We find that the spreading of the COVID-19 pandemic in Italy can be described as the propagation of a wave packet in a dispersive medium where the effect of lockdown is simulated by the dispersion relation of the medium. We start expanding a previous statistical analysis based on the official data provided by the Italian Civil Protection during 100 days, from March 2nd to June 9th. As the total number of people infected with the virus is uncertain, we have considered the trend of ICU patients and the sum of hospitalized patients and the deceased. Both the corresponding curves are well approximated by the same function depending on four free parameters. The model allows to predict the short-term behavior of the pandemic and to estimate the benefits due to lockdown measures.