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 question regarding the pressure difference from regular atmospheric pressure for an organ pipe of length \( L \) closed at one end, we need to understand the concepts of standing waves in such pipes.

### Step-by-Step Solution:

1. **Understand the Structure of the Organ Pipe**:
- An organ pipe that is closed at one end has one end (the closed end) where the pressure is maximum (a node) and the other end (the open end) where the pressure is minimum (an antinode).

2. **Identify the Fundamental Frequency**:
- For the fundamental frequency (first harmonic), the pattern of pressure variation shows one node at the closed end and one antinode at the open end. This can be represented as:
- Node (closed end) - Antinode (open end).

3. **Determine the First Overtone**:
- The first overtone (second harmonic) for a closed organ pipe introduces an additional node and antinode. The pattern will now have:
- Node (closed end) - Antinode - Node (somewhere in the middle) - Antinode (open end).
- This means there will be one more node and one more antinode compared to the fundamental frequency.

4. **Visual Representation**:
- In a graphical representation, the first overtone will show:
- A node at the closed end,
- An antinode in the middle,
- A node before reaching the open end,
- An antinode at the open end.

5. **Analyze the Given Options**:
- Review the provided figures to find one that matches the pressure variation pattern described for the first overtone. The correct figure should show:
- A node at the closed end,
- An antinode,
- Another node,
- An antinode at the open end.

6. **Select the Correct Figure**:
- Based on the analysis, the correct figure that represents the pressure difference for the first overtone in a closed organ pipe will be the one that matches the described pattern.

### Conclusion:
The correct option that corresponds to the first overtone for the organ pipe closed at one end is option A, which shows the arrangement of nodes and antinodes correctly.