Condition for a standing wave on a string fixed at both ends

Statement

$L = n\lambda/2$, $n = 1, 2, 3, \ldots$ — the string length L must equal an integer number of half-wavelengths for a stable standing wave.

Why is it true?

Both fixed ends must be nodes; only wavelengths that 'fit' exactly between two fixed nodes produce a stable oscillation pattern that doesn't die out over time.

Topics that use this theorem

Step-by-step proofs