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<snip>
"RAUBER, JOEL" wrote:
executing SHM, at
METHOD A:
Note that a small mass element of mass dm of the string is
any instant of time its energy isSHM of a mass
1) dE = 1/2 dm* omega^2 *A^2 in the usual notation for a
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
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.
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.
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.
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.