An open organ pipe of length $l$ is sounded together with another organ pipe of length $l + x$ in their fundamental tones $(x < < l)$. The beat frequency heard will be (speed of sound is $v$) :
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The frequency of an organ pipe with length $l$
$f_{1}=\frac{v}{2 l}$
The frequency of an organ pipe with length $(l+x)$
$f_{2}=\frac{v}{2(l+x)}$
where $v$ is the speed of sound in air
Then, for the beat frequency heard,
$f_{1}-f_{2}=\frac{v}{2 l}-\frac{v}{2(l+x)}$
$\Rightarrow f_{1}-f_{2}=\frac{v}{2 l}\left[1-\left(1+\frac{x}{l}\right)^{-1}\right]$
$\Rightarrow f_{1}-f_{2}=\frac{v}{2 l}\left[1-1+\frac{x}{l}\right]=\frac{v x}{2 l^{2}}$
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