Owen Duffy wrote in
:

....
I think the ARRL graph is based on a well known, but apparently not
well understood formula.

The only text book that I can recall spelling out the assumptions that
underly the integral that produces the formula is Philip Smith's 'The
Electronic Applications of the Smith Chart'.

The formula is developed by integrating I^2 over an electrical half wave
of line on one side or the other from the observation point, according
to PS. (I don't know the origin of the formula, I am not suggesting that
PS invented it, not that he didn't.)

Straight away, that tells you that the VSWR must be almost the same at
both ends for it to not matter which end is the observation point, so
therefore the first assumption is that VSWR is approximately equal at
both ends of the half wave. A requirement for this is that line loss
must be relatively low, that the exponential real term in the
transmission line equations is close to zero.

If the line section is not exactly a half wave, then the real loss
factor might be higher or lower depending on the location of the current
and voltage maxima and minima and the relative contribution of R and G
to loss. So, the formula may have significant error for short lines that
are not exactly a half wave.

For a line that is many half waves, the formula is fine so long as VSWR
is approximately constant (now a very low loss line). If the line is
longer than many half waves, but not an exact integral number of half
waves, then the error in the partial section will be somewhat diminished
relatively by the loss in the complete half wave sections.

If a practical line is very long, it cannot qualify as having a constant
VSWR (unless it is 1, in which case the formula is unnecessary), so the
formula is not suited.

So, in summary, the formula is good for low loss half wave lines, or
even longish random length low loss lines, but not good for short random
length lines or very long lines.

So, why is the formula so popular?

Could it be that it underpins one of the popular myths of ham radio,
that VSWR necessarily increases line loss?

Modern computation tools are better than the 70 year old graphical
method. Publication of the formula without qualification with the
underlying assumptions treats the reader as a dummy.

Owen