DNS data. In the present analysis all the drag reduction datasets available in the literature are employed to compute a new, and more general, least-squares fit. A useful by-product of this procedure is that each dataset (both from experiments and DNS) can be discussed in terms of its difference from the fit. Since the literature data show a considerable scatter, comparing the actual measurements of drag reduction with the predictions via $S$ permits an indirect assessment of their accuracy.

The linear fit allows us to identify and study easily two distinct periods of wall oscillation that are optimal in terms of drag reduction. Although in DNS studies the oscillation parameters can be chosen freely, $T_{opt,W}$, the optimal period of oscillation for fixed maximum wall velocity, has usually been the quantity of interest. On the other hand, an experimental campaign is likely to reveal $T_{opt,D}$, the optimal period at fixed maximum wall displacement. Indeed, in a laboratory the wall motion is usually produced by a crank-slider mechanism, which allows varying $T$ for fixed $D_m$. QR however noted that $T_{opt,D}$ had never been observed by experimentalists, probably because the flow at these high frequencies of oscillation is difficult to test. The difference 