A wave travelling along positive x-axis is given by `y = A sin(omegat-kx)`. If it is reflected from rigid boundary such that 80% amplitude is reflected, then equation of reflected wave is

Options

A. `y=A sin (omegat+kx)`
B. `y=-0.8A sin (omegat +kx)`
C. `y=Asin 0.8A (omega t+0.8 kx)`
D. `y=A sin (omegat+0.8kx)`

Answer

To find the equation of the reflected wave when a wave traveling along the positive x-axis is reflected from a rigid boundary with 80% of its amplitude reflected, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Incident Wave Equation**:
The incident wave is given by the equation:
\[ y = A \sin(\omega t - kx) \]
where \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( k \) is the wave number.

**Hint**: Recognize that the wave travels in the positive x-direction due to the negative sign in the argument of the sine function.

2. **Determine the Amplitude of the Reflected Wave**:
Since 80% of the amplitude is reflected, the amplitude of the reflected wave, denoted as \( A' \), can be calculated as:
\[ A' = 0.8A \]

**Hint**: Remember that the amplitude of the reflected wave is a fraction of the incident wave's amplitude.

3. **Consider the Phase Change upon Reflection**:
When a wave reflects off a rigid boundary, it undergoes a phase change of \( \pi \) radians (or 180 degrees). This means that the reflected wave will have a phase shift added to its equation.

**Hint**: A phase change of \( \pi \) indicates that the wave will be inverted.

4. **Write the Equation for the Reflected Wave**:
The general form of the reflected wave traveling in the negative x-direction can be expressed as:
\[ y_r = A' \sin(\omega t + kx + \pi) \]
Substituting \( A' = 0.8A \):
\[ y_r = 0.8A \sin(\omega t + kx + \pi) \]

**Hint**: The sine function has a property: \( \sin(x + \pi) = -\sin(x) \).

5. **Simplify the Equation**:
Using the property of sine:
\[ y_r = 0.8A \sin(\omega t + kx + \pi) = -0.8A \sin(\omega t + kx) \]

**Hint**: The negative sign indicates that the reflected wave is inverted compared to the incident wave.

### Final Answer:
The equation of the reflected wave is:
\[ y_r = -0.8A \sin(\omega t + kx) \]