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Waves in Random Media

This project concerns three problems related to waves propagating in random media: refocalization effect for time-reversed acoustic waves, propagation of pulses in the regime of parabolic and white noise approximation and the use of wave automata in the numerical simulations of these phenomena.

Time-Reversal Mirrors (TRM's) are piezoelectric devices which can convert acoustical pressures into electric signals and vice-versa. They are coupled to memories which permit to monitor an acoustical pressure and send it back into the medium in the reverse direction of time. If the initial source is like a pulse, localized in time and space, it has been observed experimentally that few TRM's are enough to refocalize at the source in a coherent way. Surprisingly the presence of desorder (or randomness) in the medium "seen" by the wave improves this refocalization effect. Applications of this technique to inverse problems are very promising (medical imaging, nondestructive industrial control,...). Also mathematical understanding of this phenomenum is essential in domains like Geophysics where techniques such as "migration" are based on the same ideas.

Pulse propagation in the parabolic approximation:
The parabolic and white noise approximation is widely used in domains such as underwater acoustic waves or optics in atmospheric turbulence where the fluctuations of the index of refraction are small, the frequency is high and the distances of propagation are long. The monochromatic (or stationary) wave field, in this limit, satisfies a stochastic Schrodinger equation which leads to an open system of moment equations. This technique will be generalized to multi-frequency cases to derive the equations of propagation of a correctly rescaled pulse and use its shape modification in inverse problems.

Simulations using wave automata:
Wave automata give an efficient (parallel) way to simulate waves propagating on a lattice in the time domain. This is achieved for any type of waves and in any dimension of space. The introduction of varying coefficients (like random coefficients for random media) is very simple. The fact that this simulation is done in the time domain makes it well-adapted to the study of the refocalization effect by TRM's or to the study of the transmitted pulse.












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