Proposal on Pulse Shaping for Waves in Multiscale Media

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Solna0093992 The investigator analyzes how a propagating wave pulsebehave in a medium with rough variations. Predicting theevolution of the wave pulse from information about the`roughness' of the medium is a problem of deep mathematical andpractical interest, as is the inverse problem of deducinginformation about the medium from measurements of a transmittedpulse. Most often a detailed description of the (fine scale)variations of the medium is not possible, only a description oftheir statistical character. Hence, the investigator considersthe fine scale medium variations as a realization of a stochasticprocess. This entails analysis of differential equations withrandomly varying coefficients. The project has three specificobjectives: First, obtain a characterization of thetransformation of sound pulses propagating in a medium with acontinuum of length scales, a medium defined in terms of a randomfractal. Second, extend and use theory about the transformationof a wave pulse in some specific inverse problems, in particular,high contrast seismic imaging and time reversal mirrors. Third,derive strict bounds for how an electromagnetic pulse can beaffected by the medium heterogeneity without knowing the detailsof these variations. The main question the investigator addresses is: How does apropagating wave behave in a medium that varies when thevariations are not small? For instance, when a laser beampropagates through the turbulent atmosphere how is it affected byintense, turbulent, variations in the local speed of propagation?When such medium variations are smooth, say with a constantspeed of propagation, the problem is well understood. For roughmedium variations, as in the above example, the interaction of awave pulse with the medium heterogeneities is a fundamental yetlargely open question. This problem, predicting the evolution ofsignal or wave pulse that propagates in a rough medium, is ofdeep mathematical and practical interest. In addition to wavepropagation in the turbulent atmosphere there are many otherapplications. In reflection seismology the earth's crust isprobed with a wave pulse and the reflected pulses are used toinfer knowledge about macroscale variations in the medium. It isthen important to know how the rough microscale variations in themedium affect the probing pulse. Remote sensing, ocean acoustics,and medical imaging provide similar examples. In communicationsystems, it is important to be able to describe how mediumimperfections and variations affect the propagating signal. Inthe theory of composite materials, it is important to optimallyconstruct a complex material so that it has some desiredpropagation properties.