Result

Normal modes of a fixed string

f_{n} = \frac{n v}{2 L} (n = 1, 2, 3, \dots)

Statement

The discrete set of frequencies at which a string of length L, fixed at both ends, can vibrate.

Symbols and units

$f_{n}$	nth normal-mode frequency	Hz
$v$	wave speed	m/s
$L$	string length	m
$n$	harmonic number	—

When it applies

A string (or medium) fixed — or matching boundary conditions — at both ends.

Don't use when

Applies to a string fixed at both ends (or a pipe open at both ends) — a pipe closed at one end supports only odd harmonics and needs a different formula.

Derivation

A standing wave requires a node at each fixed end, which restricts the fitting wavelengths to:

L = n \frac{λ _{n}}{2}

Combined with v = f λ, this gives the allowed frequencies:

f_{n} = \frac{n v}{2 L}

Worked examples

The second harmonic

f_{2} = \frac{2 v}{2 L} = 2 f_{1}

Exactly twice the fundamental frequency — an octave higher, in musical terms.

Watch out for

This series applies to a string fixed at both ends, or a pipe open at both ends. A pipe closed at one end supports only odd harmonics.