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Mathematical analysis of the energy concentration in waves traveling through a rectangular material structure in space-time

Daniel Onofrei
We consider propagation of waves through a spatio-temporal doubly periodic material structure with checkerboard microgeometry in one spatial dimension and time. Both spatial and temporal periods in this dynamic material are assumed to be the same order of magnitude. Mathematically the problem is governed by a standard wave equation $(\rho
u_t)_t-(ku_z)_z=0$ with variable coefficients. The rectangles in a space-time checkerboard are assumed filled with materials differing in the values of phase velocities $\sqrt{\frac{k}{\rho}}$ but having equal wave impedance
Within certain parameter ranges, the existence of distinct and stable limiting characteristic paths, i.e., limit cycles, was observed in \cite{Lurie-Weekes}; such paths attract neighboring characteristics after a few time periods. The average speed of propagation along the limit cycles remains the same throughout certain ranges of structural parameters, and this was called in \cite{Lurie-Weekes} a plateau effect.
Based on numerical evidence, it was conjectured in \cite{Lurie-Weekes} that a checkerboard structure is on a plateau if and only if it yields stable limit cycles and that there may be energy concentrations over certain time intervals depending on material parameters.
In the present work we give a more detailed analytic
characterization of these phenomena and provide a set of sufficient conditions for the energy concentration that was predicted numerically in \cite{Lurie-Weekes}.
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