A pipe open at both ends has a fundamental frequency f in air. The pipe is dippoed vertically in water so that half of it is in water. The fundamental frequency of the air column is now :-

Options

A. f
B. `(f)/(2)`
C. `(3f)/(4)`
D. 2f

Answer

To solve the problem, we need to analyze the situation of the pipe before and after it is dipped in water.

### Step-by-Step Solution:

1. **Identify the Initial Conditions**:
- The pipe is open at both ends and has a fundamental frequency \( f \) in air.
- The fundamental frequency \( f \) is related to the length of the pipe \( L \) and the speed of sound in air \( v \) by the formula:
\[ f = \frac{v}{\lambda} \]
- For a pipe open at both ends, the fundamental frequency corresponds to a wavelength \( \lambda \) that is twice the length of the pipe:
\[ \lambda = 2L \]

2. **Determine the Length of the Air Column When Dipped in Water**:
- When the pipe is dipped vertically in water so that half of it is submerged, the length of the air column \( L' \) becomes:
\[ L' = \frac{L}{2} \]

3. **Calculate the New Wavelength**:
- For the new length of the air column, the wavelength \( \lambda' \) for the fundamental frequency in the new configuration (open at one end and closed at the water surface) is given by:
\[ \lambda' = 4L' \]
- Substituting \( L' \):
\[ \lambda' = 4 \left(\frac{L}{2}\right) = 2L \]

4. **Determine the New Fundamental Frequency**:
- The new fundamental frequency \( f' \) can now be calculated using the speed of sound in air \( v \):
\[ f' = \frac{v}{\lambda'} \]
- Substituting \( \lambda' \):
\[ f' = \frac{v}{2L} \]

5. **Relate the New Frequency to the Original Frequency**:
- We know that the original frequency \( f \) was:
\[ f = \frac{v}{2L} \]
- Therefore, we can conclude that:
\[ f' = f \]

### Final Answer:
The fundamental frequency of the air column when the pipe is half submerged in water remains \( f \).