A progressive wave y = A sin (k x - `omega`t) is reflected by a rigid wall at x = 0 . Then the reflected wave can be represented by -

Options

A. y = A sin (kx + `omega`t)
B. y = A cos (kx + `omega`t)
C. y = - A sin (kx - `omega`t)
D. y = - A sin (kx + `omega`t)

Explanation

To find the equation of the reflected wave when a progressive wave \( y = A \sin(kx - \omega t) \) is reflected by a rigid wall at \( x = 0 \), we will follow these steps:

### Step-by-Step Solution:

1. **Understand the Original Wave**: The given wave is \( y = A \sin(kx - \omega t) \). This wave is traveling in the positive x-direction.

2. **Reflection at a Rigid Wall**: When a wave reflects off a rigid wall, it undergoes a phase change of \( \pi \) (or 180 degrees). This means the reflected wave will have an inverted amplitude.

3. **Change in Direction**: After reflection, the wave will travel in the negative x-direction. Therefore, we need to change the sign of \( x \) in the wave equation.

4. **Write the Reflected Wave Equation**: The reflected wave can be expressed as:
\[ y = A \sin(k(-x) - \omega t + \pi) \]
Simplifying this, we have:
\[ y = A \sin(-kx - \omega t + \pi) \]

5. **Using the Sine Function Properties**: We know that \( \sin(\theta + \pi) = -\sin(\theta) \). Thus, we can rewrite the equation as:
\[ y = -A \sin(-kx - \omega t) \]

6. **Simplifying Further**: Using the property \( \sin(-\theta) = -\sin(\theta) \), we can simplify this to:
\[ y = -A \sin(kx + \omega t) \]

### Final Answer:
The equation of the reflected wave is:
\[ y = -A \sin(kx + \omega t) \]