Which of the following travelling wave will produce standing wave , with nodes at `x = 0`, when superimosed on ` y = A sin ( omega t - kx)`

Options

A. `A sin ( omega t + kx)`
B. `A sin ( omega t + kx + pi)`
C. `A cos (omega t + kx)`
D. `A cos ( omega t + kx + pi)`

Explanation

To determine which traveling wave will produce a standing wave with nodes at \( x = 0 \) when superimposed on the wave \( y = A \sin(\omega t - kx) \), we can follow these steps:

### Step 1: Understand the Requirement for Standing Waves
For two waves to produce a standing wave, they must have the same amplitude and frequency. Additionally, one wave must travel in the positive x-direction while the other travels in the negative x-direction.

### Step 2: Identify the Given Wave
The given wave is:
\[ y_1 = A \sin(\omega t - kx) \]
This wave travels in the positive x-direction.

### Step 3: Formulate the Second Wave
Let the second wave traveling in the negative x-direction be represented as:
\[ y_2 = A \sin(\omega t + kx + \phi) \]
where \( \phi \) is the phase constant we need to determine.

### Step 4: Superimpose the Two Waves
The resultant wave \( y_r \) when these two waves are superimposed is:
\[ y_r = y_1 + y_2 = A \sin(\omega t - kx) + A \sin(\omega t + kx + \phi) \]

### Step 5: Use the Sine Addition Formula
Using the sine addition formula, we can rewrite the sum:
\[ y_r = 2A \sin\left(\omega t + \frac{\phi}{2}\right) \cos\left(kx + \frac{\phi}{2}\right) \]

### Step 6: Set Condition for Nodes at \( x = 0 \)
For there to be a node at \( x = 0 \), the displacement must be zero when \( x = 0 \):
\[ 0 = 2A \sin\left(\omega t + \frac{\phi}{2}\right) \cos\left(\frac{\phi}{2}\right) \]
This equation will hold true if either \( \sin\left(\omega t + \frac{\phi}{2}\right) = 0 \) or \( \cos\left(\frac{\phi}{2}\right) = 0 \).

### Step 7: Solve for Phase Constant \( \phi \)
To ensure there is a node at \( x = 0 \), we focus on the second condition:
\[ \cos\left(\frac{\phi}{2}\right) = 0 \]
This implies:
\[ \frac{\phi}{2} = \frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z}) \]
Taking the simplest case \( n = 0 \):
\[ \frac{\phi}{2} = \frac{\pi}{2} \implies \phi = \pi \]

### Step 8: Write the Equation of the Second Wave
The equation of the second wave becomes:
\[ y_2 = A \sin(\omega t + kx + \pi) = A \sin(\omega t + kx) \cdot (-1) = -A \sin(\omega t + kx) \]
This indicates that the second wave is \( -A \sin(\omega t + kx) \).

### Conclusion
The traveling wave that will produce a standing wave with nodes at \( x = 0 \) when superimposed on \( y = A \sin(\omega t - kx) \) is:
\[ y = -A \sin(\omega t + kx) \]
Thus, the correct option is the one that represents this wave.