Two identical sinusoidal waves each of amplitude 5 mm with a phase difference of `pi//2` are traveling in the same direction in a string. The amplitude of the resultant wave (in mm) is

Options

A. zero
B. `5sqrt(2)`
C. `(5)/(sqrt(2))`
D. `2.5

Explanation

To find the amplitude of the resultant wave when two identical sinusoidal waves interfere with a phase difference, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Amplitude of each wave, \( A_1 = A_2 = 5 \, \text{mm} \)
- Phase difference, \( \phi = \frac{\pi}{2} \)

2. **Use the Formula for Resultant Amplitude**:
The formula for the resultant amplitude \( A \) when two waves with amplitudes \( A_1 \) and \( A_2 \) interfere with a phase difference \( \phi \) is given by:
\[ A = \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos(\phi)} \]

3. **Substitute the Values into the Formula**:
Since both waves have the same amplitude:
\[ A = \sqrt{5^2 + 5^2 + 2 \cdot 5 \cdot 5 \cdot \cos\left(\frac{\pi}{2}\right)} \]

4. **Calculate the Cosine of the Phase Difference**:
We know that:
\[ \cos\left(\frac{\pi}{2}\right) = 0 \]
Therefore, the equation simplifies to:
\[ A = \sqrt{5^2 + 5^2 + 2 \cdot 5 \cdot 5 \cdot 0} \]

5. **Simplify the Expression**:
This further simplifies to:
\[ A = \sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50} \]

6. **Final Calculation**:
We can express \( \sqrt{50} \) as:
\[ A = \sqrt{25 \cdot 2} = 5\sqrt{2} \]

7. **State the Result**:
Therefore, the amplitude of the resultant wave is:
\[ A = 5\sqrt{2} \, \text{mm} \]

### Final Answer:
The amplitude of the resultant wave is \( 5\sqrt{2} \, \text{mm} \).

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