Two travelling waves `y_1=Asin[k(x-ct)]` and `y_2=Asin[k(x+ct)]` are superimposed on string. The distance between adjacent nodes is

Options

A. `ct//pi`
B. `ct//2pi`
C. `pi//2k`
D. `pi//k`

Explanation

To find the distance between adjacent nodes for the given traveling waves, we can follow these steps:

### Step 1: Understand the wave equations
The two waves are given by:
- \( y_1 = A \sin[k(x - ct)] \)
- \( y_2 = A \sin[k(x + ct)] \)

These waves are traveling in opposite directions.

### Step 2: Determine the resultant wave
When two waves superimpose, the resultant wave can be expressed as:
\[ y = y_1 + y_2 = A \sin[k(x - ct)] + A \sin[k(x + ct)] \]

### Step 3: Use the principle of superposition
Using the trigonometric identity for the sum of sines:
\[ \sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \]
we can rewrite the resultant wave:
\[ y = 2A \sin\left[kx\right] \cos\left[ct\right] \]

### Step 4: Identify the nodes
Nodes occur where the amplitude of the wave is zero. From the resultant wave equation:
\[ y = 2A \sin(kx) \cos(ct) \]
The nodes occur when:
\[ \sin(kx) = 0 \]
This happens at:
\[ kx = n\pi \quad (n = 0, 1, 2, \ldots) \]
Thus,
\[ x = \frac{n\pi}{k} \]

### Step 5: Calculate the distance between adjacent nodes
The distance between adjacent nodes can be found by calculating the difference between the positions of two consecutive nodes:
For \( n \) and \( n + 1 \):
\[ x_{n+1} - x_n = \frac{(n + 1)\pi}{k} - \frac{n\pi}{k} = \frac{\pi}{k} \]

### Step 6: Conclusion
The distance between adjacent nodes is:
\[ \frac{\pi}{k} \]