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 see any Doppler effect and will receive the same frequency \( N \) as produced by the source, we can analyze the conditions under which the Doppler effect occurs. The Doppler effect is influenced by the relative motion between the source of sound and the observer.

### Step-by-Step Solution:

1. **Understand 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 observed frequency \( f' \) is given by:
\[ f' = f \left( \frac{v + v_o}{v + v_s} \right) \]
where:
- \( f \) = original 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. **Conditions for No Doppler Effect**: The observer will not experience any Doppler effect and will receive the same frequency \( N \) as produced by the source if:
- Both the source and observer are stationary.
- The distance between the source and observer remains constant.

3. **Analyze the Given Cases**:
- **Case 1**: Both the source \( S \) and observer \( O \) remain stationary, but a wind blows with a constant speed in an arbitrary direction.
- Here, since both the source and observer are stationary, the frequency received will be the same as the emitted frequency \( N \). **(No Doppler effect)**

- **Case 2**: The observer remains stationary, but the source \( S \) moves parallel to and in the same direction as the wind.
- In this case, the source is moving, which will change the frequency received by the observer. **(Doppler effect occurs)**

- **Case 3**: The source remains stationary, but the observer and wind have the same speed away from the source.
- Here, the observer is moving away from the source, which will also result in a change in frequency. **(Doppler effect occurs)**

- **Case 4**: The source and observer move directly against the wind, but both at the same speed and in the same direction.
- Since both the source and observer are moving at the same speed in the same direction, the relative distance between them does not change. Thus, the frequency received will be the same as emitted. **(No Doppler effect)**

4. **Conclusion**: The cases where the observer will NOT see any Doppler effect and will receive the same frequency \( N \) are:
- Case 1: Both the source and observer are stationary.
- Case 4: Both the source and observer move directly against the wind at the same speed.

### Final Answer:
The observer will NOT see any Doppler effect and will receive the same frequency \( N \) in the following cases:
- Case 1: Both the source \( S \) and observer \( O \) remain stationary.
- Case 4: The source and observer move directly against the wind, but both at the same speed.