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Fields satisfying a wave equation on a curved space-time can be accompanied by tails propagating inside the light cone. A formal characterization of tails is given, and the relationship of the tail-free property with the characteristic propagation property (CPP) discussed in the literature is clarified. The scalar wave equation satisfies the CPP independently of the tail-free property only in the case of minimal coupling in two space-time dimensions. The peculiar feature of electromagnetic radiation, for which pure gauge tails in the potential can occur, is pointed out. [ABSTRACT FROM AUTHOR]