A wave representing by the equation `y = A cos(kx - omegat)` is suerposed with another wave to form a stationary wave such that point `x = 0` is a node. The equation for the other wave is

Options

A. `y_(2)=-A sin(kx-omegat)`
B. `y_(2)=-A cos(kx+omegat)`
C. `y_(2)=A sin(kx-omegat)`
D. `y_(2)=A cos(kx-omegat)`

Answer

To solve the problem, we need to find the equation of the second wave that, when superposed with the first wave \( y_1 = A \cos(kx - \omega t) \), results in a stationary wave with a node at \( x = 0 \).

### Step-by-Step Solution:

1. **Understand the Concept of Stationary Waves**:
- A stationary wave is formed when two waves of the same frequency and amplitude travel in opposite directions. This means we will have one wave moving in the positive direction and the other in the negative direction.

2. **Identify the Given Wave**:
- The first wave is given by the equation:
\[ y_1 = A \cos(kx - \omega t) \]

3. **Formulate the Second Wave**:
- The second wave can be represented as:
\[ y_2 = A \cos(kx + \omega t) \quad \text{or} \quad y_2 = -A \cos(kx + \omega t) \]
- Since we need to ensure that there is a node at \( x = 0 \), we will explore both options.

4. **Check for Node Condition**:
- A node occurs when the resultant displacement \( y \) is zero at a specific position. Therefore, we need to check the resultant wave at \( x = 0 \).

5. **Calculate the Resultant Wave for \( y_2 = A \cos(kx + \omega t) \)**:
- The resultant wave \( y \) is:
\[ y = y_1 + y_2 = A \cos(kx - \omega t) + A \cos(kx + \omega t) \]
- Using the trigonometric identity:
\[ \cos A + \cos B = 2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
- We can rewrite the resultant as:
\[ y = 2A \cos(kx) \cos(\omega t) \]
- Now, substituting \( x = 0 \):
\[ y(0) = 2A \cos(0) \cos(\omega t) = 2A \cos(\omega t) \]
- This is not zero unless \( A = 0 \), which means this does not satisfy the node condition.

6. **Calculate the Resultant Wave for \( y_2 = -A \cos(kx + \omega t) \)**:
- The resultant wave becomes:
\[ y = A \cos(kx - \omega t) - A \cos(kx + \omega t) \]
- Using the identity again:
\[ y = -2A \sin(kx) \cos(\omega t) \]
- Now, substituting \( x = 0 \):
\[ y(0) = -2A \sin(0) \cos(\omega t) = 0 \]
- This satisfies the node condition.

7. **Conclusion**:
- Therefore, the equation for the other wave that forms a stationary wave with a node at \( x = 0 \) is:
\[ y_2 = -A \cos(kx + \omega t) \]

### Final Answer:
The equation for the other wave is:
\[ y_2 = -A \cos(kx + \omega t) \]