The light waves from two independent monochromatic light sources are given by, `y_(1)=2 sin omega t` and `y_(2)=3 cos omegat`. Then the correct statement is

Options

A. both the wave are coherent
B. both the waves are incoherent
C. both the waves are in the same phase.
D. both the waves have same time period.

Explanation

To solve the problem, we need to analyze the two given wave equations and determine their characteristics, particularly focusing on coherence. The wave equations are:

1. \( y_1 = 2 \sin(\omega t) \)
2. \( y_2 = 3 \cos(\omega t) \)

### Step 1: Identify the nature of the waves
Both waves are monochromatic, meaning they have a single frequency.

### Step 2: Convert the cosine function to a sine function
We can express the second wave \( y_2 \) in terms of sine:
\[ y_2 = 3 \cos(\omega t) = 3 \sin\left(\frac{\pi}{2} + \omega t\right) \]
This shows that \( y_2 \) can be written as a sine wave with a phase shift of \( \frac{\pi}{2} \).

### Step 3: Determine the frequency and time period
For both waves:
- The angular frequency \( \omega \) is the same.
- The time period \( T \) for both waves is given by:
\[ T = \frac{2\pi}{\omega} \]
- The frequency \( f \) is:
\[ f = \frac{\omega}{2\pi} \]
Since both waves have the same angular frequency, they also have the same time period and frequency.

### Step 4: Analyze the phase difference
The first wave \( y_1 \) has a phase of \( 0 \) (since it is \( \sin(\omega t) \)), while the second wave \( y_2 \) has a phase of \( \frac{\pi}{2} \) (since it is \( \sin(\omega t + \frac{\pi}{2}) \)). Thus, the phase difference \( \Delta \phi \) between the two waves is:
\[ \Delta \phi = \frac{\pi}{2} - 0 = \frac{\pi}{2} \]
This indicates that there is a constant phase difference between the two waves.

### Step 5: Determine coherence
For two waves to be coherent, they must have the same frequency and a constant phase difference. Since both conditions are satisfied (same frequency and a constant phase difference of \( \frac{\pi}{2} \)), we can conclude that the two waves are coherent.

### Conclusion
Based on the analysis, the correct statement regarding the two light waves is that they are coherent due to having the same frequency and a constant phase difference.