Question

Show that all harmonics are present on a stretched string between two rigid supports.

Show that all harmonics are present in case of a stretched string.

Answer 1

Consider a string stretched between two rigid supports and plucked. Due to plucking, the string vibrates and loops are formed in the string. Vibrations of the string are as shown in the figure.

Let,

p = number of loops

l = length of string

∴ Length of one loop = `l /p`   ......(1)                                 

Two successive nodes form a loop. Distance between two successive nodes is `λ/2`.

∴ Length of one loop = `λ/2` .......(2) 

From equations (1) and (2),

`λ/2 = l/p`

∴ λ = `(2l)/p`  ....(3)

Velocity of transverse wave is given by,

`v=sqrt(T/m)`

Frequency of string is given by,

n = `v/λ`

Substituting λ from equation (3),

`n=sqrt(T/m)/((2l)/p)`

`thereforen=p/(2l)sqrt(T/m)`  ......(4)

Fundamental mode or first harmonic:

In this case, p = 1

∴ From equation (4),

`n=1/(2l)sqrt(T/m)`

This frequency is called the fundamental frequency.

First overtone or second harmonic:

In this case, p = 2

∴ From equation (4),

`n_1=2/(2l)sqrt(T/m)`

`n_1=2xx1/(2l)sqrt(T/m)`

∴ n₁ = 2n

Second overtone or third harmonic:

In this case, p = 3

Using equation (4),

`n_2=3/(2l)sqrt(T/m)`

`n_2=3xx1/(2l)sqrt(T/m)`

∴ n₂ = 3n

(p - 1)^th overtone or p^th harmonic:

`n_((p-1))=pxx1/(2l)sqrt(T/m)=pn`

For the overtone,

`n_p=(p+1)/(2l)sqrt(T/m)=(p+1)n`

Thus, in the vibration, a stretched string's frequencies of vibrations are n, 2n, 3n, ........ so on.

Hence, all harmonics (even as well as odd) are present in the vibrations of a stretched string.