139 Standing Wave Harmonics

This video shows the standing waves that can be formed on a string fixed at both ends.

Since the string is fixed at both ends, any standing wave that forms on the string must have nodes at both ends.

The simplest standing wave that has nodes at both ends is the NAN. (Node-Antinode-Node) The next standing wave that can be formed is the NANAN, followed by

  1. NAN,
  2. NANAN,
  3. NANANAN,
  4. NANANANAN,
  5. NANANANANAN, and so on.

Notice that each NAN corresponds to one half-wavelength segment. This means

  1. NAN packs 1 half-wavelength along the length of the string,
  2. NANAN packs 2 half-wavelengths along the length of the string,
  3. NANANAN packs 3 half-wavelengths along the length of the string,
  4. NANANANAN packs 4 half-wavelengths along the length of the string,
  5. NANANANANAN packs 5 half-wavelengths along the length of the string, and so on.

Which means that

  1. NAN’s wavelength is called the fundamental wavelength,
  2. NANAN’s wavelength is 2x as short as that of NAN’s,
  3. NANANAN’s wavelength is 3x as short as that of NAN’s,
  4. NANANANAN’s wavelength is 4x as short as that of NAN’s,
  5. NANANANANAN’s wavelength is 5x as short as that of NAN’s, and so on.

Which means that

  1. NAN’s frequency is called the fundamental frequency, or 1st
  2. NANAN’s frequency is 2x that of NAN’s, hence called the 2nd harmonic,
  3. NANANAN’s frequency is 3x that of NAN’s, hence called the 3rd harmonic,
  4. NANANANAN’s frequency is 4x that of NAN’s, hence called the 4th harmonic,
  5. NANANANANAN’s frequency is 5x that of NAN’s, hence called the 5th harmonic, and so on.

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