A transverse waves is travelling in a string. Study following statement.
(i) Equation of the wave is equal to the shape of the string at an instant t. (ii) Equation of thhe wave is general equation for displacement of a particle of the string (iii) Equation of the wave must be sinusoidal equation (iv) Equation of the wave is an equation for displacement of the particle at one end only.correct statement are

Options

A. (i) and (ii)
B. (ii) and (iii)
C. (i) and (iii)
D. (ii) and (iv)

Explanation

To solve the question, we need to analyze each of the four statements regarding the transverse wave traveling in a string and determine which ones are correct.

### Step-by-Step Solution:

1. **Statement (i): "Equation of the wave is equal to the shape of the string at an instant t."**
- The wave equation can be expressed as \( y = A \sin(\omega t - kx) \).
- If we fix time \( t \) (e.g., \( t = 2 \) seconds), we can express the equation as \( y = A \sin(2\omega - kx) \).
- In this form, \( y \) is a function of \( x \) for a fixed \( t \), which represents the shape of the wave at that instant.
- **Conclusion:** This statement is **true**.

2. **Statement (ii): "Equation of the wave is general equation for displacement of a particle of the string."**
- The wave equation \( y = A \sin(\omega t - kx) \) describes the displacement of particles in the string as a function of both time and position.
- If we consider a specific position \( x \) (e.g., \( x = 1 \)), the equation becomes \( y = A \sin(\omega t - k) \), showing that the displacement at that position varies with time.
- **Conclusion:** This statement is **true**.

3. **Statement (iii): "Equation of the wave must be sinusoidal equation."**
- While many waves are sinusoidal, the general form of a wave can also be non-sinusoidal. For example, wave equations can take forms like \( y = (x - vt)^2 \) or \( y = (x - vt)^3 \), which are valid waveforms but not sinusoidal.
- **Conclusion:** This statement is **false**.

4. **Statement (iv): "Equation of the wave is an equation for displacement of the particle at one end only."**
- The wave equation describes the displacement of all particles along the string, not just at one end. Each point on the string has its own displacement as a function of time and position.
- **Conclusion:** This statement is **false**.

### Final Summary:
The correct statements are:
- Statement (i) is true.
- Statement (ii) is true.
- Statement (iii) is false.
- Statement (iv) is false.

Thus, the correct statements are **1 and 2**.