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A pipe open at both ends and a pipe closed at one end have save same length. The ratio of frequencies of air columns in their $\mathrm{p}^{\text {th }}$ overtone respectively is
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Verified Answer
The correct answer is:
$\frac{2(p+1)}{2 p+1}$
Let $l$ be the length of the pipe and $v$ the speed of the sound. The frequency of the open organ pipe of $\mathrm{p}^{\text {th }}$ overtone is,
$\mathrm{f}_0=(\mathrm{p}+1) \frac{\mathrm{v}}{21}$
And frequency of closed organ pipe of nth overtone is,
$\mathrm{f}_{\mathrm{c}}=(\mathrm{p}+2) \frac{\mathrm{v}}{41}$
The desired ratio is thus,
$\frac{\mathrm{f}_0}{\mathrm{f}_{\mathrm{c}}}=\frac{2(\mathrm{p}+1)}{2 \mathrm{p}+1}$
$\mathrm{f}_0=(\mathrm{p}+1) \frac{\mathrm{v}}{21}$
And frequency of closed organ pipe of nth overtone is,
$\mathrm{f}_{\mathrm{c}}=(\mathrm{p}+2) \frac{\mathrm{v}}{41}$
The desired ratio is thus,
$\frac{\mathrm{f}_0}{\mathrm{f}_{\mathrm{c}}}=\frac{2(\mathrm{p}+1)}{2 \mathrm{p}+1}$
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