Abstract. Hydrodynamic jets, underdense with respect to their environment by a factor of up to 10 4 , were computed in axisymmetry as well as in 3D. They finally reached a size of up to 220 jet radii, corresponding to a 100 kpc sized radio galaxy. The simulations are "bipolar", involving both jets. These are injected into a King type density profile with small stochastic density variations. The back-reaction of the cocoons on the beams in the center produces armlength asymmetries of a few percent, with the longer jets on the side with the higher average density. Two distinguishable bow shock phases were observed: an inner elliptical part, and a later cylindrical, cigar-like phase, which is known from previous simulations. The sideways motion of the inner elliptical bow shock part is shown to follow the law of motion for spherical blast waves also in the late phase, where the aspect ratio is high, with good accuracy. X-ray emission maps are calculated and the two bow shock phases are shown to appear as rings and elongated or elliptical regions, depending on the viewing angle. Such structures are observed in the X-ray data of several radio galaxies (e.g. in Abell 2052 and Hercules A), the best example being Cygnus A. In this case, an elliptical bow shock is infered from the observations, a jet power of 10 47 erg/s is derived, and the Lorentz factor can be limited to Γ > 10. Based on the simulation results and the comparison to the observations, the emission line gas producing the alignment effect in radio galaxies at high redshift is suggested to be cooled gas entrained over the cocoon boundary.