First slide

Stationary waves

Question

A closed organ pipe of length L and an open organ pipe contain gases of densities $ρ_{1}$ and $ρ_{2}$ respectively. The compressibility of gases are equal in both the pipes. Both the pipes are vibrating in their first overtone with same frequency. The length of the open organ pipe is

Moderate

A

$\frac{L}{3}$
B

$\frac{4 L}{3}$
C

$\frac{4 L}{3} \sqrt{\frac{ρ_{1}}{ρ_{2}}}$
D

$\frac{4 L}{3} \sqrt{\frac{ρ_{2}}{ρ_{1}}}$

Solution

Frequency of first over tone of closed pipe = Frequency of first over tone of open pipe
$\Rightarrow \frac{3 v}{4 L_{1}} = \frac{v}{L_{2}} \Rightarrow \frac{3}{4 L_{1}} \sqrt{\frac{γP}{ρ_{1}}} = \frac{1}{L_{2}} \sqrt{\frac{γP}{ρ_{2}}} [∵ v = \sqrt{\frac{γP}{ρ}}]$
$\Rightarrow L_{2} = \frac{4 L_{1}}{3} \sqrt{\frac{ρ_{1}}{ρ_{2}}} = \frac{4 L}{3} \sqrt{\frac{ρ_{1}}{ρ_{2}}}$

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