\overline{v' w'}^+/\partial y^+|_{t^+}$. The barred quantities are averaged along $x$ and $z$, $w$ is the turbulent spanwise velocity, and $v'$ and $w'$ are the wall-normal and spanwise velocity components fluctuating about their corresponding space-averaged quantities. The equation describing the laminar Stokes flow is obtained by replacing $\overline{w}$ with $w_s$, and by setting $R_{vw}=0$. Fig. \[fig:figure\_2\] shows $R_{vw}$ at different oscillation phases from the start-up of the wall motion for $W_m^+=18$ and $T^+=125$. It initially increases, reaches a maximum when $t \approx T/2$, and then decreases as the transient elapses after a few oscillation periods. As $R_{vw}$ is negligible once the new modified turbulent flow is established, it follows that $\overline{w}^+ \approx w^+_s$ as $t^+ \rightarrow \infty$. The fact that $\overline{w}^+$ does not coincide with $w^+_s$ during the initial transient is reflected in the time history of the space-averaged spanwise wall-shear stress $\tau_z$. It is observed in Fig. 8 at page 12 in @quadrio-ricco-2003 that this quantity deviates slightly from the laminar value at $t \approx T/2$, when $R_{vw}$ reaches its maximum value.

The scaling parameter $S$ is now introduced. It has been expressed by @choi-xu-sung-2002 as the product of $a_m^+$, the maximum acceleration of the Stokes layer at a specified 