Question 235: Waves

Question

The closed and open organ pipes have same length. When they are vibrating simultaneously in first overtone, they produce four beats. The length of open pipe is made half and that of the closed pipe is made two times the original. Now the number of beats produced if the two pipes are vibrating in their fundamental modes simultaneously is

Accepted Answer

For open pipe first overtone, $\mathrm{n}_1=\frac{\mathrm{v}}{\mathrm{L}}$
For closed pipe first overtone, $\mathrm{n}_1^{\prime}=\frac{3 \mathrm{v}}{4 \mathrm{~L}}$
$	herefore \quad$ Number of beats produced are,
$$ \begin{array}{ll}  & \mathrm{n}_1-\mathrm{n}_1^{\prime}=\frac{\mathrm{v}}{\mathrm{~L}}-\frac{3 \mathrm{v}}{4 \mathrm{~L}}=4 \ 	herefore & \frac{\mathrm{v}}{4 \mathrm{~L}}=4 \ 	herefore & \frac{\mathrm{v}}{\mathrm{~L}}=16 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...(i)\end{array} $$
When length of open pipe is made $\frac{L}{2}$, the fundamental frequency becomes,
$$ \mathrm{n}=\frac{\mathrm{v}}{2\left(\frac{\mathrm{~L}}{2}ight)}=\frac{\mathrm{v}}{\mathrm{~L}} $$
When length of closed pipe is made 2 times, the fundamental frequency becomes,
$$ \mathrm{n}^{\prime}=\frac{\mathrm{v}}{4(2 \mathrm{~L})}=\frac{\mathrm{v}}{8 \mathrm{~L}} $$
$$ \begin{aligned}  	herefore & 	ext { Beats produced }=n-n^{\prime} \ & =\frac{v}{L}-\frac{v}{8 L} \ & =\frac{7}{8} 	imes \frac{v}{L}=\frac{7}{8} 	imes 16 \,\,\,\,\,\,\,\,	ext { ...[From (i)] }\ & =14 \end{aligned} $$

Answer

8

Answer

10

Answer

14

Answer

16

Waves

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