Statistical stability in time reversal

When a signal is emitted from a source, recorded by an array of transducers, time reversed and re-emitted into the medium, it will refocus approximately on the source location. We analyze the refocusing resolution in a high frequency, remote sensing regime, and show that, because of multiple scattering, in an inhomogeneous or random medium it can improve beyond the diffraction limit. We also show that the back-propagated signal from a spatially localized narrow-band source is self-averaging, or statistically stable, and relate this to the self-averaging properties of functionals of the Wigner distribution in phase space. Time reversal from spatially distributed sources is self-averaging only for broad-band signals. The array of transducers operates in a remote-sensing regime so we analyze time reversal with the parabolic or paraxial wave equation

Bulk Burning Rate in Passive - Reactive Diffusion

We consider a passive scalar that is advected by a prescribed mean zero divergence-free velocity field, diffuses, and reacts according to a KPP-type nonlinear reaction. We introduce a quantity, the bulk burning rate, that makes both mathematical and physical sense in general situations and extends the often ill-defined notion of front speed. We establish rigorous lower bounds for the bulk burning rate that are linear in the amplitude of the advecting velocity for a large class of flows. These "percolating" flows are characterized by the presence of tubes of streamlines connecting distant regions of burned and unburned material and generalize shear flows. The bound contains geometric information on the velocity streamlines and degenerates when these oscillate on scales that are finer than the width of the laminar burning region. We give also examples of very different kind of flows, cellular flows with closed streamlines, and rigorously prove that these can produce only sub-linear enhancement of the bulk burning rate.Comment: 32 pages, 4 figure