Re: Waves on a slightly dispersive cable

[John Denker <jsd@MONMOUTH.COM>]

At 11:28 PM 7/3/00 +0200, Mark Sylvester wrote:
> A few years ago I was hiking in the Alps. At a refuge where we spent the
> night there was a very long steel cable for hauling goods up from the
> valley. It was a single span, goodness knows how long. I couldn't resist
> whacking it with a piece of firewood from a stack nearby. The pulse came
> back from the valley after what seemed like a very long time. The
> interesting part is that for a few seconds before the main pulse arrived
> the cable sang with high frequency noise.

Cool.  Very cool.

At 09:51 AM 7/4/00 -0700, Leigh Palmer wrote:
> the tension being much
> higher at the upper end than at the lower. 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.

The longitudinal hypothesis seems not very likely.  There is no theoretical
or experimental reason to favor it, and several reasons to disfavor it,
relative to the more-obvious hypothesis that Mark was observing dispersion
of _transverse_ waves due to the stiffness in the cable.

  1a) Transverse waves (with small stiffness and large tension) have a
wave speed that depends on density and tension.  Meanwhile, longitudinal
waves depend on density and compressibility.  For these speeds to be at all
comparable requires a tension on the order of the compressibility, which
would correspond to something like 100% elongation, which is implausible
for a  cable of this kind.

  1b) Similarly, transverse waves driven by flexion are slower than
compressional waves by a factor that depends on wavelength over
diameter.  Again, Mark's observation that the high-frequency components
arrived at a time even remotely associated with the main pulse is powerful
evidence against the longitudinal hypothesis.

 2) The method of excitation (whacking) couples much more strongly
strongly to longitudinal waves than compressional waves.

 3) The method of detection (listening) couples much more strongly to
longitudinal than compressional waves.

 4a) Standard theory predicts very little dispersion in longitudinal waves
in such circumstances.  (If somebody has a non-standard theory, please
spell it out for us.)

 4b) In contrast, there is every reason to expect that a slight amount of
stiffness will add a slight amount of dispersion to transverse waves, as
discussed a few months ago on this list.  Fourth order Green function and
all that.

At 10:29 AM 7/4/00 -0700, Leigh Palmer wrote:
> If you are fortunate enough to live
> where you can skate outdoors on natural ice, observe the sound which
> comes through the ice from the hard slap of a hockey stick on the
> ice a hundred meters or more away from you. The sound starts with
> higher partials and descends with a "BEEEOOOUUUP" sort of chirp. The
> farther one is from the source, the longer the duration of the chirp.

Previous arguments #2, #3, and #4 apply here.  I betcha it's transverse.

At 02:22 PM 7/5/00 -0700, Leigh Palmer wrote:
> I'm not basing my statement on theory, I'm stating an empirical
> observation. Remember my ice example?  This result may be simply
> expressed in a theory of sound transmission in solids, but I am
> unaware of it. I have never subscribed to the dictum "Never trust
> an experimental result until it has been confirmed by theory".

The observations are undisputed.  But the interpretation of the waves as
longitudinal is a separate statement.  The latter is not an observation ...
it is not data ... it is just an interpretation of the data.