A pipe closed at end vibrating in the fourth overtone is in unison with the open pipe vibrating in its fourth overtone. The ratio of l c : l o is { l c = vibrating length of closed pipe, l o =

Question

A pipe closed at end vibrating in the fourth overtone is in unison with the open pipe vibrating in its fourth overtone. The ratio of l c ​ : l o ​ is { l c ​ = vibrating length of closed pipe, l o ​ = vibrating length of open pipe}

TestHub · Accepted Answer

For a closed pipe, the frequency of the $p$ -th overtone is given by $f_{c} = \frac{( 2 p + 1 ) v}{4 l _{c}}$ .

For the fourth overtone ( $p = 4$ ), the frequency is $f_{c} = \frac{9 v}{4 l _{c}}$ .

For an open pipe, the frequency of the $p$ -th overtone is given by $f_{o} = \frac{( p + 1 ) v}{2 l _{o}}$ .

For the fourth overtone ( $p = 4$ ), the frequency is $f_{o} = \frac{5 v}{2 l _{o}}$ .

Since the two pipes are in unison, their frequencies are equal:

$\frac{9 v}{4 l _{c}} = \frac{5 v}{2 l _{o}}$

$\frac{l _{c}}{l _{o}} = \frac{9 \times 2}{4 \times 5} = \frac{18}{20} = \frac{9}{10}$

The ratio $l_{c} : l_{o}$ is $9 : 10$ .

Answer: $9 : 10$

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