The third overtone of a closed organ pipe is in unison with 3 r d overtone of an open pipe. The ratio of length of closed to open pipe is

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TestHub · Accepted Answer

For a closed organ pipe, the frequency of the $p^{t h}$ overtone is given by $f_{c} = \frac{( 2 p + 1 ) v}{4 L _{c}}$ .

For the $3^{r d}$ overtone ( $p = 3$ ), the frequency is $f_{c} = \frac{7 v}{4 L _{c}}$ .

For an open organ pipe, the frequency of the $p^{t h}$ overtone is given by $f_{o} = \frac{( p + 1 ) v}{2 L _{o}}$ .

For the $3^{r d}$ overtone ( $p = 3$ ), the frequency is $f_{o} = \frac{4 v}{2 L _{o}}$ .

Given that the two overtones are in unison, their frequencies are equal:

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

$\frac{7}{4 L _{c}} = \frac{2}{L _{o}}$

$\frac{L _{c}}{L _{o}} = \frac{7}{8}$

Answer: $\frac{7}{8}$

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