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

Options

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

Explanation

To solve the problem, we need to analyze the situation step by step.

### Step 1: Understand the initial conditions
The pipe is open at both ends and has a fundamental frequency \( f \) in air. The length of the pipe is \( L \).

**Hint:** Remember that for an open pipe, the fundamental frequency is determined by the length of the pipe and the speed of sound in air.

### Step 2: Determine the fundamental frequency of the open pipe
For an open pipe, the fundamental frequency \( f \) is given by the formula:
\[ f = \frac{v}{2L} \]
where \( v \) is the speed of sound in air.

**Hint:** The fundamental frequency is inversely proportional to the length of the pipe.

### Step 3: Analyze the situation when the pipe is dipped in water
When the pipe is dipped vertically in water so that half of it is submerged, the effective length of the air column in the pipe becomes \( \frac{L}{2} \).

**Hint:** Consider how the length of the air column changes when the pipe is partially submerged.

### Step 4: Determine the new frequency for the closed pipe
Now, the pipe behaves like a closed pipe (one end closed, the other open). The fundamental frequency for a closed pipe is given by:
\[ f_c = \frac{v}{4L'} \]
where \( L' \) is the new length of the air column, which is \( \frac{L}{2} \).

**Hint:** Remember that for a closed pipe, the fundamental frequency is based on the quarter wavelength.

### Step 5: Substitute the new length into the frequency formula
Substituting \( L' = \frac{L}{2} \) into the formula for \( f_c \):
\[ f_c = \frac{v}{4 \cdot \frac{L}{2}} = \frac{v}{2L} \]

**Hint:** Simplifying the expression will help you see the relationship between the frequencies.

### Step 6: Compare the frequencies
Notice that:
\[ f_c = \frac{v}{2L} = f \]
This shows that the fundamental frequency of the air column when half of the pipe is submerged in water remains the same as the original frequency \( f \).

**Hint:** Look for relationships between the original and new frequencies to confirm your findings.

### Conclusion
The fundamental frequency of the air column when the pipe is half submerged in water is still \( f \).

**Final Answer:** \( f \)