  We present the results of a numerical analysis of the propagation and
interaction of a supersonic jet with the external medium. We discuss the motion
of the head of the jet into the ambient in different physical conditions,
carrying out calculations with different Mach numbers and density ratios of the
jet to the exteriors. Performing the calculation in a reference frame in motion
with the jet head, we can follow in detail its long term dynamics. This
numerical scheme allows us also to study the morphology of the cocoon for
different physical parameters. We find that the propagation velocity of the jet
head into the ambient medium strongly influences the morphology of the cocoon,
and this result can be relevant in connection to the origin and structure of
lobes in extragalactic radiosources.
