  Extended Air Showers produced by cosmic rays impinging on the earth
atmosphere irradiate radio frequency radiation through different mechanisms.
Upon certain conditions, the emission has a coherent nature, with the
consequence that the emitted power is not proportional to the energy of the
primary cosmic rays, but to the energy squared. The effect was predicted in
1962 by Askaryan and it is nowadays experimentally well established and
exploited for the detection of ultra high energy cosmic rays.
  In this paper we discuss in details the conditions for coherence, which in
literature have been too often taken for granted, and calculate them
analytically, finding a formulation which comprehends both the coherent and the
incoherent emissions. We apply the result to the Cherenkov effect, obtaining
the same conclusions derived by Askaryan, and to the geosynchrotron radiation.
