Standing waves are produced by superposition of two waves
`y_(1) = 0.05 sin(3pi t - 2x),`
`y_(2) = 0.05 six(3pi t +2x)`
where x and y are measured in metre and t in second. Find the amplitude of the particle at x = 0.5m.

Answer

To find the amplitude of the particle at \( x = 0.5 \, \text{m} \) for the given standing waves, we can follow these steps:

### Step 1: Write down the given wave equations
The two waves are given as:
\[ y_1 = 0.05 \sin(3\pi t - 2x) \]
\[ y_2 = 0.05 \sin(3\pi t + 2x) \]

### Step 2: Find the resultant wave equation
The resultant wave \( y_r \) is the sum of the two waves:
\[ y_r = y_1 + y_2 = 0.05 \sin(3\pi t - 2x) + 0.05 \sin(3\pi t + 2x) \]

### Step 3: Use the sine addition formula
We can use the sine addition formula:
\[ \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
Let \( A = 3\pi t - 2x \) and \( B = 3\pi t + 2x \). Then:
\[ A + B = 2(3\pi t) = 6\pi t \]
\[ A - B = -4x \]
Thus, we can rewrite \( y_r \):
\[ y_r = 0.05 \cdot 2 \sin\left(3\pi t\right) \cos\left(-2x\right) \]
Since \( \cos(-\theta) = \cos(\theta) \), we have:
\[ y_r = 0.1 \sin(3\pi t) \cos(2x) \]

### Step 4: Identify the amplitude of the resultant wave
The amplitude \( A \) of the resultant wave is:
\[ A = 0.1 \cos(2x) \]

### Step 5: Substitute \( x = 0.5 \, \text{m} \)
Now, we substitute \( x = 0.5 \):
\[ A = 0.1 \cos(2 \times 0.5) = 0.1 \cos(1 \, \text{radian}) \]

### Step 6: Calculate \( \cos(1 \, \text{radian}) \)
Using a calculator or trigonometric tables, we find:
\[ \cos(1 \, \text{radian}) \approx 0.5403 \]

### Step 7: Calculate the final amplitude
Thus, the amplitude at \( x = 0.5 \, \text{m} \) is:
\[ A \approx 0.1 \times 0.5403 \approx 0.05403 \, \text{m} \approx 5.4 \times 10^{-2} \, \text{m} \]

### Final Answer
The amplitude of the particle at \( x = 0.5 \, \text{m} \) is approximately \( 5.4 \times 10^{-2} \, \text{m} \).

---