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(D) For an open organ pipe of length $\ell$,the frequency of the $p^{th}$ mode is given by $f_{open} = p \frac{v}{2\ell}$,where $v$ is the speed of so

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$\frac{2p}{2p - 1}$

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(D) For an open organ pipe of length $\ell$,the frequency of the $p^{th}$ mode is given by $f_{open} = p \frac{v}{2\ell}$,where $v$ is the speed of sound.For a closed organ pipe of length $\ell$,the frequency of the $p^{th}$ mode (where $p$ is the harmonic number) is given by $f_{closed} = (2p - 1) \frac{v}{4\ell}$.To find the ratio of the $p^{th}$ mode frequency of the open pipe to the closed pipe,we divide the two expressions:$\frac{f_{open}}{f_{closed}} = \frac{p \frac{v}{2\ell}}{(2p - 1) \frac{v}{4\ell}}$Simplifying the expression:$\frac{f_{open}}{f_{closed}} = \frac{p}{2\ell} 	imes \frac{4\ell}{2p - 1} = \frac{2p}{2p - 1}$Thus,the ratio is $\frac{2p}{2p - 1}$.

An open and a closed organ pipe have the same length. The ratio of the $p^{th}$ mode of frequency of vibration of the two pipes is

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