A wave travels along the positive x-direction with a speed of `20 ms^-1`. The amplitude of the wave is 0.20 cm and the wavelength 2.0 cm. (a) Write a suitable wave equation which describes this wave. (b) What is the displacement and velocity of the particle at x = 2.0 cm at time t = 0 according to the wave equation written ?

Explanation

To solve the problem step by step, we will address both parts of the question.

### Part (a): Writing the Wave Equation

1. **Identify Given Values:**
- Speed of the wave, \( V = 20 \, \text{m/s} \)
- Amplitude, \( A = 0.20 \, \text{cm} = 0.20 \times 10^{-2} \, \text{m} \)
- Wavelength, \( \lambda = 2.0 \, \text{cm} = 2.0 \times 10^{-2} \, \text{m} \)

2. **Calculate the Wave Number \( k \):**
\[ k = \frac{2\pi}{\lambda} = \frac{2\pi}{2.0 \times 10^{-2}} = \frac{2\pi}{0.02} = 100\pi \, \text{m}^{-1} \]

3. **Calculate the Angular Frequency \( \omega \):**
- First, find the frequency \( \nu \):
\[ \nu = \frac{V}{\lambda} = \frac{20 \, \text{m/s}}{2.0 \times 10^{-2} \, \text{m}} = 1000 \, \text{Hz} \]
- Now calculate \( \omega \):
\[ \omega = 2\pi\nu = 2\pi \times 1000 \, \text{s}^{-1} = 2000\pi \, \text{s}^{-1} \]

4. **Write the Wave Equation:**
The general form of the wave equation traveling in the positive x-direction is:
\[ y(x, t) = A \sin(kx - \omega t) \]
Substituting the values of \( A \), \( k \), and \( \omega \):
\[ y(x, t) = (0.20 \times 10^{-2}) \sin(100\pi x - 2000\pi t) \]

### Part (b): Finding Displacement and Velocity at \( x = 2.0 \, \text{cm} \) and \( t = 0 \)

1. **Substituting Values into the Wave Equation:**
- Convert \( x = 2.0 \, \text{cm} \) to meters:
\[ x = 2.0 \, \text{cm} = 2.0 \times 10^{-2} \, \text{m} \]
- Substitute \( x \) and \( t = 0 \) into the wave equation:
\[ y(2.0 \times 10^{-2}, 0) = (0.20 \times 10^{-2}) \sin(100\pi \times 2.0 \times 10^{-2} - 2000\pi \times 0) \]
\[ = (0.20 \times 10^{-2}) \sin(2\pi) = (0.20 \times 10^{-2}) \times 0 = 0 \]

2. **Finding the Velocity of the Particle:**
- The velocity \( v \) is given by:
\[ v = \frac{dy}{dt} = -A\omega \cos(kx - \omega t) \]
- Substitute \( A \), \( \omega \), \( k \), \( x \), and \( t = 0 \):
\[ v = - (0.20 \times 10^{-2}) (2000\pi) \cos(100\pi \times 2.0 \times 10^{-2} - 2000\pi \times 0) \]
\[ = - (0.20 \times 10^{-2}) (2000\pi) \cos(2\pi) \]
\[ = - (0.20 \times 10^{-2}) (2000\pi) \times 1 = -0.20 \times 2000\pi \times 10^{-2} \]
\[ = -4\pi \, \text{m/s} \]

### Final Answers:
- (a) The wave equation is:
\[ y(x, t) = (0.20 \times 10^{-2}) \sin(100\pi x - 2000\pi t) \]
- (b) The displacement at \( x = 2.0 \, \text{cm} \) and \( t = 0 \) is \( 0 \, \text{m} \) and the velocity is \( -4\pi \, \text{m/s} \).