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### Standing waves in strings

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Physics - Standing waves in strings

Standing waves in strings

The quickest way to gain an understanding of standing waves is to look at
a string vibrating in its fundamental mode as shown below.

In this case it is far easier to think about the situation in terms of
displacement rather than pressure variation. When the string is vibrating
in its most simple mode an amplitude antinode is located at the centre of
the string. This means that this section of the string goes through a
maximum change in its displacement over a period of one cycle. At either
end of the string there are fixed points. Hence each of these points is at
a displacement node. That is they experience no change in their
displacement value.

The diagram below shows a number of modes in which a string can vibrate.
Each diagram shows the extremes of the string's position. It is important
to remember that this string behaviour is the result of the superposition
of the many waves that are travelling up and down the string and reflecting
from each end.

The vibration of the string involves a complex set of waves, not just the
fundamental mode of vibration discussed so far. Each string is actually
able to resonate at a set of frequencies, called harmonics [1], all related
to the fundamental frequency. So far we have examined the fundamental
frequency, also called the first harmonic. The standing wave is established
such that the length of the string is equal to half a wavelength. The
relationships between wavelength and string length formed in the higher
harmonics are shown below.

Plucking a string is plucked and allowing it to vibrate naturally produces
a number of harmonics at the same time. In most circumstances it is the
fundamental mode of vibration that has the maximum amplitude. The higher
harmonics exist in much smaller proportions. It is clear from the diagram
above that the harmonics have a specific relationship to one another. The
frequency of each harmonic is a whole number multiple of the fundamental
frequency, f0, as summarised by the equation:

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