Standing Waves, Medium Fixed At Both Ends

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Here is an animation showing the standing wave patterns that are produced on a medium such as a string on a musical instrument. This type of medium is said to be fixed at both ends. That means is is held motionless at both ends.

Harmonics:
Animation:

Basic operation:

  • You can select which harmonics, 1st, 2nd, 3rd, 4th, or 5th, which you want to see by checking the appropriate checkbox.
  • The sum of all the harmonics is always shown. It is white.
  • If you click the 'See' checkbox you will see the separate harmonics drawn behind the white sum.
  • The 'Slow' checkbox makes the animation run five times slower for careful study.

 

When a string on an instrument is plucked, vibrations, or waves, travel back and forth through the medium that are reflected at each fixed end. Certain sized waves can survive on the medium. These certain sized waves will not cancel each other out as they reflect back upon themselves. These certain sized waves are called the harmonics of the vibration. They are standing waves. That means they produce patterns which do not move.

 

On a medium such as a violin string several harmonically related standing wave patterns are possible. The first five of them are illustrated above. It is important to understand that for any one given medium fixed at each end only certain sized waves can stand. We say, therefore, that the medium is tuned.

 

The first pattern has the longest wavelength and is called the first harmonic. It is also called the fundamental.

 

The second pattern, or second harmonic, has half the wavelength and twice the frequency of the first harmonic. This second harmonic is also called the first overtone. This can get confusing with the second member of the harmonic group being called the first member of the overtone group.

 

The term '1st harmonic' and 'fundamental' are usually used interchangeably in physics discussions. Although the terms '2nd harmonic' and 'first overtone' are the same (on this medium, but not all), 'overtone' terms are not used that often.

 

The third harmonic, or pattern, has one third the wavelength and three times the frequency when compared to the first harmonic. This third harmonic is called the second overtone.

The other harmonics follow this pattern regarding wavelengths, frequencies, and overtone naming conventions described above. This is summarized below.

Closed at both ends:

Harmonic Overtone group Frequency relationship
1st Fundamental Fundamental frequency
2nd 1st overtone 2 times the fundamental frequency
3rd 2nd overtone 3 times the fundamental frequency
4th 3rd overtone 4 times the fundamental frequency
5th 4th overtone 5 times the fundamental frequency

 

Depending upon how the string is plucked or bowed, different harmonics can be emphasized. In the above animation all harmonics have the same maximum amplitude. This is for purposes of illustration. Actually, the higher harmonics almost always have amplitudes much less than the fundamental, or first harmonic.

 

It is the fundamental frequency that determines the note that we hear. The way the other harmonics are emphasized determines the timbre of the instrument.

 

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