Wave propagation and energy exchange in a spatio-temporal material composite with rectangular microstructure

We consider propagation of waves through a spatio-temporal doubly periodic material structure with rectangular microgeometry in one spatial dimension and time. Both spatial and temporal periods in this dynamic material are assumed to be of the same order of magnitude. A “double Floquet” solution is obtained in the special case when the wave equation (ρut)t−(kuz)z=0 allows for the separation of variables. We also consider a checkerboard microgeometry where variables cannot be separated. The squares in a space–time checkerboard are assumed to be filled with materials having equal impedance but different phase speeds. Within certain parameter ranges, we observe numerically the formation of distinct and stable limiting characteristic paths (“limit cycles”) that attract neighbouring characteristics after a few time periods. The average speed of propagation along the limit cycles remains the same throughout certain ranges of parameters of the microgeometry (the “plateau effect”). We formulate, as a hypothesis, the statement saying that a checkerboard structure is on a plateau if and only if it yields stable limit cycles. A dynamic material is a thermodynamically open system, as it is involved in a permanent exchange of energy and momentum with the environment. Material assemblages that produce the limit cycles are special in this aspect. Specifically, to make a wave travel through such an assemblage, we find analytically that an external agent may need to supply infinite energy and this may be so regardless of the wave frequency. For spatio-temporal laminates, however, an accumulation of energy (parametric resonance) may emerge only for frequencies that are not too low relative to some characteristic frequency of the system.