Which of the figures, shows the pressure difference from regular atmospheric pressure for an organ pipe of length `L` closed at one end, corresponds to the `1st` overtime for the pipe ?

Options

A.
B.
C.
D.

Explanation

To solve the problem of identifying which figure corresponds to the pressure difference from regular atmospheric pressure for an organ pipe of length \( L \) closed at one end, we need to understand the behavior of standing waves in such a pipe.

### Step-by-Step Solution:

1. **Understanding the Organ Pipe Configuration**:
- An organ pipe closed at one end has one end (the closed end) acting as a displacement node (where the air does not move) and the other end (the open end) acting as a displacement antinode (where the air moves maximally).

2. **Identifying the Fundamental Frequency**:
- The fundamental frequency (first harmonic) for a closed organ pipe has one node at the closed end and one antinode at the open end. This configuration creates a quarter wavelength (\( \frac{\lambda}{4} \)) in the length of the pipe \( L \).

3. **Understanding Overtones**:
- The first overtone (second harmonic) for a closed organ pipe has one node at the closed end, one antinode at the open end, and an additional node and antinode in between. This results in three-quarters of a wavelength fitting in the length of the pipe, which can be represented as \( \frac{3\lambda}{4} \).

4. **Visualizing the Pressure Difference**:
- In terms of pressure difference, the closed end corresponds to a pressure node (minimum pressure), while the open end corresponds to a pressure antinode (maximum pressure). For the first overtone, we will have a pattern that shows one node and two antinodes along the length of the pipe.

5. **Analyzing the Given Figures**:
- We need to look for a figure that shows three quarter loops (or three pressure variations) along the length of the pipe. Each quarter loop corresponds to a change in pressure from maximum to minimum.

6. **Selecting the Correct Option**:
- Among the options provided, we need to identify the figure that shows three quarter loops. This can be visualized as having one node at the closed end, one antinode at the open end, and one additional node and antinode in between.

7. **Conclusion**:
- After analyzing the figures, we find that option (a) is the correct representation of the pressure difference for the first overtone in a closed organ pipe, as it shows three quarter loops.

### Final Answer:
The correct figure that corresponds to the pressure difference from regular atmospheric pressure for an organ pipe of length \( L \) closed at one end, representing the first overtone, is option (a).