Explanation:
For open pipe first overtone
\(v_{1}=\dfrac{v}{L}\)
For closed pipe first overtone,
\({v_1} = \frac{{3v}}{{4L}}\)
\(\therefore {v_1} - {v_1} = \frac{v}{L} - \frac{{3v}}{{4L}} = 3\)
\( \Rightarrow \frac{v}{{3L}} = 3\)
\(\therefore \frac{v}{L} = 9\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( 1 \right)\)
When length of open pipe is made \(\dfrac{L}{3}\), then fundamnetal frequency
\(v=\dfrac{v}{2\left(\dfrac{L}{3}\right)}=\dfrac{3 v}{2 L}\)
When length of closed pipe is made 3 times, then fundamental frequency
\(v^{\prime}=\dfrac{v}{4(3 L)}=\dfrac{3 v}{12 L}\)
Beats produced \(v = v'\),
\(=\dfrac{3 v}{2 L}-\dfrac{v}{12 L}=\dfrac{17}{12} \dfrac{v}{L}\)
\(=\dfrac{17}{12} \times 12 \quad \because[\) from eq. \((1)]\)
\(=17\)
