An open organ pipe of length L vibrates in its fundamental mode. The pressure variation is maximum

Options

A. At the two ends
B. At the middle of the pipe
C. At the distance `l/8` inside the ends
D. At the distance `l/4` inside the ends

Answer

To solve the problem of determining where the pressure variation is maximum in an open organ pipe vibrating in its fundamental mode, we can follow these steps:

### Step 1: Understand the Structure of the Open Organ Pipe
An open organ pipe has two open ends and can support standing waves. In the fundamental mode, the length of the pipe (L) is equal to half the wavelength (λ/2) of the sound wave produced.

**Hint:** Recall that in an open pipe, there are antinodes at both ends.

### Step 2: Identify the Nodes and Antinodes
In the fundamental mode of vibration:
- There are two antinodes (maximum displacement) at the open ends of the pipe.
- There is one node (minimum displacement) at the center of the pipe.

**Hint:** Nodes are points where there is no displacement, while antinodes are points of maximum displacement.

### Step 3: Relate Displacement to Pressure Variation
For sound waves, pressure variations are related to the displacement of air particles:
- At nodes, the air particles do not move, leading to minimum pressure variation.
- At antinodes, the air particles have maximum displacement, leading to maximum pressure variation.

**Hint:** Remember that pressure is highest where particles are closest together (compression) and lowest where they are farthest apart (rarefaction).

### Step 4: Analyze the Pressure Variation
In the fundamental mode:
- The maximum pressure variation occurs at the antinodes, which are located at both ends of the pipe.
- However, the pressure variation is also significant at the midpoint (L/2) of the pipe due to the compression and rarefaction occurring there.

**Hint:** Consider how pressure changes in relation to the movement of particles at different points along the pipe.

### Step 5: Conclusion
The pressure variation is maximum at the open ends of the pipe, but since the question specifically asks for the maximum pressure variation in relation to the fundamental mode, we can conclude that:

**Final Answer:** The pressure variation is maximum at the midpoint of the pipe (L/2).