of Fig. \[fig:figure\_6\] as the zero drag reduction curves. The drag reduction for a $(T^+,W_m^+)$ pair, computed by (\[eq:S\]) and (\[eq:fit\]), is represented in Fig. \[fig:figure\_6\] (top graph) for $30 \leq T^+ \leq 150$. $W_{m,min}$ grows unbounded as $T$ decreases because a stronger wall forcing is needed to affect the turbulent flow as the penetration depth $\delta_s \sim \sqrt{T}$ of the Stokes layer vanishes. $W_{m,min}^+$ becomes approximately constant at higher periods of oscillations, say $T^+>30$, and its value $\approx 1.8$ is of the order of magnitude of the near-wall spanwise velocity fluctuations. It also follows that $D_{m,min}^+ \approx D_1 T^+$, $D_1=W_{m,min}^+(T^+=150)/\pi=0.57$, for $30 \leq T^+ \leq 150$.

Fig. \[fig:figure\_6\] (bottom graph) shows the drag reduction corresponding to a $(T^+,D_m^+)$ pair. $D_{m,min}^+(T_{opt,W}^+=125)\approx 70$ compares well with the spanwise width of a low-speed streak, $\approx 20-60 \nu/u_\tau$ from flow visualizations of the near-wall turbulence [@hirata-kasagi-1979; @ricco-2004]. This result confirms that these structures need to be sufficiently swept laterally to achieve drag reduction [@baron-quadrio-1996]. The minimal conditions are in good agreement with the experimental results by Raskob & Sanderson (personal communication[^1]), who first showed that a displacement $D_m^+ \approx 80$ is needed for drag reduction when $80<T^+<1000$. The existence of the minimal conditions 