An excited tuning fork of frequency 512 Hz is held over the open end of a tall jar and water is slowly poured into it. Resonance was first observed at a level and then at a higher level. Calculate the difference in heights of the two levels. Velocity of sound in air at the time of experiment is 346 m/s.

Answer

To solve the problem, we need to calculate the difference in heights of the two levels (L2 - L1) where resonance is observed in the jar when water is poured into it.

### Step-by-Step Solution:

1. **Understanding Resonance in a Jar**:
- When the tuning fork is placed over the open end of the jar, it creates a standing wave inside the jar. The first resonance occurs when the length of the air column (L1) is equal to one-fourth of the wavelength (λ/4).
- The second resonance occurs when the length of the air column (L2) is equal to three-fourths of the wavelength (3λ/4).

2. **Finding the Difference in Lengths**:
- The difference in lengths (heights) where resonance is observed can be expressed as:
\[ L2 - L1 = \frac{3\lambda}{4} - \frac{\lambda}{4} = \frac{2\lambda}{4} = \frac{\lambda}{2} \]

3. **Calculating the Wavelength (λ)**:
- The relationship between frequency (f), velocity of sound (V), and wavelength (λ) is given by:
\[ V = f \cdot \lambda \]
- Rearranging this formula gives:
\[ \lambda = \frac{V}{f} \]

4. **Substituting the Given Values**:
- Given:
- Frequency, \( f = 512 \, \text{Hz} \)
- Velocity of sound, \( V = 346 \, \text{m/s} \)
- Substitute these values into the wavelength formula:
\[ \lambda = \frac{346 \, \text{m/s}}{512 \, \text{Hz}} \]

5. **Calculating the Wavelength**:
- Performing the calculation:
\[ \lambda = \frac{346}{512} \approx 0.676 \, \text{m} \]

6. **Finding the Difference in Heights**:
- Now, substitute λ back into the equation for the difference in heights:
\[ L2 - L1 = \frac{\lambda}{2} = \frac{0.676}{2} \approx 0.338 \, \text{m} \]

### Final Answer:
The difference in heights of the two levels where resonance is observed is approximately **0.338 meters**.