The stationary wave y = 2a sin kx cos`omega`t in a closed organ pipe is the result of the superposition of y = a sin( `omega`t — kx) and

Answer

To solve the problem, we need to find the reflected wave \( y_r \) that, when superimposed with the incident wave \( y_i = A \sin(\omega t - kx) \), results in the stationary wave equation \( y = 2A \sin(kx) \cos(\omega t) \).

### Step-by-Step Solution:

1. **Identify the Incident Wave**:
The incident wave is given as:
\[ y_i = A \sin(\omega t - kx) \]

2. **Understanding the Stationary Wave**:
The stationary wave is given by:
\[ y = 2A \sin(kx) \cos(\omega t) \]
This form suggests that it is the result of the superposition of two waves traveling in opposite directions.

3. **Use the Superposition Principle**:
The stationary wave can be expressed as the sum of the incident wave and the reflected wave:
\[ y = y_i + y_r \]

4. **Formulate the Reflected Wave**:
For a wave reflecting from a closed end, the reflected wave will have the same amplitude but will travel in the opposite direction. Thus, we can express the reflected wave as:
\[ y_r = A \sin(\omega t + kx) \]

5. **Combine the Waves**:
Now, we add the incident and reflected waves:
\[ y = A \sin(\omega t - kx) + A \sin(\omega t + kx) \]

6. **Use the Trigonometric Identity**:
We can use the identity for the sum of sine functions:
\[ \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
Here, let \( A = \omega t - kx \) and \( B = \omega t + kx \):
\[ y = 2A \sin\left(\frac{(\omega t - kx) + (\omega t + kx)}{2}\right) \cos\left(\frac{(\omega t - kx) - (\omega t + kx)}{2}\right) \]
Simplifying gives:
\[ y = 2A \sin(\omega t) \cos(kx) \]

7. **Final Result**:
Thus, we have confirmed that the stationary wave can be expressed as:
\[ y = 2A \sin(kx) \cos(\omega t) \]
Therefore, the reflected wave is:
\[ y_r = A \sin(\omega t + kx) \]

### Summary:
The reflected wave \( y_r \) that, when superimposed with the incident wave \( y_i \), produces the stationary wave is:
\[ y_r = A \sin(\omega t + kx) \]