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

Explanation

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

### Step 1: Understand the given wave
The given wave is:
\[ y = A \sin(\omega t - kx) \]
This is a traveling wave moving in the positive x-direction.

### Step 2: Identify the condition for standing waves
A standing wave with nodes at \( x = 0 \) can be represented as:
\[ y = -2A \sin(kx) \cos(\omega t) \]
This form indicates that at \( x = 0 \), the sine function will yield zero, creating a node.

### Step 3: Set up the equation for superposition
Let the wave that we need to find be \( y_1 \). The superposition of the two waves should yield the standing wave:
\[ y + y_1 = -2A \sin(kx) \cos(\omega t) \]
Substituting \( y \):
\[ A \sin(\omega t - kx) + y_1 = -2A \sin(kx) \cos(\omega t) \]

### Step 4: Isolate \( y_1 \)
Rearranging gives us:
\[ y_1 = -2A \sin(kx) \cos(\omega t) - A \sin(\omega t - kx) \]

### Step 5: Simplify \( y_1 \)
Factoring out \( -A \):
\[ y_1 = -A \left(2 \sin(kx) \cos(\omega t) + \sin(\omega t - kx)\right) \]

### Step 6: Use trigonometric identities
Using the identity for sine:
\[ \sin(\omega t - kx) = \sin(\omega t) \cos(kx) - \cos(\omega t) \sin(kx) \]
Substituting this into the equation gives:
\[ y_1 = -A \left(2 \sin(kx) \cos(\omega t) + \sin(\omega t) \cos(kx) - \cos(\omega t) \sin(kx)\right) \]
Combining terms:
\[ y_1 = -A \left((2 \sin(kx) - \sin(kx)) \cos(\omega t) + \sin(\omega t) \cos(kx)\right) \]
This simplifies to:
\[ y_1 = -A \left(\sin(kx) \cos(\omega t) + \sin(\omega t) \cos(kx)\right) \]

### Step 7: Final form of \( y_1 \)
Using the sine addition formula:
\[ y_1 = -A \sin(\omega t + kx) \]
Thus, the wave that needs to be superimposed is:
\[ y_1 = A \sin(\omega t + kx + \pi) \]
This indicates that the correct option is the one that represents this wave.

### 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:
\[ y_1 = A \sin(\omega t + kx + \pi) \]