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Re: Group velocites greater than the speed of light

On Wed, 10 Jan 2001, Eric Asselin wrote:

Thank you very much for the explanation. I have a better feeling about
this now.

Aha, I have a realistic analogy!

Imagine a playground with a "swing set" where a large number of swings are
hanging from a long horizontal beam. If children swing on all of them,
all with approximately the same frequency, the overall signal looks like
chaos. But if each child swings with a phase which is advanced slightly
from the previous neigbor, then our "illusory wave" arises. However, we
can make the wave REAL. If we use rubber ropes to couple each swing to
both its neigbors, then if ONLY the child on the far end of the row of
swings should start oscillating, a genuine "wave" will propagate from
swing to swing. Mechanical energy propagates along the swings.

Now the resulting wave will have various behaviors depending upon the
coupling between the swings. If the coupling is very high, then when only
one child is driven into oscillation, a familiar "wave" propagates rapidly
along the row of children, and the first child stops swinging immediately
as the wave passes. But if the coupling is very low, then each
progressive child swings back and forth many times before the next neigbor
builds up a significant amount of swinging. With low coupling, the
pattern of wave/oscillation which is APPARENT in the row of children can
have a very different velocity than the propagation of information. The
propagation of information takes the form of the leading edge of the
"region of swinging children", and in that case we look only at the
changes in peak amplitude of each swing, while ignoring the patterns
formed by the phase relationships between neigboring oscillating children
in the chain.

So you're right, the moving sine waves are a "mathematical trick", while
the motions of the stored energy (i.e. changes in the time-averaged peak
amplitude of each oscillator) is the "real" propagating signal that moves
along the chain.

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William J. Beaty SCIENCE HOBBYIST website
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