A wave `y = a sin (omegat - kx)` on a string meets with another wave producing a node at `x = 0`. Then the equation of the unknown wave is

Options

A. (a) `y = a sin (omegat + kx)`
B. (b) `y = -a sin (omegat + kx)`
C. ( c ) `y = a sin (omegat - kx)`
D. (d) `y = -a sin (omegat - kx)`

Explanation

To find the equation of the unknown wave that produces a node at \( x = 0 \) when it meets the wave \( y = a \sin(\omega t - kx) \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Condition for a Node**:
- A node occurs when the resultant amplitude of two overlapping waves is zero. This means that the two waves must cancel each other out at that point.

2. **Identifying the Direction of the Given Wave**:
- The wave given is \( y = a \sin(\omega t - kx) \). This wave is traveling in the positive x-direction.

3. **Determining the Nature of the Unknown Wave**:
- To create a node at \( x = 0 \), we need another wave traveling in the opposite direction (negative x-direction). This wave will interfere with the first wave.

4. **Considering Phase Change**:
- When a wave reflects off a boundary (like a fixed end), it undergoes a phase change of \( \pi \) radians (or 180 degrees). This means that the reflected wave will have a negative amplitude.

5. **Writing the Equation of the Reflected Wave**:
- The general form of a wave traveling in the negative x-direction can be written as \( y = a \sin(\omega t + kx) \).
- However, due to the phase change upon reflection, the equation of the reflected wave becomes:
\[ y = -a \sin(\omega t + kx) \]

6. **Final Equation of the Unknown Wave**:
- Thus, the equation of the unknown wave that meets the condition of producing a node at \( x = 0 \) is:
\[ y = -a \sin(\omega t + kx) \]

### Conclusion:
The equation of the unknown wave is \( y = -a \sin(\omega t + kx) \).