If two waves having amplitudes 2 A and A and same frequency and velocity, propagate in the same direction in the same phase, the resulting amplitude will be

Answer

To find the resulting amplitude when two waves with amplitudes \(2A\) and \(A\) propagate in the same direction and in phase, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Amplitudes**:
- Let the amplitude of the first wave be \(A_1 = 2A\).
- Let the amplitude of the second wave be \(A_2 = A\).

2. **Determine the Phase Relationship**:
- Since the waves are in the same phase, the phase difference \(\phi = 0\) degrees.

3. **Use the Formula for Resultant Amplitude**:
- The formula for the resultant amplitude \(A_R\) when two waves interfere is given by:
\[ A_R = \sqrt{A_1^2 + A_2^2 + 2A_1A_2\cos(\phi)} \]

4. **Substitute the Values**:
- Substitute \(A_1 = 2A\), \(A_2 = A\), and \(\cos(0) = 1\) into the formula:
\[ A_R = \sqrt{(2A)^2 + (A)^2 + 2(2A)(A)(1)} \]

5. **Calculate Each Term**:
- Calculate \((2A)^2 = 4A^2\)
- Calculate \((A)^2 = A^2\)
- Calculate \(2(2A)(A)(1) = 4A^2\)

6. **Combine the Terms**:
- Now combine the terms inside the square root:
\[ A_R = \sqrt{4A^2 + A^2 + 4A^2} = \sqrt{9A^2} \]

7. **Final Calculation**:
- Taking the square root gives:
\[ A_R = 3A \]

### Final Result:
The resulting amplitude is \(3A\).

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