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:

The correct option/answer is f.

Let us understand and solve this step-by-step.

Step 1: Fundamental frequency of an open pipe
For a pipe open at both ends of length L, the fundamental frequency $f$ of the standing wave in the air column is given by the formula:
$f=\frac{v}{2L}$
where $v$ is the speed of sound in air.

Step 2: Pipe dipped in water
When the pipe is dipped vertically in a water drum to half of its length, the water surface seals the bottom end of the pipe. As a result, the pipe behaves as a closed pipe (closed at one end by the water surface, and open at the other end).
The new length of the air column above the water is:
$L^{\prime }=\frac{L}{2}$

Step 3: Fundamental frequency of a closed pipe
The fundamental frequency of a pipe closed at one end of length $L^{\prime }$ is given by:
$f^{\prime }=\frac{v}{4L^{\prime }}$

Substituting the new length $L^{\prime }=\frac{L}{2}$ into this expression:
$f^{\prime }=\frac{v}{4\left(\frac{L}{2}\right)}=\frac{v}{2L}$

Step 4: Comparison
Comparing the new fundamental frequency $f^{\prime }$ with the original fundamental frequency $f$:
$f^{\prime }=f$

Therefore, the fundamental frequency of the air column remains equal to f.