Statement-1 : The superposition of the waves `y_(1) = A sin(kx - omega t)` and `y_(2) = 3A sin (kx + omega t)` is a pure standing wave plus a travelling wave moving in the negative direction along X-axis
Statement-2 : The resultant of `y_(1)` & `y_(2)` is `y=y_(1) + y_(2) = 2A sin kx cos omega t + 2A sin (kx + omega t)`.

Options

A. Statement-1 is true, Statement-2 is true , Statement-2 is correct explanation for Statement-7
B. Statement-1 is true,Statement-2 is true , Statement-2 is NOT a correct explanation for Statement-7
C. Statement-1 is true, Statement-2 is false
D. Statement-1 is false, Statement-2 is true

Answer

To solve the problem, we need to analyze the two statements regarding the superposition of waves \( y_1 \) and \( y_2 \).

### Step 1: Write down the given waves
The two waves are given as:
- \( y_1 = A \sin(kx - \omega t) \)
- \( y_2 = 3A \sin(kx + \omega t) \)

### Step 2: Apply the principle of superposition
The principle of superposition states that the resultant displacement \( y \) due to the two waves is the sum of their individual displacements:
\[ y = y_1 + y_2 = A \sin(kx - \omega t) + 3A \sin(kx + \omega t) \]

### Step 3: Use trigonometric identities to simplify
We can use the sine addition and subtraction formulas:
- \( \sin(a \pm b) = \sin a \cos b \pm \cos a \sin b \)

Applying this to \( y_1 \) and \( y_2 \):
1. For \( y_1 \):
\[ y_1 = A \left( \sin(kx) \cos(\omega t) - \cos(kx) \sin(\omega t) \right) \]
2. For \( y_2 \):
\[ y_2 = 3A \left( \sin(kx) \cos(\omega t) + \cos(kx) \sin(\omega t) \right) \]

### Step 4: Combine the two waves
Now, substituting back into the equation for \( y \):
\[ y = A \left( \sin(kx) \cos(\omega t) - \cos(kx) \sin(\omega t) \right) + 3A \left( \sin(kx) \cos(\omega t) + \cos(kx) \sin(\omega t) \right) \]
Combining like terms:
\[ y = (A + 3A) \sin(kx) \cos(\omega t) + (3A - A) \cos(kx) \sin(\omega t) \]
\[ y = 4A \sin(kx) \cos(\omega t) + 2A \cos(kx) \sin(\omega t) \]

### Step 5: Identify the type of wave
The term \( 4A \sin(kx) \cos(\omega t) \) represents a standing wave, while the term \( 2A \cos(kx) \sin(\omega t) \) represents a traveling wave moving in the negative direction (since it has the form \( \sin(kx + \omega t) \)).

### Conclusion
- **Statement 1** is true: The superposition results in a pure standing wave plus a traveling wave moving in the negative direction.
- **Statement 2** is true: The resultant wave \( y \) is as derived above.

Both statements are correct, and statement 2 correctly explains statement 1.