In a stationary wave that forms as a result of reflection of wave from an obstacle, the ratio of this amplitude at an antinode to the amplitude at anode is n. The ratio of energy reflected to energy incident is

Options

A. `((n-1)/(n^(2)))`
B. `((n-1)/(n+1))^(2)`
C. `((1)/(n))^(2)`
D. `((n)/(n+1))^(2)`

Answer

To solve the problem, we need to find the ratio of the energy reflected to the energy incident in a stationary wave formed due to the reflection of a wave from an obstacle. The ratio of the amplitude at an antinode to the amplitude at a node is given as \( n \).

### Step-by-Step Solution:

1. **Define the Amplitudes**:
- Let \( A_i \) be the amplitude of the incident wave.
- Let \( A_r \) be the amplitude of the reflected wave.

2. **Understand the Relationship at Antinode and Node**:
- At an antinode, the total amplitude is given by:
\[ A_{antinode} = A_i + A_r \]
- At a node, the total amplitude is given by:
\[ A_{node} = A_i - A_r \]

3. **Set Up the Ratio**:
- According to the problem, the ratio of the amplitude at an antinode to the amplitude at a node is \( n \):
\[ \frac{A_{antinode}}{A_{node}} = n \]
- Substituting the expressions for \( A_{antinode} \) and \( A_{node} \):
\[ \frac{A_i + A_r}{A_i - A_r} = n \]

4. **Cross-Multiply to Solve for Amplitudes**:
- Cross-multiplying gives:
\[ A_i + A_r = n(A_i - A_r) \]
- Expanding this:
\[ A_i + A_r = nA_i - nA_r \]
- Rearranging terms:
\[ A_r + nA_r = nA_i - A_i \]
\[ A_r(1 + n) = (n - 1)A_i \]

5. **Express the Ratio of Amplitudes**:
- Solving for \( \frac{A_r}{A_i} \):
\[ \frac{A_r}{A_i} = \frac{n - 1}{n + 1} \]

6. **Relate Energy to Amplitude**:
- The energy is proportional to the square of the amplitude. Therefore, the ratio of the energy reflected (\( E_r \)) to the energy incident (\( E_i \)) can be expressed as:
\[ \frac{E_r}{E_i} = \left(\frac{A_r}{A_i}\right)^2 \]

7. **Substituting the Amplitude Ratio**:
- Substituting the expression for \( \frac{A_r}{A_i} \):
\[ \frac{E_r}{E_i} = \left(\frac{n - 1}{n + 1}\right)^2 \]

### Final Answer:
Thus, the ratio of energy reflected to energy incident is:
\[ \frac{E_r}{E_i} = \left(\frac{n - 1}{n + 1}\right)^2 \]