Equation of a plane progressive wave is given by `y=0.6 sin 2pi(t-(x)/(2)).` On reflection from a denser medium, its amplitude becomes `2//3` of the amplitude of the incident wave. The equation of the reflected wave is

Options

A. `y=0.6sin2pi(t+(x)/(2))`
B. `y=-0.4sin2pi(t+(x)/(2))`
C. `y=0.4sin2pi(t+(x)/(2))`
D. `y=-0.4sin2pi(t-(x)/(2))`

Explanation

To find the equation of the reflected wave given the incident wave equation and the conditions of reflection, we can follow these steps:

### Step 1: Identify the parameters of the incident wave
The equation of the incident wave is given as:
\[ y = 0.6 \sin \left( 2\pi \left( t - \frac{x}{2} \right) \right) \]

From this equation, we can identify:
- Amplitude \( A_i = 0.6 \)
- Angular frequency \( \omega = 2\pi \)
- Wave number \( k = \frac{2\pi}{2} = \pi \)

### Step 2: Determine the amplitude of the reflected wave
According to the problem, the amplitude of the reflected wave becomes \( \frac{2}{3} \) of the amplitude of the incident wave. Therefore, we calculate:
\[ A_r = \frac{2}{3} \times A_i = \frac{2}{3} \times 0.6 = 0.4 \]

### Step 3: Understand the phase change upon reflection
When a wave reflects from a denser medium, it undergoes a phase reversal of \( \pi \). This means that the reflected wave will have a negative sign in front of its amplitude.

### Step 4: Write the equation of the reflected wave
The general form of the reflected wave can be written as:
\[ y = A_r \sin \left( 2\pi \left( t + \frac{x}{2} \right) \right) \]
However, since there is a phase reversal, we include the negative sign:
\[ y = -A_r \sin \left( 2\pi \left( t + \frac{x}{2} \right) \right) \]

Substituting \( A_r = 0.4 \):
\[ y = -0.4 \sin \left( 2\pi \left( t + \frac{x}{2} \right) \right) \]

### Final Equation
Thus, the equation of the reflected wave is:
\[ y = -0.4 \sin \left( 2\pi t + \pi x \right) \]

### Summary
The correct answer is:
\[ y = -0.4 \sin \left( 2\pi \left( t + \frac{x}{2} \right) \right) \]

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