Re: trivial pedagogy

[Ludwik Kowalski <KowalskiL@MAIL.MONTCLAIR.EDU>]

Mark Sylvester wrote:

> I approach this by marking two "corks" on the wave, close to each
> other, and asking which one is oscillating slightly ahead of the
> other. It's the one on the left, of course.

That is certainly correct, it tells us that a wave is a sequence of
linked harmonic oscillators. But I was teaching in terms of something
more simple. A moving shape (yes, a snap shot, if you wish) is like
a wire bent to a sine curve. At the beginning y=A*sin(2*Pi*x/L) tells
me how far (vertically) the wire is at any given x, from the axis.

The challenge is to give a formula for the next y(x) knowing that
the wire is being moved from left to right with a constant v. Yes,
the x axis is defined and they know that x is positive to the right.
Here each bit of a wire moves horizontally; there is not harmonic
motion.

It is not a wave, just an exercise of describing mathematically what
is being observed. The concept of wave (particles moving up and down
while the "shape" is moving horizontally) is next. My hope was that
by formulating the problem mathematically first we can make it easier
to accept the physics behind the wave equation. The idea that for a
moving wire y becomes a function of two variables may be introduced
before dealing with waves. If the wire is moved by a distance z=v*t
then the phase to subtract must by 2*PI*(v*t)/L. The mathematical
description becomes y=A*sin[2*Pi(x/T-v*t/L]. Periodicity in space
and periodicity in time are introduced without any reference to
harmonic motion. A simple description of what is actually seen by
watching water waves for a while.

I am not saying this is better; it is only different from what I
used to do. By the way, my favorite example for a "wave in general"
is a domino wave; it has nothing to do with the sine curve.

Nothing profound but worth sharing, I hope.
Ludwik Kowalski