Circular Guitar String and Resonance

name         Duane
status       other
grade        other
location     IL

Question -   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.
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Dear Duane,

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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