Characterization of an elastic cylinder and an elastic sphere with the time-reversal operator: application to the sub-resolution limit

Abstract

The Décomposition de l'Opérateur de Retournement Temporel method applies to scattering analysis with arrays of transducers. It comprises the study of the time-reversal invariants which correspond to the eigenvectors of the time-reversal operator or to the singular vectors of the array response matrix K. In this paper, the decomposition of the scattered pressure into normal modes of vibrations is used to determine the theoretical time-reversal invariants for a large elastic object such as a cylinder or sphere. For an N transducers one-dimensional array, the time-reversal operator is dimension N. It is shown that the dimension is reduced to 2k0a for a cylinder or a sphere, where a is the scatterer radius and k0 is the wave number in the surrounding fluid. Furthermore, this approach provides analytical expressions of symmetric and anti-symmetric singular values and vectors in the sub-resolution limit, i.e. when the scatterer diameter is smaller than the array resolution cell. These results are verified experimentally and in good agreement with the original point of view: for a small scatterer, one singular value dominates and the associated singular vector focuses isotropically on the scatterer.