= Impulsively kicked sinusoidal shear dispersion
{title2=$\kappa_e=\kappa+\frac{U_1^2}{4T}\coth(\kappa m^2T/2)$}
When velocity consists of impulses $U_1$ spaced by $T$, the periodic corrector jumps by $U_1$ and decays between kicks. With $a=\kappa m^2$, $g(t)=U_1e^{-at}/(1-e^{-aT})$ for $0<t<T$. Integrating $g^2$ in the <effective diffusivity of a periodic sinusoidal shear> formula yields the displayed result. For $aT\gg1$, independent transverse phases give longitudinal variance $U_1^2/2$ per kick and enhancement $U_1^2/(4T)$. For $aT\ll1$, the enhancement is $U_1^2/(2\kappa m^2T^2)$ to leading order, the steady-shear result for average speed $U_1/T$. The zero-diffusivity limit does not commute with the long-time limit: without transverse diffusion the displacement is ballistic.
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