Re: Energy Transmission on a string.

["RAUBER, JOEL" <JOEL_RAUBER@SDSTATE.EDU>]

John,

Thanks for the thoughtful response,
> "RAUBER, JOEL" wrote:
> > METHOD A:
> >
> > Note that a small mass element of mass dm of the string is
> executing SHM, at
> > any instant of time its energy is
> >
> > 1) dE = 1/2 dm* omega^2 *A^2 in the usual notation for a
> SHM of a mass
> > element.
> >
> > 2) one notes that dm = mu* dx   (mu = mass per unit length)
> >
> > 3) divide by an increment of time dt and write
> >
> > P = dE/dt = 1/2 mu* dx/dt *omega^2 *A^2 = 1/2 mu *v *omega^2 *A^2
<snip>
> That's not quite as bogus as it seems.  The key is this:  You aren't
> choosing just any old time interval dt.  You are choosing one just
> long enough for the wave to travel the distance dx.  You could choose
> a bigger piece of the wave (say Dx not dx) to look at.  It
> would contain
> more energy, but it would take longer to travel somewhere else, so
> the power calculation will come out the same, as it should.
>
> It's still somewhat bogus;  whatever happened to the potential energy
> embodied in the wave?  That has to get transported, too.

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?

E_mech = KE +PE = 1/2 *dm *v^2 + 1/2 *dm *omega^2 *x^2

,where x is displacement from equilibrium.

= 1/2 *dm *omega^2 *A^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!.


> Yeah, but then the energy density isn't what you think it is.
> Remember that the small-angle approximation was invoked to
> derive the wave equation itself.

This also bothers me regarding the standard introductory derivations of the
wave equation!

Of course, one can always back substitute the harmonic wave form and show
that it is a solution to the wave equation, despite the fact that the
derivation breaks down for such a wave form, in so far as it relies on the
small angle approximation.  I suppose this means that there must exist an
alternative derivation of the wave equation that doesn't rely on the small
angle approximation?

> If you want to re-derive
> the equations of motion for the string in their full nonlinear
> glory, you can do so.  The power will still be related to the
> energy density and the wavespeed, if (big if) conditions allow
> the notion of wavespeed to make sense.

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