. We have shown that at high frequencies with the use of the Kirchoff approximation together with the geometrical theory of diffraction, the far field (equivalently the total scattering cross section) is zero for a discrete set of wave numbers.…”