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Extended Air Showers produced by cosmic rays impinging on the earthatmosphere irradiate radio frequency radiation through different mechanisms.Upon certain conditions, the emission has a coherent nature, with theconsequence that the emitted power is not proportional to the energy of theprimary cosmic rays, but to the energy squared. The effect was predicted in1962 by Askaryan and it is nowadays experimentally well established andexploited for the detection of ultra high energy cosmic rays.  In this paper we discuss in details the conditions for coherence, which inliterature have been too often taken for granted, and calculate themanalytically, finding a formulation which comprehends both the coherent and theincoherent emissions. We apply the result to the Cherenkov effect, obtainingthe same conclusions derived by Askaryan, and to the geosynchrotron radiation.

On the coherent emission of radio frequency radiation from high energy particle showers