wall-normal location, and $\ell_w^+$, a distance from the wall at which the oscillating layer directly affects the turbulent flow.

At the root of this procedure lies the laminar solution for the second Stokes problem [@batchelor-1967]: $$\label{eq:stokes}
w^+_s \left( y^+,t^+;W_m^+,T^+ \right) = W_m^+  \exp \left( - y^+ \sqrt{\pi / T^+} \right)
\sin \left( \frac{2 \pi}{T^+} t^+ - y^+ \sqrt{\frac{\pi}{T^+}} \right).$$ The maximum spanwise acceleration $a_m^+$ at a wall-normal distance $y^+=\ell_a^+$ is obtained by differentiating (\[eq:stokes\]) with respect to $t^+$: $$a_m^+ = \frac{2 \pi W_m^+}{T^+} \exp \left(- \ell_a^+ \sqrt{\pi / T^+}\right).$$ It is further required that the maximum spanwise velocity of the Stokes layer at $y^+=\ell_w^+$ be larger than a typical value $W_{th}^+$ of the spanwise velocity fluctuations. The wall forcing must be intense enough for the Stokes layer to influence the turbulent fluctuations and disrupt the near-wall turbulence-producing cycle. By imposing the above condition and by using (\[eq:stokes\]), it follows $$\ell_w^+ = \sqrt{\frac{T^+}{\pi}} \ln \left(\frac{W_m^+}{W_{th}^+}\right).$$ The quantities $a_m^+$ and $\ell_w^+$ are united into the expression for $S$: $$\label{eq:S}
S = \frac{a_m^+ \ell_w^+}{W_m^+} = 2 \sqrt{\frac{\pi}{T^+}}
\ln \left( \frac{W_m^+}{W_{th}^+} \right) \exp \left(- \ell_a^+ \sqrt{\pi / T^+}\right).$$ $\ell_a^+$ and $W_{th}^+$ have been determined by maximizing the correlation coefficient $C_S$ between the drag reduction 