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 cos (kx-omegat)`
B. `-a cos (kx+omegat)`
C. `-a cos (kx-omegat)`
D. `+a cos (omegat -kx)`

Explanation

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

### Step-by-Step Solution:

1. **Understanding the Node Condition**:
A node occurs at a point where the displacement of the wave is always zero. For this to happen at \( x = 0 \), the two waves must interfere destructively.

2. **Identifying the Given Wave**:
The given wave is \( y_1 = a \sin(\omega t - kx) \). This wave travels in the positive x-direction.

3. **Forming the Reflected Wave**:
To create a node at \( x = 0 \), we need to consider a reflected wave. The reflected wave will travel in the opposite direction (negative x-direction) and will have a phase change of \( \pi \) (which corresponds to a half-wavelength shift).

4. **Writing the Equation for the Reflected Wave**:
The reflected wave can be expressed as:
\[ y_2 = -a \sin(\omega t - kx) \]
Since the sine function has a property that \( \sin(\theta + \pi) = -\sin(\theta) \), we can rewrite the reflected wave as:
\[ y_2 = a \sin(\omega t + kx) \]

5. **Superposition of the Waves**:
The total displacement at any point is the sum of the displacements due to both waves:
\[ y = y_1 + y_2 = a \sin(\omega t - kx) + (-a \sin(\omega t - kx)) \]
This simplifies to:
\[ y = a \sin(\omega t - kx) - a \sin(\omega t - kx) = 0 \]
Therefore, at \( x = 0 \), the total displacement is zero, confirming a node.

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

### Final Answer:
The equation of the unknown wave is:
\[ y = -a \sin(\omega t + kx) \]