A source `S` of sound wave of fixed frequency `N` and an observer `O` are located in air initially at the space points `A` and `B`, a fixed distance apart. State in which of the following cases, the observer will NOT see any Doppler effect and will receive the same frequency `N` as produced by the source.

Options

A. Both the source `S` and observer `O` remain stationary but a wind blows with a constant speed in an arbitrary direction.
B. The observer remains stationary but the source `S` moves parallel to and in the same direction and with the same speed as the wind.
C. The source remains stationary but the observer and the wind have the same speed away from the source.
D. The source and the observer move directly against the wind but both with the same speed.

Explanation

To determine in which case the observer will not experience any Doppler effect and will receive the same frequency \( N \) as produced by the source \( S \), we can analyze each scenario based on the principles of the Doppler effect.

### Step-by-Step Solution:

1. **Understanding the Doppler Effect**:
The Doppler effect describes the change in frequency of a wave in relation to an observer moving relative to the wave source. The formula for the apparent frequency \( f' \) is given by:
\[ f' = f \left( \frac{v + v_o}{v + v_s} \right) \]
where:
- \( f \) = actual frequency of the source
- \( v \) = speed of sound in the medium
- \( v_o \) = speed of the observer (positive if moving towards the source)
- \( v_s \) = speed of the source (positive if moving away from the observer)

2. **Case Analysis**:
- **Case 1**: Both the source and observer are stationary, but wind blows with a constant speed in an arbitrary direction.
- Here, \( v_o = 0 \) and \( v_s = 0 \). The wind does not affect the frequency received by the observer since the speeds in the numerator and denominator will be equal. Thus, \( f' = f \). **No Doppler effect.**

- **Case 2**: The observer remains stationary, but the source moves parallel and in the same direction as the wind.
- In this scenario, the source is moving, so \( v_s \neq 0 \) while \( v_o = 0 \). The observer will perceive a different frequency due to the motion of the source. **Doppler effect occurs.**

- **Case 3**: The source is stationary, but the observer and the wind have the same speed moving away from the source.
- Here, \( v_s = 0 \) and \( v_o \neq 0 \). The observer will perceive a different frequency because the observer is moving away from the stationary source. **Doppler effect occurs.**

- **Case 4**: Both the source and observer move directly against the wind, but both with the same speed.
- In this case, both the source and observer are moving with the same speed, resulting in no relative motion between them. Thus, \( v_o = v_s \) and the relative velocities cancel out, leading to \( f' = f \). **No Doppler effect.**

3. **Conclusion**:
The observer will not see any Doppler effect and will receive the same frequency \( N \) in **Case 1** and **Case 4**.

### Final Answer:
The observer will NOT see any Doppler effect and will receive the same frequency \( N \) in:
- Case 1: Both the source and observer are stationary, but wind blows with a constant speed in an arbitrary direction.
- Case 4: Both the source and observer move directly against the wind, but both with the same speed.