Explanation:
\({f_{o}-f_{c}=2}\)
or \({\dfrac{v}{2 l}-\dfrac{v}{4 l}=2 \Rightarrow \dfrac{v}{4 l}=2}\)
\( \Rightarrow \frac{v}{l} = 8\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( 1 \right)\)
When length of open organ pipe is halved and that of closed organ pipe is doubled, the beat frequency will be,
\({f_{o}^{\prime}-f_{c}^{\prime}=\dfrac{v}{l}-\dfrac{v}{8 l}=\dfrac{7}{8} \dfrac{v}{l}=\dfrac{7}{8} \times 8=7}\)