Three values of $D_m$ have been considered, $D_m^+ = 100, 200, 300$. At a given $D_m$, we have explored the existence of $T_{opt,D}$ by varying $T$ along the hyperbola $W_m=D_m \pi/T$ in the $(T,W_m)$ space. The amount of drag reduction has been determined by a procedure (QR) which involves discarding the initial temporal transient, averaging over time and along the homogeneous directions, and considering both walls to enlarge the statistical sample. For the $Re_\tau=400$ case, we have carried out one simulation for the canonical flow ($C_f = 6.495 \times 10^{-3}$, only 1% smaller than the value given by (\[eq:cf-re-tau\])) and three simulations with $W_m^+$=12 and $T^+$=30, 125 and 200.

Laminar analysis {#sec:laminar-analysis}
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The analysis is based on the close agreement between the space-averaged (along the homogeneous directions) spanwise flow and the laminar solution for the second Stokes problem, defined by $w^+_s \left( y^+,t^+;W_m^+,T^+ \right)$ [@quadrio-sibilla-2000; @choi-xu-sung-2002; @quadrio-ricco-2003]. At relatively high frequencies, the space-averaged spanwise flow uncouples from the space-averaged streamwise flow, although the spanwise turbulent fluctuations are significantly altered by the wall motion [@quadrio-ricco-2003]. We can explain this agreement as follows. The space-averaged spanwise momentum equation is $$\frac{\partial \overline{w}^+}{\partial t^+}\Bigg{|}_{y^+} = \frac{\partial^2 \overline{w}^+}{\partial y^{+ 2}}\Bigg{|}_{t^+} - R_{vw},$$ where $R_{vw}=\partial 