# 4.9.11 Harmonics

### 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 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 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:

#### 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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