Two waves
`y_1 = A sin (omegat - kx)`
and ` y_2 = A sin (omegat + kx)`
superimpose to produce a stationary wave, then

Options

A. `x =0` is a node
B. `x = 0` is an antinode.
C. `x= pi/k` is a node.
D. `pi = (2pi)/k` is an antinode

Explanation

To solve the problem, we need to analyze the two given waves and their superposition to determine the positions of nodes and antinodes.

### Step-by-Step Solution:

1. **Identify the Waves**:
We have two waves given by:
\[ y_1 = A \sin(\omega t - kx) \]
\[ y_2 = A \sin(\omega t + kx) \]

2. **Superimpose the Waves**:
To find the resultant wave \(y\), we add \(y_1\) and \(y_2\):
\[ y = y_1 + y_2 = A \sin(\omega t - kx) + A \sin(\omega t + kx) \]

3. **Use the Trigonometric Identity**:
We can use the trigonometric identity for the sum of sines:
\[ \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
Here, let \(A = \omega t - kx\) and \(B = \omega t + kx\):
\[ y = 2A \sin\left(\omega t\right) \cos\left(kx\right) \]

4. **Identify Nodes and Antinodes**:
The resultant wave \(y = 2A \sin(\omega t) \cos(kx)\) indicates that:
- The factor \(\cos(kx)\) determines the positions of nodes and antinodes.
- Nodes occur where \(\cos(kx) = 0\), which gives:
\[ kx = \frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z}) \]
This implies:
\[ x = \frac{\pi}{2k} + \frac{n\pi}{k} \]

- Antinodes occur where \(\cos(kx) = \pm 1\), which gives:
\[ kx = n\pi \quad (n \in \mathbb{Z}) \]
This implies:
\[ x = \frac{n\pi}{k} \]

5. **Evaluate Specific Points**:
- For \(x = 0\):
\[ \cos(0) = 1 \quad \text{(Antinode)} \]
- For \(x = \frac{\pi}{2k}\):
\[ \cos\left(\frac{\pi}{2}\right) = 0 \quad \text{(Node)} \]
- For \(x = \pi\):
\[ \cos(\pi) = -1 \quad \text{(Antinode)} \]
- For \(x = \frac{2\pi}{k}\):
\[ \cos(2\pi) = 1 \quad \text{(Antinode)} \]

6. **Conclusion**:
- At \(x = 0\) is an **Antinode**.
- At \(x = \frac{\pi}{2k}\) is a **Node**.
- At \(x = \pi\) is an **Antinode**.
- At \(x = \frac{2\pi}{k}\) is an **Antinode**.

### Final Answer:
- \(x = 0\) is an **Antinode**.
- \(x = \frac{\pi}{2k}\) is a **Node**.
- \(x = \pi\) is an **Antinode**.
- \(x = \frac{2\pi}{k}\) is an **Antinode**.