  The conservation of the energy flux in turbulent jets that propagate in the
intergalactic medium (IGM) allows us to deduce the law of motion in the
classical and relativistic cases. Four types of IGM are considered: constant
density, hyperbolic decrease of density, inverse power law decrease of density
and a Lane--Emden ($n=5$) profile. The conservation of the relativistic flux
for the energy allows us to derive, to the first order, an analytical
expression for the velocity. It also allows us to numerically determine the
trajectory for the four types of medium. In the case of a Lane--Emden ($n=5$)
profile, the back-reaction due to the radiative losses for the trajectory is
evaluated both in the classical and the relativistic case. Astrophysical
applications are made to the centerline intensity of the synchrotron emission
and to the evolution of the magnetic field in the case of the radio-galaxy
3C31.
