Two sine waves travel in the same direction in a medium. The amplitude of each wave is `A` and the phase difference between the two waves is `120^@`. The resultant amplitude will be

Options

A. A
B. 2A
C. 4A
D. `sqrt2A`

Answer

To find the resultant amplitude of two sine waves traveling in the same direction with a given amplitude and phase difference, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values**:
- Amplitude of each wave, \( A_1 = A \) and \( A_2 = A \).
- Phase difference, \( \phi = 120^\circ \).

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

3. **Substitute the Known Values**:
Since both amplitudes are equal, we can substitute \( A_1 = A \) and \( A_2 = A \):
\[ A_r = \sqrt{A^2 + A^2 + 2 A \cdot A \cos(120^\circ)} \]

4. **Calculate \( \cos(120^\circ) \)**:
The cosine of \( 120^\circ \) is:
\[ \cos(120^\circ) = -\frac{1}{2} \]

5. **Substitute \( \cos(120^\circ) \) into the Formula**:
Now, substituting \( \cos(120^\circ) \) into the equation:
\[ A_r = \sqrt{A^2 + A^2 + 2 A^2 \left(-\frac{1}{2}\right)} \]

6. **Simplify the Expression**:
Simplifying the expression inside the square root:
\[ A_r = \sqrt{A^2 + A^2 - A^2} \]
\[ A_r = \sqrt{A^2} \]

7. **Final Result**:
Therefore, the resultant amplitude is:
\[ A_r = A \]

### Conclusion:
The resultant amplitude of the two waves is \( A \).