Two sinusoidal waves have the same angular frequeny the same amplitude `y_m`, and travel in the same direction in the same medium. If they differ in phase by `50^@` the amplitude the resultant wave is given by

Options

A. `0.64 y_(m)`
B. `1.3y_(m)`
C. `0.91 y_(m)`
D. `1.8 y_(m)`

Answer

To find the amplitude of the resultant wave formed by the interference of two sinusoidal waves that have the same angular frequency and amplitude but differ in phase by \(50^\circ\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Amplitude of both waves: \(y_m\)
- Phase difference (\(\phi\)): \(50^\circ\)

2. **Use the Formula for Resultant Amplitude:**
The resultant amplitude \(A\) of two coherent waves can be calculated using the formula:
\[ A = \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos(\phi)} \]
Here, \(A_1\) and \(A_2\) are the amplitudes of the two waves, and \(\phi\) is the phase difference.

3. **Substitute the Values:**
Since both waves have the same amplitude \(y_m\):
\[ A_1 = y_m, \quad A_2 = y_m \]
Substitute these into the formula:
\[ A = \sqrt{y_m^2 + y_m^2 + 2 y_m y_m \cos(50^\circ)} \]

4. **Simplify the Expression:**
This simplifies to:
\[ A = \sqrt{2y_m^2 + 2y_m^2 \cos(50^\circ)} \]
Factor out \(2y_m^2\):
\[ A = \sqrt{2y_m^2(1 + \cos(50^\circ))} \]

5. **Calculate \(\cos(50^\circ)\):**
Using a calculator, we find:
\[ \cos(50^\circ) \approx 0.643 \]
Substitute this value into the equation:
\[ A = \sqrt{2y_m^2(1 + 0.643)} = \sqrt{2y_m^2(1.643)} \]

6. **Final Calculation:**
\[ A = \sqrt{3.286y_m^2} = \sqrt{3.286} \cdot y_m \approx 1.81y_m \]

Thus, the amplitude of the resultant wave is approximately:
\[ A \approx 1.81y_m \]

### Final Answer:
The amplitude of the resultant wave is \( \approx 1.81y_m \).