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(C) For a closed organ pipe,the frequency of the $n$-th overtone is given by $f_{c} = (2n + 1) \frac{v}{4\ell}$,where $n$ is the overtone number.For t

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$16$

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(C) For a closed organ pipe,the frequency of the $n$-th overtone is given by $f_{c} = (2n + 1) \frac{v}{4\ell}$,where $n$ is the overtone number.For the $7$-th overtone $(n=7)$,$f_{c} = (2 	imes 7 + 1) \frac{v}{4\ell} = \frac{15v}{4\ell}$.For an open organ pipe,the frequency of the $n$-th overtone is given by $f_{o} = (n + 1) \frac{v}{2\ell}$.For the $7$-th overtone $(n=7)$,$f_{o} = (7 + 1) \frac{v}{2\ell} = \frac{8v}{2\ell} = \frac{4v}{\ell} = \frac{16v}{4\ell}$.The ratio of the frequencies is $\frac{f_{c}}{f_{o}} = \frac{15v/4\ell}{16v/4\ell} = \frac{15}{16}$.Given the ratio is $\left(\frac{a-1}{a}ight) = \frac{15}{16}$,comparing the terms,we get $a = 16$.

$A$ closed and an open organ pipe have the same lengths. If the ratio of the frequencies of their seventh overtones is $\left(\frac{a-1}{a}\right)$,then the value of $a$ is:

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