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

Question

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

TestHub · Accepted Answer

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

For the sixth overtone ( $n = 6$ ), the frequency is $f_{c} = (2 (6) + 1) \frac{v}{4 l _{c}} = 13 \frac{v}{4 l _{c}}$ .

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

For the sixth overtone ( $n = 6$ ), the frequency is $f_{o} = (6 + 1) \frac{v}{2 l _{o}} = 7 \frac{v}{2 l _{o}}$ .

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

$13 \frac{v}{4 l _{c}} = 7 \frac{v}{2 l _{o}}$

$\frac{13}{2 l _{c}} = \frac{7}{l _{o}}$

$\frac{l _{c}}{l _{o}} = \frac{13}{14}$

The ratio $l_{c} : l_{o}$ is $13 : 14$ .

Answer: $13 : 14$

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