On the radiation field of a perfectly conducting base insulated cylindrical antenna over a perfectly conducting plane earth, and the calculation of radiation resistance and reactance

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Abstract
The art of radio-broadcasting has in recent years directed attention to the radiation characteristics of the vertical antenna in circumstances where the height is comparable to the wave-length, X. Existing theory generally assumes a sinusoidal distribution of current along the antenna. By integrating Hertz’s expression for the radiation field of a current element it is indeed possible to obtain expressions for the electric and magnetic vectors of the entire antenna, and ultimately, by making use of Poynting’s theorem over a very large hemisphere, to arrive at the well-known expression for the radiation resistance. Such an antenna, known as asine-wave antenna, gives results in tolerably good agreement with observation as long as / / X does not exceed J. This theory, admittedly imperfect, predicts sharp nodes of current along the antenna when / / X > with the result that the radiation resistance calculated with respect to the current at the base of the antenna tends to infinite values at //X = |, f, f, ..., etc. Then again the theory of the sine-wave antenna ignores a prescribed boundary condition over the surface of the conductor. In the case of a perfectly conducting antenna this requires that the component E of the electric field along the antenna should be zero over a cylinder of radius p =. The sine-wave solution, however, gives rise to two components of E? in phase quadrature, neither of which is zero along the antenna, while one of them tends to infinite values at the foot and at the top of the antenna.*

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