For the second overtone which option shows correct mode for displacement `y` and pressure variation `DeltaP`

Options

A.
B.
C.
D.

Answer

To determine the correct mode for displacement \( y \) and pressure variation \( \Delta P \) for the second overtone (which corresponds to the third harmonic), we can follow these steps:

### Step 1: Understand the Harmonics
The second overtone refers to the third harmonic. In a standing wave pattern, the harmonics are defined as follows:
- First harmonic (fundamental frequency) has 1 antinode.
- Second harmonic has 2 antinodes.
- Third harmonic (second overtone) has 3 antinodes.

### Step 2: Identify Antinodes and Nodes
In a standing wave:
- Displacement antinodes correspond to pressure nodes.
- Displacement nodes correspond to pressure antinodes.

For the third harmonic:
- There will be 3 displacement antinodes and 2 displacement nodes.

### Step 3: Sketch the Displacement Graph
For the third harmonic:
- The displacement graph starts with an antinode at the open end (top).
- The pattern will look like this:
- Antinode (A) at \( 0 \)
- Node (N) at \( \frac{\lambda}{4} \)
- Antinode (A) at \( \frac{\lambda}{2} \)
- Node (N) at \( \frac{3\lambda}{4} \)
- Antinode (A) at \( \lambda \)

### Step 4: Sketch the Pressure Variation Graph
Using the relationship between displacement and pressure:
- The pressure variation graph will have:
- Node (N) at the top (where the displacement antinode is).
- Antinode (A) at \( \frac{\lambda}{4} \) (where the first displacement node is).
- Node (N) at \( \frac{\lambda}{2} \) (where the second displacement antinode is).
- Antinode (A) at \( \frac{3\lambda}{4} \) (where the second displacement node is).
- Node (N) at \( \lambda \) (where the third displacement antinode is).

### Step 5: Compare with Given Options
Now, we should compare our derived graphs with the provided options:
- **Option A**: Incorrect, does not match the expected pattern.
- **Option B**: Correct, matches the expected displacement and pressure variation.
- **Option C**: Incorrect, does not match the expected pattern.
- **Option D**: Incorrect, does not match the expected pattern.

### Conclusion
The correct option that shows the correct mode for displacement \( y \) and pressure variation \( \Delta P \) for the second overtone is **Option B**.

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