Re: A wave or not a wave ?

[John Denker <jsd@MONMOUTH.COM>]

> I wrote:
>
> > ... d) Standing waves are a red herring.  They can be considered
> >  a fortuitous superposition of leftward plus rightward propagating
> > waves.  In the space of all possible rightward and leftward wave
> > patterns, standing waves are a subset of measure zero.
> > Pedagogically speaking, I would not emphasize them and would
> > not cite them as an example of typical waves.

whereupon at 03:03 PM 7/2/00 -0400, Ludwik Kowalski wrote:

> Are you aware that in most elementary physics labs students are
> introduced to waves via a vibrating string. On one side a vibrator,
> on the other (about 2 m away) a pulley and the load to control
> tension. Students try to create standing waves with different
> number of loops by changing the load or frequency.
>
> 1) Do you think that this may be a source of misconceptions
>     about waves?

Yes.  Misconceptions and frustrations.  Almost the only waves they can
ponder outside the lab are surface waves in the ocean/lake/pond, which are
not standing waves.

If I were doing it, I would
 a) Lose the vibrator.
 b) Replace it with a motorized wheel with a quill to pluck the string.
 c) Minimize reflections by making the string much longer than 2m and then
if necessary attaching dampers (e.g. pom-poms) near the far end.

> 2) If so then why is this experiment used nearly everywhere?

That's a fair question, and deserves a better answer than I can
give.  Non-answers include:

a) I'm not responsible for what other people do.  I'm only responsible for
what I do.  I wouldn't do it that way.

b) It would be bad luck (and bad manners) for me to speculate on _why_
other people do what they do.  Better to let them say for themselves.

> Is it really necessary to define waves in "the most general
> mathematical way" when they are introduced for the first time.

Not necessarily necessary.  One must weigh the advantages against the costs
and risks of a less-than-general definition.

> What is wrong with saying "many kinds of waves exist but
> TO BEGIN WITH we will limit our attention to harmonic
> waves"? That is good enough for ordinary acoustics, for
> explaining diffraction phenomena, for Fourier synthesis
> or analysis, etc. In other words, what is wrong with the
> traditional approach used in textbooks?

That's changing the subject a little bit.
 --) Regarding standing waves versus running waves, I still think standing
waves are a red herring.
 --) I don't see what Fourier has to do with it;  you can Fourier analyze
a running wave just as easily as a standing wave.
 --) Regarding dispersion, doing the general case is hardly any more work
than doing the nondispersive case.
 --) Regarding nonlinearity, I agree it should not be emphasized in an
introductory course.

In any case, one needs to be up-front about the fact that dispersion and
nonlinearity exist.  One should not define waves in such a way as to
exclude them.  In particular one should not call
  (d/dt)^2 q + (d/dx)^2 q = 0
"the" wave equation as if no other wave equations existed.

===================

The general question as to what can be simplified and what cannot is a
monumental question.  It's one of the reasons why teaching is hard.

To illustrate how general the problem is, here are a couple of extreme
examples:

1) Consider the H--O--H bond angle in water.  It's about 104.5
degrees.  Such questions were not covered in my high-school chemistry
course, but I scrounged a book from somewhere and read that this angle is
larger than the "obvious" 90 degree angle because the hydrogens push each
other out of the way a little bit.  Then I get to college and they tell me
that the "obvious" bond angle is the tetrahedral angle (109.47 degrees) and
that the lone-pair electrons are bigger than the hydrogens, and push the
hydrogens together.
  http://www.shef.ac.uk/~chem/vsepr/chime/H2O.html
You can't have it both ways.  Either hydrogens are big or they aren't.  If
they're not big, don't tell me fairy stories about it.  Deal me straight or
deal me out.  This is an example of completely boneheaded bad pedagogy,
because the right answer is not at all more complicated than the fairy story.

2) Pilot training (including textbooks as well as oral tradition) is full
of fairy stories about how airplanes respond in various
circumstances.  Some of these fairy stories put lives at risk, by leading
people to use the controls inappropriately in emergencies.

====

As I said before, and as Bob S. said, nit-picking a precise definition of
"wave" isn't worth the trouble.  But students demand definitions, and we
have to give them something.  And the discussion here isn't really about
definitions;  the larger issue is what topics to cover.

In any halfway complicated subject, you can't teach the whole subject all
at once.

Good pedagogy involves
 -- Finding the win/win solution where possible (simplicity AND generality).
 -- Otherwise making astute tradeoffs between simplicity and generality,
saying things that are true to a good approximation, saying things that are
true in the appropriate limits.
 -- Being honest about the limits of validity of whatever stories are told.