The condition for destructive interference between two waves is that their phase difference should be `"…………."`

Options

A. `0,pi,2pi"…………."`
B. `0,2pi,4pi "…………."`
C. `pi,3pi,5pi,"…….."`
D. `(pi)/(2),(3pi)/(2),(5pi)/(2) "……"`

Explanation

To determine the condition for destructive interference between two waves, we need to understand the concept of phase difference. Here’s a step-by-step solution:

### Step 1: Understanding Phase Difference
Phase difference (\( \Delta \phi \)) is the difference in the phase of two waves at a given point in time. It is measured in radians.

### Step 2: Condition for Destructive Interference
For two waves to interfere destructively, they must be out of phase. This means that when one wave is at its maximum displacement, the other wave should be at its minimum displacement.

### Step 3: Mathematical Representation
The condition for destructive interference can be mathematically expressed as:
\[ \Delta \phi = (2m + 1) \pi \]
where \( m \) is any integer (0, 1, 2, ...). This means that the phase difference can be \( \pi, 3\pi, 5\pi, \) etc.

### Step 4: Explanation of the Condition
- When \( \Delta \phi = \pi \), the waves are perfectly out of phase.
- When \( \Delta \phi = 3\pi \) or \( 5\pi \), the waves are still out of phase but have completed additional cycles.
- The odd multiples of \( \pi \) ensure that the peaks of one wave coincide with the troughs of the other, leading to cancellation of the resultant wave.

### Conclusion
Thus, the condition for destructive interference between two waves is that their phase difference should be:
\[ \Delta \phi = (2m + 1) \pi \]