OCR AS Physics

Revision Notes

4.9.11 Harmonics


  • Stationary waves have different wave patterns depending on the frequency of the vibration and the situation in which they are created

Two Fixed Ends

  • When a stationary wave, such as a vibrating string, is fixed at both ends, the simplest wave pattern is a single loop made up of two nodes and an antinode
    • This is called the fundamental mode of vibration or the first harmonic
  • The particular frequencies (i.e. resonant frequencies) of standing waves possible in the string depend on its length L and its speed v
  • As the frequency is increased, the higher harmonics begin to appear
  • The frequencies can be calculated from the string length and wave equation

Fixed end wavelengths and harmonics, downloadable AS & A Level Physics revision notes

Diagram showing the first three modes of vibration of a stretched string with corresponding frequencies

  • The nth harmonic has n antinodes and n + 1 nodes

One or Two Open Ends in an Air Column

  • When a stationary wave is formed in an air column with one or two open ends, slightly different wave patterns are observed in each


Closed and open ends, downloadable AS & A Level Physics revision notes

Diagram showing modes of vibration in pipes with one end closed and the other open or both ends open

  • In Image 1: only one end of the air column is open, so, the fundamental mode is now made up of a quarter of a wavelength with one node and one antinode
    • Every harmonic after that adds on an extra node or antinode
  • In Image 2: the column is open on both ends, so, the fundamental mode is made up of one node and two antinodes
  • In summary, a column length L for a wave with wavelength λ and resonant frequency f for stationary waves to appear is as follows:

Table of length and corresponding resonant frequency, downloadable AS & A Level Physics revision notes

Worked Example

A standing wave is set up in a loudspeaker emits sound with frequency f and is placed at one end of a pipe with length L. The pipe is closed at the other end.

The speed of sound is 340 m s-1.

With a sound wave of wavelength of 10 m, what is the frequency of the second lowest note produced?

Exam Tip

The fundamental counts as the first harmonic or n = 1 and is the lowest frequency with half or quarter of a wavelength.

A full wavelength with both ends open or both ends closed is the second harmonic. Make sure to match the correct wavelength with the harmonic asked for in the question!


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