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[Physltest] [Phys-L] Re: pulse on a vertical rope



Carl Mungan wrote:

On the one hand, as the pulse goes up the string, it speeds up. This
will cause its amplitude y to decrease even without damping. It's
like the tsunami, which in fast deep water is only 1 foot high, but
piles up into a tall wave when it slows down near the coast.

Right.

The
front of the pulse rushes forward, the rear of the pulse lags.

That statement suffers from an ambiguous antecedent. Does "the pulse"
refer to the tsunami, or the rope?

In the case of the tsunami approaching shore, it is the front that
slows down while the back rushes forward, which is why the wave gets
taller and sharper as it moves into shallow water.

But
later when the pulse goes back down the rope, it will mostly
reconstitute itself, thus explaining why experimentally (from the
previous reference I gave), up to 10 returns of the pulse were
observed.

I've never done careful experiments on this system, but my intuition
and analysis are screaming at me, saying that there is a wide class
of pulses that will not reconstitute themselves -- especially for
narrow square pulses on a long rope.

But, even absent air drag, nonidealities of the string, and a loose
attachment to the top point, I still think the pulse will suffer some
dissipation. As it speeds up, it's like a light pulse passing from
one refractive index zone to another. Hence there will be some
reflection losses everywhere along the length of the whole string.
Soon the whole string will be wiggling and you'll no longer have a
coherent wave pulse, right?

I wouldn't call it dissipation. To my ears, the word dissipation
refers to some sort of thermalization process. But in the absence
of air drag, internal friction in the rope, and similar nonidealities,
there is no thermalization in this system. This is what I call a
nano-canonical system, by which I mean that it is even more restricted
than a micro-canonical system. Not only is total energy conserved,
but the energy of each mode is conserved on a mode-by-mode basis.
So if we start out with a non-equilibrium (non-Boltzmann) distribution
of energy in the nodes, we will retain it forever ... no thermalization.

Dispersion may disaggregate the initial pulse into unrecognizability,
but this is not what is normally called dissipation. It does not
require adding an imaginary part to the wavenumber. It is a natural
consequence of the equation of motion.

Real dissipation is part-and-parcel of the fluctuation/dissipation
theorem, and there is no hint of thermal fluctuations in the
equation of motion we are considering. Real dissipation involves
coupling to the heat bath, and we don't have a heat bath in the
scenario we are considering.

Does the imaginary part of k include both of these effects (change in
y as wave spreads out without loss, and loss in energy of the pulse
due to continuous reflection impedance mismatch)?

Reflection from an impedance mismatch is not dissipative. On the Smith
chart, it shows up as a purely reactive impedance.

More generally, the pulse does not lose energy. Hint: resolve the pulse
into a superposition of modes, i.e. the generalized Fourier representation,
as we have been discussing. Observe that the amplitude of each mode
is constant ... they're stationary, after all. Calculate the energy of
the original pulse by squaring the Fourier series and applying Parseval's
theorem.

Another argument leading to the same conclusion: The equation of motion
was derived by application of Newton's laws. Newton's laws are known
to conserve energy. If this equation of motion describes dissipation,
there's something seriously wrong.

==============================

Remarks addressing previous parts of the discussion:

1) The whole discussion of dispersion relations and group velocity is
a bit fishy. The usual derivation that gives dw/dk as the group
velocity is predicated on the equation of motion being translationally
invariant ... which is not the case with the equations we have been
considering.

Certainly there is no such thing as "the" group velocity for the whole
rope. The best we can do is to consider a short segment of the rope,
not too near the free end, and approximate the tension as being
locally approximately constant there. That leads to a local notion
of group velocity.

2) That has been called the "naive" approximation, but a better name
would be the WKB approximation. None of those guys (Wenzel, Kramers,
Brillouin) was particularly known for his naiveté.

Some people first encounter WKB in connection with evanescent waves,
but it works just fine with propagating waves such as we have here.
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