A travelling wave `y=A sin (kx- omega t +theta )` passes from a heavier string to a lighter string. The reflected wave has amplitude `0.5A` . The junction of the strings is at `x=0`. The equation fo the refelcted wave is `:`

Options

A. `y'=0.5 A sin (kx+omegat+theta)`
B. `y'=-0.5 A sin (kx+omegat + theta )`
C. `y'=-0.5 A sin ( omega t -kx - theta )`
D. `y'=0.5 A sin (kx+omega t -theta)`

Explanation

To find the equation of the reflected wave when a traveling wave passes from a heavier string to a lighter string, we can follow these steps:

### Step 1: Understand the incident wave
The incident wave is given by the equation:
\[ y = A \sin(kx - \omega t + \theta) \]
where:
- \( A \) is the amplitude,
- \( k \) is the wave number,
- \( \omega \) is the angular frequency,
- \( t \) is time,
- \( x \) is the position,
- \( \theta \) is the phase constant.

### Step 2: Identify the properties of the reflected wave
When the wave reflects at the boundary between the two strings, the amplitude of the reflected wave is given as \( 0.5A \). This means the amplitude of the reflected wave is half of the incident wave's amplitude.

### Step 3: Determine the direction of the reflected wave
Since the wave is traveling from a heavier string to a lighter string, the reflected wave will travel in the opposite direction. This means that the wave equation will change from \( kx - \omega t \) to \( -kx + \omega t \).

### Step 4: Write the equation of the reflected wave
The amplitude of the reflected wave is \( 0.5A \), and since it travels in the opposite direction, the equation of the reflected wave can be written as:
\[ y' = -0.5A \sin(-kx + \omega t + \theta) \]

### Step 5: Simplify the equation
Using the property of sine that \( \sin(-x) = -\sin(x) \), we can rewrite the equation:
\[ y' = -0.5A \sin(-kx + \omega t + \theta) = 0.5A \sin(kx - \omega t - \theta) \]

### Final Equation
Thus, the equation of the reflected wave is:
\[ y' = -0.5A \sin(kx + \omega t - \theta) \]