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Re: Energy Transmission on a string.



"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

it seems a little spurious to necessarily equate dx/dt with the wave
velocity, as it implies you chose your element of length dx to be the
distance the wave travels in an element of time dt. I suppose one can
choose it that way, put it is putting a constraint on how you choose your
infinitesimal elements.

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.

METHOD B:

Note that instantaneous power exerted by an element on the adjacent element
(in direction of wave propagation) is F dot v

and

F dot v = F_y *v (where y is the transverse coordinate.)

P = v *F *sin (theta )

where theta is the angle between the slope of the sinusuodial curve and the
horizontal (wave propagation direction).

The texts then invoke the small angle approximation by approximating
sin(theta) with tan(theta).

This strikes me as demonstrably incorrect (in general) as there are places
on the sinusuodial curve where theta is not a small angle, indeed it is
sometimes as large as 45 degrees (Pi /4 radians)

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.