  We consider the time-dependent Navier-Stokes equations in a half-space with
boundary data on the line $(x,y)=(x_0,y)$ assumed to be time-periodic (or
stationary) with a fixed asymptotic velocity ${\bf u}_{\infty}=(1,0)$ at
infinity. We show that there exist (locally) unique solutions for all data
satisfying a compatibility condition in a certain class of fuctions.
Furthermore, we prove that asymptotically the vorticity decompose itself in a
dominant stationary part on the parabolic scale $y\sim\sqrt{x}$ and corrections
of order $x^{-{3/2}+\epsilon}$, while the velocity field decompose itself in a
dominant stationary part in form of an explicit multiscale expansion on the
scales $y\sim\sqrt{x}$ and $y\sim x$ and corrections decaying at least like
$x^{-{9/8}+\epsilon}$. The asymptotic fields are made of linear combinations of
universal functions with coefficients depending mildly on the boundary data.
The asymptotic expansion for the component parallel to ${\bf u}_{\infty}$
contains `non-trivial' terms in the parabolic scale with amplitude
$\ln(x)x^{-1}$ and $x^{-1}$. To first order, our results also imply that
time-periodic wakes behave like stationary ones as $x\to\infty$.
  The class of functions used is `natural' in the sense that `Physically
Reasonable' (in the sense of Finn & Smith) stationary solutions of the N.-S.
equations around an obstacle are covered if the half-space is choosen
sufficiently far downstream. The coefficients appearing in the asymptotics may
then be linearly related to the net force acting on the obstacle. To our
knowledge, it is the first time that estimates uncovering the $\ln(x)x^{-1}$
correction are proved in this setting.
