Two waves of same amplitude a and frequency v and having a phase difference of `pi//2` radian, are superposed. The amplitude of resultant wave is

Options

A. 2a
B. a
C. `(a)/(sqrt(2))`
D. `sqrt(2)a`

Answer

To find the amplitude of the resultant wave when two waves of the same amplitude and frequency are superposed with a phase difference of \( \frac{\pi}{2} \) radians, we can follow these steps:

### Step 1: Identify the Given Data
We are given:
- Amplitude of the first wave, \( A_1 = A \)
- Amplitude of the second wave, \( A_2 = A \)
- Phase difference, \( \phi = \frac{\pi}{2} \) radians

### Step 2: Use the Formula for Resultant Amplitude
The formula for the resultant amplitude \( R \) when two waves are superposed is given by:
\[ R = \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos \phi} \]

### Step 3: Substitute the Values into the Formula
Substituting the values into the formula:
\[ R = \sqrt{A^2 + A^2 + 2 A \cdot A \cdot \cos\left(\frac{\pi}{2}\right)} \]

### Step 4: Simplify the Expression
Now, simplify the expression:
\[ R = \sqrt{A^2 + A^2 + 2 A^2 \cos\left(\frac{\pi}{2}\right)} \]
Since \( \cos\left(\frac{\pi}{2}\right) = 0 \), the equation simplifies to:
\[ R = \sqrt{A^2 + A^2 + 0} \]
\[ R = \sqrt{2A^2} \]

### Step 5: Final Calculation
Taking the square root:
\[ R = \sqrt{2} A \]

### Conclusion
The amplitude of the resultant wave is:
\[ R = \sqrt{2} A \]

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