Two pipes of lengths L 1 and L 2 , open at both ends are joined in series. If ' f 1 ' and ' f 2 ' are the fundamental frequencies of two pipes, then the fundamental frequen

Question

Two pipes of lengths L 1 ​ and L 2 ​ , open at both ends are joined in series. If ' f 1 ​ ' and ' f 2 ​ ' are the fundamental frequencies of two pipes, then the fundamental frequency of series combination will be (neglect end correction)

TestHub · Accepted Answer

The fundamental frequency of a pipe open at both ends is given by $f = \frac{v}{2 L}$ .

For the first pipe, $f_{1} = \frac{v}{2 L _{1}} \Rightarrow L_{1} = \frac{v}{2 f _{1}}$ .

For the second pipe, $f_{2} = \frac{v}{2 L _{2}} \Rightarrow L_{2} = \frac{v}{2 f _{2}}$ .

When the two pipes are joined in series, the total length of the combined open pipe becomes $L = L_{1} + L_{2}$ .

The fundamental frequency of the series combination is $f = \frac{v}{2 L} = \frac{v}{2 ( L _{1} + L _{2} )}$ .

Substituting the values of $L_{1}$ and $L_{2}$ :

$f = \frac{v}{2 ( \frac{v}{2 f _{1}} + \frac{v}{2 f _{2}} )}$

$f = \frac{v}{v ( \frac{1}{f _{1}} + \frac{1}{f _{2}} )}$

$f = \frac{1}{\frac{f _{1} + f _{2}}{f _{1} f _{2}}}$

$f = \frac{f _{1} f _{2}}{f _{1} + f _{2}}$

Answer: $\frac{f _{1} f _{2}}{f _{1} + f _{2}}$

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