TIME REVERSAL OF WAVES IN A PERTURBED RANDOM MEDIUM DANIEL G. ALFARO VIGO AND KNUT SØLNA Abstract. In a time reversal experiment, a signal recorded by an array of transducers and sent back time reversed into the same medium approximately refocuses on the original source center. The refocusing resolution is improved in an inhomogeneous medium. In this work we study the effect of changes in the medium, namely, the case when back propagation takes place in a perturbed medium. Under the paraxial approximation assumption for a medium with weak inhomogeneities we consider a high frequency white noise regime. We show that relatively small perturbations do not affect the stable refocusing (self- averaging) for a localized source, but produces an interesting blurring of the refocused time signal. In some simple situations this effect can be explicitly quantified and related to the statistical model for the medium. Contents 1. Introduction 3 2. Time reversal of waves in a perturbed medium 5 2.1. Parabolic approximation and asymptotic regime 5 2.2. Time-reversal of waves 6 3. Asymptotics for the back-propagated wave 10 3.1. Limiting back-propagated wave 10 3.2. Wigner transform and the high frequency limit 10 3.3. Diffusion limit for the Wigner transform 13 3.4. Statistical stability of the back-propagated wave 17 4. Time-reversal super-focusing and stability 18 4.1. Perturbation effects on the refocused wave 18 4.2. Numerical results 19 4.3. Lateral diversity and stability 21 4.4. Concluding remarks 22 Acknowledgments 22 Appendix A. Diffusion limit of the characteristic ODEs 22 A.1. Introduction 22 A.2. One-particle problem 24 A.3. Two-particle problem 34 This work was partially supported by DARPA grant N00014-02-1-0603, ONR grant N00014-02- 1-0090, NSF grant 0307011 and the Sloan Foundation, DAV was also supported by the National Council for Scientific and Technological Development (CNPq, Brazil) through the Instituto do Milˆ enio. 1