Two waves of equal amplitude A, and equal frequency travel in the same direction in a medium. The amplitude of the resultant wave is

Options

A. 0
B. A
C. 2A
D. between 0 and 2A

Answer

To find the amplitude of the resultant wave when two waves of equal amplitude \( A \) and equal frequency travel in the same direction, we can follow these steps:

### Step 1: Understand the formula for resultant amplitude
The amplitude of the resultant wave \( A_R \) when two waves of amplitudes \( A_1 \) and \( A_2 \) interfere can be expressed using the formula:
\[ A_R = \sqrt{A_1^2 + A_2^2 + 2A_1A_2 \cos \phi} \]
where \( \phi \) is the phase difference between the two waves.

### Step 2: Substitute the values of \( A_1 \) and \( A_2 \)
Since both waves have equal amplitude \( A \), we can set \( A_1 = A \) and \( A_2 = A \). Thus, the formula becomes:
\[ A_R = \sqrt{A^2 + A^2 + 2A \cdot A \cos \phi} \]

### Step 3: Simplify the expression
This simplifies to:
\[ A_R = \sqrt{2A^2 + 2A^2 \cos \phi} \]
or
\[ A_R = \sqrt{2A^2(1 + \cos \phi)} \]

### Step 4: Analyze the maximum and minimum values
1. **Maximum Amplitude**: The maximum value of \( A_R \) occurs when \( \cos \phi = 1 \) (i.e., when the waves are in phase). Therefore:
\[ A_R = \sqrt{2A^2(1 + 1)} = \sqrt{4A^2} = 2A \]

2. **Minimum Amplitude**: The minimum value of \( A_R \) occurs when \( \cos \phi = -1 \) (i.e., when the waves are out of phase). Therefore:
\[ A_R = \sqrt{2A^2(1 - 1)} = \sqrt{0} = 0 \]

### Step 5: Conclusion
Thus, the amplitude of the resultant wave varies between 0 and \( 2A \). Therefore, the resultant amplitude can be expressed as:
\[ 0 \leq A_R \leq 2A \]