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(B) For a closed organ pipe,the first overtone is the $3^{rd}$ harmonic. The frequency is given by $f = \frac{3v_1}{4L} = \frac{3}{4L} \sqrt{\frac{1}{

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$\frac{4 L}{3} \sqrt{\frac{\rho_1}{\rho_2}}$

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(B) For a closed organ pipe,the first overtone is the $3^{rd}$ harmonic. The frequency is given by $f = \frac{3v_1}{4L} = \frac{3}{4L} \sqrt{\frac{1}{\beta \rho_1}}$,where $\beta$ is the compressibility.For an open organ pipe,the first overtone is the $2^{nd}$ harmonic. The frequency is given by $f = \frac{2v_2}{2L'} = \frac{v_2}{L'} = \frac{1}{L'} \sqrt{\frac{1}{\beta \rho_2}}$.Since the compressibility $\beta$ is the same for both gases,and assuming the bulk modulus $K = 1/\beta$ is the same,we equate the frequencies:$\frac{3}{4L} \sqrt{\frac{1}{\beta \rho_1}} = \frac{1}{L'} \sqrt{\frac{1}{\beta \rho_2}}$$\frac{3}{4L \sqrt{\rho_1}} = \frac{1}{L' \sqrt{\rho_2}}$$L' = \frac{4L}{3} \sqrt{\frac{\rho_1}{\rho_2}}$.

$A$ closed organ pipe of length $L$ and an open organ pipe contain gases of densities $\rho_1$ and $\rho_2$ respectively. The compressibility of the gases is equal in both the pipes. If the frequencies of their first overtones are the same,then the length of the open organ pipe is:

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