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For $u=U\cos (\omega t)\sin (\pi y/L)$ on $|y|<L/2$, positive diffusivity $\kappa$, nonzero frequency and insulating walls, a transversely uniform zeroth moment remains constant. Its first moment is proportional to $g(t)\sin (\pi y/L)$, where $g^{\prime }+ag=U\cos (\omega t)$. The periodic solution is $g=U[a\cos (\omega t)+\omega \sin (\omega t)]/(a^2+{\omega }^2)$. The longitudinal moment hierarchy for shear transport then gives $m_2=2Kt{m}_0+O(1)$ after both transverse and periodic transients. The spatial average of ${\sin }^2(\pi y/L)$ is $1/2$ and the time average of ${\cos }^2(\omega t)$ is $1/2$, producing the factor $1/4$. At exactly zero frequency the steady enhancement is instead $U^2/(2a)$; averaging indefinitely over slow cycles does not commute with setting the frequency to zero.

Take $\kappa >0$ and initially $\omega \ne 0$. Write $m_j(y,t)={\int }_{\mathbb{R}}{x}^j\chi (x,y,t)dx$, and set $k=\pi /L$, $a=\kappa k^2$. The advection-diffusion equation is ${\chi }_t+u{\chi }_x=\kappa ({\chi }_{xx}+{\chi }_{yy})$. Integration by parts in the unbounded coordinate gives the longitudinal moment hierarchy for shear transportAll three moments obey Neumann boundary conditions at the walls. The initial values are $m_0=L$, $m_1=0$, $m_2=L^3/12$. These manipulations require finite moments and vanishing boundary terms at $x=\pm \infty$; diffusion of the compact initial distribution supplies the required decay for positive time.

The zeroth moment remains exactly $L$. Since $\sin (ky)$ is a Neumann eigenfunction, put $m_1=Lg(t)\sin (ky)$. Then $g^{\prime }+ag=U\cos (\omega t)$ and $g(0)=0$, givingFor the second moment, $2u{m}_1=LUg\cos (\omega t)[1-\cos (2ky)]$. Thus $m_2=A(t)+B(t)\cos (2ky)$, withThe equation for $B$ has positive damping and bounded forcing, so $B$ remains bounded. In the equation for $A$, the decaying term in $g$ contributes only a bounded accumulated transient. Over a complete nonzero-frequency cycle, $⟨{\cos }^2(\omega t){⟩}_t=1/2$ and $⟨\sin (\omega t)\cos (\omega t){⟩}_t=0$. ConsequentlyThis proves oscillating shear dispersion in an insulating channel, including the local-in-$y$ leading term, not just a cross-channel average. The instantaneous derivative retains bounded periodic oscillations; its cycle-averaged slope is $2K{m}_0$. Thus $2Kt{m}_0$ is the leading second moment, rather than a literal rate proportional to $t$.

The asymptotic statement holds for fixed positive diffusivity and fixed nonzero frequency, after times long compared with $a^{-1}$ and the oscillation period. If $|\omega |L^2/\kappa \ll 1$, transverse diffusion adjusts rapidly to the changing shear flow, andThis is the cycle average of the instantaneous steady-shear enhancement $U^2{\cos }^2(\omega t)/(2a)$. At exactly $\omega =0$, there is no cycle average: $g\to U/a$ and $K=\kappa +U^2/(2a)$. Hence the zero-frequency and long-time limits do not commute. In the intermediate range $a^{-1}\ll t\ll |\omega {|}^{-1}$, the shear is approximately steady at its initial amplitude and that latter enhancement is appropriate.