The second overtone of an open organ pipe has | vedclass

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(D) For an open organ pipe of length $L'$,the frequency of the $n^{th}$ harmonic is given by $f_n = n \frac{v}{2L'}$,where $n = 1, 2, 3, \dots$. The s

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$2L \; m$

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(D) For an open organ pipe of length $L'$,the frequency of the $n^{th}$ harmonic is given by $f_n = n \frac{v}{2L'}$,where $n = 1, 2, 3, \dots$. The second overtone corresponds to the $3^{rd}$ harmonic $(n=3)$. Thus,$f_{open} = 3 \frac{v}{2L'}$.For a closed organ pipe of length $L$,the frequency of the $n^{th}$ harmonic is given by $f_n = n \frac{v}{4L}$,where $n = 1, 3, 5, \dots$. The first overtone corresponds to the $3^{rd}$ harmonic $(n=3)$. Thus,$f_{closed} = 3 \frac{v}{4L}$.According to the problem,the frequencies are equal:$3 \frac{v}{2L'} = 3 \frac{v}{4L}$Canceling $3v$ from both sides:$\frac{1}{2L'} = \frac{1}{4L}$Solving for $L'$:$2L' = 4L \implies L' = 2L \; m$.

The second overtone of an open organ pipe has the same frequency as the first overtone of a closed organ pipe $L$ metre long. The length of the open pipe will be:

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