Assertion: In a standing wave `x=0` is a node. Then, total mechanical energy lying between `x=0` and `x=(lambda/8)` is not equal to the energy lying between `x=lambda/8` and `x=lambda/4`.
Reason: In standing waves different particles oscillate with different amplitudes.

Options

A. If both Assertion and Reason are true and the Reason is correct explanation of the Assertion.
B. If both Assertion and Reason are true but Reason is not the correct explanation of Assertion.
C. If Assertion is true, but the Reason is false.
D. If Assertion is false but the Reason is true.

Explanation

To solve the question, we need to analyze the assertion and the reason provided regarding standing waves.

### Step 1: Understanding the Assertion
The assertion states that in a standing wave, the total mechanical energy between \( x = 0 \) and \( x = \frac{\lambda}{8} \) is not equal to the energy between \( x = \frac{\lambda}{8} \) and \( x = \frac{\lambda}{4} \).

### Step 2: Analyzing the Energy Distribution
In a standing wave, the energy is proportional to the square of the amplitude of oscillation. The amplitude varies along the length of the wave. At \( x = 0 \), there is a node (point of no displacement), and as we move towards \( x = \frac{\lambda}{4} \), the amplitude increases.

### Step 3: Identifying the Regions
- The region from \( x = 0 \) to \( x = \frac{\lambda}{8} \) contains particles with lower amplitudes compared to those in the region from \( x = \frac{\lambda}{8} \) to \( x = \frac{\lambda}{4} \).
- The amplitude at \( x = 0 \) is zero, and it gradually increases to a maximum value at \( x = \frac{\lambda}{4} \).

### Step 4: Calculating Energy in Each Region
- The total mechanical energy \( E \) in a segment of a standing wave is given by \( E \propto A^2 \), where \( A \) is the amplitude.
- In the first segment (from \( x = 0 \) to \( x = \frac{\lambda}{8} \)), the average amplitude is lower than in the second segment (from \( x = \frac{\lambda}{8} \) to \( x = \frac{\lambda}{4} \)).
- Therefore, the energy in the first segment will be less than that in the second segment.

### Step 5: Conclusion on the Assertion
Since the energy in the segment from \( x = 0 \) to \( x = \frac{\lambda}{8} \) is indeed less than that from \( x = \frac{\lambda}{8} \) to \( x = \frac{\lambda}{4} \), the assertion is true.

### Step 6: Understanding the Reason
The reason states that in standing waves, different particles oscillate with different amplitudes. This is true because the amplitude varies along the wave, with nodes having zero amplitude and antinodes having maximum amplitude.

### Step 7: Conclusion on the Reason
The reason correctly explains the assertion because the difference in energy between the two segments is due to the variation in amplitude of the particles in those segments. Thus, the reason is also true.

### Final Conclusion
Both the assertion and the reason are true, and the reason is the correct explanation for the assertion.