It is known that the current distribution on a thin dipole antenna is always sinusoidal:
How can I prove this?
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You can't prove this, it's not true. It is only a very good approximation: it fits with the boundary conditions. For example, in a half wave dipole, by symmetry, the current needs to be maximum in the middle at the feeder inputs and nought at the ends of the antenna (no current flows out into space!). It's also found that calculations grounded on this assumption fit well with experimental results and that numerical models of the antenna show something very lika a sinusoidal distribution. Similar boundary conditions apply for antennas other than the half wave dipole, but the principle is the same: the sinusoid is an approximation.
Currents at all different points in the antenna act, through the retarded potential, on the currents at all other parts of the antenna. Charge also is shuttled between different points in the antenna. The system is highly coupled, and, as such, one can really only find the true current density either numerically or experimentally. A simple numerical model assumes the antenna comprises short current lengths (wherein the current is approximately constant) spaced by time varying charges. Models like this one tell us that the current distribution is, to a good approximation, sinusoidal.