The homogenization procedure developed here is conducted on a laminate with periodic space-time modulation on the fine scale: at leading order, this modulation creates convection in the low-wavelength regime if both parameters are modulated. However, if only one parameter is modulated, which is more realistic, this convective term disappears and one recovers a standard diffusion equation with effective homogeneous parameters; this does not describe the non-reciprocity and the propagation of the field observed from exact dispersion diagrams. This inconsistency is corrected here by considering second-order homogenization which results in a non-reciprocal propagation term that is proved to be non-zero for any laminate and verified via numerical simulation. The same methodology is also applied to the case when the density is modulated in the heat equation, leading therefore to a corrective advective term which cancels out non-reciprocity at the leading order but not at the second order.

翻译：本文发展的均匀化方法应用于细观尺度上具有周期性时空调制的层状材料：在主导阶，若两个参数均被调制，这种调制会在长波区域产生对流效应。然而，若仅调制单一参数（此情形更符合实际），该对流项将消失，此时得到具有等效均匀参数的标准扩散方程；但该方程无法解释精确色散图所观测到的场传播现象与非互易特性。本文通过引入二阶均匀化理论修正了此不一致性，所得的非互易传播项被证明对任意层状结构均非零，并经由数值模拟验证。该方法同样适用于热传导方程中密度被调制的情形，此时产生的修正对流项虽在一阶近似下抵消了非互易性，但在二阶近似中仍然存在。