how to prove that the frequencies produced in the string are in the ratio 1:2:3

When a stretched string vibrates in fundamenetal mode or first harmonic , wavelength λ₁ = 2 l  as shown in figure ,

where l is length of string. If v is the speed of wave then frequency n₁ of fundamental mode , n₁ = v / λ₁ = 1 × ( v / 2l ) .

 

When a stretched string vibrates in second harmonic , wavelength λ₂ =  l  as shown in figure ,

Then frequency n₂ of second harmonic , n₂ = v / λ₂ = 2 × ( v / 2l ) .

 

Similarly , frequency n₃ of third harmonic , n₃ = v / λ₃ = 3 × ( v / 2l ) .

 

Frequency n₄ of fourth harmonic , n₄ = v / λ₄ = 4 × ( v / 2l ) .

 

Hence frequencies of vibration in stretched string = n₁ : n₂ : n₃ : n₄  = 1 × ( v / 2l ) : 2 × ( v / 2l ) : 3 × ( v / 2l ) : 4 × ( v / 2l ) .

 

n₁ : n₂ : n₃ : n₄  = 1  : 2  : 3  : 4  .