For a standing wave formed within an open ended air column, which of the following statements describe the condition at both ends?

Question

BlackTom AI · Accepted Answer

At an open end the air is free to move, so displacement reaches a maximum (displacement antinode) while the pressure fluctuation is minimal (pressure node).

1. Relevant knowledge points and concepts
- Sound in air involves oscillations of particle displacement and corresponding pressure variations.  
- In a standing wave, nodes are points of zero amplitude and antinodes are points of maximum amplitude for a given quantity.  
- Displacement and pressure variations are complementary: where displacement is large, pressure variation tends to be small, and vice versa.  
- Boundary conditions at an open end require the local pressure to equal atmospheric pressure (no net pressure fluctuation), while the air particles are free to oscillate.

2. Step-by-step reasoning
- Consider the open end of a tube: it is not sealed, so the gas there can move freely in and out. That freedom means the amplitude of particle motion (displacement) is maximal at the open end → displacement antinode.  
- Pressure fluctuations correspond to local compressions and rarefactions relative to atmospheric pressure. At the open boundary these fluctuations must vanish because the air outside the tube stays at atmospheric pressure. Therefore the pressure fluctuation is zero at the open end → pressure node.  
- The displacement–pressure relationship in standing acoustic waves implies a 90° phase difference and spatial complementarity: maxima of displacement align with minima of pressure variation.

3. (No numeric calculations required) Intermediate logical steps shown above identify how boundary conditions force displacement and pressure extrema.

4. Why the correct answer is correct
- Option (a) states a pressure node and a displacement antinode at both open ends, which matches the physical boundary conditions and the complementary nature of displacement and pressure in standing sound waves.

5. Why the other options are incorrect
- (b) pressure antinode & displacement antinode: impossible since both quantities cannot be maximal at the same open boundary.  
- (c) both nodes: would mean neither displacement nor pressure vary at the end, contradicting the free motion at an open end.  
- (d) pressure antinode & displacement node: describes a closed end, where particles cannot move but pressure can peak—opposite of an open end.

Conclusion: The correct option is (a): a pressure node and displacement antinode is formed, matching the provided answer.

For a standing wave formed within an open ended air column, which of the following statements describe the condition at both ends?单项选择题

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