The wave reaching the receiving antenna will have maximum intensity if the direct wave & wave after reflection from earth's surface

Options

A. are out of phase by `90^(@)`
B. are out of phase by `270^(@)`
C. are in phase
D. None of these

Explanation

To solve the problem of determining the condition under which the wave reaching the receiving antenna has maximum intensity when considering both the direct wave and the wave reflected from the Earth's surface, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Maximum Intensity**:
- The maximum intensity at the receiving antenna occurs due to constructive interference between the direct wave and the reflected wave.

**Hint**: Remember that constructive interference happens when two waves combine to produce a wave of greater amplitude.

2. **Condition for Constructive Interference**:
- For constructive interference to occur, the path difference between the direct wave and the reflected wave must be an integer multiple of the wavelength (λ). This can be expressed mathematically as:
\[ \text{Path Difference} = n\lambda \quad (n = 0, 1, 2, \ldots) \]

**Hint**: Think about how waves interact when they meet: they can either add up (constructive) or cancel out (destructive).

3. **Calculating Phase Difference**:
- The phase difference (Δφ) corresponding to a path difference of λ is given by:
\[ \Delta \phi = \frac{2\pi}{\lambda} \times \text{Path Difference} \]
- For a path difference of λ, we substitute:
\[ \Delta \phi = \frac{2\pi}{\lambda} \times \lambda = 2\pi \]

**Hint**: Remember that a phase difference of 2π means the waves are in phase.

4. **Interpreting the Phase Difference**:
- A phase difference of 2π indicates that the two waves are in phase, meaning they reach their maximum and minimum values at the same time.

**Hint**: Visualize the waves: if they start at the same point and move together, they are in phase.

5. **Conclusion**:
- Therefore, for maximum intensity at the receiving antenna, the direct wave and the reflected wave must be in phase.

**Final Answer**: The answer is "in phase".