Direct and inverse acoustic scattering problems involving smart obstacles

Abstract

Direct and inverse acoustic scattering problems involving smart obstacles are proposed and some ideas to study them are suggested. A smart obstacle is an obstacle that when hit by an incoming acoustic wave reacts circulating on its boundary a pressure current, that is a quantity dimensionally given by a pressure divided by a time, in order to generate a scattered wave that pursues a preassigned goal. In our models (see [6–10,12]) the smart obstacle pursues one of the following goals: i) to be undetectable, ii) to appear with a shape and/or acoustic boundary impedance different from its actual ones, iii) to appear with a shape and/or acoustic boundary impedance and in a location in space different from its actual ones. That is, in the first case the smart obstacle tries to be furtive, in the second case it tries to be masked that is it tries to appear as another obstacle that we call the mask and finally in the third case it tries to appear as another obstacle in a location in space different from its actual one. We refer to this last apparent obstacle as the ghost. The direct scattering problem considered is the following: given the incoming acoustic field, the obstacle, its acoustic impedance and its goal formulate an adequate mathematical model for the problems previously considered and find the optimal strategy to pursue the assigned goal within the proposed model. The inverse scattering problem considered is the following: given the knowledge of several far fields generated by the smart obstacle when hit by known incident acoustic fields it reacts with the optimal strategy and the knowledge of the goal pursued by the obstacle find the obstacle (i.e. find the shape, acoustic impedance and spatial location of the obstacle). For simplicity in this paper we limit our attention to the case of the obstacle that tries to be masked when the incoming acoustic field is time harmonic. Moreover in the inverse problem we assume that the acoustic boundary impedance of the obstacle and of the mask are known. In this case the direct scattering problem is translated in a constrained optimization problem and its solution is characterized as the solution of a set of auxiliary equations. The inverse scattering problem is translated in a two steps optimization procedure. Finally in a test case the inverse problem is solved numerically.