A pipe open at both ends has a fundamental frequency in air. The pipe is now dipped vertically in a water drum to half of its length. The fundamental frequency of the air column is now equal to [2025]

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Solution

Q.
A pipe open at both ends has a fundamental frequency $f$ in air. The pipe is now dipped vertically in a water drum to half of its length. The fundamental frequency of the air column is now equal to [2025]

1 $\frac{3 f}{2}$

2 $2 f$

3 $\frac{f}{2}$

4 $f$

Ans.
(4)

The fundamental frequency $(f)$ is related to the length of the pipe $(L)$ and speed of sound in air $(v)$ as

   $f = \frac{v}{λ}$

For a pipe open at both ends, the fundamental frequency corresponding to a wavelength $λ$ that is twice the length of the pipe is

   $f = \frac{v}{2 L}$ ...(i)

When the pipe is dipped vertically in water so that half of its length is submerged, the new length of the air column becomes

   $L' = \frac{L}{2}$ ...(ii)

For the new length of the air column, the wavelength $λ'$ is given by

   $λ' = 4 L'$

$λ' = 4 \times \frac{L}{2}$    $(Using (ii))$

$λ' = 2 L$

The new fundamental frequency,

   $f' = \frac{v}{λ'} = \frac{v}{2 L}$ ...(iii)

Comparing eq. (i) and (iii), we get

   $f' = f$

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