Explain how a standing wave is formed on a stretched string that is fixed at both ends, and outline the conditions that determine which harmonics can be sustained.
Written & reviewed by James Millett — Biology (Imperial College London), PGCE Science (University of Cambridge).
A guitar string is plucked so that it vibrates transversely. The string has a fixed length L and is attached firmly at both ends.
Model answer (4 marks)
Two waves of the same frequency and amplitude travel in opposite directions along the string and superpose. The fixed ends are nodes because the string cannot move there, so the reflected wave changes phase by π (half a wavelength). A standing wave is formed when a whole number of half‑wavelengths fit exactly into the length L, i.e. L = nλ/2, where n is a positive integer. Because both ends are nodes, all harmonics (n = 1, 2, 3, …) are allowed, giving a fundamental frequency and all integer multiples of it.
Examiner tips
- Use the phrase ‘superpose’ and ‘phase change of π’ to show understanding of wave superposition and reflection.
- State the node condition at both ends and the quantisation condition L = nλ/2 explicitly.
- Mention that n can be any positive integer to cover all harmonics.
- Keep the answer concise – 4 marks are enough for a short explanation.
Common mistakes
- Saying the ends are antinodes instead of nodes.
- Forgetting that the phase change is π, not 0.
- Claiming only odd harmonics are allowed, which is incorrect for a string fixed at both ends.
Mark scheme (4 marks)
- Two waves of the same frequency (and amplitude) travel in opposite directions along the string and superpose.
- The fixed ends must be nodes because the string cannot displace at a fixed point, so the reflected wave undergoes a phase change of π (half a wavelength).
- A whole number of half-wavelengths must fit exactly into the length L, i.e. L = nλ/2, where n is a positive integer.
- Because both ends are nodes, all harmonics (n = 1, 2, 3, …) are permitted, so the string supports a fundamental frequency and all integer multiples of it.
Key terms in this question
Related
- All IB DP Physics Standard Level (2023 syllabus) revision notes →
- How to answer a "Explain" question →
- Decode the mark scheme abbreviations →
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