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Q.
The third overtone of an open organ pipe of length $l_o$ has the same frequency as the third overtone of a closed pipe of length $l_c$. The ratio $\frac{l_o}{l_c}$ is equal to
The frequency of the third overtone (or fourth harmonic) of an open pipe of length $l_o$ is
$\upsilon_o = \frac{4 v}{2l_o}$
The frequency of the third overtone (or seven harmonic) of a closed pipe of length $l_c$ is
$\upsilon_c = \frac{7 v}{4 l_c}$
As per question $\upsilon_o = \upsilon_c$
$\therefore \:\:\: \frac{4 v}{2l_o} = \frac{7 v}{4 l_c} $ or $\frac{l_o}{l_c} = \frac{8}{7}$