Two open organ pipes of fundamental frequencies $n_{1}$ and $n_{2}$ are joined in series. The fundamental frequecny of the new pipe so obtained will be
NEET 2017, Medium
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Fundamental frequency of an open pipe of length $L$ is given by $n =\frac{ v }{2 L }$ $\Longrightarrow L =\frac{ v }{2 n }$
So, we get lengths of two open pipes as $L _{1}=\frac{ v }{2 n _{1}}$ and $L _{2}=\frac{ v }{2 n _{2}}$ Now the pipes are join in series.
Fundamental frequency of new open pipe of length $L _{1}+ L _{2},$
$n =\frac{ v }{2\left( L _{1}+ L _{2}\right)}$
$n =\frac{ v }{2\left(\frac{ v }{2 n _{1}}+\frac{ v }{2 n _{2}}\right)}$
$n=\frac{1}{\frac{1}{n_{1}}+\frac{1}{n_{2}}}$
$\Rightarrow n =\frac{ n _{1} n _{2}}{ n _{1}+ n _{2}}$
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