Following are equations of four waves :
(i) `y_(1) = a sin omega ( t - (x)/(v))`
(ii) `y_(2) = a cos omega ( t + (x)/(v))`
(iii) `z_(1) = a sin omega ( t - (x)/(v))`
(iv) `z_(1) = a cos omega ( t + (x)/(v))`
Which of the following statements are correct ?

Options

A. On superposition of wave `(i)` and `(iii)`, a travelling wave having amplitude `asqrt(2)` will be formed
B. Superposition of waves `(ii)` and `(iii)` is not possible
C. On superposition of `(i)` and `(ii)`, a stationary wave having amplitude `asqrt(2)` will be formed
D. On superposition of `(iii)`, and `(iv)`, a transervers stationary wave will be formed

Explanation

To analyze the given wave equations and determine the correctness of the statements, we will break down the problem step by step.

### Step 1: Identify the wave equations
The equations of the waves are:
1. \( y_1 = a \sin\left(\omega t - \frac{x}{v}\right) \)
2. \( y_2 = a \cos\left(\omega t + \frac{x}{v}\right) \)
3. \( z_1 = a \sin\left(\omega t - \frac{x}{v}\right) \)
4. \( z_2 = a \cos\left(\omega t + \frac{x}{v}\right) \)

### Step 2: Determine the direction of wave propagation
- For \( y_1 \) and \( z_1 \) (both have the form \( \sin(\omega t - \frac{x}{v}) \)):
- The negative sign indicates that these waves are traveling in the positive x-direction.

- For \( y_2 \) and \( z_2 \) (both have the form \( \cos(\omega t + \frac{x}{v}) \)):
- The positive sign indicates that these waves are traveling in the negative x-direction.

### Step 3: Analyze the statements
1. **Statement A**: "Waves 1 and 3 will form a traveling wave having amplitude \( a\sqrt{2} \)."
- **Analysis**:
- Both \( y_1 \) and \( z_1 \) are traveling in the same direction (positive x-direction).
- When two waves of the same amplitude \( a \) travel in the same direction, the resultant amplitude is \( \sqrt{a^2 + a^2} = a\sqrt{2} \).
- **Conclusion**: This statement is **correct**.

2. **Statement B**: "Superposition of waves 2 and 3 is not possible."
- **Analysis**:
- \( y_2 \) (cosine wave) and \( z_1 \) (sine wave) can superpose since they are independent of each other and can exist in the same space.
- **Conclusion**: This statement is **incorrect**.

3. **Statement C**: "Superposition of waves 1 and 2 represents a stationary wave having amplitude \( a\sqrt{2} \)."
- **Analysis**:
- \( y_1 \) travels in the positive x-direction and \( y_2 \) travels in the negative x-direction.
- The superposition of two waves traveling in opposite directions can form a stationary wave, but the amplitude will not be \( a\sqrt{2} \) since they are not at 90 degrees to each other.
- **Conclusion**: This statement is **incorrect**.

4. **Statement D**: "Superposition of waves 3 and 4 will form a transverse stationary wave."
- **Analysis**:
- \( z_1 \) and \( z_2 \) are traveling in opposite directions (one in positive x-direction and the other in negative x-direction).
- Their superposition will indeed form a stationary wave.
- **Conclusion**: This statement is **correct**.

### Final Conclusion
- The correct statements are A and D.