that an analysis through $S$ is not needed at these high wall velocities, (iv) it will be shown in Section \[sec:minimal\] that the net energy saving may be positive only in the range $W_m^+<$7.

The fact that the prediction for $T_{op t,D}$ is good over most of the explored range is further confirmed by Fig. \[fig:figure\_5\], which shows that the optima computed by the DNS data compare satisfactorily with $T_{opt,D}=T_{opt,D}(D_m)$ obtained by (\[eq:Topt-Dm\]).

Minimal oscillating conditions and estimates of $DR$(%) and $P_{net}$(%) as functions of $W_m^+, D_m^+, T^+$ {#sec:minimal}
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The idea that a finite intensity of the forcing is needed to affect the turbulent friction is contained in the definition of $S$, where a threshold velocity $W_{th}$ is introduced. The fact that the regression line in Fig. \[fig:figure\_3\] crosses the abscissa at $S=S_{min}=0.0063>0$ implies that the wall must oscillate with a minimal velocity $W_{m,min}>W_{th}$ (or with a minimal displacement $D_{m,min}>W_{th}T/\pi$) to achieve drag reduction. From (\[eq:S\]) it follows that $$\label{eq:minimal}
\left\{
\begin{array}{l l}
W_{m,min}^+ \\
D_{m,min}^+ \\
\end{array}
\right\}
=
\left\{
\begin{array}{l l}
1 \\
T^+/\pi \\
\end{array}
\right\}
 W_{th}^+
\exp \left[ \frac{S_{min}}{2} \sqrt{\frac{T^+}{\pi}} \exp \left( \ell_a^+ \sqrt{\frac{\pi}{T^+}} \right) \right].$$ These minimal quantities are displayed in the contours plots 