When two waves of the same amplitude and frequency but having a phase difference of `phi`, travelling with the same speed in the same direction (positive x), interfere, then

Options

A. their resultant amplitude will be twice that of a single wave but the frequency will be same
B. their resultant amplitude will both be twice that of a single wave
C. their resultant amplitude will depends on the phase angle while the frequency will be the same
D. the frequency and amplitude of the resultant wave will depends upon the phase angle.

Explanation

To solve the problem of interference of two waves with the same amplitude and frequency but with a phase difference of \( \phi \), we can follow these steps:

### Step 1: Understand the Given Information
We have two waves:
- Same amplitude \( A \)
- Same frequency \( f \)
- Phase difference \( \phi \)
- Traveling in the same direction

### Step 2: Write the General Equation for Each Wave
The general form of a wave can be expressed as:
- Wave 1: \( y_1 = A \sin(kx - \omega t) \)
- Wave 2: \( y_2 = A \sin(kx - \omega t + \phi) \)

Where:
- \( A \) is the amplitude
- \( k \) is the wave number
- \( \omega \) is the angular frequency
- \( x \) is the position
- \( t \) is the time

### Step 3: Find the Resultant Wave
To find the resultant wave \( y \), we add the two waves:
\[ y = y_1 + y_2 \]
Using the sine addition formula, we can express this as:
\[ y = A \sin(kx - \omega t) + A \sin(kx - \omega t + \phi) \]

### Step 4: Use the Formula for Resultant Amplitude
The resultant amplitude \( A_R \) of two interfering waves can be calculated using the formula:
\[ A_R = \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos(\phi)} \]
Since both waves have the same amplitude \( A \):
\[ A_R = \sqrt{A^2 + A^2 + 2A \cdot A \cos(\phi)} \]
This simplifies to:
\[ A_R = \sqrt{2A^2(1 + \cos(\phi))} \]
Using the identity \( 1 + \cos(\phi) = 2 \cos^2(\phi/2) \):
\[ A_R = \sqrt{2A^2 \cdot 2 \cos^2(\phi/2)} \]
Thus:
\[ A_R = 2A \cos(\phi/2) \]

### Step 5: Determine the Frequency of the Resultant Wave
The frequency of the resultant wave remains the same as that of the individual waves since they are both of the same frequency:
\[ f_R = f \]

### Conclusion
- The resultant amplitude \( A_R \) depends on the phase difference \( \phi \) and is given by \( A_R = 2A \cos(\phi/2) \).
- The frequency of the resultant wave is the same as the frequency of the individual waves.

### Final Answer
The correct conclusion is that the resultant amplitude depends on the phase difference while the frequency remains the same.

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