Plucked strings

[William Beaty <billb@ESKIMO.COM>]

On Mon, 3 Jul 2000, David Abineri wrote:

> Am I correct in explaining string instruments, to say that a string
> actually vibrates in many modes simultaneously when bowed or plucked?
> That is, is it producing its fundamental along with several harmonics
> all at the same time?

Yes, but the phase of the various frequencies is important, with the
result that the shape of the vibrating string is very interesting, and
seeing the shape explains a lot.

I remember seeing frames from high-speed movie of a plucked string a
couple of years ago. The upper harmonics arise because the string is not
vibrating sinusoidally (as with standing waves.)  Instead, a pair of
pulses are racing back and forth along the string.  The pulses arise
because the string is initially deflected by a force applied at one point,
and when the force suddenly vanishes, it takes time for the changing
forces to propagate to all parts of the string.  Propagating waves are
created, and before the wave arrives at a particular point on the string,
the string is stretched taut, but after the pulse has passed, the string
is loose, but moving rapidly.  The waves race to and fro and invert their
sign when they reflect from the ends of the string. If these brief,
repetitive pulses should drive a sounding board (as in a guitar), then the
spectrum of the outgoing sound would be very different that if the same
sounding board was driven by a pure sine wave.

The movie of the plucked string looked something like this below.  Note
the odd "flat top" of the string's shape which grows and shrinks.  The
article also had some photos which showed the same string plucked from an
off-center point.  I don't remember where I saw the article.  Perhaps in
Science News?  Or an old issue of SciAm?


Initial position
                __--__
            __--      --__
        __--              --__
    __--                      --__
__--                              --__




2nd frame
            __----------__
        __--              --__
    __--                      --__
__--                              --__




3rd
        __------------------__
    __--                      --__
__--                              --__



4th

    __--------------------------__
__--                              --__




5th

__---------------------------------__




6th
__                                 __
  ---------------------------------



7th
__                                  __
  --__                          __--
      --------------------------



8th
__                                  __
  --__                          __--
      --__                  __--
          ------------------


9th
__                                  __
  --__                          __--
      --__                  __--
          --__          __--
              --__  __--
                  --


That's one half of a cycle.  The endpoints of the string will probably
experience a large transient force between times #5 and #6 above, as the
travelling "kink" is reflected.  Note that in #9 the string has recovered
its initial shape, but as a mirror image!

Perhaps you could use a microphone and PC to capture the waveform of the
initial sound from a plucked string, and verify that it resembles a series
of repeated pulses rather than a sine wave.

Another place I've heard a similar effect: with tuning forks.  Hit a
tuning fork with a leather-covered rod, and you hear a nice sinewave tone.
But hit it with a metal rod, and it goes "BLAAANNNGGG".  I imagine that
the difference is analogous to the plucked string: pulses are racing
around on the metal forks, and this generates lots of higher frequencies.
The hammers inside pianos are made of felt, which would tend to eliminate
these high-frequency pulses (the felt would "push upon" the string for a
long-ish time, rather than striking it briefly.)

Heh.  Ever hear of Franklin's Glass Armonica?  It was an "organ" made from
slowly rotating glass bowls, played with wetted fingers.  It's one of the
few instruments which lacks these "plucking harmonics", and people of
Franklin's time did not know what to make of it's sound.  The first time
Franklin's wife heard it, she feared she had died.  The device aquired the
reputation of driving people insane, and both Mozart and Anton Mesmer were
great fans.  I wonder if the missing "plucking harmonics" made all the
difference?  http://www.glassharmonica.com/armonica/franklin/

> Is it possible to show this behavior (if the notion is correct) with a
> string vibrator or other equipment, rather than just postulating to
> students that this does in fact occur?

A multi-frame video capture system might allow you to SEE the sequence of
positions of the string as it is plucked.  Or, with proper trigger-delay,
a PC camera might be used to grab a single frame at any desired time (for
example, grab a frame when the string has made one excursion, and the odd
straight-lines shape has reappeared as in #9 above.)

It would also be interesting to see what happens over a longer time, since
a plucked string eventually settles down into a sinusoidal motion.  What
happens to the pulses?  Maybe the string acts as a nonlinear medium where
different frequencies travel at different velocities.  If so, maybe the
pulses become "chirped" into a bunch of short, out-of-phase sine waves
which make sounds resembling white-noise.

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William J. Beaty                                  SCIENCE HOBBYIST website
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