A closed and an open organ pipe have same lengths. If the ratio of frequencies of their seventh overtones is $\left(\frac{a-1}{a}\right)$ then the value of $a$ is
JEE MAIN 2024, Diffcult
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For closed organ pipe
$\mathrm{f}_{\mathrm{c}}=(2 \mathrm{n}+1) \frac{\mathrm{v}}{4 \ell}=\frac{15 \mathrm{v}}{4 \ell}$
For open organ pipe
$\mathrm{f}_{\mathrm{o}}=(\mathrm{n}+1) \frac{\mathrm{v}}{2 \ell}=\frac{8 \mathrm{v}}{2 \ell}$
$\frac{\mathrm{f}_{\mathrm{c}}}{\mathrm{f}_{\mathrm{o}}}=\frac{15}{16}=\frac{\mathrm{a}-1}{\mathrm{a}}$
$\Rightarrow \mathrm{a}=16$
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