An observer `A` is moving directly towards a stationary sound source while another observer `B` is moving away from the source with the same velocity. Which of the following statements are correct?

Options

A. (a)Average of freqeuncies recorded by `A` and `B` is equal to natural frequency of the source
B. (b)Wavelength of wave received by `A` is less than that of waves received by `B`.
C. (c)Wavelength of waves received by two observers will be same.
D. (d)Both the observers will observe the wave travelling with same speed.

Explanation

To solve the problem, we need to analyze the situation of two observers, A and B, in relation to a stationary sound source.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- Observer A is moving towards a stationary sound source with velocity \( u \).
- Observer B is moving away from the same stationary sound source with the same velocity \( u \).

2. **Frequency Received by Observer A**:
- The frequency received by observer A, denoted as \( n_1 \), can be calculated using the Doppler effect formula for a moving observer towards a stationary source:
\[ n_1 = n_0 \left( \frac{v + u}{v} \right) \]
- Here, \( n_0 \) is the natural frequency of the source, \( v \) is the velocity of sound, and \( u \) is the velocity of observer A.

3. **Frequency Received by Observer B**:
- The frequency received by observer B, denoted as \( n_2 \), can be calculated using the Doppler effect formula for a moving observer away from a stationary source:
\[ n_2 = n_0 \left( \frac{v - u}{v} \right) \]

4. **Comparing Frequencies**:
- We can analyze the two frequencies:
- For observer A:
\[ n_1 = n_0 \left( \frac{v + u}{v} \right) \]
- For observer B:
\[ n_2 = n_0 \left( \frac{v - u}{v} \right) \]

5. **Average Frequency**:
- The average frequency received by both observers can be calculated as:
\[ \text{Average Frequency} = \frac{n_1 + n_2}{2} \]
- Substituting the expressions for \( n_1 \) and \( n_2 \):
\[ \text{Average Frequency} = \frac{n_0 \left( \frac{v + u}{v} \right) + n_0 \left( \frac{v - u}{v} \right)}{2} \]
- Simplifying this gives:
\[ \text{Average Frequency} = n_0 \left( \frac{v}{v} \right) = n_0 \]

6. **Conclusion**:
- The frequency received by observer A is higher than the natural frequency \( n_0 \), while the frequency received by observer B is lower than \( n_0 \).
- However, the average frequency received by both observers is equal to the natural frequency \( n_0 \).
- The wavelengths of the sound waves received by both observers will also be the same since they are both derived from the same source.

### Final Statements:
- The frequency received by A and B is equal to the natural frequency of the source.
- The wavelength of the wave received by both observers will be the same.

### Correct Options:
- Therefore, options A and C are correct.