Explain how resonance occurs in a tube open at both ends when a sound source is placed near one end, and explain why only certain frequencies cause resonance in such a tube.

IB DP Physics Higher Level (2023 syllabus) — C.4 Standing waves and resonance · Explain · 4 marks · View as Markdown

Written & reviewed by James Millett — Biology (Imperial College London), PGCE Science (University of Cambridge).

A loudspeaker producing a continuous sinusoidal tone of variable frequency is placed near one open end of a tube that is open at both ends. As the frequency is slowly increased from a low value, resonance is observed at several distinct frequencies.

Model answer (4 marks)

The loudspeaker emits a sinusoidal sound wave that travels down the tube. At the far open end the wave is reflected, creating a wave travelling back towards the source. The incident and reflected waves superpose, giving a standing wave inside the tube. Because both ends are open, each end must be a displacement antinode. Therefore resonance can only occur when the tube length contains an integer number of half‑wavelengths:
L = nλ/2 (n = 1, 2, 3 …).
At these frequencies the standing wave is sustained with maximum amplitude, so energy from the source is efficiently transferred to the air column and the tube resonates.

Examiner tips

  • Use the word ‘superpose’ or ‘interfere’ to show understanding of standing waves.
  • State the boundary condition (antinode at open ends) and give the formula L = nλ/2.
  • Explain that only these frequencies give maximum amplitude and sustained resonance.
  • Mention the source is continuously supplying energy at the natural frequency.

Common mistakes

  • Confusing nodes with antinodes at the open ends.
  • Giving the wrong formula (e.g. L = nλ instead of nλ/2).
  • Failing to explain why only those frequencies produce resonance.

Mark scheme (4 marks)

  1. The sound wave from the loudspeaker travels along the tube and reflects at the far open end, producing a wave travelling in the opposite direction.
  2. The incident and reflected waves superpose (interfere) to form a standing wave inside the tube.
  3. A displacement antinode must be present at each open end, so resonance occurs only when the tube length is a whole number of half-wavelengths (L = nλ/2, where n = 1, 2, 3, …).
  4. Resonance occurs because at these specific frequencies the standing wave is sustained with maximum amplitude, as energy is continuously transferred from the source to the oscillating air column at the natural frequency of the tube.

Key terms in this question

resonance

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