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Waves & soundSchool & A-level

Standing-wave modes on a string

Displays the first three normal modes of a uniform string fixed at both ends, including nodes, wavelengths and harmonic frequencies.

How to read this diagram

Each fixed end is a node, so an integer number of half-wavelengths must fit into length L.
The fundamental contains one loop; the second and third harmonics contain two and three loops.
Internal zero-displacement points are nodes and the centres of loops are antinodes.
At fixed wave speed v, mode frequencies are integer multiples of v/(2L).

Model notes

The model assumes a uniform flexible string, small transverse displacement, constant tension and perfectly fixed ends. The vertical amplitude is exaggerated.

ReferenceOpenStax — University Physics Volume 1 ↗

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A growing reference from school fundamentals through undergraduate and selected advanced physics. No finite collection contains every specialised equation or theory; use topic and level filters, and always check assumptions.

Definitions use original wording, and each reference links to source material. Study labels are guides, not exam-board checklists. Worked check questions are educational examples, not an exam question bank.

Equations use SI quantities unless stated otherwise. Numerical values retain the units, precision and conditions of their source. Diagrams are explanatory models; their captions identify simplifications.