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

Options

A. `v/4`
B. `v/2`
C. `v`
D. `2v`

Answer

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

### Step 1: Understand the Initial Condition
The cylindrical tube is open at both ends and has a fundamental frequency \( f \) in air. The length of the tube is \( L \). The fundamental frequency corresponds to the first harmonic, where the length of the air column is equal to half the wavelength (\( \lambda \)) of the sound wave.

**Hint:** Remember that for a tube open at both ends, the fundamental frequency is determined by the length of the tube and the wavelength.

### Step 2: Determine the Wavelength in Air
From the fundamental frequency condition, we know:
\[ \frac{\lambda}{2} = L \implies \lambda = 2L \]
This means the wavelength of the sound in the tube when it is fully in air is \( 2L \).

**Hint:** The relationship between wavelength and tube length is crucial for understanding how sound behaves in different conditions.

### Step 3: Analyze the Tube When Half is in Water
When the tube is dipped vertically in water so that half of it is submerged, the effective length of the air column is now \( \frac{L}{2} \). This changes the boundary conditions, making the tube effectively closed at one end (the end submerged in water) and open at the other end.

**Hint:** Identify how the change in boundary conditions affects the harmonics of the tube.

### Step 4: Determine the New Wavelength
For a tube closed at one end and open at the other, the fundamental frequency corresponds to:
\[ \frac{\lambda'}{4} = \frac{L}{2} \implies \lambda' = 2L \]
Thus, the wavelength of the sound in the tube when half is submerged in water remains \( 2L \).

**Hint:** The wavelength remains the same because the speed of sound in air does not change with the water level.

### Step 5: Relate Frequency and Wavelength
The frequency of a wave is related to its speed and wavelength by the equation:
\[ f = \frac{v}{\lambda} \]
Since the speed of sound in air \( v \) remains constant and the wavelength \( \lambda \) does not change, the frequency will also remain the same.

**Hint:** Consider how the speed of sound and wavelength interact to determine frequency.

### Step 6: Conclusion
Since the wavelength remains unchanged and the speed of sound in air remains constant, the fundamental frequency of the air column when half of the tube is submerged in water is still \( f \).

**Final Answer:** The fundamental frequency of the air column is still \( f \).

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