A travelling wave represented by
`y=Asin (omegat-kx)`
is superimposed on another wave represented by
`y=Asin(omegat+kx).` The resultant is

Options

A. standing wave having nodes at x=`(n+1/2)lamda/2`, n=0,1,2,….
B. satnding wave having nodes at `x=(nlamda)/2`,n=0,1,2..
C. wave travelling along +x direction
D. wave travelling along-x direction

Explanation

To solve the problem of finding the resultant wave when two traveling waves are superimposed, we can follow these steps:

### Step 1: Write the equations of the given waves
The two waves are given as:
1. \( y_1 = A \sin(\omega t - kx) \)
2. \( y_2 = A \sin(\omega t + kx) \)

### Step 2: Superimpose the two waves
The resultant wave \( y \) can be found by adding the two waves:
\[ y = y_1 + y_2 = A \sin(\omega t - kx) + A \sin(\omega t + kx) \]

### Step 3: Factor out the common amplitude
We can factor out \( A \):
\[ y = A \left( \sin(\omega t - kx) + \sin(\omega t + kx) \right) \]

### Step 4: Use the sine addition formula
Using the sine addition formula, which states that:
\[ \sin a + \sin b = 2 \sin\left(\frac{a + b}{2}\right) \cos\left(\frac{a - b}{2}\right) \]
we can apply it here:
- Let \( a = \omega t - kx \)
- Let \( b = \omega t + kx \)

Calculating \( a + b \) and \( a - b \):
\[ a + b = 2\omega t, \quad a - b = -2kx \]
Thus,
\[ y = A \left( 2 \sin\left(\frac{2\omega t}{2}\right) \cos\left(\frac{-2kx}{2}\right) \right) \]
This simplifies to:
\[ y = 2A \sin(\omega t) \cos(kx) \]

### Step 5: Identify the resultant wave
The resultant wave can be expressed as:
\[ y = 2A \sin(\omega t) \cos(kx) \]
This equation represents a standing wave.

### Step 6: Determine the nodes of the standing wave
The nodes of the standing wave occur when \( \cos(kx) = 0 \). This happens when:
\[ kx = \frac{(2n + 1)\pi}{2} \quad \text{where } n \text{ is an integer} \]
From this, we can express \( x \):
\[ x = \frac{(2n + 1)\pi}{2k} \]
Since \( k = \frac{2\pi}{\lambda} \), we can substitute:
\[ x = \frac{(2n + 1)\pi}{2 \cdot \frac{2\pi}{\lambda}} = \frac{(2n + 1)\lambda}{4} \]

### Final Result
The resultant wave is a standing wave given by:
\[ y = 2A \sin(\omega t) \cos(kx) \]
with nodes at:
\[ x = \frac{(2n + 1)\lambda}{4} \quad \text{where } n \text{ is an integer} \]

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