Explain how resonance occurs in a closed pipe and why the pipe supports only odd harmonics.
Written & reviewed by James Millett — Biology (Imperial College London), PGCE Science (University of Cambridge).
A closed pipe is one that is sealed at one end and open at the other. When a sound source is placed near the open end, the pipe can resonate at certain frequencies.
Model answer (4 marks)
A closed pipe must have a displacement node (pressure antinode) at the sealed end and a displacement antinode (pressure node) at the open end.
When the source frequency equals one of the pipe’s natural frequencies, the incident wave and the wave reflected from the closed end interfere constructively, forming a standing wave – this is resonance.
The shortest resonant length contains one quarter of a wavelength (λ/4). Hence the fundamental wavelength is four times the pipe length (λ₁=4L) and the fundamental frequency is f₁=v/λ₁.
Only odd multiples of the fundamental satisfy the node‑at‑closed‑end and antinode‑at‑open‑end conditions. Therefore the pipe supports the 1st, 3rd, 5th… harmonics; even harmonics cannot form.
When the source frequency equals one of the pipe’s natural frequencies, the incident wave and the wave reflected from the closed end interfere constructively, forming a standing wave – this is resonance.
The shortest resonant length contains one quarter of a wavelength (λ/4). Hence the fundamental wavelength is four times the pipe length (λ₁=4L) and the fundamental frequency is f₁=v/λ₁.
Only odd multiples of the fundamental satisfy the node‑at‑closed‑end and antinode‑at‑open‑end conditions. Therefore the pipe supports the 1st, 3rd, 5th… harmonics; even harmonics cannot form.
Examiner tips
- Use the exact boundary‑condition terms (node/antinode, pressure/velocity).
- Show the λ/4 reasoning for the fundamental.
- Explain why only odd multiples meet both boundary conditions.
Common mistakes
- Confusing displacement and pressure nodes/antinodes.
- Forgetting that the fundamental is λ=4L, not 2L.
Mark scheme (4 marks)
- A displacement node (pressure antinode) must form at the closed end and a displacement antinode (pressure node) must form at the open end.
- When the frequency of the source matches a natural frequency of the pipe, the incident and reflected waves superpose to form a standing wave, producing resonance.
- The shortest resonating length contains one quarter-wavelength (λ/4), so the fundamental has a wavelength four times the pipe length.
- Only odd multiples of the fundamental frequency (1st, 3rd, 5th harmonics…) satisfy the node-at-closed-end and antinode-at-open-end boundary conditions simultaneously, so even harmonics cannot form.
Key terms in this question
resonance · harmonics · closed pipe
Related
- All IB DP Physics Standard Level (2023 syllabus) revision notes →
- How to answer a "Explain" question →
- Decode the mark scheme abbreviations →
More Standing waves and resonance questions
- Explain how the amplitude of a standing wave varies along its length, and distin…
- Explain how a standing wave is formed on a stretched string that is fixed at bot…
- Explain why a stretched string fixed at both ends can only vibrate at certain di…
- Explain how the superposition of two waves produces a standing wave, and state t…