Circular Guitar String and Resonance
Assume one had a circular guitar string
(continuous) tightly strung around three fixed pulleys, one of
which is driven by a motor. If the string was plucked between
the non-motorized two fixed pulleys if would vibrate with a given
wave form and frequency (motor off). What would the wave form
and frequency (still confined between these two fixed pulleys)
look like if the motor was then started causing the string to
circulate at a fixed velocity.
The frequency of the vibration of the guitar string would be the
same since the density of the string and the tension would be
unchanged, but the wave form would be quite different. The exact
form would be complex and difficult to calculate.
For example if the string speed were the same as the speed of wave
propagation along the string, the wave moving opposite to the string
motion would actually be stationary and the wave moving in the
direction of the string would appear to move twice as fast. In this
case, there would be no standing wave so the sound would be much muted.
If the string were moving at 2/3 the speed of the wave along the
string, the wave moving in the direction of the string would have
twice the wavelength of the wave moving in the other direction (work
it out!). The result might be some strange sounds!
The standing wave depends on the constructive interference of the
waves moving in opposite directions after being reflected from the
ends of the string. If a wave does not move along the string, it
cannot interfere with the other wave.
It might be fun to build such an apparatus and try to make music
with it. If you should be so ambitious, I would greatly appreciate
hearing about what you have done and learned!
Best, Dick Plano, Professor of Physics emeritus, Rutgers University
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Update: June 2012