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Do *not* underestimate the power of "operational mathematics". Oliver Heaviside was correctly solving problems that the mathematicians couldn't solve for years before some of them were intrepid enough to prove why it worked. In the end, the development of the LaPlace transform eclipsed Heaviside. But Heaviside was not wrong and Churchill's text is very much an homage to that. It's also important to remember that "Maxwell's equations" as we know them are really "Heavyside's equations". Maxwell used quaternions. It was Heaviside who introduced Faraday's work in vector notation with the addition of Maxwell's inspired assumption that another equation was needed for symmetry.

In this case, consider a series of wires at uniform spacing orthogonal to the direction of propagation which are of uniform length with a dipole in place of the single elements at some point. The dipole and the lengths of the other elements are unspecified except that that they are all resonant at the same frequency. Such a line will divide the input equally in both directions when the dipole is excited.

This constitutes a transmission line. Rather different from normal models, but you can plug all the values in and calculate impedances, etc. If one then perturbs the elements on one side of the dipole such that it is highly reflective it starts to get interesting.

If one can show that the radiation off axis to the direction of propagation along such a transmission line is a minimum at the resonant frequency, then one has proved that it is the "sharpest beam". I don't know how to do that, but I suspect someone who is better at EM than I can. I suspect it is obvious to a very select few.

From there it is simply a matter of approximating the effects of truncation. That in the 1920's could *only* be done by empirical methods. But the insight needed to recognize that such a structure constitutes a transmission line and could be used in truncated form as an antenna is without question world class. Or in Reg speak, "mondo cool!"

A big thanks to Kaz for including this paper in the issue. This has truly been great fun to contemplate. I'd been looking for a paper in QEX about which I had something at least semi-sensible to say. This was just close enough to my home turf of elastic wave propagation.

If my conjectures above are correct, then it seems likely that there may be some obscure papers by Yagi, Uda or both which are only available in Japanese. If anyone can read Japanese and has access to the Japanese literature, I think it would be fitting and proper to have that translated and published for more general access and to properly credit what is without question an exceptional piece of work.

Have Fun!
Reg