A mechanism is proposed for the formation of radio components in strong double sources such as Cygnus A. Relativistic plasma generated in an active galactic nucleus cannot escape isotropically if the nucleus is surrounded by too much dense thermal gas. There is, however, a possible equilibrium flow in which the plasma escapes along two oppositely-directed channels or ‘exhausts’. At all points on the boundary of these channels, the pressure of the relativistic (possibly magnetized) plasma must balance the pressure of the static thermal gas cloud. The outflow velocity becomes sonic (implying, for ultra-relativistic plasma, a velocity |$c/\sqrt3$|) where the external pressure is |$\approx \frac 12$| its central value. The channel cross-section reaches a minimum value at this point. The channel then widens again as the external pressure drops still further, and, as in a de Laval nozzle, the flow becomes supersonic. Relativistic plasma can thus be collimated into two relativistic beams. Hargrave & Ryle's high resolution maps of Cygnus A reveal ‘hot spots’ ≲ 2 kpc in size at the outer edge of each individual component into which (it is believed) energy is being continuously supplied. We identify these hot spots with the regions where the beams impinge on the intergalactic medium. The dimensions and radio luminosity of Cygnus A imply that the central galactic nucleus must have maintained a power output |$\sim 10^{46}$| erg s−1 for 106–107 yr. This outflow could have been collimated into two sufficiently narrow beams if the galactic nucleus were surrounded by gas with |$T\simeq10^8$| K, density |$\sim 10^3$| particles cm−3 and scale height ∼ 200 pc. Some aspects of the detailed morphology of Cygnus A are also interpreted on the basis of this general model. The possible role of instabilities in the flow pattern, and the influence of magnetic fields, is discussed. Applications to other sources (which generally require less extreme parameters than Cygnus A) are briefly considered.