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I often see solutions to radiating, center fed dipoles with "approximations" of the current distribution being sinusoidal. When describing this, people often refer to the "Boundary Conditions" as no current on the ends of the dipole, and peak current at the center, where the source is. But this seems hand wavy, I've never actually seen a mathematical description of this current distribution. Are these actual boundary conditions, or just intuitive ones?

Moreover, if these are actual boundary conditions, I would expect the solution to include "modes" similar to a cavity resonator. These might reflect the solutions of the lambda/4, lambda/2, lambda, 2/lambda ... etc. dipole antenna. Instead most people just assume a current distribution and find the radiation from it. Is there a rigorous way to find the current distribution on the conductor? I imagine it's some solution of the diffusion equation?

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  • $\begingroup$ the derivation of the voltage and current distribution along a center-fed dipole is mathematically rigorous and the boundary conditions needed to solve the so-called "antenna equation" are indeed the imposition of the no-current condition at the ends of the wire and peak current at the center point. In real life, there is always a little current leakage at the ends into the air and nonvanishing voltage at the center point, plus capacitive coupling to the ground under the antenna and resistive losses in the wire, but these effects are manageable from a design standpoint. $\endgroup$
    – niels nielsen
    Oct 31, 2017 at 3:29
  • $\begingroup$ regarding the modes in the antenna itself, you are right. an antenna which is resonant at one frequency will also exhibit resonances at multiples of the fundamental resonance. This characteristic allows one antenna to function satisfactorily at more than one wavelength, a trick that antenna designers use to make multiband antennas. the derivation of the current distribution along a center-fed dipole can be found in any treatise dealing with radio-frequency (RF) antenna theory. $\endgroup$
    – niels nielsen
    Oct 31, 2017 at 3:33
  • $\begingroup$ It's the "antenna" equation you mention I can't seem to find. Every antenna textbook I have includes a rigorous derivation of the fields from a given current distribution, but the current distribution itself is always assumed (constant for infinitesimal dipoles, triangular for small ones, and of course half and full wave distributions). I have many antenna engineering books with me, but none of them actually solve this current distribution in any rigorous way, unless I'm not looking at the right place. $\endgroup$
    – kthaxt
    Nov 3, 2017 at 23:11
  • $\begingroup$ try looking at ham radio literature for antenna design, that may help. I just got my ham license and the ham guys deal with this stuff all the time; in fact, they now have freeware programs that solve for feedpoint impedance and 3D propagation patterns as functions of frequency- and one would think this requires knowledge of both the voltage and current distributions in the dipole. $\endgroup$
    – niels nielsen
    Nov 4, 2017 at 0:01

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I found the rigorous solution I was looking for in the Theory of Characteristic Modes, which solves for the current modes of a given conductive body. It's apparently becoming a popular computational electromagnetic tool and can be used to numerically solve for the current modes of arbitrary structures. Analytical solutions only seem to exist for very canonical solutions, like spheres. One can find more information here: https://en.wikipedia.org/wiki/Characteristic_mode_analysis

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