Consider a souce of sound `S`, and an observer/detector `D`. The source emits a sound wave of frequency `f_0`. The frequency observed by `D` is found to be (i) `f_1`, if `D` approaches `S` and `S` is stationary
(ii) `f_2`, if `S` approaches `S` and `S` is stationary
(iii) `f_3`,if both `S` and `D` and `D` is stationary Speed
In all three cases, relative velocity of `S` wrt `D` is the same. For this situation which is incorrect?

Options

A. `n_1=n_2=n_3`
B. `n_1ltn_2`
C. `n_3gtn_0`
D. `n_3` lies between `n_1` and `n_2`

Explanation

To solve the problem, we need to analyze the three scenarios involving the source of sound \( S \) and the observer \( D \) in terms of the Doppler effect.

### Step-by-Step Solution:

1. **Understanding the Doppler Effect**:
The Doppler effect describes the change in frequency or wavelength of a wave in relation to an observer moving relative to the wave source. The frequency observed depends on the relative motion between the source and the observer.

2. **Case (i): Observer Approaching Source**:
- When the observer \( D \) approaches the stationary source \( S \), the observed frequency \( f_1 \) can be calculated using the formula:
\[ f_1 = f_0 \left( \frac{v + v_D}{v} \right) \]
Here, \( v \) is the speed of sound, and \( v_D \) is the speed of the observer towards the source.

3. **Case (ii): Source Approaching Observer**:
- When the source \( S \) approaches the stationary observer \( D \), the observed frequency \( f_2 \) is given by:
\[ f_2 = f_0 \left( \frac{v}{v - v_S} \right) \]
Here, \( v_S \) is the speed of the source towards the observer.

4. **Case (iii): Both Source and Observer Moving**:
- If both \( S \) and \( D \) are moving towards each other, the observed frequency \( f_3 \) can be expressed as:
\[ f_3 = f_0 \left( \frac{v + v_D}{v - v_S} \right) \]
In this case, both the observer and the source contribute to the change in frequency.

5. **Analyzing the Frequencies**:
- We need to compare \( f_1 \), \( f_2 \), and \( f_3 \):
- In case (i), \( f_1 \) will be greater than \( f_0 \) since the observer is moving towards the source.
- In case (ii), \( f_2 \) will also be greater than \( f_0 \) since the source is moving towards the observer.
- In case (iii), \( f_3 \) will be the highest because both the source and observer are moving towards each other.

6. **Conclusion**:
- The incorrect statement among the options given is that \( f_1 \), \( f_2 \), and \( f_3 \) are equal. This is not true as we have shown that \( f_1 < f_3 \) and \( f_2 < f_3 \).

### Final Answer:
The incorrect statement is that \( f_1 = f_2 = f_3 \).

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