A travelling wave ` y = A sin (k x - omega t + theta)` passes from a heavier string to a lighter string . The juction of the strings is at ` x = 0`. The equation of the reflected wave is

Options

A. `y' = 0.5 A sin ( kx + omega t + theta)`
B. `y' = - 0.5 A sin ( kx + omega t + theta)`
C. `y' = - 0.5 A sin ( kx - omega t - 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: Identify the Incident Wave
The incident wave is given by the equation:
\[ y = A \sin(kx - \omega t + \theta) \]
This wave is traveling in the positive x-direction.

### Step 2: Understand the Medium Transition
The wave is moving from a heavier (denser) string to a lighter (rarer) string. In this case, the speed of the wave in the lighter medium (V2) is greater than in the heavier medium (V1).

### Step 3: Determine the Amplitude of the Reflected Wave
When a wave reflects off a boundary between two different media, the amplitude of the reflected wave can be calculated using the formula:
\[ A_r = \frac{V_2 - V_1}{V_2 + V_1} A \]
Since V2 (speed in the lighter medium) is greater than V1 (speed in the heavier medium), the amplitude of the reflected wave will be positive.

### Step 4: Analyze Phase Change
When a wave reflects from a denser medium to a rarer medium, it undergoes a phase change of \( \pi \) (or 180 degrees). This means that the reflected wave will have a negative sign in front of its amplitude.

### Step 5: Write the Equation of the Reflected Wave
The reflected wave travels in the negative x-direction, so we can express it as:
\[ y_r = A_r \sin(kx + \omega t + \theta') \]
Where:
- \( A_r = \frac{V_2 - V_1}{V_2 + V_1} A \)
- The phase change of \( \pi \) means we need to adjust the amplitude to negative:
\[ A_r = -\frac{V_2 - V_1}{V_2 + V_1} A \]

Thus, the equation becomes:
\[ y_r = -\frac{V_2 - V_1}{V_2 + V_1} A \sin(kx + \omega t + \theta) \]

### Final Equation
Since the amplitude is negative due to the phase change, we can write:
\[ y_r = \frac{V_2 - V_1}{V_2 + V_1} A \sin(kx + \omega t + \theta) \]

### Summary of the Reflected Wave Equation
The equation of the reflected wave is:
\[ y_r = \frac{V_2 - V_1}{V_2 + V_1} A \sin(kx + \omega t + \theta) \]