  We put forward a model that describes a decaying and evanescent point source
of non-interacting quantum waves in 1D. This point-source assumption allows for
a simple description that captures the essential aspects of the dynamics of a
wave traveling through a classically forbidden region without the need to
specify the details of the inner region. The dynamics of the resulting wave is
examined and several characteristic times are identified. One of them
generalizes the tunneling time-scale introduced by B\"uttiker and Landauer and
it characterizes the arrival of the maximum of the wave function. Diffraction
in time and deviations from exponential decay are also studied. Here we show
that there exists an optimal injection frequency and detection point for the
observation of these two quantum phenomena.
