The equation of a transverse wave travelling along a coil spring is
`y=4.0 sin pi (0.010 x-2.0t)`
where y and x are in cm and t in s. Find the (i)amplitude (ii)wavelength (iii)initial phase at the origin (iv)speed and (v)frequency on the wave.

Answer

To solve the problem step by step, we will analyze the given wave equation:

**Given Wave Equation:**
\[ y = 4.0 \sin(\pi (0.010 x - 2.0t)) \]

Where \( y \) and \( x \) are in cm, and \( t \) is in seconds.

### Step 1: Find the Amplitude
The amplitude \( A \) of a wave is the coefficient in front of the sine function in the wave equation.

From the equation:
\[ A = 4.0 \, \text{cm} \]

### Step 2: Find the Wavelength
The general form of the wave equation is:
\[ y = A \sin(kx - \omega t) \]
where \( k \) is the wave number.

From the given equation, we can identify:
\[ k = \pi \times 0.010 = 0.01\pi \]

The wavelength \( \lambda \) can be calculated using the formula:
\[ \lambda = \frac{2\pi}{k} \]
Substituting the value of \( k \):
\[ \lambda = \frac{2\pi}{0.01\pi} = \frac{2}{0.01} = 200 \, \text{cm} \]

### Step 3: Find the Initial Phase at the Origin
To find the initial phase at the origin, we substitute \( x = 0 \) and \( t = 0 \) into the wave equation:
\[ y = 4.0 \sin(\pi(0.010 \cdot 0 - 2.0 \cdot 0)) = 4.0 \sin(0) = 0 \]

The initial phase at the origin is:
\[ \text{Initial Phase} = 0 \]

### Step 4: Find the Speed of the Wave
The speed \( v \) of the wave is given by the formula:
\[ v = \frac{\omega}{k} \]

From the equation, we can identify:
\[ \omega = 2\pi \]
Substituting the values:
\[ v = \frac{2\pi}{0.01\pi} = \frac{2}{0.01} = 200 \, \text{cm/s} = 2 \, \text{m/s} \]

### Step 5: Find the Frequency of the Wave
The frequency \( f \) can be calculated using the relationship:
\[ f = \frac{\omega}{2\pi} \]

Substituting the value of \( \omega \):
\[ f = \frac{2\pi}{2\pi} = 1 \, \text{Hz} \]

### Summary of Results:
1. **Amplitude:** \( 4.0 \, \text{cm} \)
2. **Wavelength:** \( 200 \, \text{cm} \)
3. **Initial Phase at the Origin:** \( 0 \)
4. **Speed of the Wave:** \( 2 \, \text{m/s} \)
5. **Frequency of the Wave:** \( 1 \, \text{Hz} \)