  We study the time-dependent scattering of a quantum mechanical wave packet at
a barrier for energies larger than the barrier height, in the semi-classical
regime. More precisely, we are interested in the leading order of the
exponentially small scattered part of the wave packet in the semiclassical
parameter when the energy density of the incident wave is sharply peaked around
some value. We prove that this reflected part has, to leading order, a Gaussian
shape centered on the classical trajectory for all times soon after its birth
time. We give explicit formulas and rigorous error bounds for the reflected
wave for all of these times.
