A source which is emitting sound of frequency `f` is initially at `(-r,0)` and an observer is situated initally at `(2r, 0)`. If observer and source both are moving with velocities `overset(vec)(v)_("observer") = -sqrt(2)Vhat(i) - sqrt(2)hat (j)` and `overset(vec)(v)_("source") = (V)/(sqrt(2))hat(i) + (V)/(sqrt(2))hat(j)`, then which of the following is correct option ?

Options

A. Apparent frequency first increases, then decreases and observer the original frequency once during the motion.
B. Apparent frequency first increases, then decreases and observer the original frequency twice during the motion.
C. Apparent frequency first increases, then decreases during the motion and observer never observer the initial frequency.
D. Apparent frequency continously decreases and one during the motion, observer hears the original frequency.

Explanation

To solve the problem, we need to analyze the motion of both the source and the observer, and how their velocities affect the apparent frequency of the sound they perceive.

### Step-by-Step Solution:

1. **Identify Initial Positions and Velocities**:
- The source is at position \((-r, 0)\) and emits sound with frequency \(f\).
- The observer is at position \((2r, 0)\).
- The velocity of the observer is given as \(\vec{v}_{\text{observer}} = -\sqrt{2}V \hat{i} - \sqrt{2}V \hat{j}\).
- The velocity of the source is given as \(\vec{v}_{\text{source}} = \frac{V}{\sqrt{2}} \hat{i} + \frac{V}{\sqrt{2}} \hat{j}\).

2. **Determine the Direction of Motion**:
- The observer is moving towards the origin (left and down), while the source is moving towards the first quadrant (up and right).
- This means that the observer and the source are moving towards each other.

3. **Calculate the Relative Velocities**:
- The relative velocity of approach between the observer and the source can be calculated by adding their velocities since they are moving towards each other.
- The relative velocity \(v_{\text{relative}} = \vec{v}_{\text{observer}} - \vec{v}_{\text{source}}\).

4. **Determine the Apparent Frequency**:
- The apparent frequency \(f'\) can be calculated using the Doppler effect formula:
\[ f' = f \left( \frac{v + v_{\text{observer}}}{v - v_{\text{source}}} \right) \]
- Here, \(v\) is the speed of sound, \(v_{\text{observer}}\) is the component of the observer's velocity towards the source, and \(v_{\text{source}}\) is the component of the source's velocity towards the observer.

5. **Analyze the Change in Apparent Frequency**:
- As the observer approaches the source, the apparent frequency will increase.
- Once the observer reaches the closest point to the source, the frequency will start to decrease as the observer moves away.

6. **Conclusion**:
- The apparent frequency first increases as the observer approaches the source and then decreases as the observer moves away. The observer will hear the original frequency once during this motion when they are at the closest point to the source.

### Final Answer:
The correct option is **D**: The apparent frequency continuously decreases, and once during the motion, the observer hears the original frequency.