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

[John Denker <jsd@av8n.com>]

On 06/22/2014 09:02 AM, rjensen@ualberta.ca wrote:

> Is "energy proportional to the wave amplitude" stated in FLP a
> reasonable simplification for the level of the student FLP is 
> endeavoring to educate?

That's a really interesting question.  Let me expand a
bit upon my previous answer.

   As before, I assume we are talking about energy as
   related to amplitude /squared/.

As I said previously, questions about the appropriate 
level of detail versus simplification are matters of 
taste.  De gustibus non disputandum.

On the other hand, we need to be CONSISTENT.  Let me
explain what I mean, by giving an example of inconsistency:

Sometimes we complain that students take things too 
literally, when they should be generalizing.  I quote 
from Bruce Sherwood's blog, expressing an idea that 
was also discussed on this list:

> > His problem was that he knew a canned procedure that if you have an
> > x, and there’s an exponent, you put the exponent in front and reduce
> > the exponent by one, and that thing is called “dy/dx” but has no
> > meaning. There is no way to evaluate dS/dE starting from aE^{0.5},
> > because there is no x, there is no y, and nowhere in calculus is
> > there a thing called dS/dE.

In other words, the student was too literal, analyzing
things at the lexical level, not the semantic level.
For context and additional details, see
  http://matterandinteractions.wordpress.com/

In the same vein, a lot of people (including me!) 
like to emphasize the unity, power, and grandeur
of physics.  We quote Feynman's dictum:
 *The same equations have the same solutions.*

ON THE OTHER HAND ... Suppose a student takes some
formula -- such as the one that says wave energy 
is proportional to the square of the amplitude --
and applies it outside its range of validity.  Then
we accuse them of "equation hunting".

=====

The twin problems of over-generalization and under-
generalization go away if the student /understands/
what's going on.

However, the square-law energy formula was presented
without proof, without derivation, without explanation,
without any expressed limits on its range of validity.
Just a bold assertion.

In such a case, we are in no position to criticize the 
student for not fully understanding the formula.  We 
are in no position to complain about over-generalization 
or under-generalization.

========

Here is a constructive suggestion:  I suggest a 
different law, one that is simple /and/ general 
/and/ powerful:
  In any given medium, the energy of a wave 
  scales like the square of the amplitude ... 
  /for waves of the same shape/.
  (Sometimes the shape doesn't even matter, but
  sometimes it does.)

This cannot be used as a segue into Feynman's discussion
of the Parseval identity, but maybe that's a good thing.
I don't see the Parseval identity as being all that
closely related to energy.  Referring to it as the
"energy theorem" seems like yet another deception.

================================
Bottom line:

The question of "more generality" versus "more simplicity"
is an interesting question, but sometimes other questions
are even more interesting.  I am super-especially interested
in:
  a) promoting understanding, and
  b) being fair to the students.

When is the energy proportional to the square of the
amplitude, and when is it not?  How do we know?  How
do we teach people to think about such things properly?