Exact Treatment of Antenna Current Wave Reflection at the End of a Tube-Shaped Cylindrical Antenna

Antennas, whose radiating body has a cross section which is small in comparison to the length of the antenna and to the wave length, i. e. antennas consisting of wires and rods, are to a very wide extent treated with the help of a linearized integral equation, the invention of the author (Hallén 1930, eqn. 20a, b; Hallén 1938, eqn. 24; Hallén 1953, eqn. 35.31). In this equation the distance between two points on the antenna is normally represented by the distance between the corresponding points on some central line. Only when the distance is small this is not permitted and from such regions arises the only term which contains the dimension of the cross section, which is a parameter mainly consisting of a logarithm. The equation therefore has a certain limited degree of accuracy which is such that the ratio of the radius of cross section to the length of the antenna or to the wave length is neglected compared with unity. The results which can be drawn from the linearized integral equation thus also should have this limited accuracy which is a normal one in electrotechnics in all kinds of devices, where wires are involved. Nevertheless much discussion has gone on about this accuracy. The only way of finding definite numerical answers to this question is to solve exactly the antenna integral equations, both the linearized one and the exact one, for some special case. Nowadays this can be done for a straight cylindrical tube-shaped antenna.