  We study existence, uniqueness, regularity and asymptotic spatial behavior of
a Navier-Stokes flow past a body moving by a time-periodic translational motion
of period $T$, and with zero average. For example, $\mathscr B$ moves in an
oscillating fashion. The flow is also time-periodic with same period $T$.
However, sufficiently ``far" from the body, the oscillatory component decays
faster than the averaged component, so that the flow shows there a distinctive
steady-state character. This provides a rigorous proof of the ``steady
streaming" phenomenon.
