Equations of two progressive wave are given by `y_(1) = asin (omega t + phi_(1))` and `y_(2) = a sin (omegat + phi_(2))`. IF amplitude and time period of resultant wave is same as that of both the waves, then `(phi_(1)-phi_(2))` is

Options

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

Explanation

To solve the problem, we need to find the difference in phase angles \((\phi_1 - \phi_2)\) given the equations of two progressive waves and the condition that the resultant wave has the same amplitude and time period as the individual waves.

### Step-by-Step Solution:

1. **Identify the Given Waves**:
The equations of the two progressive waves are:
\[ y_1 = a \sin(\omega t + \phi_1) \]
\[ y_2 = a \sin(\omega t + \phi_2) \]

2. **Resultant Wave**:
The resultant wave \(y\) due to the superposition of \(y_1\) and \(y_2\) can be expressed as:
\[ y = y_1 + y_2 = a \sin(\omega t + \phi_1) + a \sin(\omega t + \phi_2) \]

3. **Using the Phasor Representation**:
The two waves can be represented as vectors (phasors) in a complex plane. The amplitude of the resultant wave can be found using the formula:
\[ R = \sqrt{A_1^2 + A_2^2 + 2A_1A_2 \cos(\phi_2 - \phi_1)} \]
where \(A_1 = A_2 = a\). Therefore,
\[ R = \sqrt{a^2 + a^2 + 2a \cdot a \cos(\phi_2 - \phi_1)} \]
Simplifying this gives:
\[ R = \sqrt{2a^2 + 2a^2 \cos(\phi_2 - \phi_1)} = a \sqrt{2 + 2 \cos(\phi_2 - \phi_1)} \]

4. **Condition for Resultant Amplitude**:
According to the problem, the resultant amplitude is the same as the original amplitude \(a\). Therefore, we set:
\[ a \sqrt{2 + 2 \cos(\phi_2 - \phi_1)} = a \]
Dividing both sides by \(a\) (assuming \(a \neq 0\)):
\[ \sqrt{2 + 2 \cos(\phi_2 - \phi_1)} = 1 \]

5. **Squaring Both Sides**:
Squaring both sides gives:
\[ 2 + 2 \cos(\phi_2 - \phi_1) = 1 \]
Rearranging this results in:
\[ 2 \cos(\phi_2 - \phi_1) = 1 - 2 \]
\[ 2 \cos(\phi_2 - \phi_1) = -1 \]
\[ \cos(\phi_2 - \phi_1) = -\frac{1}{2} \]

6. **Finding the Phase Difference**:
The cosine of an angle is \(-\frac{1}{2}\) at angles of \(120^\circ\) or \(240^\circ\). Therefore, we can write:
\[ \phi_2 - \phi_1 = 120^\circ \quad \text{or} \quad \phi_2 - \phi_1 = 240^\circ \]

7. **Calculating \(\phi_1 - \phi_2\)**:
From the above, we can express:
\[ \phi_1 - \phi_2 = -120^\circ \quad \text{or} \quad \phi_1 - \phi_2 = -240^\circ \]
Thus, the difference \((\phi_1 - \phi_2)\) can be written as:
\[ \phi_1 - \phi_2 = -120^\circ \quad \text{(or equivalently, } 240^\circ \text{)} \]

### Final Answer:
\[ \phi_1 - \phi_2 = -120^\circ \quad \text{(or } 240^\circ \text{)} \]