An analytic relationship that predicts the Reynolds number effect on turbulent drag reduction by active means is developed in analogy with riblets. It is applicable to all control techniques whose action result in an upward shift ∆B of the logarithmic region of the turbulent velocity profile. In particular, we use it to address the Re-effect affecting streamwise-traveling waves of spanwise wall velocity \cite{quadrio-ricco-viotti-2009}, aided by a new large dataset of Direct Numerical Simulations of turbulent channel flows at increasing Re. The main outcome of this study is that the control-induced upward-shift of the logarithmic region ∆B does not vary with Re along a large part of the wave parameter space, also where high drag reduction is achieved. Here, the analytical relationship allows to extrapolate low-Re drag reduction information to high-Re flows. In the narrow regions where ∆B does vary with Re, an additional Re-effect is deemed to exist, which depends on the present control technique only and which is investigated with a three-dimensional phase conditional averaging procedure.