A closed organ pipe has

Question

BlackTom AI · Accepted Answer

At a rigidly closed end the air cannot oscillate, so there must be a displacement node; at an open end the air is free to move, so there is a displacement antinode.

1. Relevant knowledge points and concepts
- Standing sound waves in pipes have boundary conditions set by the ends: a closed (rigid) end forces zero particle displacement; an open end allows maximum particle displacement.  
- Pressure fluctuations and particle displacement are complementary: displacement antinodes correspond to pressure nodes, and displacement nodes correspond to pressure antinodes.  
- For a pipe closed at one end and open at the other, allowed modes are quarter-wave resonances (odd harmonics): L = (2n − 1)λ/4.

2. Step-by-step reasoning
- At the closed end the wall prevents air motion, so particle displacement u = 0 there → displacement node.  
- Because displacement is zero where compression/rarefaction cannot relieve, pressure fluctuation is maximal at that point → pressure antinode at closed end.  
- At the open end the air is free to move, so particle displacement reaches a maximum → displacement antinode.  
- The open end is connected to the outside atmosphere, which fixes the pressure fluctuation to zero → pressure node at the open end.

3. Intermediate logical steps (no numeric calculation needed)
- Apply boundary condition at closed end: u = 0 → displacement node.  
- Apply boundary condition at open end: Δp = 0 (pressure equal to atmosphere) → pressure node → hence displacement antinode.

4. Why the correct answer is correct
- Option b states precisely: "a displacement node at the closed end and a displacement antinode at the open end," which matches the physical boundary conditions for a closed–open pipe and the displacement–pressure relationship.

5. Why other options are incorrect
- a. "a pressure node at each end" is false — closed end is a pressure antinode, not a node.  
- c. "a pressure antinode at each end" is false — open end is a pressure node.  
- d. "a displacement antinode at the closed end and a displacement node at the open end" is the reverse of the true situation.

6. Conclusion
The correct answer is b. This matches the provided answer: a displacement node at the closed end and a displacement antinode at the open end.

A closed organ pipe has单项选择题

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