Question

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

Answer 1

For first harmonic in stretched string ,
we can see , λ/2 = L, here L is length of string .
so, λ = 2L
Now, frequency of first harmonic , f₁ = v/2L , here v is speed of wave in string

Similarly , for 2nd harmonic ,
λ = L ∴ frequency of 2nd harmonic , f₂ = v/L = 2v/2L
For 3rd harmonic ,
3λ/2 = L⇒λ = 2L/3 ∴ frequency of 3rd harmonic , f₃ = 3v/2L
for 4th harmonic ,
2λ = L ⇒λ = L/2 , frequency of 4th harmonic , f₄ = 2v/L = 4v/2L
Now, f₁ : f₂ : f₃ : f₄ = v/2L : 2v/2L : 3v/2L : 4v/2L = 1 : 2 : 3 : 4
Hence proved