Re: Energy Transmission on a string.

["John S. Denker" <jsd@MONMOUTH.COM>]

At 12:53 PM 12/4/01 -0600, RAUBER, JOEL wrote:
> When you write the total energy of the SHO as 1/2 *dm *omega^2 *A^2
> doesn't that include the potential energy part of the total mechanical
> energy as well as the kinetic energy piece?

Yes, it does.  My previous thinking (or semi-thinking :-) was off
by a factor of two.  It's hard to say for sure what happened;
possibly I mistook peak KE for average KE or some such.

        E equals KE + PE, but
        peak(E) is not peak(KE) + peak(PE).
        1 = cos^2 + sin^2

> > Also note that this method (A) has nothing to do with wave mechanics.
> > It could equally well describe the transport of energetic harmonic
> > oscillators in a truck.
>
> This is the part that worries me about derivation A.  Well said!.

OTOH I don't think there is anything really to worry about here.
Similar logic is applied all the time to energy, momentum, mass,
entropy, lepton number, etc. being carried (convected) by a fluid.
It's not always the whole story, but you can pretty easily decide
whether it is or not.

In particular, imagine an isolated wave packet propagating
in a non-dispersive medium.  We are !!not!! assuming sine waves
here.  It's pretty clear that the energy moves along with the
wave packet, just like the aforementioned harmonic oscillators
in a truck.  Method A is not bogus;  in some sense it is
actually more general than the argument based on the detailed
wave-mechanical analysis.


> Would it be fair to interpret what you are saying here as implying that it
> is probably mis-placed mathematical rigour on the part of introductory
> authors, who use method B, to arrive at the standard result
>
> 1/2 *mu * v *omega^2 * A^2

Absolutely that is not a rigorous result.  In the real world
there are all sorts of nonlinearities, and even in a linear
system there is usually dispersion that calls into question
the very notion of a definite speed of propagation.