# Modes of Vibration in a Stretched String

We know that a stretched string can produce a wave of different frequencies. Some of the mode of vibrations are given below.

First mode of vibration:

When the string is plucked in the middle, an antinode is formed at the middle between two nodes in the string as shown in first figure above. At the simplest mode of vibration, the distance between two consecutive nodes is  λ/2 where λ is the wavelength of the transverse wave in the string. Now, the frequency of the wave is given by:

$$f = \frac{v}{\lambda} = \frac{v}{2l} = \frac{1}{2l} \frac{T}{ \mu }$$

Here V is the velocity of the transverse wave. This is the fundamental frequency or first harmonics. It is the lowest frequency produced by the vibrating string.

Second mode of vibration:

When the string is plucked at one fourth of the length of the string, the string vibrates with loops having two nodes and two antinodes.  The figure for second mode of vibration is shown in the second figure above. The frequency of the second mode is:

$$f_1 = \frac{v}{\lambda _1} = \frac{v}{l} = \frac{1}{l} \frac{T}{ \mu } = 2f_o$$

This is the frequency of the second mode of vibration or first overtone. This is also called the second harmonics.

Third mode of vibration:

When the string is plucked at the one sixth of the length of the string, the string begins to vibrate with three loops having three nodes and three antinodes.  The figure for the third mode of vibration is shown in the third figure above. The frequency of the third mode is:

$$f_2 = \frac{v}{\lambda _2} = \frac{3v}{2l} = 3f_o$$

This is the frequency of the third mode of vibration of second overtone.  This is also called third harmonics.

Nth mode of the vibration:

In general, the frequency of nth harmonic or (n-1)th overtone is given by:

fn-1 = nfo

where n is any integer (1,2,3…..etc)

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