The amplitude factor of resulting wave, formed by superposition of two sinusoidal waves of equal amplitude a and constant phase difference is given by

Options

A. `2acosphi`
B. `2acos(phi/2)`
C. `2acos(phi/3`
D. `Zero`

Answer

To find the amplitude factor of the resulting wave formed by the superposition of two sinusoidal waves of equal amplitude \( A \) and constant phase difference \( \phi \), we can follow these steps:

### Step-by-Step Solution:

1. **Define the Two Waves**:
Let the two waves be:
\[ y_1 = A \sin(\omega t) \]
\[ y_2 = A \sin(\omega t + \phi) \]

2. **Superposition of the Waves**:
The resulting wave \( y \) from the superposition of these two waves is given by:
\[ y = y_1 + y_2 = A \sin(\omega t) + A \sin(\omega t + \phi) \]

3. **Factor Out the Amplitude**:
We can factor out \( A \):
\[ y = A \left( \sin(\omega t) + \sin(\omega t + \phi) \right) \]

4. **Use the Sine Addition Formula**:
We can use the sine addition formula:
\[ \sin C + \sin D = 2 \sin\left(\frac{C+D}{2}\right) \cos\left(\frac{C-D}{2}\right) \]
Here, let \( C = \omega t \) and \( D = \omega t + \phi \):
\[ y = A \left( 2 \sin\left(\frac{2\omega t + \phi}{2}\right) \cos\left(\frac{\phi}{2}\right) \right) \]

5. **Simplify the Result**:
This simplifies to:
\[ y = 2A \sin\left(\omega t + \frac{\phi}{2}\right) \cos\left(\frac{\phi}{2}\right) \]

6. **Identify the Amplitude Factor**:
The amplitude factor of the resulting wave is:
\[ \text{Amplitude} = 2A \cos\left(\frac{\phi}{2}\right) \]

### Final Answer:
The amplitude factor of the resulting wave formed by the superposition of two sinusoidal waves of equal amplitude \( A \) and constant phase difference \( \phi \) is:
\[ 2A \cos\left(\frac{\phi}{2}\right) \]