Re: Energy Transmission on a string.

[John Mallinckrodt <ajmallinckro@CSUPOMONA.EDU>]

On Tue, 4 Dec 2001, Bob Sciamanda wrote:

> You might be interested in looking at the eye-opener:
> Reuben Benumof, "Simple harmonic motion in harmonic waves", AJP
> 48, No 5, 387-392, may 1980.

Bob,

I have only begun reading this article, but my progress has been
severely impeded by what I see as a serious error on the very
first page on which most of his later results seem to depend.
Specifically, Benumof derives the potential energy in the string
by implicitly assuming that pieces of the string engage in *no*
longitudinal motion.  The result of this assumption is that a
portion of the string that normally has length dx is stretched by
the waveform to a length dx*secant(theta), where theta is the
instantaneous angle of the piece of string wrt the "at rest"
string.  Accordingly he derives a *local* increase in the elastic
potential energy that is directly proportional to
(tangent(theta))^2.

He provides no justification for this assumption.  Indeed, he does
not seem even to be aware that he is making it.

Moreover, the assumption goes very strongly against my intuition;
I would anticipate that the additional stretch that *is*
necessitated by the string's curvature would be accommodated on
*globally*.  For instance, I don't see any reason to assume, as
Benumof *does*, that portions of the string that are
instantaneously at the tops and bottoms of a large amplitude wave
do not share in the overall stretch of the string.  I'll go one
step further:  Since the overall length of the string is strictly
*constant* in the presence of an extended, unidirectional harmonic
wave, it would be my intuition that the stretch factor should be
essentially constant everywhere.

Can you or someone else comment?

John Mallinckrodt      mailto:ajm@csupomona.edu
Cal Poly Pomona          http://www.csupomona.edu/~ajm