  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.
