Two coherent waves are represented by `y_(1)=a_(1)cos_(omega)` t and `y_(2)=a_(2)sin_(omega)`t. The resultant intensity due to interference will be

Options

A. `(a_(1) + a_(2))`
B. `(a_(1)-a_(2))`
C. `(a_(1)^(2) + a_(2)^(2))`
D. `(a_(1)^(2) + a_(2)^(2))`

Answer

To solve the problem of finding the resultant intensity due to interference of two coherent waves represented by \( y_1 = a_1 \cos(\omega t) \) and \( y_2 = a_2 \sin(\omega t) \), we can follow these steps:

### Step 1: Understand the Waves
We have two coherent waves:
- \( y_1 = a_1 \cos(\omega t) \)
- \( y_2 = a_2 \sin(\omega t) \)

### Step 2: Convert the Second Wave
We can express \( y_2 \) in terms of cosine:
\[ y_2 = a_2 \sin(\omega t) = a_2 \cos\left(\omega t - \frac{\pi}{2}\right) \]
This shows that \( y_2 \) is a cosine wave with a phase shift of \( -\frac{\pi}{2} \).

### Step 3: Resultant Wave Equation
The resultant wave \( y \) can be expressed as:
\[ y = y_1 + y_2 = a_1 \cos(\omega t) + a_2 \cos\left(\omega t - \frac{\pi}{2}\right) \]

### Step 4: Use the Amplitude Formula
The resultant amplitude \( A \) of two waves can be calculated using the formula:
\[ A = \sqrt{a_1^2 + a_2^2 + 2a_1a_2 \cos(\phi)} \]
where \( \phi \) is the phase difference between the two waves. Here, \( \phi = -\frac{\pi}{2} \), and \( \cos\left(-\frac{\pi}{2}\right) = 0 \).

### Step 5: Calculate the Resultant Amplitude
Substituting the values into the amplitude formula:
\[ A = \sqrt{a_1^2 + a_2^2 + 2a_1a_2 \cdot 0} = \sqrt{a_1^2 + a_2^2} \]

### Step 6: Calculate the Resultant Intensity
The intensity \( I \) is proportional to the square of the amplitude:
\[ I \propto A^2 = (A)^2 = (a_1^2 + a_2^2) \]

### Final Result
Thus, the resultant intensity due to interference is:
\[ I \propto a_1^2 + a_2^2 \]