Mathematics Colloquia and Seminars

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We consider two problems related to propagation of wave pulses in a heterogeneous environment with rapidly changing medium parameters. The shape of pulse and also its speed of propagation may be affected by the heterogeneities and we want to characterize how. First we consider propagation of elecromagnetic signals through a composite structure made up of two phases and derive bounds for the group velocity of the signal. We show that the group velocity of a pulse in fact can be higher in the composite structure than in either of the two phases and give the microstructures with the maximum relative increase and decrease in velocity. Next we consider propagation of acoustic waves through a multiscale or fractal medium with medium parameters modeled in terms of fractional Brownian motion. We sho whow the modification in the pulse shape depends on the roughness of the medium and can be described in a deterministic way when the pulse is observed at its random arrival time. For very rough media the wave is localized to a surface layer.