Explain why a string fixed at both ends can only sustain standing waves at specific frequencies.

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 string of length L is fixed at both ends and is set into vibration. Only certain frequencies produce sustained, stable patterns of vibration on the string.

Model answer (4 marks)

A standing wave is produced by the superposition of two waves travelling in opposite directions along the string.
The fixed ends must be nodes – the displacement at the ends is always zero because the string cannot move there.
Only wavelengths for which an integer number of half‑wavelengths fit exactly between the two fixed ends satisfy these node conditions.
Since f = v/λ, each allowed wavelength gives a single frequency; thus only discrete, quantised frequencies – the harmonic series – can be sustained.

Examiner tips

State the superposition of counter‑propagating waves first.
Explain the node requirement at the fixed ends.
Show that λ = 2L/n (n = 1,2,3…) and link to f = v/λ.
Use the term ‘harmonic series’ to demonstrate understanding of quantisation.

Common mistakes

Failing to mention that the ends are nodes.
Using the wrong relationship between wavelength and length (e.g., λ = L/n instead of 2L/n).
Not linking the allowed wavelengths to discrete frequencies via f = v/λ.

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 must be nodes (zero displacement) because the string cannot move at a fixed boundary.
Only wavelengths for which a whole number of half-wavelengths fit exactly between the two fixed ends satisfy the boundary conditions.
Since frequency and wavelength are related by f = v/λ, only discrete (quantised) frequencies corresponding to the allowed wavelengths are possible, forming the harmonic series.

Key terms in this question

standing wave