Harmonics on a String

String fixed at both ends: standing-wave modes have wavelengths λ_n = 2L/n, frequencies f_n = nf₁ (n = 1, 2, 3, …).

Fundamental (n = 1): one antinode in middle, nodes at both ends. Lowest frequency.

n-th harmonic: n antinodes, n+1 nodes.

All integer multiples of f₁ are PRESENT — string can produce 1st, 2nd, 3rd, 4th, … harmonics. (Difference from closed pipe.)

Frequency: f_n = nv/(2L) = (n/2L)·√(T/μ).

Higher harmonics give a 'richer' sound (timbre depends on harmonic mix).

Plucking position determines WHICH harmonics get excited. Plucking center excites odd harmonics; plucking off-center excites both.

Tuning: change T (peg) or L (finger) or μ (different string).