The pulsed electromagnetic radiation from a vertical electric dipole above an atmospheric surface duct is investigated theoretically. The duct model used is that of Kahan and Eckart  and consists of a discontinuous drop of the otherwise constant relative permittivity at the upper duct boundary. The Earth is assumed to be an ideal conductor and planar. The modified Cagniard method is used to derive closed form expressions for the Hertzian vector anywhere above the duct. From the physical point of view, Cagniard's idea is applicable as it is based on evaluating the field in a series of image sources of the primary source. The step - function solution of the problem can then be determined as an infinite sum of definite integrals over finite intervals. Two cases would be distinguished on the basis of the distance between the receiving and transmitting ends and whether it is greater or lesser than the total reflection distance.