Tom Donaly, KA6RUH wrote:
"Actually, it`s supposed to be impossible to represent the current
distribution along a dipole using simple mathematical formulas because
integral equations have to be solved that are impervious to any solution
other than numerical approximation."

How many places do you attach to pi?

First, what is linearity? It is the absence of nonlinearity.

Millman and Seely wrote on page 525 of the 1951 edition of "Electronics"
(one of my old textbooks):
"Because of this nonlinear characteristic of the dynamic curve over the
operating range, the wave form of the output wave differs slightly from
that of the grid-exciting-voltage waveshape. Distortion of this type is
called "nonlinear" or "amplitude" distortion.."

All of the antennas I`ve worked with had no noticeable amplitude
distortion. They caused no harmonics or mixing products.

On page 235 of Kraus` 1950 edition of "Antennas" he sets out to solve
Hallen`s equation for current distribution. On page 239, Kraus writes:
"It is generally assumed that the current distribution of an
infinitesimally thin antenna is sinusoidal, and that the phase is
constant over a 1/2-wavelength interval, changing abruptly by
180-degrees between intervals."

You can take what Kraus says to the bank.

Best regards, Richard Harrison, KB5WZI