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[Phys-L] error in Feynman § I-50-5 : wave energy "theorem"

Hi --

Executive summary: There's a bug. I mostly know the
right answer, but perhaps somebody could suggest a
more elegant way of saying what needs to be said.

Background: For the last 15 years or so, people
have been collecting and correcting errors in
_The Feynman Lectures on Physics_. Hundreds of trivial
punctuation and spelling errors have been caught.

On the other hand, it is astonishing how few /physics/
errors there are. In his preface to _The Definitive Edition_
Kip Thorne wrote "It is remarkable that the errata included
only two inadvertent errors in physics."

This stands in contrast to the textbooks published
nowadays, where a typical 1000-page book contains many
hundreds of nontrivial physics errors.


I claim there is a third physics error in the Feynman
lectures. Volume I chapter 50 section 5 starts by saying:

The energy in a wave is proportional to the square of its amplitude.
For a wave of complex shape, the energy in one period will be
proportional to ∫ f^2(t) dt. [1]

See for yourself:

The problem is, statement [1] is not reliably true. You
can get away with it for plane waves in the electromagnetic
field, or ideal plane waves in air ... but it's not correct
for waves on a string, or waves in the electromagnetic potential.

There are presumably other counterexamples.

Not coincidentally, my two counterexamples have the property
that you can shift the ordinate of the wavefunction by a
gauge transformation that doesn't change the physics. This
alone is sufficient to guarantee that the simple square
law [1] cannot possibly be correct.

For details, see

If anybody has any clever ideas about how to understand this
bit of physics -- or a more elegant way to explain it --
please let us know.