is also confirmed by the analysis of @quadrio-ricco-2003. They have found that the wall-shear stress is not affected at the beginning of the oscillation, when the wall velocity is small. Figs. 2 and 3 in their paper show that the space-averaged $\tau_x$ changes by less than 1$\%$ when $W^+ < 2$. This value compares well with $W_{m,min}^+$ and it remains essentially unchanged as $T^+$ varies for $50<T^+<200$, similarly to the behaviour of $W_{m,min}^+$ in the same $T^+$ range (see Fig. \[fig:figure\_6\], top graph).

An analogous behaviour emerges from the experimental study by [@mao-hanratty-1986] and by the numerical works by [@ismael-cotton-1996] and [@cotton-2007], who studied the variation of wall-shear stress in turbulent pipe flows subjected to streamwise oscillations of the pressure gradient. They found that for small forcing amplitudes the flow responds linearly and the mean value of the velocity gradient at the wall is unaffected.

The expressions (\[eq:minimal\]) contain the parameter $S_{min}$. Table \[tab:Smin\] shows that this quantity varies substantially when $W_{th}^+$ and $\ell_a^+$ change by a small amount. However, the minimal conditions remain almost unvaried, as revealed by the values of $W_{m,min}^+$ at $T^+=125$ in Table \[tab:Wmin\].

We now turn to the prediction of the net power saving $P_{net}$, 