Re: Waves

[Mark Sylvester <msylvest@SPIN.IT>]

At 09:51 AM 7/4/00 -0700, Leigh Palmer wrote:
> Bill Beaty's note may have led some to believe that standing waves are
> not important in the understanding of the motion of a plucked string
> which is stopped at both ends. Nothing could be further from the
> truth. The motion of such a string in the ideal case can be decomposed
> entirely into harmonic standing waves with nodes at the ends of the
> string. The travelling pulses described by Bill and Mark can be
> reproduced by the superposition of many such standing waves. If you
> wish to do so you may choose to treat these standing waves as a mere
> mathematical fiction, but the basilar membrane in your cochlea will
> readily decompose the auditory results and produce their fourier
> decomposition on your auditory cortex. That is real enough for me!
>
> Mark, the cable you played with is not an ideal taut string for a
> couple of reasons. An ideal taut string is perfectly limp; it does not
> provide a restorative stiffness when it is bent, as wire rope does.
> The ideal taut string moves under the influence of restorative forces
> due to a hypothetically constant tension in the string. Your wire rope
> had a tension which varied due to gravity, the tension being much
> higher at the upper end than at the lower.


Indeed. I used to take people caving. In one particular cave we'd descend a
70 metre rope hanging done in a huge bell-shaped chamber, open to the sky.
If there were physics students in the group we'd be sure to send a short
wave-train up the rope by shaking the bottom end a few times. Very
instructive to watch it going up and reflecting back down, with tension
changing continuously.

>  The high pitched precursor
> to the main pulse is likely due to the transmission of longitudinal
> modes in the cable at speeds higher than those of the transverse mode.

Yes, but why is "modes" in the plural for longitudinal? Would one expect
the cable to transmit longitudinal waves with a range of speeds? Why?

> Your action in whacking the cable with a piece of firewood is entirely
> understandable and appropriate; I would have done the same thing, and
> I encourage my students to seek out acoustic phanomena everywhere in
> just that manner. I once almost got into trouble doing this in the
> Statue of Liberty while standing in line. Did you know that the Statue
> is a fairly high-Q structure? It hangs by some highly elastic cables
> from a base structure, and it can be set into vertical oscillation by
> periodically pushing sideways on any one of them with the proper
> frequency and phase. It was very easy to get an amplitude of a couple
> of millimeters, but my wife made me stop when other people on the
> stairs began to notice the motion. Another of my favorite resonance
> demos involves tall light standards or tall, thin, straight dead or
> defoliated trees.

Flagpoles are amusing too, and can get people quite annoyed, for some reason.

Mark