The equation of a transverse wave travelling along x-axis is given by y = 10 sin `pi=` (0.01 x- 2.0t) where y and x are represented in centimeters and, in seconds. Find the amplitude, frequency velocity and wavelength of the wave.

Answer

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

### Step 1: Identify the given wave equation
The wave equation provided is:
\[ y = 10 \sin \left( \pi \left( 0.01 x - 2.0 t \right) \right) \]

### Step 2: Compare with the general wave equation
The general form of a transverse wave is given by:
\[ y = a \sin(kx - \omega t) \]
where:
- \( a \) is the amplitude,
- \( k \) is the wave number,
- \( \omega \) is the angular frequency.

### Step 3: Find the amplitude
From the given equation, we can see that:
- The amplitude \( a = 10 \) cm.

### Step 4: Find the angular frequency \( \omega \)
In the equation, we have:
\[ \omega = 2\pi \text{ (since it is the coefficient of } t) \]

### Step 5: Calculate the frequency \( f \)
Using the relationship between angular frequency and frequency:
\[ \omega = 2\pi f \]
Substituting \( \omega = 2\pi \):
\[ 2\pi = 2\pi f \]
Dividing both sides by \( 2\pi \):
\[ f = 1 \text{ Hz} \]

### Step 6: Find the wave number \( k \)
From the equation, we have:
\[ k = \pi \times 0.01 = 0.01\pi \text{ (since it is the coefficient of } x) \]

### Step 7: Calculate the velocity \( v \)
The velocity of the wave is given by the formula:
\[ v = \frac{\omega}{k} \]
Substituting the values of \( \omega \) and \( k \):
\[ v = \frac{2\pi}{0.01\pi} \]
The \( \pi \) cancels out:
\[ v = \frac{2}{0.01} = 200 \text{ cm/s} \]

### Step 8: Calculate the wavelength \( \lambda \)
The wavelength is related to the wave number by the formula:
\[ \lambda = \frac{2\pi}{k} \]
Substituting \( k = 0.01\pi \):
\[ \lambda = \frac{2\pi}{0.01\pi} \]
Again, the \( \pi \) cancels out:
\[ \lambda = \frac{2}{0.01} = 200 \text{ cm} \]

### Summary of Results
- Amplitude \( a = 10 \) cm
- Frequency \( f = 1 \) Hz
- Velocity \( v = 200 \) cm/s
- Wavelength \( \lambda = 200 \) cm