Two waves having equations
`x_1=asin(omegat+phi_1)`, ` x_2=asin(omegat+phi_2)`
If in the resultant wave the frequency and amplitude remain equal to those of superimposing waves. Then phase difference between them is

Options

A. `(pi)/(6)`
B. `(2pi)/(3)`
C. `(pi)/(4)`
D. `(pi)/(3)`

Explanation

To solve the problem, we need to analyze the given wave equations and the conditions provided. Let's break it down step by step.

### Step 1: Write down the wave equations
The two waves are given by:
- \( x_1 = a \sin(\omega t + \phi_1) \)
- \( x_2 = a \sin(\omega t + \phi_2) \)

### Step 2: Identify the parameters
Here, \( a \) is the amplitude, \( \omega \) is the angular frequency, and \( \phi_1 \) and \( \phi_2 \) are the phase constants of the two waves.

### Step 3: Understand the condition for resultant wave
The problem states that the resultant wave has the same frequency and amplitude as the individual waves. This means that when these two waves superimpose, the resultant amplitude \( A_r \) must equal \( a \).

### Step 4: Use the formula for resultant amplitude
The formula for the resultant amplitude \( A_r \) of two waves with the same amplitude \( a \) and a phase difference \( \theta \) is:
\[ A_r = \sqrt{a^2 + a^2 + 2a \cdot a \cos(\theta)} \]
This simplifies to:
\[ A_r = \sqrt{2a^2(1 + \cos(\theta))} = a \sqrt{2(1 + \cos(\theta))} \]

### Step 5: Set the resultant amplitude equal to the individual amplitude
Since the resultant amplitude must equal \( a \):
\[ a \sqrt{2(1 + \cos(\theta))} = a \]

### Step 6: Divide both sides by \( a \) (assuming \( a \neq 0 \))
\[ \sqrt{2(1 + \cos(\theta))} = 1 \]

### Step 7: Square both sides to eliminate the square root
\[ 2(1 + \cos(\theta)) = 1 \]

### Step 8: Solve for \( \cos(\theta) \)
\[ 1 + \cos(\theta) = \frac{1}{2} \]
\[ \cos(\theta) = \frac{1}{2} - 1 = -\frac{1}{2} \]

### Step 9: Determine the phase difference
The value of \( \theta \) for which \( \cos(\theta) = -\frac{1}{2} \) is:
\[ \theta = \frac{2\pi}{3} \text{ or } \theta = \frac{4\pi}{3} \]
However, since we are looking for the phase difference between the two waves, we can take:
\[ \theta = \frac{2\pi}{3} \]

### Conclusion
Thus, the phase difference between the two waves is:
\[ \theta = \frac{2\pi}{3} \]