Decomposition of the time-reversal operator applied to quantitative characterization of small elastic cylinders

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Recent experiments showed how the DORT method can be used for the characterization of elastic cylinders imbedded in water (Minonzio et al., J. Acoust. Soc. Am. 117 (2), pp 789-798, 2005). Here, the small cylinder limit (k0a < 0.5) is considered. The singular values of the array response matrix are studied. It is show that the first singular value is proportional to k0(α + β)a 2 and the second one is proportional to k0βa 2 where a is the radius, α is the compressibility contrast, β is the density contrast between the cylinder and the fluid. Thus, the linear frequency dependence of the two singular values provides two equations with three unknowns, α, β and a. If one of these three parameters is known (for example, α is about 1 for metals), the other two can be determined. Measurements carried out for materials of α ranging from 0.6 to 0.99 and β between 0.1 and 1.6 are presented. A good agreement between calculated and experimental singular values was observed. Generalized expressions of the two first singular values are also given for k0a < 10. The analysis of acoustic scattering is an important tool for object identification. It has applications among non-destructive evaluation, medical imaging or underwater acoustics. The decomposition of the time-reversal operator (DORT method in French) is a new approach to scattering analysis that was developed since 1994 and is applied to detection and selective focusing through non-homogeneous media with array of transducers (1). The principle is the analysis of the complete set of pulse echo responses of the array. The eigenvalues and the eigenvectors of the time-reversal operator (i.e., singular values and singular vectors of the array response matrix) provide information on the scatterers in the insonified medium. Recent experiments showed how the DORT method can also be used for the characterization of sub-wavelength scatterers (2),(3), as elastic cylinders (4), imbedded in water. It has been shown that a single elastic scatterer is generally associated to several singular values and that the singular vectors are combinations of the projection on the array of the normal modes of vibrations (monopolar, dipolar, quadripolar...). After a brief presentation of these recent results, we focus on the case of small cylinder (k0a < 0.5), for which the two first normal modes are preponderant. The singular values and the singular vectors of the array response matrix are calculated and experimentally observed. We show that there are two preponderant singular values and study their dependence on the geometrical parameters, the cylinder radius a, the compressibility contrast α, and the density contrast β between the cylinder and the fluid. We present experimental results obtained in the MHz range for cylinders of different material and diameters using 96 transducers array. We also propose generalized expressions of the two first singular values, valid for k0a < 10.