Question

Discuss the formation of harmonics in a stretched string. Show that in case of a stretched string the first four harmonics are in the ratio 1 : 2 : 3 : 4.

Answer 1

For first harmonic in stretched string ,

we can see , \(\frac{\lambda}{2}\) = L, here L is length of string .

so, λ = 2L

Now, frequency of first harmonic , f₁ = \(\frac{v}{2L}\) , here v is speed of wave in string

Similarly , for 2^nd harmonic ,

λ = L ∴ frequency of 2nd harmonic , f₂ = \(\frac{v}{L} = \frac{2v}{2L}\)

For 3^rd harmonic ,

\(\frac{3\lambda}{2}\) = L⇒λ = \(\frac{2L}{3}\) ∴ frequency of 3rd harmonic , f₃ = \(\frac{3v}{2L}\)  

for 4th harmonic ,

2λ = L ⇒λ = \(\frac{L}{2}\) , frequency of 4th harmonic , f₄ = \(\frac{2v}{L}\) = \(\frac{4v}{2L}\)

Now, f₁ : f₂ : f₃ : f₄ = \(\frac{v}{2L}:\frac{2v}{2L}:\frac{3v}{2L}:\frac{4v}{2L}\) = 1 : 2 : 3 : 4

Hence proved