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.

Explanation

To solve the problem step by step, we will analyze the given wave equation and extract the required parameters: loop length, frequency, velocity, and maximum amplitude.

### Given Wave Equation:
\[ y = 10 \sin(2\pi(100t - 0.02x)) + 10 \sin(2\pi(100t + 0.02x)) \]

### Step 1: Combine the Sine Terms
Using the trigonometric identity for the sum of sine functions:
\[ \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
Let \( A = 2\pi(100t - 0.02x) \) and \( B = 2\pi(100t + 0.02x) \).

Calculating \( A + B \) and \( A - B \):
\[ A + B = 2\pi(100t) \]
\[ A - B = -0.04\pi x \]

Now substituting back:
\[ y = 10 \sin A + 10 \sin B = 20 \sin(2\pi(100t)) \cos(0.02\pi x) \]

### Step 2: Identify Maximum Amplitude
The maximum amplitude of the stationary wave is given by the coefficient of the sine function:
\[ \text{Maximum Amplitude} = 20 \]

### Step 3: Determine Frequency
The angular frequency \( \omega \) can be identified from the term \( 2\pi(100t) \):
\[ \omega = 2\pi \times 100 \]
The frequency \( f \) is given by:
\[ f = \frac{\omega}{2\pi} = 100 \text{ Hz} \]

### Step 4: Calculate Wave Number \( k \)
From the cosine term \( \cos(0.02\pi x) \), we can identify the wave number \( k \):
\[ k = 0.02\pi \text{ rad/m} \]

### Step 5: Calculate Velocity
The velocity \( v \) of the wave can be calculated using the formula:
\[ v = \frac{\omega}{k} \]
Substituting the values:
\[ v = \frac{2\pi \times 100}{0.02\pi} = \frac{200\pi}{0.02\pi} = 10000 \text{ m/s} \]

### Step 6: Calculate Loop Length (Wavelength)
The loop length (or wavelength \( \lambda \)) is related to the wave number:
\[ \lambda = \frac{2\pi}{k} \]
Substituting for \( k \):
\[ \lambda = \frac{2\pi}{0.02\pi} = 100 \text{ m} \]
Since one loop corresponds to half the wavelength:
\[ \text{Loop Length} = \frac{\lambda}{2} = \frac{100}{2} = 50 \text{ m} \]

### Summary of Results:
- Maximum Amplitude: **20**
- Frequency: **100 Hz**
- Velocity: **10000 m/s**
- Loop Length: **50 m**