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 given wave equation
The traveling wave is given by the equation:
\[ y = A \sin(kx - \omega t + \theta) \]
Here, \( A \) is the amplitude, \( k \) is the wave number, \( \omega \) is the angular frequency, \( t \) is time, and \( \theta \) is the phase constant.

### Step 2: Identify the reflection conditions
When the wave travels from a denser medium (heavier string) to a rarer medium (lighter string), the following conditions apply:
- The amplitude of the reflected wave is less than the incident wave.
- There is no phase change upon reflection.

### Step 3: Determine the amplitude of the reflected wave
According to the problem, the amplitude of the reflected wave is given as:
\[ \text{Amplitude of reflected wave} = 0.5A \]

### Step 4: Write the equation for the reflected wave
Since the wave reflects back towards the denser medium (from lighter to heavier), the direction of the wave changes. The general form of the reflected wave will be:
\[ y' = A' \sin(kx + \omega t + \theta) \]
Where \( A' \) is the amplitude of the reflected wave. Substituting the amplitude:
\[ y' = 0.5A \sin(kx + \omega t + \theta) \]

### Step 5: Final equation for the reflected wave
Thus, the equation for the reflected wave is:
\[ y' = 0.5A \sin(kx + \omega t + \theta) \]