Consider a standing wave formed on a string . It results due to the superposition of two waves travelling in opposite directions . The waves are travelling along the length of the string in the `x` - direction and displacements of elements on the string are along the `y` - direction . Individual equations of the two waves can be expressed as
`Y_(1) = 6 (cm) sin [ 5 (rad//cm) x - 4 ( rad//s)t]`
`Y_(2) = 6(cm) sin [ 5 (rad//cm)x + 4 (rad//s)t]` Here `x` and `y` are in `cm`.
Answer the following questions.
If one end of the string is at `x = 0` , positions of the nodes can be described as

Options

A. `x = n pi//5 cm , where n = 0 , 1, 2,…`
B. `x = n 2pi//5 cm , where n = 0 , 1, 2,…`
C. `x = n pi//5 cm , where n = 0 , 1, 3, 5,…`
D. `x = n pi//10 cm , where n = 0 , 1, 3,5,…`

Explanation

To find the positions of the nodes in the standing wave formed by the superposition of two waves traveling in opposite directions, we can follow these steps:

### Step 1: Write the equations of the two waves
The equations of the two waves traveling along the string are given as:
- \( Y_1 = 6 \, \text{cm} \, \sin(5 \, \text{rad/cm} \, x - 4 \, \text{rad/s} \, t) \)
- \( Y_2 = 6 \, \text{cm} \, \sin(5 \, \text{rad/cm} \, x + 4 \, \text{rad/s} \, t) \)

### Step 2: Find the resultant wave equation
The resultant wave \( Y \) can be found by adding the two waves:
\[ Y = Y_1 + Y_2 = 6 \, \text{cm} \, \sin(5x - 4t) + 6 \, \text{cm} \, \sin(5x + 4t) \]
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 equation as:
\[ Y = 2 \times 6 \, \text{cm} \, \sin(5x) \cos(4t) = 12 \, \text{cm} \, \sin(5x) \cos(4t) \]

### Step 3: Identify the condition for nodes
In a standing wave, nodes occur where the amplitude is zero. The amplitude of the resultant wave is given by:
\[ A = 12 \, \text{cm} \, \sin(5x) \]
For nodes, we set the amplitude to zero:
\[ 12 \, \text{cm} \, \sin(5x) = 0 \]

### Step 4: Solve for \( x \)
The sine function is zero at integer multiples of \( \pi \):
\[ \sin(5x) = 0 \implies 5x = n\pi \quad (n = 0, 1, 2, \ldots) \]
Solving for \( x \):
\[ x = \frac{n\pi}{5} \quad (n = 0, 1, 2, \ldots) \]

### Step 5: Final answer
The positions of the nodes on the string are given by:
\[ x = \frac{n\pi}{5} \quad \text{for } n = 0, 1, 2, \ldots \]