Two waves travelling in opposite directions produce a standing wave . The individual wave functions are given by `y_(1) = 4 sin ( 3x - 2 t)` and `y_(2) = 4 sin ( 3x + 2 t) cm` , where `x` and `y` are in cm

Options

A. The maximum displacement of the motion at `x = 2.3 cm is 4.63 cm`.
B. The maximum displacement of the motion at `t = 2.3 s is 4.63 cm`.
C. Nodes are formed at `x` values given by
` 0 , pi//3 , 2 pi//3 , 4 pi//3 , ….`
D. Antinodes are formed at `x` values given by
`pi//6 , pi//2 , 5 pi//6 , 7 pi//6,…`

Explanation

To solve the problem of finding the resultant wave from the superposition of two waves traveling in opposite directions, we will follow these steps:

### Step 1: Write down the wave functions
The individual wave functions are given as:
- \( y_1 = 4 \sin(3x - 2t) \)
- \( y_2 = 4 \sin(3x + 2t) \)

### Step 2: Apply the principle of superposition
According to the principle of superposition, the resultant wave \( y \) is the sum of the individual waves:
\[ y = y_1 + y_2 = 4 \sin(3x - 2t) + 4 \sin(3x + 2t) \]

### Step 3: Use the sine addition formula
We can use the sine addition formula:
\[ \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
Here, let \( A = 3x - 2t \) and \( B = 3x + 2t \).

Calculating \( A + B \) and \( A - B \):
- \( A + B = (3x - 2t) + (3x + 2t) = 6x \)
- \( A - B = (3x - 2t) - (3x + 2t) = -4t \)

Substituting back into the sine addition formula:
\[ y = 4 \sin(3x - 2t) + 4 \sin(3x + 2t) = 2 \cdot 4 \sin\left(3x\right) \cos\left(-2t\right) = 8 \sin(3x) \cos(2t) \]

### Step 4: Identify maximum displacement
The maximum displacement occurs when \( \cos(2t) = 1 \):
\[ y_{\text{max}} = 8 \sin(3x) \]

### Step 5: Find maximum displacement at \( x = 2.3 \, \text{cm} \)
Substituting \( x = 2.3 \):
\[ y_{\text{max}} = 8 \sin(3 \cdot 2.3) = 8 \sin(6.9) \]

Calculating \( \sin(6.9) \):
Using a calculator, we find:
\[ \sin(6.9) \approx 0.999 \]
Thus,
\[ y_{\text{max}} \approx 8 \cdot 0.999 \approx 7.992 \, \text{cm} \]

### Step 6: Determine nodes and antinodes
- **Nodes** occur where the amplitude is zero, which happens when \( \sin(3x) = 0 \):
\[ 3x = n\pi \quad \Rightarrow \quad x = \frac{n\pi}{3} \]
For \( n = 0, 1, 2, \ldots \), the positions of nodes are:
\[ x = 0, \frac{\pi}{3}, \frac{2\pi}{3}, \ldots \]

- **Antinodes** occur between the nodes, at positions:
\[ x = \frac{(n + 0.5)\pi}{3} \]

### Final Result
The maximum displacement at \( x = 2.3 \, \text{cm} \) is approximately \( 7.992 \, \text{cm} \). The positions of nodes and antinodes can be calculated using the above formulas.