  This paper considers plane channel flow modified by waves of spanwise
velocity applied at the wall and travelling along the streamwise direction.
Laminar and turbulent regimes for the streamwise flow are both studied.
  When the streamwise flow is laminar, it is unaffected by the spanwise flow
induced by the waves. This flow is a thin, unsteady and streamwise-modulated
boundary layer that can be expressed in terms of the Airy function of the first
kind. We name it the generalized Stokes layer because it reduces to the
classical oscillating Stokes layer in the limit of infinite wave speed.
  When the streamwise flow is turbulent, the laminar generalized Stokes layer
solution describes well the space-averaged turbulent spanwise flow, provided
that the phase speed of the waves is sufficiently different from the turbulent
convection velocity, and that the time scale of the forcing is smaller than the
life time of the near-wall turbulent structures. Under these conditions, the
drag reduction is found to scale with the Stokes layer thickness, which renders
the laminar solution instrumental for the analysis of the turbulent flow.
  A classification of the turbulent flow regimes induced by the waves is
presented by comparing parameters related to the forcing conditions with the
space and time scales of the turbulent flow.
