  We derive an exact analytical solution to the time-dependent Schr\"odinger
equation for transmission of a Gaussian wave packet through an arbitrary
potential of finite range. We consider the situation where the initial Gaussian
wave packet is sufficiently broad in momentum space to guarantee that the
resonance structure of the system is included in the dynamical description. We
demonstrate that the transmitted wave packet exhibits a transient behavior
which at very large distances and long times may be written as the free
evolving Gaussian wave packet solution times the transmission amplitude of the
system and hence it reproduces the resonance spectra of the system. This is a
novel result that predicts the ultimate fate of the transmitted Gaussian wave
packet. We also prove that at a fixed distance and very long times the solution
goes as $t^{-3/2}$ which extends to arbitrary finite range potentials previous
analysis on this issue. Our results are exemplified for single and multibarrier
systems.
