A student performed the experiment to measure the speed of sound in air using resonance air-column method. Two resonances in the air-column were obtained by lowering the water level. The resonance with the shorter air-column is the first resonance and that with the longer air-column is the second resonance. Then,

Options

A. (a)the intensity of the sound heard at the first reson was more than that at the second resonance.
B. (b)the prongs of the tuning fork were kept in a horizontal plane above the resonance tube.
C. (c)the amplitude of vibration of the ends of the prongs is typically around `1 cm`
D. (d)the length of the air-column at the first resonance somewhat shorter than `1//4 th` of the wavelength sound in air.

Explanation

To solve the problem of measuring the speed of sound in air using the resonance air-column method, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Resonance in Air Column**:
- In the resonance air-column method, the resonances are produced by adjusting the length of the air column. The first resonance occurs at a shorter length of the air column (L1), while the second resonance occurs at a longer length (L2).

2. **Relationship Between Resonance Lengths**:
- It is known that for the first and second resonances in an open air column, the lengths of the air column are related to the wavelength (λ) of the sound. The first resonance (L1) corresponds to a quarter wavelength, while the second resonance (L2) corresponds to three-quarters of the wavelength.
- Mathematically, we can express this as:
- \( L1 = \frac{\lambda}{4} \)
- \( L2 = \frac{3\lambda}{4} \)

3. **Finding the Wavelength**:
- From the above equations, we can derive the wavelength:
- \( L2 - L1 = \frac{3\lambda}{4} - \frac{\lambda}{4} = \frac{2\lambda}{4} = \frac{\lambda}{2} \)
- Therefore, the difference in lengths of the air column between the second and first resonance is equal to half the wavelength:
- \( L2 - L1 = \frac{\lambda}{2} \)

4. **Speed of Sound Calculation**:
- The speed of sound (v) can be calculated using the formula:
- \( v = f \cdot \lambda \)
- Where \( f \) is the frequency of the tuning fork used in the experiment. If we know the frequency and have measured the wavelength (from the difference in lengths), we can calculate the speed of sound.

5. **Intensity of Sound**:
- The intensity of sound at the first resonance (I1) is greater than at the second resonance (I2). This is due to the fact that the first resonance corresponds to a position where the sound waves constructively interfere more effectively than at the second resonance.
- Therefore, we can conclude:
- \( I1 > I2 \)

### Final Results:
- The lengths of the air column at the first and second resonance are related as \( L2 > L1 \).
- The wavelength can be determined from the difference in lengths of the resonances.
- The speed of sound can be calculated using the frequency and wavelength.
- The intensity of sound at the first resonance is greater than that at the second resonance.