An air column is of length 17 cm . The ratio of frequencies of 5 th overtone if the air column is closed at one end to that open at both ends is (velocity of sound in air = 340 ms − 1 )

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TestHub · Accepted Answer

For a pipe closed at one end, the frequency of the $p^{th}$ overtone is given by $n_{c} = (2 p + 1) \frac{v}{4 L}$ .

For the $5^{th}$ overtone ( $p = 5$ ), the frequency is $n_{c} = (2 (5) + 1) \frac{v}{4 L} = 11 \frac{v}{4 L}$ .

For a pipe open at both ends, the frequency of the $p^{th}$ overtone is given by $n_{o} = (p + 1) \frac{v}{2 L}$ .

For the $5^{th}$ overtone ( $p = 5$ ), the frequency is $n_{o} = (5 + 1) \frac{v}{2 L} = 6 \frac{v}{2 L} = 12 \frac{v}{4 L}$ .

The ratio of the frequencies is $\frac{n _{c}}{n _{o}} = \frac{11 \frac{v}{4 L}}{12 \frac{v}{4 L}} = \frac{11}{12}$ .

Answer: $\frac{11}{12}$

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