Standing Waves, Resonance, and Harmonics

I. Vibrating String

  1. When you pluck a fixed string, you get reflections and the waves start to interfere. The result is destructive interference for all frequencies except those that are harmonics of the fundamental frequency of the string.
  2. The harmonics reflect in a way that they are never destroyed but instead create a complex interaction of standing waves. See animation.

      Harmonics: series of frequencies that are related by an integral value

  1. The fundamental frequency of this interaction is the note we hear or the pitch.
  1. The interaction of all the harmonics provide the timbre.

 

Solving for Harmonics of a String:

  1. Creates standing waves with fixed boundaries at each end. 
  2. The fixed ends must be nodes. 
  3. All string instruments work this way.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

II. Air Column Open on Both Ends

  1. When you blow across an open pipe, you get reflections and the waves start to interfere as they move down the pipe. The result is destructive interference for all frequencies except those that are harmonics of the fundamental frequency of the string.
  2. The harmonics reflect in a way that they are never destroyed but instead create a complex interaction of standing waves.  This is very similar to strings but with a twist. See animation.

 

Solving for Harmonics of an Air Column that is open on both ends:

  1. Creates standing waves with free boundaries at each end. 
  2. The free ends must be antinodes. 
  3. Wind instruments such as a flute or pipe organ work this way.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

III. Air Column Open on one end and closed on one end.

  1. When waves travel down a tube with a closed end, you get reflections and the waves start to interfere as they move down the pipe.  The result is destructive interference for all frequencies except those that are harmonics of the fundamental frequency of the tube.
  2. The harmonics reflect in a way that they are never destroyed but instead create a complex interaction of standing waves. This is similar to the first two but it has a definite difference we will see in the resulting formula.  See animation.

 

Solving for Harmonics of an Air Column that has one closed end and one open end:

  1. Creates standing waves with free boundary at one end and fixed at the other. 
  2. The free end must be an antinode; the fixed a node. 
  3. Wind instruments such as a trumpet, clarinet work this work this way.

 

           

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

End correction:  If you are trying to achieve a certain frequency you need to add an end correction to the length of pipe you calculate from the above equations.  The open end changes things slightly so you have to add a small amount of length to your calculated value.  To find the corrected length, you add 60% of the radius of the air column to the measured length of the pipe.    

 

IV.  Beats

      The perceived difference in volume of two waves that have frequencies very close to each other. 

 

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