A wave-packet is a solution of the wave equation, generated as a coherent superposition  of free propagating waves, which describes  a  short pulse of localized wave-action that travels as a unit,  at a constant speed. Recently, ultra – short  wave packets of electromagnetic radiation became experimentally available, and they are used to explore the scattering of light by atomic and solid systems with high temporal resolution. In the present talk I shall discuss the modification of the wave packet by the scattering process, which causes a broadening of the wave-packet. This modified shape can be interpreted as the delay-time distribution due to the interaction.  I shall  illustrate the phenomenon by  studying the scattering of wave-packets from a random potential on the real half-line - a paradigm one dimensional systems which displays Anderson Localization in the stationary case.