A stretched string of length L, is fixed at both ends and then set to vibrate in its allowed modes. Derive an expression for frequency of the second overtone in terms of fundamental frequency

A stretched string of length L, is fixed at both ends and then set to vibrate in its allowed modes. Derive an expression for frequency of the second overtone in terms of fundamental frequency

When the string is vibrating with fundamental frequency, the number of waves between A and B is 1/2

L =  1/2λ  or  λ =2L

Let the velocity of the wave be v, then frequency of fundamental note, f_{0 = v/λ = v/2L}

When vibrating with 2^nd overtone, the number of waves made by the length of the string between A and B is  (3/2)λ

L = 3/2_{λ i.e. λ = 2/3L}

Then f₂=  (v/_{λ) =v/(2/3L) = 3(v/2L)}

f₂= 3f₀

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