The equation of a plane wave travelling along positive direction of `x-`axis is `y = asin"(2pi)/(lambda)(vt-x)` When the wave is reflected at a rigid surface and its amplitude becomes `80%`, then find the equation of the reflected wave.

Explanation

To find the equation of the reflected wave when a plane wave traveling along the positive x-axis reflects off a rigid surface and its amplitude becomes 80%, we can follow these steps:

### Step 1: Write the equation of the original wave
The equation of the wave traveling along the positive x-axis is given as:
\[ y = a \sin\left(\frac{2\pi}{\lambda}(vt - x)\right) \]

### Step 2: Determine the amplitude of the reflected wave
When the wave is reflected at a rigid surface, the amplitude of the reflected wave becomes 80% of the original amplitude. Therefore, the amplitude of the reflected wave is:
\[ A_{\text{reflected}} = 0.8a \]

### Step 3: Determine the direction of the reflected wave
The reflected wave travels in the opposite direction to the original wave. Therefore, we need to change the sign of the \(x\) term in the wave equation.

### Step 4: Write the equation of the reflected wave
The general form of the wave traveling in the negative x-direction can be written as:
\[ y = A \sin\left(\frac{2\pi}{\lambda}(vt + x + \phi)\right) \]
where \(\phi\) is the phase difference due to reflection. Since the wave reflects off a rigid boundary, the phase difference is \(\frac{\lambda}{2}\).

Thus, substituting the amplitude and the phase difference, the equation of the reflected wave becomes:
\[ y = 0.8a \sin\left(\frac{2\pi}{\lambda}(vt + x + \frac{\lambda}{2})\right) \]

### Final Equation
Therefore, the final equation of the reflected wave is:
\[ y = 0.8a \sin\left(\frac{2\pi}{\lambda}(vt + x + \frac{\lambda}{2})\right) \]

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