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: (neglect the end correction)

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

Given
Length of closed organ pipe= L
Density of gases, $ρ_{1} and ρ_{2}$ respectively
Frequency of first over tone of closed organ pipe = Frequency of first over tone of open organ pipe
$\begin{array}{l} \Rightarrow \frac{3 v_{1}}{4 L_{1}} = \frac{2 v_{2}}{2 L_{2}} \\ \Rightarrow \frac{3}{4 L_{1}} \sqrt{\frac{γ P}{ρ_{1}}} = \frac{1}{L_{2}} \sqrt{\frac{γ P}{ρ_{2}}} [As v = \sqrt{\frac{γ P}{ρ}}] \\ \Rightarrow L_{2} = \frac{4 L_{1}}{3} \sqrt{\frac{ρ_{1}}{ρ_{2}}} = \frac{4 L}{3} \sqrt{\frac{ρ_{1}}{ρ_{2}}} \end{array}$

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