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Propagation of a directed pulse of electromagnetic radiation in a vacuum and in a rarefied, temporally dispersive medium is investigated. The pulse belongs to the class of broadband directed radiation beams termed electromagnetic ‘‘missiles’’ and is superposed on a carrier wave. First the propagation characteristics of the pulse in a vacuum are reviewed. In a vacuum the far‐field time‐integrated Poynting flux U(z) of the pulse drops off as 1/z1+4ν, with ∥ν∥≤1/4. In the corresponding expression for U(z) in a dispersive medium the leading term also varies as 1/z1+4ν since the high‐frequency end of the spectrum experiences very little dispersion. In a dispersive medium the propagation of the precursors and of the carrier signal is examined in certain limits and expressions for the spatial variation of the vector potential are given.