A wave is given by the equation
` y = 10 sin 2 pi (100 t - 0.02 x) + 10 sin 2 pi (100 t + 0.02 x)`
Find the loop length , frequency , velocity and maximum amplitude of the stationary wave produced.

Options

A. 20 units and 30 units
B. 20 units and 25 units
C. 30 units and 20 units
D. 25 units and 20 units

Answer

To solve the problem, we need to analyze the given wave equation and extract the required parameters: loop length, frequency, velocity, and maximum amplitude of the stationary wave produced.

### Step-by-Step Solution:

1. **Identify the Wave Equation**:
The given wave equation is:
\[ y = 10 \sin(2\pi(100t - 0.02x)) + 10 \sin(2\pi(100t + 0.02x)) \]
This represents two traveling waves moving in opposite directions.

2. **Determine the Frequency**:
The angular frequency \(\omega\) is given by the term \(2\pi \times 100\). We can relate angular frequency to frequency \(f\) using the formula:
\[ \omega = 2\pi f \]
Thus, we can find \(f\):
\[ f = 100 \text{ Hz} \]

3. **Calculate the Wavenumber \(k\)**:
The wavenumber \(k\) can be determined from the term \(0.02x\):
\[ k = 2\pi \times 0.02 = 0.04\pi \text{ rad/m} \]

4. **Find the Wavelength \(\lambda\)**:
The relationship between wavenumber and wavelength is given by:
\[ k = \frac{2\pi}{\lambda} \]
Rearranging gives:
\[ \lambda = \frac{2\pi}{k} = \frac{2\pi}{0.04\pi} = 50 \text{ m} \]

5. **Determine the Velocity \(v\)**:
The wave velocity \(v\) can be calculated using the formula:
\[ v = f \lambda \]
Substituting the values:
\[ v = 100 \times 50 = 5000 \text{ m/s} \]

6. **Calculate the Maximum Amplitude**:
The maximum amplitude of the stationary wave is given by the superposition of the two waves. Since both waves have an amplitude of 10, the maximum amplitude \(A_{max}\) is:
\[ A_{max} = 2 \times 10 = 20 \text{ units} \]

7. **Determine the Loop Length**:
The loop length is half the wavelength:
\[ \text{Loop Length} = \frac{\lambda}{2} = \frac{50}{2} = 25 \text{ m} \]

### Summary of Results:
- **Loop Length**: 25 m
- **Frequency**: 100 Hz
- **Velocity**: 5000 m/s
- **Maximum Amplitude**: 20 units