Explain why a stretched string fixed at both ends can only vibrate at certain discrete frequencies.
Written & reviewed by James Millett — Biology (Imperial College London), PGCE Science (University of Cambridge).
A musician plucks a stretched string that is fixed at both ends. The string is observed to vibrate in several different patterns, each at a specific frequency.
Model answer (4 marks)
A standing wave is produced by the superposition of two waves travelling in opposite directions along the string.
Because the string is fixed at both ends, the ends must be nodes – points of zero displacement. Therefore only whole numbers of half‑wavelengths can fit between the two fixed ends.
This boundary condition limits the allowed wavelengths to a discrete set: λ = 2L/n, where n = 1, 2, 3… is the mode number.
The wave speed on the string is fixed for a given tension and mass per unit length, so each allowed wavelength corresponds to a unique frequency f = v/λ. Hence the string can vibrate only at a discrete set of resonant frequencies.
Because the string is fixed at both ends, the ends must be nodes – points of zero displacement. Therefore only whole numbers of half‑wavelengths can fit between the two fixed ends.
This boundary condition limits the allowed wavelengths to a discrete set: λ = 2L/n, where n = 1, 2, 3… is the mode number.
The wave speed on the string is fixed for a given tension and mass per unit length, so each allowed wavelength corresponds to a unique frequency f = v/λ. Hence the string can vibrate only at a discrete set of resonant frequencies.
Examiner tips
- Show the standing‑wave picture and explain the node condition; use the formula λ = 2L/n; link λ to f via f = v/λ; keep the answer concise and use the exact terminology.
- Use the word "discrete" and emphasise the boundary condition at the fixed ends.
Mark scheme (4 marks)
- A standing wave is formed by the superposition (interference) of two waves travelling in opposite directions along the string.
- The fixed ends of the string must be nodes (zero displacement), so only whole numbers of half-wavelengths fit between the two fixed ends.
- This boundary condition restricts the allowed wavelengths to a discrete set (λ = 2L/n), so only specific wavelengths can form stable standing waves.
- Since wave speed on the string is fixed (for given tension and mass per unit length), each allowed wavelength corresponds to a unique frequency (f = v/λ), giving a discrete set of resonant frequencies.
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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