Problem — Identifying the third normal mode

Problem

Standing wave. A taut string is fixed at both ends and vibrates in its third normal mode ($n=3$). Which configuration in the figure represents this mode?

Solution

Setup. In the $n$-th mode of a string fixed at both ends, exactly $n$ antinodes (bellies) and $n+1$ nodes form, and the string contains $n$ half-wavelengths: $L = n\,\tfrac{\lambda}{2}$.

Counting the antinodes in the figures.

• A: a single belly → $n=1$ (fundamental).
• B: two bellies → $n=2$ (second harmonic).
• C: three bellies → $n=3$ (third harmonic).

The third normal mode has three antinodes and four nodes (the two ends plus two internal ones):

$\ev{\text{Configuration C}}$

Linked atoms

Argomenti: Waves Concetti: Standing waves Competenze: Graph reading