Analysis of the time-reversal operator for scatterers of finite size

Recently, it was shown that the time-reversal operator for a single, small spherical scatterer could have up to four distinguishable eigenstates [Chambers and Gautesen, J. Acoust. Soc. Am. 109, 2616-2624 (2001)]. In this paper, that analysis is generalized for scatterers of arbitrary shape and larger size. It is shown that the time-reversal operator may have an indefinitely large number of distinguishable eigenstates, with the exact number depending on the nature of the scatterer and the geometry of the time-reversal mirror. In addition, the case of a multiple number of well-separated scatterers is investigated, with the result that the total spectrum is the direct combination of the eigenstates associated with each scatterer. As an example, the singular value spectrum of the time-reversal operator for a linear array is calculated explicitly for bubbles and hard rubber spheres of finite size. Both resonance peaks and apparent crossing points can be observed in the spectrum as the size of the scatterer increases.