Standing Waves, Medium Fixed At One End, Open At The Other End

Here is an animation showing the standing wave patterns that are produced on a medium such as the metal strip on a thumb piano. This type of medium would be said to be fixed at one end and open at the other end. The medium is held motionless at one fixed end, but allowed to move at the other open end.


Basic operation:

  • You can select which harmonics, 1st, 3rd, 5th, 7th, or 9th, 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 thumb piano is plucked, vibrations, or waves, travel back and forth through the medium and 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 is, they produce patterns which do not move.


On a mediums like those found on a thumb piano 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 with fixed and open ends like this 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 third harmonic, has one third the wavelength and triple the frequency of the first harmonic. This third harmonic is also called the first overtone. This can get confusing with the third member of the harmonic group being called the first member of the overtone group.


The fifth harmonic, or pattern, has one fifth the wavelength and five times the frequency when compared to the first harmonic. This fifth 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 one end, open at the other:

Harmonic Overtone group Frequency relationship
1st Fundamental Fundamental frequency
3rd 1st overtone 3 times the fundamental frequency
5th 2nd overtone 5 times the fundamental frequency
7th 3rd overtone 7 times the fundamental frequency
9th 4th overtone 9 times the fundamental frequency


Depending upon how the thumb piano is plucked, 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 maximum amplitudes much less than the fundamental, or first harmonic.


It is the fundamental frequency that determines the note that we hear. It is the higher harmonics that determine the timbre of the instrument.


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