defined as the difference between the power saved thanks to the wall motion, i.e. the drag reduction when $U_b$ is constant, and the power $P_{sp}$ spent to move the walls against the viscous stresses. These quantities are expressed as percentage of the power spent to drive the fluid through the unmanipulated channel, $2 \tau_x^+ U_b^+$, where $U_b^+=15.88$ at $Re_\tau=200$. $P_{sp}$ over an interval $t_f - t_i$ is $$\label{eq:power-spent}
P_{sp}= \frac{L_x L_z}{t_f - t_i} \int_{t_i}^{t_f} \left( \tau^{\ell}_z + \tau^{u}_z \right) W \mathrm{d} t,$$ where $\ell$ and $u$ denote the lower and upper walls. Inasmuch as $\tau_z(t)$ is well predicted by the laminar Stokes solution (see Section \[sec:laminar-analysis\]), (\[eq:power-spent\]) becomes

$$\label{eq:power-spent-percent}
P_{sp}(\%)=\frac{100}{\tau_x^+ U_b^+ T^+} \int_0^{T^+} W^+ \frac{\partial w^+_s}{\partial y^+}\Bigg{|}_{y^+=0} \mathrm{d} t =
\frac{100 (W_m^{+})^2}{2 U_b^+}\sqrt{\frac{\pi}{T^+}}=\frac{100 (W_m^{+})^3}{2 U_b^+ Re_\delta}.$$

$P_{net}(T^+,W_m^+)$ and $P_{net}(T^+,D_m^+)$ are displayed in Fig. \[fig:figure\_7\]. Steep changes occur when $P_{net}<0$, whereas regions with $P_{net}>0$ present more gradual variations. When compared with DNS results by QR, the period $T^+ \approx 60$ at $W_m^+=4.5$ needed for $P_{net}>0$ is well captured. For $T^+=150$, $P_{net}>0$ for $2 < W_m^+ < 7$, while in QR this interval is $1.5 < W_m^+ < 9$.

By setting $$\frac{\partial P_{net}}{\partial W_m^+}\Bigg{|}_{T^+} = 0, \hspace{0.25 in}
\frac{\partial^2 P_{net}}{\partial W_m^{+2}}\Bigg{|}_{T^+} < 