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 \( y = 0.6 \sin \left( 2\pi \left( t - \frac{x}{2} \right) \right) \), we will follow these steps:

### Step 1: Identify the amplitude of the incident wave
The amplitude of the incident wave is given as:
\[ A_i = 0.6 \]

### Step 2: Determine the amplitude of the reflected wave
The amplitude of the reflected wave is given as \( \frac{2}{3} \) of the amplitude of the incident wave:
\[ A_r = \frac{2}{3} \times A_i = \frac{2}{3} \times 0.6 = 0.4 \]

### Step 3: Identify the phase change upon reflection
When a wave reflects off a denser medium, it undergoes a phase change of \( \pi \) radians (or 180 degrees). This means we need to adjust the sine function accordingly.

### Step 4: Write the equation for the reflected wave
The general form of the wave equation is:
\[ y = A \sin(kx - \omega t + \phi) \]
For the reflected wave, the amplitude is \( 0.4 \) and the phase change of \( \pi \) will be added to the argument of the sine function. The wave number \( k \) and angular frequency \( \omega \) remain the same as in the incident wave.

Thus, the equation becomes:
\[ y = 0.4 \sin\left(2\pi \left(t + \frac{x}{2}\right) + \pi\right) \]

### Step 5: Simplify the equation
Using the property of sine that \( \sin(x + \pi) = -\sin(x) \), we can simplify:
\[ y = 0.4 \sin\left(2\pi t + \frac{x}{2} + \pi\right) = -0.4 \sin\left(2\pi t + \frac{x}{2}\right) \]

### Final Answer
The equation of the reflected wave is:
\[ y = -0.4 \sin\left(2\pi t + \frac{x}{2}\right) \]