data and $S$. In QR, $C_S=0.99$, $\ell_a^+=6.3$ and $W_{th}^+=1.2$ (which remarkably agrees with the maximum r.m.s. of the turbulent spanwise velocity fluctuations, $w^+_{rms} \approx 1.1$ [@kim-moin-moser-1987]) and $$\label{eq:fit}
DR(\%) = S_1 S + S_2,$$ where $DR(\%)$ is the drag reduction, $S_1$=130.6 and $S_2$=-2.7.

An expression for $T^+_{opt,D}$, the period of oscillation which guarantees the maximum drag reduction for fixed $D_m$, is found by setting: $$\frac{\partial DR}{\partial T^+}\Bigg{|}_{D_m^+} = S_1 \frac{\partial S}{\partial T^+}\Bigg{|}_{D_m^+} = 0, \hspace{0.25 in}
\frac{\partial^2 DR}{\partial T^{+ 2}}\Bigg{|}_{D_m^+} = S_1 \frac{\partial^2 S}{\partial T^{+ 2}}\Bigg{|}_{D_m^+} < 0.$$ The second condition has been verified graphically via (\[eq:S\]). After eliminating $W_m^+ = \pi D_m^+ / T^+$ in (\[eq:S\]), it follows: $$\label{eq:Topt-Dm}
\left( \ell_a^+\sqrt{\frac{\pi}{T^+_{opt,D}}}-1 \right) \ln\left( \frac{\pi D_m^+}{T^+_{opt,D} W_{th}^+} \right) = 2.$$ Differently from $T^+_{opt,W}$, which does not depend on $W_m^+$, $T^+_{opt,D}$ increases monotonically with $D_m^+$ and is smaller than $T^+_{opt,W}$. The latter is obtained by $\partial S / \partial T^+ \big{|}_{W_m^+}= 0$, i.e. $T^+_{opt,W}=\pi (\ell_a^+)^2 \approx 125$, which agrees with the values in @jung-mangiavacchi-akhavan-1992, @dhanak-si-1999, @quadrio-sibilla-2000 and QR.

QR have also shown that $S$ scales linearly with the drag reduction only for $T^+ \leq 150$. Fortunately, it occurs that $T^+_{opt,D}, T^+_{opt,W} <$150, so that the prediction of the optimal periods based 