Question

A pipe open at both ends has a fundamental frequency \(f\) in air. The pipe is now dipped vertically in water drum to half of its length. The fundamental frequency of the air column is now equal to:

• A. \( f \)
• B. \( \frac{3}{4} f \)
• C. \( 2f \)
• D. \( \frac{1}{2} f \)

Hint:

Remember the fundamental frequencies for open and closed pipes. For an open pipe of length \(L\), \(f = v/(2L)\). For a closed pipe of length \(L\), \(f = v/(4L)\). When an open pipe is half-submerged, it effectively becomes a closed pipe of half the original length.

Answer 1

The Correct Option is A

Consider a pipe open at both ends, which has a fundamental frequency \( f \) in air. The fundamental frequency of a pipe open at both ends is given by the equation:

\[ f = \frac{v}{2L} \]

where \( v \) is the speed of sound in air, and \( L \) is the length of the pipe.

When the pipe is dipped in water to half its length, it becomes a closed pipe (one end is closed) with a new effective length \( \frac{L}{2} \). The fundamental frequency of a closed pipe is given by:

\[ f' = \frac{v}{4L'} \]

Substituting the effective length \( L' = \frac{L}{2} \) into the equation:

\[ f' = \frac{v}{4 \times \frac{L}{2}} = \frac{v}{2L} = f \]

Thus, the fundamental frequency of the air column in the pipe when it is dipped to half its length in water is the same as the original frequency \( f \).

Therefore, the correct answer is \( f \).