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

Explanation

To analyze the given equations of the waves and determine which statements are correct, we will go through the equations step by step.

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

### Step 2: Analyze the Waves
- **Wave 1 and Wave 3**: Both are sine waves traveling in the positive x-direction.
- **Wave 2 and Wave 4**: Both are cosine waves traveling in the negative x-direction.

### Step 3: Superposition of Waves
1. **Superposition of \( y_1 \) and \( y_2 \)**:
- \( y_1 \) travels in the positive x-direction and \( y_2 \) travels in the negative x-direction.
- Since they are traveling in opposite directions, they will form a standing wave.
- The resultant amplitude can be calculated using the Pythagorean theorem since they are 90 degrees out of phase (sine and cosine).
- Resultant Amplitude: \( A_{resultant} = \sqrt{a^2 + a^2} = a\sqrt{2} \).

2. **Superposition of \( y_2 \) and \( z_3 \)**:
- \( y_2 \) (cosine) and \( z_3 \) (sine) cannot be superimposed directly as they are not traveling in opposite directions. They will not form a standing wave.

3. **Superposition of \( y_1 \) and \( z_2 \)**:
- Similar to the first case, \( y_1 \) and \( z_2 \) will also form a standing wave since they are traveling in opposite directions.
- The resultant amplitude will also be \( a\sqrt{2} \).

4. **Superposition of \( z_1 \) and \( z_2 \)**:
- \( z_1 \) and \( z_2 \) are both sine and cosine waves traveling in opposite directions.
- They will form a standing wave as well.

### Step 4: Conclusion
Based on the analysis:
- **Statement A**: True, a traveling wave is formed from \( y_1 \) and \( y_2 \) with the resultant amplitude \( a\sqrt{2} \).
- **Statement B**: False, superposition of \( y_2 \) and \( z_3 \) is possible.
- **Statement C**: False, the superposition of \( y_1 \) and \( y_2 \) does not yield a traveling wave.
- **Statement D**: True, a transverse wave is formed from \( z_1 \) and \( z_2 \).

### Final Answer
The correct statements are A and D.